diff --git a/statistics/Makefile b/statistics/Makefile new file mode 100644 index 0000000..a8dbb44 --- /dev/null +++ b/statistics/Makefile @@ -0,0 +1,24 @@ +DOTSOURCES = $(wildcard figs/*.dot) + + + +all: $(DOTSOURCES:dot=pdf) + python figs/generate.py + python figs/generate03.py + python figs/generateTPlots.py + pdflatex talk*.tex + pdflatex talk*.tex + pdflatex talk*.tex + + +figs/prob%.pdf : figs/prob%.dot + dot -Tpdf -o $@ $< + +figs/test%.pdf : figs/test%.dot + dot -Tpdf -o $@ $< + +figs/fig%.pdf : figs/fig%.dot + dot -Tpdf -o $@ $< + +clean: + rm -rf *.dvi *.pdf *.aux *.out *.log auto *.nav *.snm *.toc *.vrb diff --git a/statistics/beamercolorthemetuebingen.sty b/statistics/beamercolorthemetuebingen.sty new file mode 120000 index 0000000..7648eb4 --- /dev/null +++ b/statistics/beamercolorthemetuebingen.sty @@ -0,0 +1 @@ +../latex/beamercolorthemetuebingen.sty \ No newline at end of file diff --git a/statistics/certificate.lyx b/statistics/certificate.lyx new file mode 100644 index 0000000..f502b73 --- /dev/null +++ b/statistics/certificate.lyx @@ -0,0 +1,218 @@ +#LyX 2.0 created this file. For more info see http://www.lyx.org/ +\lyxformat 413 +\begin_document +\begin_header +\textclass g-brief2 +\begin_preamble +\fenstermarken % prints address window marks +\faltmarken % prints folding marks +%\lochermarke % prints puncher marks +\trennlinien % prints striplines +%\unserzeichen % prints "our ref" instead of "my ref" +\end_preamble +\use_default_options false +\maintain_unincluded_children false +\language english +\language_package default +\inputencoding auto +\fontencoding global +\font_roman palatino +\font_sans default +\font_typewriter default +\font_default_family default +\use_non_tex_fonts false +\font_sc false +\font_osf false +\font_sf_scale 100 +\font_tt_scale 100 + +\graphics default +\default_output_format default +\output_sync 0 +\bibtex_command default +\index_command default +\paperfontsize 12 +\spacing onehalf +\use_hyperref false +\papersize default +\use_geometry false +\use_amsmath 1 +\use_esint 1 +\use_mhchem 1 +\use_mathdots 1 +\cite_engine basic +\use_bibtopic false +\use_indices false +\paperorientation portrait +\suppress_date false +\use_refstyle 0 +\index Index +\shortcut idx +\color #008000 +\end_index +\secnumdepth 4 +\tocdepth 4 +\paragraph_separation skip +\defskip medskip +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle empty +\tracking_changes false +\output_changes false +\html_math_output 0 +\html_css_as_file 0 +\html_be_strict false +\end_header + +\begin_body + +\begin_layout Standard +\begin_inset Note Note +status open + +\begin_layout Plain Layout +Note also the document preamble settings. +\end_layout + +\end_inset + + +\end_layout + +\begin_layout NameRowA +Dr. + rer. + nat. + Fabian Sinz +\end_layout + +\begin_layout NameRowB + +\end_layout + +\begin_layout NameRowC + +\end_layout + +\begin_layout NameRowD + +\end_layout + +\begin_layout NameRowE + +\end_layout + +\begin_layout NameRowF + +\end_layout + +\begin_layout NameRowG + +\end_layout + +\begin_layout AddressRowA +University Tübingen +\end_layout + +\begin_layout AddressRowB + +\end_layout + +\begin_layout AddressRowC +Auf der Morgenstelle 28 +\end_layout + +\begin_layout AddressRowD +72076 Tübingen +\end_layout + +\begin_layout InternetRowA +http:/ +\begin_inset Formula $\!$ +\end_inset + +/www.epagoge.de +\end_layout + +\begin_layout InternetRowB +fabian.sinz@epagoge.de +\end_layout + +\begin_layout ReturnAddress +University Tübingen +\begin_inset Formula $\cdot$ +\end_inset + + Auf der Morgenstelle 28 +\begin_inset Formula $\cdot$ +\end_inset + + 72076 Tübingen +\end_layout + +\begin_layout Date +\begin_inset ERT +status collapsed + +\begin_layout Plain Layout + + +\backslash +today +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Reference + +\end_layout + +\begin_layout Opening +To whom it may concern, +\end_layout + +\begin_layout Closing +Best regards +\end_layout + +\begin_layout Signature +Dr. + Fabian Sinz +\end_layout + +\begin_layout Encl. + +\end_layout + +\begin_layout Letter +this letter certifies that +\emph on +Lakshmi Channappa +\emph default + attended the course +\emph on +Statistics in a Nutshell +\emph default + held at the Neurochip research group at the +\emph on +Naturwissenschaftliches und Medizinisches Institut Reutlingen +\emph default +in 2013 +\emph on +. + +\emph default +The course was organized in two lectures of four hours each and covered + topics such as basics of probability theory, errorbars and confidence intervals +, statistical tests, p-values, multiple hypothesis testing, basics of study + design, and basics of ANOVA. + Small calculation and programming exercises were used to clarify selected + material. +\end_layout + +\end_body +\end_document diff --git a/statistics/environments.tex b/statistics/environments.tex new file mode 100644 index 0000000..bdf8aab --- /dev/null +++ b/statistics/environments.tex @@ -0,0 +1,123 @@ + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: +\definecolor{crimson}{HTML}{DC143C} +\definecolor{cornflowerblue}{HTML}{6495ED} +\definecolor{dodgerblue}{HTML}{1E90FF} +\definecolor{deepskyblue}{HTML}{00BFFF} +\definecolor{gainsboro}{HTML}{DCDCDC} +\definecolor{ghostwhite}{HTML}{F8F8F8} +\definecolor{lightgray}{HTML}{D3D3D3} + +\newenvironment<>{emphasize}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=orange!100} + \setbeamercolor{block body}{fg=black,bg=cornflowerblue!70} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{solution}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=dodgerblue!100} + \setbeamercolor{block body}{fg=black,bg=lightgray!70} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{question}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=dodgerblue!100} + \setbeamercolor{block body}{fg=black,bg=lightgray!70} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\renewenvironment<>{definition}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=dodgerblue!100} + \setbeamercolor{block body}{fg=black,bg=lightgray!70} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + + +\newenvironment<>{description}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=white,bg=gray} + \setbeamercolor{block body}{fg=black,bg=gray!30} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{family=\sffamily, series=\bfseries} + \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{task}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=dodgerblue!100} + \setbeamercolor{block body}{fg=black,bg=deepskyblue!80} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{summary}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=blue!40} + \setbeamercolor{block body}{fg=black,bg=blue!20} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} +%%%%%%%%%%%%%%%%%%% PROGRESSBAR %%%%%%%%%%%%%%%%%%%%%%%%%% +\definecolor{pbblue}{HTML}{0A75A8}% filling color for 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[fontsize=12, shape=rectangle, style=filled, nodesep=0.95,ranksep=0.95]; + edge [penwidth=2, fontsize=10 ]; + + + subgraph cluster_IR { + label = "interval/ ratio"; + bgcolor=lightblue; + + IR[label="data normal distributed\nor n large?"]; + + normal[label="1, 2, or >2 groups?"]; + notnormal[label="transform into\nnormal?"]; + trulynotnormal[label="1, 2, or >2 groups?"]; + + IR->normal[label="normal"]; + IR->notnormal[label="not normal"]; + notnormal->normal[label="yes"]; + notnormal->trulynotnormal[label="no"]; + + onesamp[label="one-sample\nt-test"]; + normal->onesamp[label="1 group"]; + + ttest[label="t-test"]; + twosamp->ttest[label="independent"]; + + + twosamp[label="paired or\nindependent?"]; + pairedttest[label="paired\nt-test"]; + normal->twosamp[label="2 groups"]; + twosamp->pairedttest[label="paired"]; + + twosampNN[label="paired or\nindependent?"]; + + trulynotnormal->twosampNN[label="2 groups"]; + + signrank[label="Wilcoxon signed\nrank test"]; + + trulynotnormal->signrank[label="1 group\n(fix other\ngroup to\n one value)"]; + + + } + + trulynotnormal->signtest[label="1 group\n(fix other\ngroup to\n one value)"]; + + twosampNN->signrank[label="paired"]; + twosampNN->indepTwosampNN[label="independent"]; + + data->ordinal[label="ordinal"]; + + subgraph cluster_O { + label = "ordinal"; + bgcolor=lightblue; + indepTwosampNN[label="Wilcoxon-Mann-Whitney\ntest"]; + + + + twosampOrd[label="paired or\nindependent?"]; + ordinal->twosampOrd[label="2 groups"]; + twosampOrd->indepTwosampNN[label="independent"]; + + signtest[label="sign test"]; + ordinal[label="1, 2, or >2 groups?"]; + ordinal->signtest[label="1 group\n(fix other\ngroup to\n one value)"]; + } + + + subgraph cluster_ND { + label = "nominal/discrete"; + bgcolor=lightblue; + nd_test_type[label="1, 2, or >2 variables",color="green"]; + onesampND[label="chi square for\ngoodness of fit"]; + + nd_test_type->onesampND[label="1 variable"]; + } + + + + + + data[label="type of data?"]; + + data->IR[label="interval/ratio"]; + data->nd_test_type[label="nominal/discrete",color="red"]; + + + + twosampNN->signtest[label="paired"]; + twosampOrd->signtest[label="paired"]; + + +} + + diff --git a/statistics/figs/fig02.pdf b/statistics/figs/fig02.pdf new file mode 100644 index 0000000..5c50172 Binary files /dev/null and b/statistics/figs/fig02.pdf differ diff --git a/statistics/figs/fig03.dot b/statistics/figs/fig03.dot new file mode 100644 index 0000000..16af84d --- /dev/null +++ b/statistics/figs/fig03.dot @@ -0,0 +1,98 @@ +digraph G { + rankdir=TB; + ranksep=0.2; + node [fontsize=12, shape=rectangle, style=filled, nodesep=0.95,ranksep=0.95]; + edge [penwidth=2, fontsize=10 ]; + + + subgraph cluster_IR { + label = "interval/ ratio"; + bgcolor=lightblue; + + IR[label="data normal distributed\nor n large?"]; + + normal[label="1, 2, or >2 groups?"]; + notnormal[label="transform into\nnormal?"]; + trulynotnormal[label="1, 2, or >2 groups?"]; + + IR->normal[label="normal"]; + IR->notnormal[label="not normal"]; + notnormal->normal[label="yes"]; + notnormal->trulynotnormal[label="no"]; + + onesamp[label="one-sample\nt-test"]; + normal->onesamp[label="1 group"]; + + ttest[label="t-test"]; + twosamp->ttest[label="independent"]; + + + twosamp[label="paired or\nindependent?"]; + pairedttest[label="paired\nt-test"]; + normal->twosamp[label="2 groups"]; + twosamp->pairedttest[label="paired"]; + + twosampNN[label="paired or\nindependent?"]; + + trulynotnormal->twosampNN[label="2 groups"]; + + signrank[label="Wilcoxon signed\nrank test"]; + + trulynotnormal->signrank[label="1 group\n(fix other\ngroup to\n one value)"]; + + + } + + trulynotnormal->signtest[label="1 group\n(fix other\ngroup to\n one value)"]; + + twosampNN->signrank[label="paired"]; + twosampNN->indepTwosampNN[label="independent"]; + + data->ordinal[label="ordinal"]; + + subgraph cluster_O { + label = "ordinal"; + bgcolor=lightblue; + indepTwosampNN[label="Wilcoxon-Mann-Whitney\ntest"]; + + + + 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pairedttest[label="paired\nt-test"]; + normal->twosamp[label="2 groups"]; + twosamp->pairedttest[label="paired"]; + + twosampNN[label="paired or\nindependent?"]; + + trulynotnormal->twosampNN[label="2 groups"]; + + signrank[label="Wilcoxon signed\nrank test"]; + + trulynotnormal->signrank[label="1 group\n(fix other\ngroup to\n one value)"]; + + + } + + trulynotnormal->signtest[label="1 group\n(fix other\ngroup to\n one value)"]; + + twosampNN->signrank[label="paired"]; + twosampNN->indepTwosampNN[label="independent"]; + + data->ordinal[label="ordinal"]; + + subgraph cluster_O { + label = "ordinal"; + bgcolor=lightblue; + indepTwosampNN[label="Wilcoxon-Mann-Whitney\ntest"]; + + + + twosampOrd[label="paired or\nindependent?"]; + ordinal->twosampOrd[label="2 groups"]; + twosampOrd->indepTwosampNN[label="independent"]; + + signtest[label="sign test"]; + ordinal[label="1, 2, or >2 groups?"]; + ordinal->signtest[label="1 group\n(fix other\ngroup to\n one value)"]; + } + + + subgraph cluster_ND { + label = "nominal/discrete"; + bgcolor=lightblue; + nd_test_type[label="1, 2, or >2 variables",color="green"]; + onesampND[label="chi square for\ngoodness of fit"]; + twosampND[label="chi square for\nindependence"]; + + nd_test_type->onesampND[label="1 variable"]; + nd_test_type->onesampND[label="n variables",color="red"]; + nd_test_type->twosampND[label="2 variables"]; + } + + + + + + data[label="type of data?"]; + + data->IR[label="interval/ratio"]; + data->nd_test_type[label="nominal/discrete"]; + + + + twosampNN->signtest[label="paired"]; + twosampOrd->signtest[label="paired"]; + + +} + + diff --git a/statistics/figs/fig04.pdf b/statistics/figs/fig04.pdf new file mode 100644 index 0000000..e1fa183 Binary files /dev/null and b/statistics/figs/fig04.pdf differ diff --git a/statistics/figs/fig05.dot b/statistics/figs/fig05.dot new file mode 100644 index 0000000..154be46 --- /dev/null +++ b/statistics/figs/fig05.dot @@ -0,0 +1,99 @@ +digraph G { + rankdir=TB; + 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data->nd_test_type[label="nominal/discrete"]; + + + + twosampNN->signtest[label="paired"]; + twosampOrd->signtest[label="paired"]; + + +} + + diff --git a/statistics/figs/fig06.pdf b/statistics/figs/fig06.pdf new file mode 100644 index 0000000..278d70e Binary files /dev/null and b/statistics/figs/fig06.pdf differ diff --git a/statistics/figs/fig1.dot b/statistics/figs/fig1.dot new file mode 100644 index 0000000..13bc03f --- /dev/null +++ b/statistics/figs/fig1.dot @@ -0,0 +1,22 @@ +digraph G { + rankdir=TB; + ranksep=0.2; + node [fontsize=12, shape=rectangle, style=filled]; + edge [penwidth=2, fontsize=10 ]; + + + data[label="type of data?"]; + IR[label="data normal distributed\nor n large?"]; + ordinal[label="?"]; + nominal[label="?"]; + + data->IR[label="interval/ratio"]; + data->nominal[label="nominal/discrete"]; + data->ordinal[label="ordinal"]; + + normal[label="?"]; + + IR->normal[label="normal"]; + + +} diff --git a/statistics/figs/fig1.pdf b/statistics/figs/fig1.pdf new file 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ttest[label="t-test"]; + twosamp->ttest[label="independent"]; + + + twosamp[label="paired or\nindependent?"]; + pairedttest[label="paired\nt-test"]; + normal->twosamp[label="2 groups"]; + twosamp->pairedttest[label="paired"]; + + twosampNN[label="paired or\nindependent?"]; + + trulynotnormal->twosampNN[label="2 groups"]; + + signrank[label="Wilcoxon signed\nrank test"]; + + trulynotnormal->signrank[label="1 group\n(fix other\ngroup to\n one value)",color="red"]; + + + } + + trulynotnormal->signtest[label="1 group\n(fix other\ngroup to\n one value)",color="red"]; + + twosampNN->signrank[label="paired"]; + twosampNN->indepTwosampNN[label="independent"]; + + data->ordinal[label="ordinal"]; + + subgraph cluster_O { + label = "ordinal"; + bgcolor=lightblue; + indepTwosampNN[label="Wilcoxon-Mann-Whitney\ntest"]; + + + + twosampOrd[label="paired or\nindependent?"]; + ordinal->twosampOrd[label="2 groups"]; + twosampOrd->indepTwosampNN[label="independent"]; + + signtest[label="sign test"]; + ordinal[label="1, 2, or >2 groups?"]; + ordinal->signtest[label="1 group\n(fix other\ngroup to\n one value)",color="red"]; + } + + data[label="type of data?"]; + nominal[label="?"]; + + data->IR[label="interval/ratio"]; + data->nominal[label="nominal/discrete"]; + + + + twosampNN->signtest[label="paired"]; + twosampOrd->signtest[label="paired"]; + + +} + + diff --git a/statistics/figs/fig12.pdf b/statistics/figs/fig12.pdf new file mode 100644 index 0000000..4552386 Binary files /dev/null and b/statistics/figs/fig12.pdf differ diff --git a/statistics/figs/fig2.dot b/statistics/figs/fig2.dot new file mode 100644 index 0000000..b2c6d31 --- /dev/null +++ b/statistics/figs/fig2.dot @@ -0,0 +1,28 @@ +digraph G { + rankdir=TB; + ranksep=0.2; + node [fontsize=12, shape=rectangle, style=filled]; + edge [penwidth=2, fontsize=10 ]; + + + data[label="type of data?"]; + IR[label="data normal distributed\nor n large?"]; + ordinal[label="?"]; + nominal[label="?"]; + + 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normal->twosamp[label="2 groups"]; + twosamp->pairedttest[label="paired"]; + + twosampNN[label="paired or\nindependent?",color="lightblue"]; + indepTwosampNN[label="Wilcoxon-Mann-Whitney\ntest",color="lightblue"]; + + trulynotnormal->twosampNN[label="2 groups"]; + twosampNN->indepTwosampNN[label="independent"]; + + twosampOrd[label="paired or\nindependent?",color="lightblue"]; + ordinal->twosampOrd[label="2 groups"]; + twosampOrd->indepTwosampNN[label="independent"]; + + + +} + + diff --git a/statistics/figs/fig6.pdf b/statistics/figs/fig6.pdf new file mode 100644 index 0000000..bb6f7a4 Binary files /dev/null and b/statistics/figs/fig6.pdf differ diff --git a/statistics/figs/fig7.dot b/statistics/figs/fig7.dot new file mode 100644 index 0000000..c6f3bed --- /dev/null +++ b/statistics/figs/fig7.dot @@ -0,0 +1,52 @@ +digraph G { + rankdir=TB; + ranksep=0.2; + node [fontsize=12, shape=rectangle, style=filled, nodesep=0.75,ranksep=0.75]; + edge [penwidth=2, fontsize=10 ]; + + + 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twosampNN->pairedTwosampNN[label="paired"]; + twosampOrd->pairedTwosampNN[label="paired"]; +} + + diff --git a/statistics/figs/fig8.pdf b/statistics/figs/fig8.pdf new file mode 100644 index 0000000..88b042e Binary files /dev/null and b/statistics/figs/fig8.pdf differ diff --git a/statistics/figs/fig9.dot b/statistics/figs/fig9.dot new file mode 100644 index 0000000..9af76f5 --- /dev/null +++ b/statistics/figs/fig9.dot @@ -0,0 +1,54 @@ +digraph G { + rankdir=TB; + ranksep=0.2; + node [fontsize=12, shape=rectangle, style=filled, nodesep=0.75,ranksep=0.95]; + edge [penwidth=2, fontsize=10 ]; + + + data[label="type of data?"]; + IR[label="data normal distributed\nor n large?"]; + ordinal[label="1, 2, or >2 groups?"]; + nominal[label="?"]; + + data->IR[label="interval/ratio"]; + data->nominal[label="nominal/discrete"]; + data->ordinal[label="ordinal"]; + + normal[label="1, 2, or >2 groups?"]; + notnormal[label="transform into\nnormal?"]; + trulynotnormal[label="1, 2, or >2 groups?"]; + + 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index 0000000..5d721bd Binary files /dev/null and b/statistics/figs/fig9.pdf differ diff --git a/statistics/figs/frequentistsvsbayesians.png b/statistics/figs/frequentistsvsbayesians.png new file mode 100644 index 0000000..735ee5b Binary files /dev/null and b/statistics/figs/frequentistsvsbayesians.png differ diff --git a/statistics/figs/generate.py b/statistics/figs/generate.py new file mode 100644 index 0000000..1c14641 --- /dev/null +++ b/statistics/figs/generate.py @@ -0,0 +1,106 @@ +from __future__ import division +import seaborn as sns +import sys +sys.path.append('/home/fabee/code/') +from matplotlib.pyplot import * +from fabee.Plotting import * +from scipy import stats +from numpy import * + +sns.set_context("talk", font_scale=1.5, rc={"lines.linewidth": 2.5}) + +# --------------------------------------------------------------------------- +fig, ax = subplots() +fig.subplots_adjust(bottom=.3, left=.3) +n = 50 +x = loadtxt('scripts/thymusglandweights.dat')[:n] +ax.bar([0,1],[mean(x),mean(x)],yerr = [std(x,ddof=1), std(x,ddof=1)/sqrt(n)], + facecolor='dodgerblue', alpha=.8,width=.7, align='center', + error_kw={'color':'k','lw':2}, capsize=10, ecolor='k') +ax.set_title('standard deviation or standard error?',fontsize=14, fontweight='bold') + +ax.set_xlim([-.5,1.5]) +box_off(ax) +#disjoint_axes(ax) +ax.set_xticks([0,1]) +ax.set_xticklabels([r'$\hat\sigma$', r'$\frac{\hat\sigma}{\sqrt{n}}$'], fontsize=30) + +ax.set_ylabel(r'$\frac{1}{n}\sum_{i=1}^n x_i$',fontsize=30, fontweight='bold') + +fig.savefig('figs/StandardErrorOrStandardDeviation.pdf') + +# --------------------------------------------------------------------------- +fig, ax = subplots() + +t = linspace(-5,5,1000) +t2 = linspace(stats.laplace.ppf(0.025),stats.laplace.ppf(1-0.025),1000) + +ax.fill_between(t,stats.laplace.pdf(t),color='dodgerblue') +ax.set_xticks([]) +ax.text(5,-0.05, r'$\hat m$',fontsize=30) +ax.text(0,0.7, r'$m$',fontsize=30) +ax.set_yticks([]) +#disjoint_axes(ax) +box_off(ax) + +ax.set_title('putative sampling distribution of the median',fontsize=14, fontweight='bold') +ax.axis([-5,5,0,.8]) +ax.plot([0,0],[0,.7],'--k',lw=2) + +fig.savefig('figs/samplingDistributionMedian00.pdf') + +ax.fill_between(t2,stats.laplace.pdf(t2),color='crimson') + +fig.savefig('figs/samplingDistributionMedian01.pdf') + +# --------------------------------------------------------------------------- +fig, ax = subplots() +k = 7 +N = 21 +F = stats.f +t = linspace(1e-6,8,1000) +t2= linspace(F.ppf(0.95,k-1,N-k),8,1000) + +ax.fill_between(t,F.pdf(t,k-1,N-k),color='dodgerblue') +ax.fill_between(t2,F.pdf(t2,k-1,N-k),color='crimson') +ax.set_xlabel('group MS/ error MS') +ax.set_ylabel(r'p(group MS/ error MS| $H_0$)') +ax.set_title('F-distribution',fontsize=14, fontweight='bold') +ax.set_ylim((0,0.8)) +box_off(ax) +fig.savefig('figs/Fdistribution00.pdf') + +# --------------------------------------------------------------------------- +fig, ax = subplots() +n = 5 +p = stats.t.pdf +t = linspace(-5,8,1000) +t0 = 1.5 +t00 = 1. + +mu0 = 3 +t1 = linspace(-5,t00,1000) +t2 = linspace(t0,8,1000) +t3 = linspace(-5,-t0,1000) +ax.fill_between(t,p(t,n-1),color='dodgerblue',alpha=1) +ax.fill_between(t2,p(t2,n-1),color='indigo',alpha=1) +ax.fill_between(t3,p(t3,n-1),color='indigo',alpha=1) +ax.set_xlabel('t') +ax.set_ylabel(r'sampling distribution') +ax.set_ylim((0,0.8)) +box_off(ax) +fig.savefig('figs/experimentalDesign00.pdf') + + +ax.fill_between(t,p(t,n-1,loc=mu0),color='lime',alpha=.5) +ax.fill_between(t1,p(t1,n-1,loc=mu0),color='magenta',alpha=1) +ax.arrow(0,.4,mu0,0,head_width=0.05) +ax.arrow(mu0,.4,-mu0,0,head_width=0.05) +ax.text(mu0/2,.45,r'$\delta$',fontsize=20) +ax.set_xlabel('t') +ax.set_ylabel(r'sampling distribution') +ax.set_ylim((0,0.8)) +box_off(ax) +fig.savefig('figs/experimentalDesign01.pdf') + + diff --git a/statistics/figs/generate03.py b/statistics/figs/generate03.py new file mode 100644 index 0000000..56137d2 --- /dev/null +++ b/statistics/figs/generate03.py @@ -0,0 +1,265 @@ +import sys +import seaborn as sns +sys.path.append('/home/fabee/code/') +from matplotlib.pyplot import * +from fabee.Plotting import * +from scipy import stats +from numpy import * + +sns.set_context("talk", font_scale=1.5, rc={"lines.linewidth": 2.5}) + +def hinton(matrix, max_weight=None, ax=None): + """Draw Hinton diagram for visualizing a weight matrix.""" + ax = ax if ax is not None else gca() + + if not max_weight: + max_weight = 2**np.ceil(np.log(np.abs(matrix).max())/np.log(2)) + + ax.patch.set_facecolor('gray') + ax.set_aspect('equal', 'box') + ax.xaxis.set_major_locator(NullLocator()) + ax.yaxis.set_major_locator(NullLocator()) + + for (x,y),w in np.ndenumerate(matrix): + color = 'white' if w > 0 else 'black' + size = np.sqrt(np.abs(w)) + rect = Rectangle([x - size / 2, y - size / 2], size, size, + facecolor=color, edgecolor=color) + ax.add_patch(rect) + + ax.autoscale_view() + ax.invert_yaxis() + +# --------------------------------------------------------------------------- +fig, ax = subplots() +fig.subplots_adjust(bottom=.2) + +ax.bar([0,1],[.2,.8],facecolor='dodgerblue', alpha=.8,width=.7, align='center') +ax.set_title('Bernoulli distribution',fontsize=16, fontweight='bold') + +ax.set_xlim([-.5,1.5]) +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('outcomes',fontsize=14, fontweight='bold') +ax.set_ylabel('P(outcome)',fontsize=14, fontweight='bold') +ax.set_xticks([0,1]) +ax.set_xticklabels([0,1]) +ax.set_ylim((0,1)) + +fig.savefig('figs/Bernoulli.pdf') + +# --------------------------------------------------------------------------- +fig, ax = subplots() +fig.subplots_adjust(bottom=.2) +n = 5 +k = arange(0,n) +ax.bar(k,0*k+1./n,facecolor='dodgerblue', alpha=.8,width=.7, align='center') +ax.set_title('uniform distribution',fontsize=16, fontweight='bold') + +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('k',fontsize=14, fontweight='bold') +ax.set_ylabel('P(X=k)',fontsize=14, fontweight='bold') +ax.set_xticks(k) +ax.set_xticklabels(k+1) +ax.set_ylim((0,1)) +fig.savefig('figs/Uniform.pdf') + +# --------------------------------------------------------------------------- + +for i,(n,p) in enumerate(zip([10,20],[.5,.8])): + fig, ax = subplots() + + fig.subplots_adjust(bottom=.2) + k = arange(n+1) + + ax.bar(k,stats.binom.pmf(k,n,p),facecolor='dodgerblue', alpha=.8,width=.7, align='center') + ax.set_title(r'binomial distribution $B\left(%.2f, %i\right)$' % (p,n),fontsize=16, fontweight='bold') + + box_off(ax) + #disjoint_axes(ax) + ax.set_xlabel('k',fontsize=14, fontweight='bold') + ax.set_ylabel('P(k)',fontsize=14, fontweight='bold') + ax.set_xticks(k) + ax.set_xticklabels(k) + ax.set_xlim((-1,n+1)) + ax.set_ylim((0,1)) + fig.savefig('figs/Binomial%02i.pdf' % (i,)) + + +# --------------------------------------------------------------------------- +n = 20 +for i, lam in enumerate([5, 0.05]): + fig, ax = subplots() + + fig.subplots_adjust(bottom=.2) + k = arange(n+1) + + ax.bar(k,stats.poisson.pmf(k,lam),facecolor='dodgerblue', alpha=.8,width=.7, align='center') + ax.set_title(r'Poisson distribution $\lambda=%.2f$' % (lam,),fontsize=16, fontweight='bold') + + box_off(ax) + #disjoint_axes(ax) + ax.set_xlabel('k',fontsize=14, fontweight='bold') + ax.set_ylabel('P(k)',fontsize=14, fontweight='bold') + ax.set_xticks(k) + ax.set_xticklabels(k) + ax.set_xlim((-1,n+1)) + ax.set_ylim((0,1)) + fig.savefig('figs/Poisson%02i.pdf' % (i,)) + +# --------------------------------------------------------------------------- +fig, ax = subplots() + +fig.subplots_adjust(bottom=.2) +t = linspace(-3,3,200) +ax.fill_between(t,stats.norm.pdf(t),facecolor='dodgerblue', alpha=.8) +ax.set_title(r'Gaussian/Normal distribution $N(\mu,\sigma)$',fontsize=16, fontweight='bold') + +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('x',fontsize=14, fontweight='bold') +ax.set_ylabel('p(x)',fontsize=14, fontweight='bold') +fig.savefig('figs/Gaussian00.pdf') + +# --------------------------------------------------------------------------- + +fig, ax = subplots() +n = 10 +kk = 5 +p = .5 +fig.subplots_adjust(bottom=.2) +k = arange(n+1) + +ax.bar(k,stats.binom.pmf(k,n,p),facecolor='dodgerblue', alpha=.8,width=.7, align='center') +ax.bar(k[:kk+1],stats.binom.pmf(k[:kk+1],n,p),facecolor='crimson', alpha=.5,width=.7, align='center') +ax.set_title(r'binomial distribution $B\left(\frac{1}{2}, %i\right)$' % (n,), fontsize=16, fontweight='bold') + +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('k',fontsize=14, fontweight='bold') +ax.set_ylabel('P(k)',fontsize=14, fontweight='bold') +ax.set_xticks(k) +ax.set_xticklabels(k) +ax.set_xlim((-1,n+1)) +ax.set_ylim((0,1)) +fig.savefig('figs/BinomialCdf00.pdf' ) + + +fig, ax = subplots() +n = 10 +kk = 5 +p = .5 +fig.subplots_adjust(bottom=.2) +k = arange(n+1) + +ax.bar(k,stats.binom.pmf(k,n,p),facecolor='dodgerblue', alpha=.8,width=.7, align='center',label='p.m.f.') +ax.bar(k[:kk+1],stats.binom.pmf(k[:kk+1],n,p),facecolor='crimson', alpha=.5,width=.7, align='center') +ax.plot(k,stats.binom.cdf(k,n,p),'ok',mfc='crimson', alpha=1.,label='c.d.f.', ms=15) + +ax.set_title(r'binomial distribution $B\left(\frac{1}{2}, %i\right)$' % (n,), fontsize=16, fontweight='bold') +ax.legend(frameon=False, loc='best') +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('k',fontsize=14, fontweight='bold') +ax.set_ylabel('P(k)',fontsize=14, fontweight='bold') +ax.set_xticks(k) +ax.set_xticklabels(k) +ax.set_xlim((-1,n+1)) +ax.set_ylim((0,1.1)) +fig.savefig('figs/BinomialCdf01.pdf' ) + +fig, ax = subplots() +n = 10 +kk = 2 +p = .5 +fig.subplots_adjust(bottom=.2) +k = arange(n+1) + +ax.bar(k,stats.binom.pmf(k,n,p),facecolor='dodgerblue', alpha=.8,width=.7, align='center',label='p.m.f.') +ax.bar(k[:kk+1],stats.binom.pmf(k[:kk+1],n,p),facecolor='crimson', alpha=.5,width=.7, align='center') +ax.bar(k[-kk-1:],stats.binom.pmf(k[-kk-1:],n,p),facecolor='crimson', alpha=.5,width=.7, align='center') + +ax.set_title(r'binomial distribution $B\left(\frac{1}{2}, %i\right)$' % (n,), fontsize=16, fontweight='bold') +ax.legend(frameon=False, loc='best') +box_off(ax) +#disjoint_axes(ax) +ax.set_xlabel('k',fontsize=14, fontweight='bold') +ax.set_ylabel('P(k)',fontsize=14, fontweight='bold') +ax.set_xticks(k) +ax.set_xticklabels(k) +ax.set_xlim((-1,n+1)) +ax.set_ylim((0,1.1)) +fig.savefig('figs/BinomialExample00.pdf' ) + + +#------------------------------------------------------ +fig = figure(figsize=(10,3.5)) +ax = fig.add_axes([.1,.13,.6,.3]) + +n = 10 +p = [.5,.8] +q = [.7, .3] +fig.subplots_adjust(bottom=0.2) +k = arange(n+1) +P = vstack((stats.binom.pmf(k,n,p[0])*q[0], stats.binom.pmf(k,n,p[1])*q[1])).T + +hinton(P, ax = None) + +#disjoint_axes(ax) +ax.set_xticks(k) +ax.set_xticklabels(k) +ax.set_yticks([0,1]) +ax.set_ylim((-.5,1.5)) +ax.set_xlim((-.5,n+.5)) +ax.set_yticklabels(['subject #1', 'subject #2']) +fig.savefig('figs/Joint00.pdf' ) + +ax = fig.add_axes([.75,.13,.2,.3]) +ax.barh([0,1],q, facecolor='dodgerblue',alpha=.8, align='center') +box_off(ax) +#disjoint_axes(ax) +ax.set_xticks([0,.5,1.]) +ax.set_yticks([]) +ax.set_ylim((-.5,1.5)) +fig.savefig('figs/Joint01.pdf' ) + +ax = fig.add_axes([.1,.6,.6,.2]) +ax.bar(k,sum(P,axis=1), facecolor='dodgerblue',alpha=.8, align='center') +a = .7 +ax.axis([-a,n-a+1.5,0,1]) +box_off(ax) +#disjoint_axes(ax) +ax.set_xticks([]) +ax.set_yticks([0,.3]) +ax.set_ylim((0,.3)) +fig.savefig('figs/Joint02.pdf' ) + + +#------------------------------------------------------ +n = 10 +k = arange(n+1) +p = [.5,.8] +q = [.7, .3] +P = vstack((stats.binom.pmf(k,n,p[0])*q[0], stats.binom.pmf(k,n,p[1])*q[1])) +Pk = sum(P,axis=0) + +fig = figure() +for i,kk in enumerate(k): + ax = fig.add_subplot(3,4,i+1) + fig.subplots_adjust(bottom=0.2) + + ax.bar([0,1],P[:,i]/Pk[i], facecolor='dodgerblue',alpha=.8, align='center') + + + #disjoint_axes(ax) + ax.set_xticks([0,1]) + ax.set_xticklabels(['#1','#2'], fontsize=8) + ax.set_yticks([0,.5,1]) + ax.set_yticklabels([0,.5,1],fontsize=8) + ax.set_xlim((-.5,1.5)) + ax.set_ylim((0,1)) + ax.set_title('P({#1,#2}| %i successes)' % (i,), fontsize=8) + +fig.subplots_adjust(wspace=.8, hspace=.8) +fig.savefig('figs/Posterior00.pdf') diff --git a/statistics/figs/generatePlots.py b/statistics/figs/generatePlots.py new file mode 100644 index 0000000..309eba7 --- /dev/null +++ b/statistics/figs/generatePlots.py @@ -0,0 +1,216 @@ +import sys +sys.path.append('/home/fabee/code/') +import seaborn as sns +from matplotlib.pyplot import * +from scipy import stats +from numpy import * + +from matplotlib.ticker import NullFormatter + +sns.set_context("talk", font_scale=1.5, rc={"lines.linewidth": 2.5}) + +# --------------- PLOT 1 ------------------------- +# the random data +distr = stats.uniform +col = '+*0<>v' + +for k,distr in enumerate([stats.laplace, stats.norm, stats.expon,stats.uniform]): + col = [col[i] for i in random.permutation(6)] + x = random.randn(5000) + + nullfmt = NullFormatter() # no labels + + # definitions for the axes + left, width = 0.1, 0.65 + bottom, height = 0.1, 0.65 + bottom_h = left_h = left+width+0.02 + + rect_scatter = [left + 0.22, bottom + 0.22 , width, height] + rect_histx = [left + 0.22, bottom, width, 0.2] + rect_histy = [left, bottom + 0.22 , 0.2, height] + + # start with a rectangular Figure + fig = figure(figsize=(8,8)) + + axQQ = axes(rect_scatter) + axHistx = axes(rect_histx) + axHisty = axes(rect_histy) + + # no labels + axHistx.yaxis.set_major_formatter(nullfmt) + axHisty.xaxis.set_major_formatter(nullfmt) + axQQ.xaxis.set_major_formatter(nullfmt) + axQQ.yaxis.set_major_formatter(nullfmt) + + + + # the scatter plot: + z = distr.ppf(stats.norm.cdf(x)) + y = linspace(amin(z),amax(z),1000) + + + z = distr.ppf(stats.norm.cdf(x)) + if distr != stats.norm: + if distr == stats.uniform: + axQQ.plot(x, z,'ok',marker=col[0],ms=5,label='c.d.f.') + else: + axQQ.plot(x, z,'ok',marker=col[0],ms=5,label='correct') + + if distr != stats.expon: + axQQ.plot((z-amin(z))/(amax(z)-amin(z))*(amax(x)-amin(x)) + amin(x),\ + (x-amin(x))/(amax(x)-amin(x))*(amax(z)-amin(z)) + amin(z),'ok',marker=col[1],ms=5) + axQQ.plot(x, (x-amin(x))/(amax(x)-amin(x))*(amax(z)-amin(z)) + amin(z),'ok',marker=col[2],ms=5) + + + # now determine nice limits by hand: + axHistx.hist(x, bins=100,normed=True) + if distr != stats.expon: + axHisty.plot(distr.pdf(y),y) + z2 = distr.pdf(y) + y = hstack((y[0],y,y[-1])) + z2 = hstack((0,z2,0)) + axHisty.fill(z2,y,color=(.0,.0,1.)) + + axQQ.set_xlim(axHistx.get_xlim()) + axQQ.set_ylim(axHisty.get_ylim()) + + if distr == stats.uniform: + axQQ.set_ylim((-.1,1.1)) + axHisty.set_ylim((-.1,1.1)) + axHisty.set_xlim((.0,1.1)) + + axHistx.set_xlabel('x',fontsize=16) + axHistx.set_ylabel('p(x)',fontsize=16) + axHisty.set_ylabel('y',fontsize=16) + axHisty.set_xlabel('p(y)',fontsize=16) + + fig.savefig('figs/HE%i.png' % (k,)) + if distr == stats.norm: + axQQ.plot(x, z,'ok',marker=col[0],ms=5) + elif distr == stats.expon: + axHisty.plot(distr.pdf(y),y) + z2 = distr.pdf(y) + y = hstack((y[0],y,y[-1])) + z2 = hstack((0,z2,0)) + axHisty.fill(z2,y,color=(.0,.0,1.)) + + else: + axQQ.legend(loc=2) + fig.savefig('figs/HE%iSolution.png' % (k,)) + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) +xx = linspace(-3.,stats.norm.ppf(1-0.2),1000) + +x = linspace(-3.,3.,1000) +y = stats.norm.pdf(x,scale=1) +yy = stats.norm.pdf(xx,scale=1) +yy[0] = 0 +yy[-1] = 0 + +ax.plot(x,y,'k-',lw=2) +ax.plot(x,stats.norm.pdf(x),'k-',lw=1) +ax.set_xlabel('x',fontsize=16) +ax.set_ylabel('pdf',fontsize=16) +ax.fill(xx,yy,'b') + +ax.set_xlim(-3.,3.) + +ax.text(xx[-1],-.1,'b'); + +ax.text(xx[-1],.4,'p(x)',color='k'); +ax.text(xx[0],.3,'F(b) = P(x <= b)',color='b'); + +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) + +fig.savefig('figs/cdf.png') + +#----------------------------- +fig = figure() + + +ax = fig.add_subplot(111) +xx = linspace(-3.,stats.norm.ppf(1-0.2),1000) + +x = linspace(-3.,3.,1000) +y = stats.norm.pdf(x,scale=1) +yy = stats.norm.pdf(xx,scale=1) +yy[0] = 0 +yy[-1] = 0 + +ax.plot(x,y,'k-',lw=2) +ax.plot(x,stats.norm.cdf(x),'b-',lw=1) +ax.set_xlabel('x/b',fontsize=16) +ax.set_ylabel('pdf/cdf',fontsize=16) + +ax.set_xlim(-3.,3.) + +ax.text(xx[-1],.4,'p(x)',color='k'); +ax.text(xx[0],.3,'F(b) = P(x <= b)',color='b'); + +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) + +fig.savefig('figs/cdf2.png') + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) + +x = hstack((linspace(-3.,stats.norm.ppf(0.13),1000),\ + linspace(stats.norm.ppf(1-0.13),3.,1000))) + +xx = hstack((linspace(-3.,stats.norm.ppf(0.2),1000),\ + linspace(stats.norm.ppf(1-0.2),3.,1000))) + +y = stats.norm.pdf(x,scale=1) +yy = stats.norm.pdf(xx,scale=1) + +y[[0,999,1000,-1]] = 0 +yy[[0,999,1000,-1]] = 0 + +t = linspace(-3.,3.,1000) +ax.plot(t,stats.norm.pdf(t),'k-',lw=2) + +ax.fill(xx[:1000],yy[:1000],'b') +ax.fill(xx[1000:],yy[1000:],'b') +ax.text(xx[1000],-.1,'b') +ax.text(xx[999],-.1,'-b') +ax.text(.2,.7,'P(|x|>b) =$\\alpha$',color='b'); + +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) + +fig.savefig('figs/pval0.png') + +#--------------------------------------------------- +fig = figure() + +ax = fig.add_subplot(111) + +t = linspace(-3.,3.,1000) +ax.plot(t,stats.norm.pdf(t),'k-',lw=2) + + +ax.fill(x[:1000],y[:1000],'r') +ax.fill(x[1000:],y[1000:],'r') + + +ax.text(x[1000],-.1,'t') +ax.text(x[999],-.1,'-t') + +ax.text(.2,.5,'P(|x| > t) = p-value',color='r'); + +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) + +fig.savefig('figs/ pval1.png') + + + +# show() + + +#----------------------------- diff --git a/statistics/figs/generateTPlots.py b/statistics/figs/generateTPlots.py new file mode 100644 index 0000000..36f10dd --- /dev/null +++ b/statistics/figs/generateTPlots.py @@ -0,0 +1,184 @@ +import sys +import seaborn as sns +sys.path.append('/home/fabee/code') +from matplotlib.pyplot import * +from scipy import stats +from numpy import * + +sns.set_context("talk", font_scale=1.5, rc={"lines.linewidth": 2.5}) + +# define the curves +x = np.linspace(2, 20, 200) +n = 16. + + +X =random.randn(n)*4.+12.5 +fig = figure() +ax = fig.add_subplot(111) +ax.set_xlim(5, 18) +#ax.set_ylim(0, .5) +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45); + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) +fig.savefig('figs/repetition0.png') + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) + +ax.plot(x,-stats.norm.pdf(x,loc=10,scale=4),'orange',label=r'Null distribution of x') + + +ax.set_xlim(5, 18) +# ax.set_ylim(0, .5) + +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45); +ax.legend() + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) + +fig.savefig('figs/repetition1.png') + + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) + +ax.plot(x,-stats.norm.pdf(x,loc=10,scale=4),'orange',label=r'Null distribution of x') +ax.plot(x,-stats.t.pdf(x,n-1,loc=10,scale=1),'b',label=r'Null distribution of $t$') + +ax.set_xlim(5, 18) +# ax.set_ylim(0, .5) + +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45); +ax.legend() + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) + +fig.savefig('figs/repetition2.png') + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) +xx = linspace(stats.norm.ppf(0.05),stats.norm.ppf(1-0.05),100) +xx += 10. + +yy = -stats.norm.pdf(xx,loc=10.,scale=1) +xx = hstack((xx[0],xx,xx[-1])) +yy = hstack((0,yy,0)) + +ax.plot(x,-stats.norm.pdf(x,loc=10,scale=4),'orange',label=r'Null distribution of x') +ax.plot(x,-stats.t.pdf(x,n-1,loc=10,scale=1),'b',label=r'Null distribution of $t$') + +ax.fill(xx,yy,'c') + +ax.set_xlim(5, 18) +# ax.set_ylim(0, .5) + +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45); +ax.legend() + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) + +fig.savefig('figs/repetition3.png') + + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) +xx = linspace(stats.norm.ppf(0.05),stats.norm.ppf(1-0.05),100) +xx += 10. + +yy = -stats.norm.pdf(xx,loc=10.,scale=1) +xx = hstack((xx[0],xx,xx[-1])) +yy = hstack((0,yy,0)) + +ax.plot(x,-stats.norm.pdf(x,loc=10,scale=4),'orange',label=r'Null distribution of x') +ax.plot(x,-stats.t.pdf(x,n-1,loc=10,scale=1),'b',label=r'Null distribution of $t$') + +ax.fill(xx,yy,'c') + +ax.set_xlim(5, 18) +# ax.set_ylim(0, .5) + +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45) + +ax.plot([xx[0],xx[-1]],[0,0],'-g',label=r'$H_0$',lw=4) +ax.plot([0,xx[0]],[0,0],'-r',label=r'$H_1$',lw=4) +ax.plot([xx[-1],20],[0,0],'-r',lw=4) + +ax.legend() + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) + +fig.savefig('figs/repetition4.png') + + +# ####################################################3 +fig = figure() + + +ax = fig.add_subplot(111) + +ax.plot(x,-stats.norm.pdf(x,loc=10,scale=4),'orange',label=r'Null distribution of x') +ax.plot(x,-stats.t.pdf(x,n-1,loc=10,scale=1),'b',label=r'Null distribution of $t$') + +xx = linspace(0,stats.norm.ppf(0.05)+10.,100) +yy = -stats.norm.pdf(xx,loc=10.,scale=1) +xx = hstack((xx[0],xx,xx[-1])) +yy = hstack((0,yy,0)) +ax.fill(xx,yy,'magenta') + +xx = linspace(stats.norm.ppf(1-0.05)+10.,20,100) +yy = -stats.norm.pdf(xx,loc=10.,scale=1) +xx = hstack((xx[0],xx,xx[-1])) +yy = hstack((0,yy,0)) +ax.fill(xx,yy,'magenta') + +ax.set_xlim(5, 18) +# ax.set_ylim(0, .5) + +ax.plot([10,10],[-.2,.2],'k-',lw=2) +ax.text(10,.3,r'stimulus position',rotation=-30); +ax.plot([12.5,12.5],[-.2,.2],'b-',lw=2) +ax.text(12.5,.3,r'$\hat\mu$',rotation=-45); +ax.legend() + +ax.set_xlabel('x eye position') +#XKCDify(ax, expand_axes=True,yaxis_loc=0,xaxis_loc=0) +ax.plot(X,0*X,'ob',label='fixations',mfc='orange',ms=10) + +fig.savefig('figs/repetition5.png') diff --git a/statistics/figs/hunger.png b/statistics/figs/hunger.png new file mode 100644 index 0000000..445ad6e Binary files /dev/null and b/statistics/figs/hunger.png differ diff --git a/statistics/figs/mensqqplot.pdf b/statistics/figs/mensqqplot.pdf new file mode 100644 index 0000000..ce49ba0 Binary files /dev/null and b/statistics/figs/mensqqplot.pdf differ diff --git a/statistics/figs/multipletesting.pdf b/statistics/figs/multipletesting.pdf new file mode 100644 index 0000000..a2c4cd2 Binary files /dev/null and b/statistics/figs/multipletesting.pdf differ diff --git a/statistics/figs/multipletesting.py b/statistics/figs/multipletesting.py new file mode 100644 index 0000000..03e019e --- /dev/null +++ b/statistics/figs/multipletesting.py @@ -0,0 +1,72 @@ +from __future__ import division +from numpy import * +from scipy import stats +from matplotlib.pyplot import * + +N = random.randn + +m = 2000 +n = 20 + +T = zeros((m,)) +R = zeros((m,)) +pT = zeros((m,)) +pR = zeros((m,)) + +for k in xrange(m): + x = N(n) + y = N(n) + + T[k], pT[k] = stats.ttest_ind(x,y) + R[k], pR[k] = stats.ranksums(x,y) + +a = stats.t.ppf([0.025,1.-0.025], n-1) +b = stats.norm.ppf([0.025,1.-0.025]) + + +fig = figure(figsize=(8,8),dpi=100) +ax = fig.add_axes([.3,.3,.6,.6]) +axb = fig.add_axes([.3,.1,.6,.2]) +axl = fig.add_axes([.1,.3,.2,.6]) + +ax.plot(T,R,'ok',mfc=(.7,.7,.7)) +axb.hist(T,bins=50,facecolor=(1.,.7,.7),normed=True) +axl.hist(R,bins=50,facecolor=(.7,.7,1.),normed=True,orientation='horizontal') +axl.axis([0,1,-5,5]) +axb.plot([a[0],a[0]],[0,1],'k--',lw=2) +axb.plot([a[1],a[1]],[0,1],'k--',lw=2) + +axl.plot([0,1],[b[0],b[0]],'k--',lw=2) +axl.plot([0,1],[b[1],b[1]],'k--',lw=2) +axl.set_ylabel('standardized U statistic', fontsize=16) +axb.set_xlabel('t statistic', fontsize=16) + +# print sum(1.*(T < a[0] ))/m + sum(1.*(T > a[1]))/m +# print sum(1.*(R < b[0] ))/m + sum(1.*(R > b[1]))/m + +ax.fill([-5,a[0],a[0],-5],[-5,-5,5,5],color=(1.,.7,.7),alpha=.5) +ax.fill([a[1],5,5,a[1]],[-5,-5,5,5],color=(1.,.7,.7),alpha=.5) +axb.fill([-5,a[0],a[0],-5],[0,0,1,1],color=(1.,.7,.7),alpha=.5) +axb.fill([a[1],5,5,a[1]],[0,0,1,1],color=(1.,.7,.7),alpha=.5) + + +ax.fill([-5,-5,5,5],[-5,b[0],b[0],-5],color=(.7,.7,1.),alpha=.5) +ax.fill([-5,-5,5,5],[b[1],5,5,b[1]],color=(.7,.7,1.),alpha=.5) +axl.fill([0,0,1,1],[-5,b[0],b[0],-5],color=(.7,.7,1.),alpha=.5) +axl.fill([0,0,1,1],[b[1],5,5,b[1]],color=(.7,.7,1.),alpha=.5) + + + +axb.axis([-5,5,0,1]) +ax.axis([-5,5,-5,5]) + +axl.set_xticks([]) +axb.set_yticks([]) +axl = axl.twiny() +axb = axb.twinx() +axl.set_xticks([0,.5,1.]) +axb.set_yticks([0,.5,1.]) + + + +fig.savefig('multipletesting.pdf') diff --git a/statistics/figs/nnqqplot.pdf b/statistics/figs/nnqqplot.pdf new file mode 100644 index 0000000..9942613 Binary files /dev/null and b/statistics/figs/nnqqplot.pdf differ diff --git a/statistics/figs/onetailed.png b/statistics/figs/onetailed.png new file mode 100644 index 0000000..155aa64 Binary files /dev/null and b/statistics/figs/onetailed.png differ diff --git a/statistics/figs/power.pdf b/statistics/figs/power.pdf new file mode 100644 index 0000000..5089bbd Binary files /dev/null and b/statistics/figs/power.pdf differ diff --git a/statistics/figs/probtree00.dot b/statistics/figs/probtree00.dot new file mode 100644 index 0000000..506c640 --- /dev/null +++ b/statistics/figs/probtree00.dot @@ -0,0 +1,29 @@ +digraph subject { + size="6,6"; + node [color=lightblue2, style=filled]; + rankdir=LR; + + + s10[label=""]; + s11[label=""]; + s1other[label="...", style=none, border=0, color=white]; + + s20[label=""]; + s21[label=""]; + s2other[label="...", style=none, border=0, color=white]; + + F1[label=""]; + F2[label=""]; + + + ""->F1; + ""->F2; + F1->s10; + F1->s11; + F1->s1other; + + F2->s20; + F2->s21; + F2->s2other; + +} diff --git a/statistics/figs/probtree00.pdf b/statistics/figs/probtree00.pdf new file mode 100644 index 0000000..c156306 Binary files /dev/null and b/statistics/figs/probtree00.pdf differ diff --git a/statistics/figs/probtree01.dot b/statistics/figs/probtree01.dot new file mode 100644 index 0000000..bdcae34 --- /dev/null +++ b/statistics/figs/probtree01.dot @@ -0,0 +1,29 @@ +digraph subject { + size="6,6"; + node [color=lightblue2, style=filled]; + rankdir=LR; + + + s10[label="k=0"]; + s11[label="k=1"]; + s1other[label="...", style=none, border=0, color=white]; + + s20[label="k=0"]; + s21[label="k=1"]; + s2other[label="...", style=none, border=0, color=white]; + + F1[label="subject #1"]; + F2[label="subject #2"]; + + + ""->F1; + ""->F2; + F1->s10; + F1->s11; + F1->s1other; + + F2->s20; + F2->s21; + F2->s2other; + +} diff --git a/statistics/figs/probtree01.pdf b/statistics/figs/probtree01.pdf new file mode 100644 index 0000000..7016fa8 Binary files /dev/null and b/statistics/figs/probtree01.pdf differ diff --git a/statistics/figs/probtree02.dot b/statistics/figs/probtree02.dot new file mode 100644 index 0000000..124f640 --- /dev/null +++ b/statistics/figs/probtree02.dot @@ -0,0 +1,29 @@ +digraph fish { + size="6,6"; + node [color=lightblue2, style=filled]; + rankdir=LR; + + + s10[label="k=0"]; + s11[label="k=1"]; + s1other[label="...", style=none, border=0, color=white]; + + s20[label="k=0"]; + s21[label="k=1"]; + s2other[label="...", style=none, border=0, color=white]; + + F1[label="fish #1"]; + F2[label="fish #2"]; + + + ""->F1[label="p=0.7"]; + ""->F2[label="p=0.3"]; + F1->s10[label="P(k=0|#1)"]; + F1->s11[label="P(k=1|#1)"]; + F1->s1other[label="..."]; + + F2->s20[label="P(k=0|#2)"]; + F2->s21[label="P(k=1|#2)"]; + F2->s2other[label="..."]; + +} diff --git a/statistics/figs/probtree02.pdf b/statistics/figs/probtree02.pdf new file mode 100644 index 0000000..cf4d6aa Binary files /dev/null and b/statistics/figs/probtree02.pdf differ diff --git a/statistics/figs/probtree03.dot b/statistics/figs/probtree03.dot new file mode 100644 index 0000000..7e0b34e --- /dev/null +++ b/statistics/figs/probtree03.dot @@ -0,0 +1,29 @@ +digraph subject { + size="6,6"; + node [color=lightblue2, style=filled]; + rankdir=LR; + + + s10[label="P(k=0, subject#1)\n=0.7 P(k=0|#1)"]; + s11[label="P(k=1, subject#1)\n=0.7 P(k=1|#1)"]; + s1other[label="...", style=none, border=0, color=white]; + + s20[label="P(k=0, subject#2)\n=0.3 P(k=0|#2)"]; + s21[label="P(k=1, subject#2)\n=0.3 P(k=1|#2)"]; + s2other[label="...", style=none, border=0, color=white]; + + F1[label="subject #1"]; + F2[label="subject #2"]; + + + ""->F1[label="p=0.7"]; + ""->F2[label="p=0.3"]; + F1->s10[label="P(k=0|#1)"]; + F1->s11[label="P(k=1|#1)"]; + F1->s1other[label="..."]; + + F2->s20[label="P(k=0|#2)"]; + F2->s21[label="P(k=1|#2)"]; + F2->s2other[label="..."]; + +} diff --git a/statistics/figs/probtree03.pdf b/statistics/figs/probtree03.pdf new file mode 100644 index 0000000..56862f6 Binary files /dev/null and b/statistics/figs/probtree03.pdf differ diff --git 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true)=1-\beta\n=power"]; + HA->FN[label="P(reject HA| HA true)=\beta\n=type II"]; + + +} diff --git a/statistics/figs/testframework00.pdf b/statistics/figs/testframework00.pdf new file mode 100644 index 0000000..a44f348 Binary files /dev/null and b/statistics/figs/testframework00.pdf differ diff --git a/statistics/figs/testframework01.dot b/statistics/figs/testframework01.dot new file mode 100644 index 0000000..845d0ba --- /dev/null +++ b/statistics/figs/testframework01.dot @@ -0,0 +1,25 @@ +digraph G { + rankdir=KR; + node [fontsize=12, shape=oval, style=filled, nodesep=0.95,ranksep=0.95, color="dodgerblue"]; + edge [penwidth=2, fontsize=10 ]; + + root[label=""]; + H0[label="H0 is true",color="green"]; + HA[label="HA is true"]; + TN[label="true negative:\naccept H0",color="green"]; + FP[label="false positive:\nreject H0",color="green"]; + TP[label="true positive:\nreject H0"]; + FN[label="false negative:\naccept H0"]; + + + root->H0[label="P(H0)"]; + root->HA[label="1-P(H0)"]; + + H0->TN[label="P(accept H0| H0 true)=1-alpha"]; + H0->FP[label="P(reject H0| H0 true)=alpha\n=type I"]; + + HA->TP[label="P(accept HA| HA true)=1-\beta\n=power"]; + HA->FN[label="P(reject HA| HA true)=\beta\n=type II"]; + + +} diff --git a/statistics/figs/testframework01.pdf b/statistics/figs/testframework01.pdf new file mode 100644 index 0000000..05070bb Binary files /dev/null and b/statistics/figs/testframework01.pdf differ diff --git a/statistics/figs/twotailed.png b/statistics/figs/twotailed.png new file mode 100644 index 0000000..382a0a2 Binary files /dev/null and b/statistics/figs/twotailed.png differ diff --git a/statistics/figs/typeingqqplot.pdf b/statistics/figs/typeingqqplot.pdf new file mode 100644 index 0000000..fdc24d4 Binary files /dev/null and b/statistics/figs/typeingqqplot.pdf differ diff --git a/statistics/material/Cumming, Fidler, Vaux - 2007.pdf b/statistics/material/Cumming, Fidler, Vaux - 2007.pdf new file mode 100644 index 0000000..34f2664 Binary files /dev/null and b/statistics/material/Cumming, Fidler, Vaux - 2007.pdf differ diff --git a/statistics/progressionbar.tex b/statistics/progressionbar.tex new file mode 100644 index 0000000..836f5d1 --- /dev/null +++ b/statistics/progressionbar.tex @@ -0,0 +1,123 @@ + +%%% Local Variables: +%%% mode: latex +%%% TeX-master: t +%%% End: + + +\newenvironment<>{description}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=white,bg=gray} + \setbeamercolor{block body}{fg=black,bg=gray!30} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{family=\sffamily, series=\bfseries} + \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{task}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=cyan!40} + \setbeamercolor{block body}{fg=black,bg=cyan!20} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} + +\newenvironment<>{summary}[1]{% + \begin{actionenv}#2% + \def\insertblocktitle{#1}% + \par% + \mode{% + \setbeamercolor{block title}{fg=black,bg=blue!40} + \setbeamercolor{block body}{fg=black,bg=blue!20} + % \setbeamercolor{itemize item}{fg=orange!20!black} + % \setbeamertemplate{itemize item}[triangle] + \setbeamerfont{block title}{series=\bfseries} + % \setbeamerfont{block body}{family=\ttfamily} + }% + \usebeamertemplate{block begin}} + {\par\usebeamertemplate{block end}\end{actionenv}} +%%%%%%%%%%%%%%%%%%% PROGRESSBAR %%%%%%%%%%%%%%%%%%%%%%%%%% + +\definecolor{pbblue}{HTML}{0A75A8}% filling color for the progress bar +\definecolor{pbgray}{HTML}{575757}% background color for the progress bar +\definecolor{pbgreen}{HTML}{57EE57}% green color for the progress bar + +\newcounter{slideminutes} +\newcounter{minutes} +\newcounter{totalminutes} +\setcounter{totalminutes}{0} +\setcounter{totalminutes}{105} + +\makeatletter +\def\progressbar@progressbar{} % the progress bar +\newcount\progressbar@tmpcounta% auxiliary counter +\newcount\progressbar@tmpcountb% auxiliary counter +\newcount\progressbar@tmpcountc% auxiliary counter +\newdimen\progressbar@pbht %progressbar height +\newdimen\progressbar@pbwd %progressbar width +\newdimen\progressbar@pbwda %progressbar width +\newdimen\progressbar@tmpdim % auxiliary dimension +\newdimen\progressbar@tmpdima % auxiliary dimension + +\progressbar@pbwd=\linewidth +\progressbar@pbht=1.5ex + + + +% the progress bar +\def\progressbar@progressbar{% + + % \progressbar@tmpcounta=\insertframenumber + % \progressbar@tmpcountb=\inserttotalframenumber + \progressbar@tmpcounta=\theminutes + \progressbar@tmpcountb=\thetotalminutes + \progressbar@tmpcountc=\theslideminutes + + \progressbar@tmpdim=\progressbar@pbwd + \divide\progressbar@tmpdim by \progressbar@tmpcountb + \multiply\progressbar@tmpdim by \progressbar@tmpcounta + + \progressbar@tmpdima=\progressbar@pbwd + \divide\progressbar@tmpdima by \progressbar@tmpcountb + \multiply\progressbar@tmpdima by \progressbar@tmpcountc + + \begin{tikzpicture}[rounded corners=2pt,very thin] + + \shade[top color=pbgray!20,bottom color=pbgray!20,middle color=pbgray!50] + (0pt, 0pt) rectangle ++ (\progressbar@pbwd, \progressbar@pbht); + + \shade[draw=pbblue,top color=pbblue!50,bottom color=pbblue!50,middle color=pbblue] % + (0pt, 0pt) rectangle ++ (\progressbar@tmpdim, \progressbar@pbht); + + \shade[draw=pbblue,top color=pbblue!50,bottom color=pbblue!50,middle color=pbgreen] % + (\progressbar@tmpdim, 0) rectangle ++ (\progressbar@tmpdima, \progressbar@pbht); + + \draw[color=normal text.fg!50] + (0pt, 0pt) rectangle (\progressbar@pbwd, \progressbar@pbht) + node[pos=0.5,color=normal text.fg] {\textnormal{\theminutes / + \thetotalminutes~ min done | the next \theslideminutes~ min will + be on } + }; + + \end{tikzpicture}% +} + +\addtobeamertemplate{headline}{} +{% + \begin{beamercolorbox}[wd=\paperwidth,ht=4ex,center,dp=1ex]{white}% + \progressbar@progressbar% + \end{beamercolorbox}% +} diff --git a/statistics/scripts/bootstrap_mean.m b/statistics/scripts/bootstrap_mean.m new file mode 100644 index 0000000..0def8fb --- /dev/null +++ b/statistics/scripts/bootstrap_mean.m @@ -0,0 +1,13 @@ +load thymusglandweights.dat +x = thymusglandweights(1:50); + +m = 500; +n = length(x); + +mu = zeros(m,1); +for i = 1:m + mu(i) = mean(x(randi(n,n,1))); +end +fprintf("bootstrap standard error: %.4f\n", std(mu)); +fprintf("standard error: %.4f\n", std(x)/sqrt(n)); + diff --git a/statistics/scripts/brainWeight.dat b/statistics/scripts/brainWeight.dat new file mode 100755 index 0000000..c1cb298 --- /dev/null +++ b/statistics/scripts/brainWeight.dat @@ -0,0 +1,185 @@ +1607 0 +1157 0 +1248 0 +1310 0 +1398 0 +1237 0 +1232 0 +1343 0 +1380 0 +1274 0 +1245 0 +1286 0 +1508 0 +1105 0 +1123 0 +1198 0 +1300 0 +1249 0 +1185 0 +915 0 +1345 0 +1107 0 +1357 0 +1227 0 +1205 0 +1435 0 +1289 0 +1093 0 +1211 0 +1260 0 +1193 0 +1330 0 +1130 0 +1357 0 +1193 0 +1232 0 +1321 0 +1260 0 +1380 0 +1230 0 +1136 0 +1029 0 +1223 0 +1240 0 +1264 0 +1020 0 +1415 0 +1410 0 +1275 0 +1230 0 +1085 0 +1048 0 +1181 0 +1103 0 +1165 0 +1547 0 +1173 0 +1660 0 +1307 0 +1535 0 +1315 0 +1257 0 +1424 0 +1309 0 +1170 0 +1412 0 +1270 0 +1230 0 +1233 0 +1561 0 +1193 0 +1272 0 +1355 0 +1137 0 +1354 0 +1110 0 +1265 0 +1407 0 +1227 0 +1330 0 +1222 0 +1305 0 +1475 0 +1177 0 +1337 0 +1145 0 +1070 0 +1305 0 +1085 0 +1303 0 +1390 0 +1532 0 +1238 0 +1233 0 +1280 0 +1245 0 +1459 0 +1157 0 +1302 0 +1385 0 +1310 0 +1342 0 +1303 0 +1248 0 +1115 0 +1365 0 +1227 0 +1353 0 +1125 1 +1027 1 +1112 1 +983 1 +1090 1 +1247 1 +1045 1 +983 1 +972 1 +1045 1 +937 1 +1245 1 +1200 1 +1270 1 +1200 1 +1145 1 +1090 1 +1040 1 +1343 1 +1010 1 +1095 1 +1180 1 +1168 1 +1095 1 +1040 1 +1235 1 +1050 1 +1038 1 +1046 1 +1255 1 +1228 1 +1000 1 +1225 1 +1220 1 +1085 1 +1067 1 +1006 1 +1138 1 +1175 1 +1252 1 +1037 1 +958 1 +1020 1 +1068 1 +1107 1 +1317 1 +952 1 +1056 1 +1203 1 +1183 1 +1392 1 +1130 1 +1284 1 +996 1 +1228 1 +1087 1 +1035 1 +1170 1 +1064 1 +1250 1 +1129 1 +1088 1 +1037 1 +1117 1 +1095 1 +1027 1 +1027 1 +1190 1 +1153 1 +1037 1 +1120 1 +1212 1 +1024 1 +1135 1 +1177 1 +1096 1 +1114 1 diff --git a/statistics/scripts/chirping.dat b/statistics/scripts/chirping.dat new file mode 100644 index 0000000..fbde300 --- /dev/null +++ b/statistics/scripts/chirping.dat @@ -0,0 +1,28 @@ +1 12 3 +2 14 1 +3 11 2 +4 13 1 +5 20 5 +6 14 3 +7 10 0 +8 12 2 +9 8 6 +10 13 3 +11 14 2 +12 15 4 +13 12 3 +14 13 2 +15 8 0 +16 18 5 +17 15 3 +18 12 2 +19 17 2 +20 15 4 +21 11 3 +22 22 4 +23 14 2 +24 18 4 +25 15 5 +26 8 1 +27 13 2 +28 16 3 diff --git a/statistics/scripts/ci_mean.m b/statistics/scripts/ci_mean.m new file mode 100644 index 0000000..0cc8e5b --- /dev/null +++ b/statistics/scripts/ci_mean.m @@ -0,0 +1,19 @@ +load thymusglandweights.dat +x = thymusglandweights(1:50); + +m = 500; +n = length(x); +x = sort(x); +me = zeros(m,1); +for i = 1:m + me(i) = mean(x(randi(n,n,1))); +end + +a1 = tinv(0.025,n-1); +a2 = tinv(1-0.025,n-1); + +se = std(x)/sqrt(n); + +fprintf('bootstrap quantiles: %.4f, %.4f \n', quantile(me,0.025), quantile(me,1-0.025)); +fprintf('analytical quantile: %.4f, %.4f \n', mean(x)+a1*se, mean(x)+a2*se); + diff --git a/statistics/scripts/ci_media.m b/statistics/scripts/ci_media.m new file mode 100644 index 0000000..ad20428 --- /dev/null +++ b/statistics/scripts/ci_media.m @@ -0,0 +1,17 @@ +load thymusglandweights.dat +x = thymusglandweights(1:50); + +m = 500; +n = length(x); +x = sort(x); +me = zeros(m,1); +for i = 1:m + me(i) = median(x(randi(n,n,1))); +end + +a1 = binoinv(0.025,n,.5)-1; +a2 = binoinv(1-0.025,n,.5); + +fprintf('bootstrap quantiles: %.4f, %.4f \n', quantile(me,0.025), quantile(me,1-0.025)); +fprintf('analytical quantile: %.4f, %.4f \n', x(a1),x(a2)); + diff --git a/statistics/scripts/menstrual.dat b/statistics/scripts/menstrual.dat new file mode 100644 index 0000000..f3d27be --- /dev/null +++ b/statistics/scripts/menstrual.dat @@ -0,0 +1,15 @@ +2.600000000000000000e+01 +2.400000000000000000e+01 +2.900000000000000000e+01 +3.300000000000000000e+01 +2.500000000000000000e+01 +2.600000000000000000e+01 +2.300000000000000000e+01 +3.000000000000000000e+01 +3.100000000000000000e+01 +3.000000000000000000e+01 +2.800000000000000000e+01 +2.700000000000000000e+01 +2.900000000000000000e+01 +2.600000000000000000e+01 +2.800000000000000000e+01 diff --git a/statistics/scripts/p_value.m b/statistics/scripts/p_value.m new file mode 100644 index 0000000..eed2074 --- /dev/null +++ b/statistics/scripts/p_value.m @@ -0,0 +1,10 @@ +m = 10000 +n = 10 + +p = zeros(m,1); + +for i = 1:m + x = randn(n,1); + [~,p(i)] = ttest(x,0); +end +hist(p,50) \ No newline at end of file diff --git a/statistics/scripts/thymusglandweights.dat b/statistics/scripts/thymusglandweights.dat new file mode 100644 index 0000000..7427de9 --- /dev/null +++ b/statistics/scripts/thymusglandweights.dat @@ -0,0 +1,10000 @@ +2.960000000000000142e+01 +2.150000000000000000e+01 +2.800000000000000000e+01 +3.460000000000000142e+01 +4.460000000000000142e+01 +3.947192304411176877e+01 +3.268980408994536901e+01 +3.529355962616484987e+01 +1.309602554891901960e+01 +3.614366143097279860e+01 +2.588788859677990217e+01 +3.114498388418756747e+01 +5.337913012275713243e+01 +3.160321867880719537e+01 +2.371044567719946627e+01 +2.046461483859459207e+01 +5.555334479438523942e+01 +4.208665608091054366e+01 +4.093979783452637378e+01 +3.106969294051364372e+01 +1.197732372739084994e+01 +2.466889848609938340e+01 +2.338564730226803334e+01 +2.744354796488791948e+01 +4.380829706014305458e+01 +4.231859121710182592e+01 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+\usepackage[scaled=.90]{helvet} +\usepackage{scalefnt} +\usepackage{tikz} +\usepackage{ textcomp } +\usepackage{soul} +\usepackage{hyperref} +\definecolor{lightblue}{rgb}{.7,.7,1.} +\definecolor{mygreen}{rgb}{0,1.,0} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\mode +{ + \usetheme{Singapore} + \setbeamercovered{opaque} + \usecolortheme{tuebingen} + \setbeamertemplate{navigation symbols}{} + \usefonttheme{default} + \useoutertheme{infolines} + % \useoutertheme{miniframes} +} + +\AtBeginSection[] +{ + \begin{frame} + \begin{center} + \Huge \insertsectionhead + \end{center} + % \frametitle{\insertsectionhead} + % \tableofcontents[currentsection,hideothersubsections] + \end{frame} +} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 + +\setbeamertemplate{blocks}[rounded][shadow=true] + +\title[]{Scientific Computing -- Statistics} +\author[Statistics]{Fabian Sinz\\Dept. Neuroethology, + University T\"ubingen\\ +Bernstein Center T\"ubingen} + +\institute[Scientific Computing]{} + \date{11/27/2013} +%\logo{\pgfuseimage{logo}} + +\subject{Lectures} + +%%%%%%%%%% configuration for code +\lstset{ + basicstyle=\ttfamily, + numbers=left, + showstringspaces=false, + language=Matlab, + commentstyle=\itshape\color{darkgray}, + keywordstyle=\color{blue}, + stringstyle=\color{green}, + backgroundcolor=\color{blue!10}, + breaklines=true, + breakautoindent=true, + columns=flexible, + frame=single, + captionpos=b, + xleftmargin=1em, + xrightmargin=1em, + aboveskip=10pt + } +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\newcommand{\mycite}[1]{ +\begin{flushright} +\tiny \color{black!80} #1 +\end{flushright} +} + +\input{environments.tex} +\makeatother + +\begin{document} + +\begin{frame} + \titlepage + +\end{frame} + +\begin{frame} + \frametitle{plan} + \setcounter{tocdepth}{1} + \tableofcontents + +\end{frame} +\begin{frame} + \frametitle{where to get information about statistics} + \begin{itemize} + \item Samuels, M. L., Wittmer, J. A., \& Schaffner, + A. A. (2010). Statistics for the Life Sciences (4th ed., + p. 668). Prentice Hall. + \item Zar, J. H. (1999). Biostatistical Analysis. (D. Lynch, + Ed.)Prentice Hall New Jersey (4th ed., Vol. 4th, p. 663). Prentice + Hall. doi:10.1037/0012764 + \item \url{http://stats.stackexchange.com} + \end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section[meta-study]{how statisticians think - the meta-study} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% ---------------------------------------------------------- +\begin{frame}[fragile] +\frametitle{statisticians are lazy} +\Large +\only<1>{ + \begin{center} + \includegraphics[width=.8\linewidth]{figs/2012-10-29_16-26-05_771.jpg} + \end{center} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} +}\pause +\only<2>{ + \begin{center} + \includegraphics[width=.8\linewidth]{figs/2012-10-29_16-41-39_523.jpg} + \end{center} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} +}\pause +\only<3>{ + \begin{center} + \includegraphics[width=.8\linewidth]{figs/2012-10-29_16-29-35_312.jpg} + \end{center} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} +} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{the (imaginary) meta-study} +\begin{center} + \only<1>{ + \framesubtitle{finite sampling introduces variation: the sampling distribution} + \includegraphics[width=.8\linewidth]{figs/samplingDistribution.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + }\pause + \only<2>{ + \framesubtitle{statistic vs. population parameter} + \includegraphics[width=.8\linewidth]{figs/statistic1.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + }\pause + \only<3>{ + \framesubtitle{statistic vs. population parameter} + \includegraphics[width=.8\linewidth]{figs/statistic2.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + }\pause + \only<4>{ + \framesubtitle{shat parts of this diagram do we have in real life?} + + \includegraphics[width=.8\linewidth]{figs/samplingDistribution.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + }\pause + \only<5>{ + \framesubtitle{what parts of this diagram do we have in real life?} + + \includegraphics[width=.8\linewidth]{figs/statistic3.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + }\pause + \only<6->{ + \framesubtitle{what statistics does } + \begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=1.\linewidth]{figs/statistic4.png} + \mycite{Hesterberg et al., Bootstrap Methods and Permutation + Tests} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{itemize} + \item it assumes, derives, or simulates the sampling + distribution\pause + \item the sampling distribution makes only sense if you think + about it in terms of the meta study\pause + \item {\color{red} the sampling distribution is the key to + answering questions about the population from the value of + the statistic} + \end{itemize} + \end{minipage} + \end{minipage} + } + +\end{center} +\end{frame} + +% % ---------------------------------------------------------- +\begin{frame} +\frametitle{illustrating examples} +\begin{question}{lung volume of smokers} + Assume you know the sampling distribution of the mean lung volume + of smokers. Would you believe that + the sample came from a group of smokers? + \begin{center} + \includegraphics[width=.6\linewidth]{figs/example01.png} + \end{center} +\end{question} +\end{frame} + +\begin{frame} +\frametitle{illustrating examples} +\begin{question}{lung volume of smokers} + What about now? How would the sampling distribution change if I + change the population to (i) athletes or (ii) old people? + \begin{center} + \includegraphics[width=.6\linewidth]{figs/example02.png} + \end{center} +\end{question} +\end{frame} + + +\begin{frame} +\frametitle{illustrating examples} +\begin{question}{Is this diet effective?} + \begin{center} + \includegraphics[width=.6\linewidth]{figs/example03.png} + \end{center} +\end{question} +\end{frame} + +\begin{frame} +\frametitle{illustrating examples} +\begin{question}{Is this diet effective?} + What do you think now? + \begin{center} + \includegraphics[width=.6\linewidth]{figs/example04.png} + \end{center} +\end{question} +\end{frame} + +\begin{frame} +\frametitle{summary} +\begin{itemize} +\item In statistics, we use finite samples from a population to reason + about features of the population. \pause +\item The particular feature of the population we are interested in is called + {\color{blue} population parameter}. We usually measure this + parameter in our finite sample as well + ({\color{blue}statistic}).\pause +\item Because of variations due to finite sampling the statistic + almost never matches the population parameter. \pause +\item Using the {\color{blue}sampling distribution} of the statistic, we make + statements about the relation between our statistic and the + population parameter. +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{outlook} +{\bf Questions to be addressed} +\begin{itemize} +\item How do we choose the statistic? +\item How do we get the sampling distribution? +\item How does statistical reasoning work in practice? +\end{itemize} +{\bf Perspective} +\begin{itemize} +\item We start by looking at a few standard distribution. +\item We will use those in the statistical tests that follow. +\item For each statistical test, I also try to provide a + non-parametric method. +\end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{probability primer} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{probability models} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{getting the model right} +In statistics/probability it is important to select the correct +distribution. Models are easier to remember if you remember a +``standard situation''. + +\begin{itemize} +\item What is the distribution corresponding to throwing a coin? \pause +\item What in neuroscience/psychology is like throwing a coin (fair or + unfair)?\pause +\item What is the distribution of counting heads in repeated + independent coin tosses?\pause +\item What in neuroscience/psychology corresponds to counting heads in + repeated independent coin tosses? +\end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{the different models} +\only<1>{ + \framesubtitle{Bernoulli distribution} + \begin{center} + \includegraphics[width=.4\linewidth]{figs/Bernoulli.pdf} + + \end{center} + +\begin{itemize} + \item single coin toss (success/ failure) + \item distribution $p(X=1)=p$ + \end{itemize} +}\pause +\only<2>{ + \framesubtitle{uniform distribution} + \begin{center} + \includegraphics[width=.4\linewidth]{figs/Uniform.pdf} + + \end{center} + +\begin{itemize} + \item $n$ items with the same probability of occurence + \item distribution $p(X=k)=\frac{1}{n}$ + \end{itemize} +}\pause +\only<3>{ + \framesubtitle{binomial distribution} + + \begin{center} + \includegraphics[width=.4\linewidth]{figs/Binomial00.pdf} + \includegraphics[width=.4\linewidth]{figs/Binomial01.pdf} + \end{center} + + \begin{itemize} + \item number of $k$ successes/heads in $n$ trials + \item distribution $P(X=k)= {n \choose + k} p^k (1-p)^{n-k}$ + \item parameters $n,p$ + \end{itemize} +}\pause +\only<4>{ + \framesubtitle{Poisson distribution} + + \begin{center} + \includegraphics[width=.4\linewidth]{figs/Poisson00.pdf} + \includegraphics[width=.4\linewidth]{figs/Poisson01.pdf} + \end{center} + + \begin{itemize} + \item successes per time unit for (very) large $n$ and small $p$ + \item distribution $P(X=k) = \frac{\lambda^k + e^{-\lambda}}{k!}$ + \item parameter: success rate $\lambda$ + \end{itemize} +} +\only<5>{ + \framesubtitle{Gaussian/ normal distribution} + + \begin{center} + \includegraphics[width=.4\linewidth]{figs/Gaussian00.pdf} + \end{center} + + \begin{itemize} + \item shows up everywhere (central limit theorem) + \item distribution $p(x) = \frac{1}{\sigma\sqrt{2\pi}}\operatorname{exp}\left\{-\frac{\left(x-\mu\right)^2}{2\sigma^2}\right\}$ + \item parameter: mean $\mu$, standard deviation $\sigma$ + \end{itemize} +} +\only<6>{ + \framesubtitle{caveat} + \begin{question}{important distinction} + \begin{itemize} + \item For {\em discrete} random variables $P(X=k)$ makes sense + (probabilities are like ``single weights''). + \item For {\em continuous} random variables $p(X=x)=0$ (probabilities + are like ``water''). + \item For {\em continuous} random variables it makes only sense to + ask for the probability that they take values in a particular + range. + \end{itemize} + \end{question} + +} + +\end{frame} + + +% ---------------------------------------------------------- + +\begin{frame} +\frametitle{example} +You place a mouse in a circular maze and place some food on the +opposite side. In each trial you record whether the mouse went {\em + left} (``L'') or {\em right} (``R'') to get the food. +\vspace{.5cm} + +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.59\linewidth} + \begin{itemize} + \item What kind of distribution would you expect for the number of + ``R'' in $10$ trials? What is the distribution of the number of + ``L''?\pause + \item Here is the result of $10$ trials: ``LLLLLLLLLL''. What is + the probability of that? + \item What do you conclude from that? + \end{itemize} + \end{minipage} + \begin{minipage}{0.4\linewidth} + \only<1->{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/Binomial00.pdf} + \end{center} + } + \end{minipage} +\end{minipage} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{cumulative distribution function} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{cumulative distribution function (c.d.f.)} + \framesubtitle{we will need that a lot in statistics} + \begin{itemize} + \item The c.d.f. is used to compute the probability that a random + variable is in a particular range. + + \item It is defined as $F(y) = P(X \le y)$ + + \item For the binomial distribution this would be + $$F(k) = P(\mbox{no. of + successes} \le k)\mbox{ in } n \mbox{ trials}$$ + + \item Where could I + see that probability in that plot for $k=5$ and $n=10$? + \begin{center} + \only<1>{ + \includegraphics[width=.5\linewidth]{figs/Binomial00.pdf} + } + \only<2>{ + \includegraphics[width=.5\linewidth]{figs/BinomialCdf00.pdf} + }\pause + \only<3>{ + \includegraphics[width=.5\linewidth]{figs/BinomialCdf01.pdf} + } + + \end{center} + \end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{cumulative distribution function (c.d.f.)} + \framesubtitle{example} + \small + You want to find out whether a subject performs significantly + different from chance in $10$ trials that either are successful or not. + \begin{itemize}[<+->] + \item What would be a good decision rule? + \item[] {\color{gray} We set thresholds on the number of successes + and decide that (s)he is performing at chance if the performance + falls within the thresholds.} + \item What is the distribution of the number of successes in $n=10$ + trials if the subject performs at chance? + \item[] {\color{gray} Binomial with $n=10$ and $p=\frac{1}{2}$} + \item Let's say we set the threshold at $k=2$ and $k=8$, what is the + probability that we think (s)he is {\em not} performing at chance, + even though (s)he is? + \end{itemize} +\end{frame} + +\begin{frame}[fragile] + \frametitle{cumulative distribution function (c.d.f.)} + \framesubtitle{example} + \small + \begin{itemize}[<+->] + \item Let's say we set the threshold at $k=2$ and $k=8$, what is the + probability that we think (s)he is {\em not} performing at chance, + even though (s)he is? + \item[] {\color{gray} The probability for that is $P(X \le 2 \mbox{ + or } X \ge 8)$. Using the c.d.f. that is + \begin{align*} + P(X \le 2 \mbox{ or } X \ge 8) &= P(X \le 2) + P(X \ge 8) + = P(X \le 2) + (1-P(X \le 7)) + \end{align*} + } + \end{itemize} + \only<2>{ + \begin{center} + \includegraphics[width=.5\linewidth]{figs/BinomialExample00.pdf} + \end{center} + } +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{joint and conditional distributions} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame}[fragile] + \frametitle{conditional and marginal $\rightarrow$ joint distribution} + \framesubtitle{Bayes' rule} + \begin{itemize} +\small + \item Assume you ran decision experiments with two subject. Subject \#1 had a success + probability of $50\%$, while subject \#2 achieved $80\%$. + \item $70\%$ of the trials were run with the first subject, $30\%$ of + the trials with the other. + \item Each trial gets saved in a file on the hard disk.\pause + \item Now, let's assume your recording software had a bug and did not + store the subject ID in the file. + \item For a given file, we have two random variables now: subject ID $X$, + number of successes $Y$. + \end{itemize} + \begin{center} + \includegraphics[height=.32\linewidth]{figs/decision01.pdf} + \end{center} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{joint and conditional distributions} + \framesubtitle{definitions} + \begin{definition}{Joint, marginal, and conditional distribution} + \begin{itemize} + \item The {\bf joint distribution $P(X,Y)$} gives the probability + that a particular combination of $X$ and $Y$ occur at the same + time. \pause + \item The {\bf marginal distributions $P(X)$ and $P(Y)$} specify + the probabilities that a particular value occurs if the value of + the other variable is ignored. \pause + \item The {\bf conditional distribution $P(X|Y)$} gives the + probability of particular values of $X$ given that $Y$ has + particular values. + \end{itemize}\pause + \end{definition} + \begin{center} {\color{blue} joint distribution + $\stackrel{\mbox{Bayes' Rule}}{\leftrightarrow}$ + marginal and conditional distribution} + \end{center} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{conditional and marginal $\rightarrow$ joint distribution} + \framesubtitle{Bayes' rule} + \begin{itemize} +\small + \item Assume you ran decision experiments with two subject. Subject \#1 had a success + probability of $50\%$, while subject \#2 achieved $80\%$. + \item $70\%$ of the trials were run with the first subject, $30\%$ of + the trials with the other. + \item What probabilities do I need to write at the edges? + \item What distribution do I use for the subjects ID ($X$)? + \item What distribution do I use for the conditional distribution $Y|X$? + \end{itemize} + \begin{center} + \only<1>{\includegraphics[height=.32\linewidth]{figs/decision01.pdf}} + \only<2>{\includegraphics[height=.32\linewidth]{figs/decision02.pdf}} + \only<3>{\includegraphics[height=.32\linewidth]{figs/decision03.pdf}} + \end{center} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{conditional and marginal $\rightarrow$ joint distribution} + \framesubtitle{Bayes' rule} + \begin{itemize} +\small + \item The joint probability are obtained by multiplying the + probabilities along the paths from the root note to the leaves. + \begin{center} + \includegraphics[height=.32\linewidth]{figs/decision03.pdf} + \end{center}\pause + \item In algebraic terms, this is known as {\em Bayes' rule} (very important!) + $$\color{red} P(Y|X)P(X) = P(X|Y)P(Y) = P(X,Y)$$\pause + \item You can remember it as ``moving variables in front of the + bar'' + $$P(X|Y) P(Y) = P(X,Y|\_)$$ + \end{itemize} + +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{Bayes' rule} + $$P(X|Y)P(Y) = P(Y|X)P(X) = P(X,Y)$$ + + \begin{task}{Independent random variables} + If two random variables are independent, the joint distribution is + the product of their marginals $$ P(X,Y) =P(X) P(Y)$$ + How can you see that from Bayes' rule? + \end{task} + \pause + + \begin{solution}{Solution} + If the variables are independent $P(X|Y) = P(X)$ and $P(Y|X) = + P(Y)$: The probability of $X$ is the same as the probability of + $X$ given that I know $Y$, because knowing $Y$ does not help. + \end{solution} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{Joint $\rightarrow$ marginal and conditional distribution} + \begin{itemize} +\small + \item The plot shows the joint distribution $P(X,Y)$, where $X$ is + the subject id and $Y$ the number of successes in $n=10$ trials. + \begin{center} + \only<-1>{\includegraphics[width=.83\linewidth]{figs/Joint00.pdf}} + \only<2>{\includegraphics[width=.83\linewidth]{figs/Joint01.pdf}} + \only<3>{\includegraphics[width=.83\linewidth]{figs/Joint02.pdf}} + \end{center} + +\only<-1>{ \vspace{2cm}} +\only<2-3>{ \item We can get the marginal distributions via {\em + marginalization} (very important!): + $$\color{red} P(Y) =\sum_{i=1}^2P(X=i, Y) \mbox{ and } P(X) = + \sum_{j=0}^{n} P(X, Y=j)$$} +\only<3->{ \item We can get the conditional distribution via Bayes' rule: + $$P(X|Y)P(Y) = P(X,Y) \Leftrightarrow P(X|Y) = \frac{P(X,Y)}{P(Y)}$$} +\only<-2>{ \vspace{2cm}} + \end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{The posterior} + \begin{itemize} + \small + \item Could we use the probability distribution to get an idea which + subject the number of successes came from?\pause + \item Use Bayes' rule to ``invert'' the conditional distribution + $$P(X|Y=k) = P(X,Y=k)/P(Y=k)$$ + \end{itemize} + \begin{center} + \only<-2>{\includegraphics[height=.28\linewidth]{figs/Joint02.pdf}} + \only<3->{\includegraphics[height=.53\linewidth]{figs/Posterior00.pdf}} + \end{center} + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{summary} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +% ---------------------------------------------------------- +\begin{frame}[fragile] + \frametitle{summary} + \begin{itemize} + \item We need to know certain distributions to use them as sampling + distribution. \pause + \item For many distributions one can use a ``standard situation'' to + remember them. \pause + \item When dealing with two or more random variables one deals with + {\color{blue}joint, marginal}, and {\color{blue}conditional + distributions}.\pause + \item Marginal and conditional distributions can be converted into + the joint distribution via {\color{blue}Bayes' rule}.\pause + \item The conversion in the other direction can be done via + {\color{blue}marginalization} and {\color{blue}Bayes' rule}. + \end{itemize} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{error bars \& confidence intervals} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% ---------------------------------------------------------- +\subsection{errorbars} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{illustrating example} + +As part of a study of the development of the thymus gland, researcher +weighed the glands of $50$ chick embyos after 14 days of +incubation. The following plot depicts the mean thymus gland weights in (mg): +\mycite{modified from SWS exercise 6.3.3.} +\pause +{\bf Which of the two bar plots is the correct way of displaying the + data?} + +\begin{columns} + \begin{column}[l]{.5\linewidth} + \includegraphics[width=\linewidth]{figs/StandardErrorOrStandardDeviation.pdf} + \end{column} + \begin{column}[r]{.5\linewidth} + \pause That depends on what you want to say + \begin{itemize} + \item To give a measure of variability in the data: use the + {\color{blue} standard deviation $\hat\sigma = + \sqrt{\frac{1}{n-1}\sum_{i=1}^n (x_i - \hat\mu)^2}$} + \item To make a statement about the variability in the mean + estimation: use {\color{blue}standard error $\frac{\hat\sigma}{\sqrt{n}}$} + \end{itemize} + \end{column} +\end{columns} + +%%%%%%%%%%%%%%% GO ON HERE %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% that depends: variability (descriptiv statistics, how variable is +% the mean -> inferential, makes only sense in the meta-study setting) +% first matlab exercise: simulate standard error +% recommend paper for eyeballing test results from standard errors +% from std of mean to confidence intervals +% introduce bootstrapping (matlab exercise), then t-statistic +% intervals +% end with standard error of the median (and the thing from wikipedia) +\end{frame} +%------------------------------------------------------------------------------ +\begin{frame} + \frametitle{standard error} + \framesubtitle{bootstrapping} + + \begin{task}{quantifying the variability in the mean} + Download \url{https://www.dropbox.com/s/20l7ptrdc4kkceq/materialNMI.zip} + + Load the dataset {\tt thymusglandweights.dat} into matlab and use + the first $50$ datapoints as your dataset. Repeat the following + steps $m=500$ times: + \begin{enumerate} + \item sample $50$ data points from $x$ with replacement + \item compute their mean and store it + \end{enumerate} + Look at the standard deviation of the computed means and compare + it to the standard error. + \end{task} +\end{frame} + +%------------------------------------------------------------------------------ +\begin{frame}[fragile] + \frametitle{standard error} + \framesubtitle{bootstrapping} + \begin{itemize} + \item The sample standard error $\frac{\hat\sigma}{\sqrt{n}}$ is + {\color{blue}an estimate of the standard deviation of the means} + in repeated experiments which is computed form a single + experiment. + \item When you want to do statistical tests on the mean, it is + better to use the standard error, because one can eyeball + significance from it + \mycite{Cumming, G., Fidler, F., \& Vaux, D. L. (2007). Error bars + in experimental biology. The Journal of Cell Biology, 177(1), + 7--11.} + \item {\color{blue}Bootstrapping} is a way to generate an estimate + of the {\color{blue}sampling distribution of any statistic}. Instead of + sampling from the true distribution, it samples from the + empirical distribution represented by your dataset. + \mycite{Efron, B., \& Tibshirani, R. J. (1994). An Introduction to the Bootstrap. Chapman and Hall/CRC} + \end{itemize} +\end{frame} + +%------------------------------------------------------------------------------ +\begin{frame}[fragile] + \frametitle{standard error of the median?} + {\bf What kind of errorbars should we use for the median?} + + It depends again: + + {\bf Descriptive statistics} + \begin{itemize} + \item As a {\color{blue}descriptive statistic} one could use the {\em median + absolute deviation}: the median of the absolute differences of + the datapoints from the median. + \item Alternatively, one could bootstrap a standard deviation of the + median. + \end{itemize} + \pause + {\bf Inferential statistics} + \begin{itemize} + \item For {\color{blue}inferential statistics} one should use + something that gives the reader {\color{blue}information about + significance}. + \item Here, {\color{blue} confidence intervals} are a better choice. + \end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\subsection{confidence intervals \& bootstrapping} +%------------------------------------------------------------------------------ +\begin{frame} +\frametitle{confidence intervals} +\begin{center} + \only<1>{ + \vspace{.1cm} + \includegraphics[width=.6\linewidth]{figs/2012-10-29_14-55-39_181.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + + }\pause + \only<2>{ + \vspace{.1cm} + \includegraphics[width=.6\linewidth]{figs/2012-10-29_14-56-59_866.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + }\pause + \only<3>{ + \vspace{.1cm} + \includegraphics[width=.4\linewidth]{figs/2012-10-29_14-58-18_054.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + }\pause + \only<4>{ + \vspace{.1cm} + \includegraphics[width=.6\linewidth]{figs/2012-10-29_14-59-05_984.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + }\pause + \only<5>{ + \vspace{.1cm} + \includegraphics[width=.6\linewidth]{figs/2012-10-29_15-04-38_517.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + }\pause + \only<6>{ + \vspace{.1cm} + \includegraphics[width=.6\linewidth]{figs/2012-10-29_15-09-25_388.jpg} + \mycite{Larry Gonick, The Cartoon Guide to Statistics} + } +\end{center} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} + \frametitle{confidence intervals for the median} + \begin{definition}{Confidence interval} + A confidence $(1-\alpha)\cdot 100\%$ interval for a statistic + $\hat\theta$ is an interval $\hat\theta \pm a$ such that the + population parameter $\theta$ is contained in that interval + $(1-\alpha)\cdot 100\%$ of the experiments. + + An alternative way to put it is that $(\hat\theta - \theta) \in + [-a,a]$ in $(1-\alpha)\cdot 100\%$ of the cases. + \end{definition} + + +\begin{columns} + \begin{column}[l]{.5\linewidth} + If we knew the sampling distribution of the median $\hat m$, could + we generate a e.g. a $95\%$ confidence interval?\pause + \vspace{.5cm} + + Yes, we could choose the interval such that $\hat m - m$ in that + interval in $95\%$ of the cases. + \end{column} + \begin{column}[r]{.5\linewidth} + \only<1>{\includegraphics[width=\linewidth]{figs/samplingDistributionMedian00.pdf}} + \only<2>{\includegraphics[width=\linewidth]{figs/samplingDistributionMedian01.pdf}} + \end{column} +\end{columns} + + + % \begin{task}{Bootstrapping a confidence interval for the median} + % \begin{itemize} + % \item Use the same dataset as before. + % \item Bootstrap $500$ medians. + % \item Compute the $2.5\%$ and the $97.5\%$ percentile of the + % $500$ medians. + % \end{itemize} + % \end{task} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} + \frametitle{confidence intervals for the median} + \framesubtitle{how to get the sampling distribution} + + \begin{task}{Bootstrapping a confidence interval for the median} + \begin{itemize} + \item Use the same dataset as before. + \item Bootstrap $500$ medians. + \item Compute the $2.5\%$ and the $97.5\%$ percentile of the + $500$ medians. + \end{itemize} + These two numbers give you $\hat m -a$ and $\hat m + a$ for + the $95\%$ confidence interval. + \end{task} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} + \frametitle{confidence intervals for the median} + \framesubtitle{how to get it analytically} + There is also an analytical estimation oft the confidence interval + for the median: Use the $\frac{\alpha}{2}$ and $1 - \frac{\alpha}{2}$ + quantile of a binomial distribution. + + + \begin{task}{Comparing the analytical interval to the bootstrapped} + \begin{itemize} + \item Get the $\frac{\alpha}{2}$ quantile minus one and $1 - + \frac{\alpha}{2}$ quantile of a binomial distribution using {\tt + binoinv}. + \item Sort you data points and use the data points at the position + corresponding to the quantiles. + \item Compare that to the bootstrapped confidence interval. + \end{itemize} + \end{task} + \tiny The idea behind this: + \begin{itemize} + \item The probability that the true median $m$ is covered by the + interval between $x_r$ and $x_{r+1}$ is binomial $${n \choose r} + \left(\frac{1}{2}\right)^r \left(\frac{1}{2}\right)^{n-r}$$ + \item No we take enough intervals in the ``middle'' of our sample + that we cover the true median with at least $1-\alpha$ + probability. + \mycite{David, H. A., \& Nagaraja, H. N. (2003). Order Statistics. MES (Vol. 1, p. 482). Wiley. doi:10.1016/j.bpj.2010.07.012} + \end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} + \frametitle{confidence intervals} + \framesubtitle{Notice the theme!} + \begin{enumerate} + \item choose a statistic + \item get a the sampling distribution of the statistic (by theory or + simulation) + \item use that distribution to reason about the relation between the + true population parameter (e.g. $m$) and the sampled statistic + $\hat m$ + \end{enumerate} + + \begin{center} + \color{blue} + This is the scaffold of most statistical techniques. Try to find + it and it can help you understand them. + \end{center} + +\end{frame} + + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{let's practice that again} +\framesubtitle{confidence interval for the mean} + +\begin{task}{Bootstrapping a confidence interval for the mean} + \begin{itemize} + \item Use the same dataset as before. + \item Use bootstrapping to get a $95\%$ confidence interval for + the mean. + \end{itemize} +\end{task} + +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{confidence interval for the mean} +\framesubtitle{confidence interval for the mean} +Getting a convenient sampling distribution is (a little bit) more +difficult: +\begin{itemize} +\item If the $x_1,...,x_n\sim \mathcal N(\mu,\sigma)$ are Gaussian, then $\hat\mu$ is Gaussian as + well +\item What is the mean of $\hat\mu$? What is its standard deviation?\pause +\item[]{\color{gray} $\langle\hat\mu\rangle_{X_1,...,X_n} = \mu$ and + $\mbox{std}(\hat\mu) = \frac{\sigma}{\sqrt{n}}$}\pause +\item The problem is, that $\hat\mu \sim \mathcal N\left(\mu, + \frac{\sigma}{\sqrt{n}}\right)$ depends on unknown population + parameters.\pause +\item However, $$\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}} \sim + \mbox{t-distribution with }n-1\mbox{ degrees of freedom}$$ +\item Therefore, +\begin{align*} + P\left(t_{2.5\%}\le\frac{\hat{\mu}-\mu}{\hat{\sigma}/\sqrt{n}}\le t_{97.5\%}\right)&=P\left(t_{2.5\%}\frac{\hat{\sigma}}{\sqrt{n}}\le\hat{\mu}-\mu\le t_{97.5\%}\frac{\hat{\sigma}}{\sqrt{n}}\right) +\end{align*} +\end{itemize} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{confidence interval for the mean} +\begin{task}{Bootstrapping a confidence interval for the mean} + Extend your script to contain the analytical confidence + interval using +\begin{align*} + P\left(t_{2.5\%}\le\frac{\hat{\mu}-\mu}{\hat{\sigma}/\sqrt{n}}\le t_{97.5\%}\right)&=P\left(t_{2.5\%}\frac{\hat{\sigma}}{\sqrt{n}}\le\hat{\mu}-\mu\le t_{97.5\%}\frac{\hat{\sigma}}{\sqrt{n}}\right) +\end{align*} +\end{task} + +\end{frame} + +% ---------------------------------------------------------- +\subsection{summary} +% ---------------------------------------------------------- + +\begin{frame} +\frametitle{summary} +\begin{emphasize}{Which errorbars should I choose?} + Always use errorbars to help the reader see your point. +\end{emphasize} +\pause + \begin{itemize} + \item Errorbars can {\color{blue} describe the variability} in a dataset + ({\color{blue}descriptive statistics}). Example: {\em standard deviation, inter-quartile + range, ...} + \item {\color{blue}Errorbars yield information about significance in testing + (inferential statistics)}. Examples: {\em standard error of the mean, confidence + intervals, ...} + \item Other possible ways of displaying variability: {\em + boxplots, violin plots, histograms, ...} + \end{itemize} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{statistical tests} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{one-sample test on the mean} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% ---------------------------------------------------------- +\begin{frame} +\frametitle{from confidence intervals to one-sample test} + +\begin{task}{example: eye movements} + \small + In an experiment you measure eye movements of subjects on the + screen. You want be sure that the subject fixates a certain target + (at $x=0$). During the fixation period, you aquire $n=16$ + measurements. The measurements have a mean of $\hat\mu=2.5$ and a + standard deviation of $\hat\sigma=4$. Assuming that the single + fixation locations are Gaussian distributed, can you be $95\%$ + confident that the subject focused the target (x-Position)? +\end{task} +\pause +\begin{solution}{use confidence intervals} + \small + Compute a $95\%$ confidence interval: Does it contain + $\mu=0$? Yes? Then we are $95\%$ confident! + + From the table we get $t_{0.025}=2.131$, the standard error is + $\frac{\hat\sigma}{\sqrt{n}} = \frac{4}{\sqrt{16}}=1$ which means + that $$0\pm t_{0.025}\frac{\hat\sigma}{\sqrt{n}} = 0 \pm 2.131$$ + is our confidence interval. Therefore we cannot be $95$\% + confident in this case. +\end{solution} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{from confidence intervals to one-sample test} +\begin{task}{example: eye movements} + Could we put the interval on $\mu=0$ as well? +\end{task} +\pause +\begin{solution}{Example: eye movements} + Yes, if the interval around $\hat\mu$ contains $\mu$, then the + interval around $\mu$ also contains $\hat\mu$. +\end{solution} + + +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{One-sample t-test} + +\begin{task}{example 2: eye movements again} + \small + Now assume that there is a fixation target at $x=0$. You are + running the experiment with a monkey and you want to discard all + trials in which the monkey was not fixating the target. + + During the trial, you aquire again $n=16$ measurements with mean + $\hat\mu=2.5$ and standard deviation $\hat\sigma=4$. How can you be + confident that the monkey did not fixate the target if you are + willing to be wrong in $5\%$ of the cases if ``wrong'' means that + you believe the subject was not fixating when in fact it was. +\end{task} +\pause +\begin{solution}{Example 2: eye movements again} + \small +The steps to the solution is exactly the same, only the logic is +different. +\begin{itemize} +\item We make a $95\%$ confidence around the fixation target + $\mu=0$. This means that if the monkey was actually fixating the + target, $95\%$ of the measured averaged positions $\hat\mu$ would + fall into that interval. +\item $5\%$ of the measured would fall outside the interval even + though the monkey fixated and we would falsely treat them as not as ``not + fixated''. +\end{itemize} +\end{solution} + +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{one-sample t-test} +\framesubtitle{Notice the theme again!} +\only<1>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition0.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \end{enumerate} +}\pause +\only<2>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition1.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \item Get a sampling distribution! Here, we get it by assuming that + the positions $x_1,...,x_{16}$ are Gaussian. + \end{enumerate} +}\pause +\only<3>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition2.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \item Get a sampling distribution! Here, we get it by assuming that + the positions $x_1,...,x_{16}$ are Gaussian. The resulting + distribution of $t$ is a t-distribution. + \end{enumerate} +}\pause +\only<4>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition3.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \item Get a {\color{blue}null distribution}! Here, we get it by assuming that + the positions $x_1,...,x_{16}$ are Gaussian. The resulting + distribution of $t$ is a t-distribution. + \item Get an interval around $\mu=0$ in which values of $\hat\mu$ + are assumed typical for $\mu=0$, the {\color{blue}null hypothesis + $H_0$}. + \end{enumerate} +} +\pause +\only<5>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition5.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean + $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \item Get a {\color{blue}null distribution}! Here, we get it by assuming that + the positions $x_1,...,x_{16}$ are Gaussian. The resulting + distribution of $t$ is a t-distribution. + \item Get an interval around $\mu=0$ in which values of $\hat\mu$ + are assumed typical for $\mu=0$, the {\color{blue}null hypothesis + $H_0$}. This is done by fixing the {\color{blue}type I error} probability. + \end{enumerate} +} +\pause +\only<6>{ + \begin{center} + \includegraphics[width=0.4\linewidth]{figs/repetition4.png} + \end{center} + \begin{enumerate} + \small + \item Choose a statistic! We take the standardized mean + $t=\frac{\hat\mu-\mu}{\hat\sigma/\sqrt{n}}$. + \item Get a {\color{blue}null distribution}! Here, we get it by assuming that + the positions $x_1,...,x_{16}$ are Gaussian. The resulting + distribution of $t$ is a t-distribution. + \item Get an interval around $\mu=0$ in which values of $\hat\mu$ + are assumed typical for $\mu=0$, the {\color{blue}null hypothesis + $H_0$}. This is done by fixing the {\color{blue}type I error} probability. + \item Outside that interval we consider $\mu=0$ as implausible and + reject $H_0$. + \end{enumerate} +} + +\end{frame} + + +% ---------------------------------------------------------- +\subsection{another one-sample test} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{another one-sample test} +\begin{task}{Fair coin?} + \small + Assume you carry out the following test to determine whether a coin + is fair or not: + + You throw the coin $n=3$ times. If the result is either $3\times$ + head or $3\times$ tail, you conclude that the coin is not fair. + + Answer the following questions (for yourself first): + \begin{enumerate} + \item What is the meta-study? \pause {\em Repeated experiments of 3 throws + with this the coin.}\pause + \item What is the statistic used? \pause {\em The number of heads (could also + be tails).}\pause + \item What is $H_0$? \pause {\em The coin is fair.}\pause + \item What is the Null distribution? \pause {\em The distribution is + binomial $$p(k \mbox{heads in }n \mbox{ throws})={n \choose k} + \left(\frac{1}{2}\right)^k \left(\frac{1}{2}\right)^{n-k} $$}\pause + \item What is the Type I error of this test? \pause {\em $p(HHH|H_0) + p(TTT|H_0) = \frac{2}{8}$} + \end{enumerate} +\end{task} +\end{frame} + +% ---------------------------------------------------------- +\subsection{paired sample t-test} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{paired sample t-test} +\begin{task}{Hunger Rating (SWS, Example 3.2.4)} + \begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} +\small During a weight loss study each of nine subjects was given either the +active drug m-chlorophenylpiperazine (mCPP) for two weeks and then a placebo +for another two weeks, or else was given the placebo for the first two weeks and +then mCPP for the second two weeks. Can we say that there was an +effect with significance level $5$\%? + \end{minipage} + \begin{minipage}{0.5\linewidth} +\begin{center} + \includegraphics[width=0.8\linewidth]{figs/hunger.png} +\end{center} + \end{minipage} + + \end{minipage} + \vspace{.5cm} + + What could we use as statistic? + What is $H_0$? + Is the difference significant? +\end{task} +\end{frame} + +\begin{frame} +\frametitle{paired sample t-test} +\begin{solution}{Hunger Rating (SWS, Example 3.2.4)} + \begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \small + \begin{enumerate} + \item The statistic is the difference between drug and placebo?\pause + \item $H_0$ is ``there is no difference'', i.e. the true mean of + the differences is zero. \pause + \item The standard error is $33/\sqrt{9}=11$.\pause + \item $n-1=8$ DoF yields (t-distribution table) $t_{0.025}=2.306$, so we + would reject $H_0$ if $\hat\mu$ in $0\pm t_{0.025}\cdot 11 = \pm + 25.366$. \pause + \item This means the difference is significant with $\alpha=0.05$. + \end{enumerate} + \end{minipage} + \begin{minipage}{0.5\linewidth} +\begin{center} + \includegraphics[width=0.8\linewidth]{figs/hunger.png} +\end{center} + \end{minipage} + + \end{minipage} +\end{solution} +\end{frame} + + +\begin{frame} +\frametitle{paired sample t-test} +\begin{itemize} +\item a paired sample consists of a number of {\em paired} + measurements (e.g. before/after)\pause +\item build the differences (either there are many and or check that + they are approx. Gaussian distributed)\pause +\item use a one-sample t-test on the differences +\end{itemize} +\end{frame} +% ---------------------------------------------------------- +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{sign rank test} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% ---------------------------------------------------------- +\begin{frame} +\frametitle{sign rank test} +\begin{task}{Hunger Rating (SWS, Example 3.2.4)} + \small + \begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \small Consider again the example data from before. Instead of + taking the difference, we consider now only whether ``drug'' was + smaller or greater than ``placebo''. We then count the number of + times for which ``drug''$<$``placebo'' and the number of times + ``drug''$>$``placebo''. + \end{minipage} + \begin{minipage}{0.5\linewidth} +\begin{center} + \includegraphics[width=0.5\linewidth]{figs/hunger.png} +\end{center} + \end{minipage} + \end{minipage} + \begin{itemize} + \item What is the statistic?\pause {\em The number $N_+$ of ``>'' + or the number $N_-$ of ``<''.} \pause + + \item What is $H_0$? \pause {\em $N_+ = N/2$} + \pause + \item What is $H_A$? \pause {\em $N+ > N/2$ or $N_+ < N/2$} + \pause + \item What is the Null distribution? \pause {\em Binomial with $p=0.5$} + \pause + \item Given $\alpha$, how is the region determined in which we + reject $H_0$? \pause {\em Choose a such that $P(k>a|H_0) + P(k{\includegraphics[width=\linewidth]{figs/testframework00.pdf}} + \only<2>{\includegraphics[width=\linewidth]{figs/testframework01.pdf}} + \end{center} +\small +\begin{columns} + \begin{column}[l]{.5\linewidth} +{\bf You want:} +\begin{itemize} +\item large power +\item small type I \& II error probability ($\alpha$ and $\beta$) +\end{itemize} + \end{column} + \begin{column}[r]{.5\linewidth} +\begin{itemize} +\item \hyperlink{sec:power}{\color{magenta}detour II: statistical power} \hypertarget{back:power}{} +\item \hyperlink{sec:bayesian}{\color{magenta}detour III: Bayes rule + and statistical tests} \hypertarget{back:bayesian}{} +\end{itemize} + \end{column} +\end{columns} + +Which of the above can {\bf you} choose? \pause {\em the type I error + probability $\alpha$} + + +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{zoo of statistical tests} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + + + + +\begin{frame} +\hypertarget{back:detourIV}{} +\frametitle{how to choose the statistical test} +\begin{center} + \includegraphics[height=.38\linewidth]{figs/fig0.pdf} +\end{center} +\begin{itemize} +\item Normality can be checked with a QQ-plot + (\hyperlink{sec:qqplots}{\color{magenta} detour IV: QQ-plots}). +\item If $n$ is large and the variance of the data distribution is + finite, the central limit theorem guarantees normality for + ``summed statistics''. +\end{itemize} +\end{frame} + + + +% ------------ + +\begin{frame} +\frametitle{} +\begin{center} + \includegraphics[height=.6\linewidth]{figs/fig2.pdf} +\end{center} + +\end{frame} +% ------------ + +\begin{frame} +\frametitle{} +\begin{center} + \includegraphics[height=.6\linewidth]{figs/fig3.pdf} +\end{center} + +\end{frame} +% ------------ + +%----------------------------------------------------------------- +%----------------------------------------------------------------- +\begin{frame} +\frametitle{tests for normal data} +\begin{task}{menstrual cycle} + The data set {\tt menstrual.dat} contains the lengths of the + menstrual cycles in a random sample of 15 women. Assume we want to + the hypothesis that the mean length of human menstrual cycle is + equal to a lunar month ($29.5$ days). Consider the data to be + sufficiently normal. + + Questions: + \begin{itemize} + \item What is $H_0$? What is $H_A$? \pause $H_0: \hat\mu=29.5$, + $H_A: \hat\mu\not=29.5$ \pause + \item What is the test statistic? \pause $t=\frac{\hat\mu - + 29.5}{\hat\sigma/\sqrt{n}}$ \pause + \item Which test should did you use and why? {\em One sample t-test: data + normal, one sample against a fixed mean.} + \end{itemize} +\end{task} + +\hyperlink{sec:twotailed}{\color{magenta}detour I: one- vs. two-tailed} +\hypertarget{back:twotailed}{} +\end{frame} + +%----------------------------------------------------------------- + +\begin{frame} +\frametitle{} +\begin{center} + \includegraphics[height=.6\linewidth]{figs/fig4.pdf} +\end{center} + +\end{frame} + + +% ---------------------------------------------------------- +\begin{frame} + \frametitle{} + \begin{task}{chirping} + A scientist conducted a study of how often her pet parakeet + chirps. She recorded the number of distinct chirps the parakeet + made in a 30-minute period, sometimes when the room was silent and + sometimes when music was playing. The data are shown in the + following table. Test whether the bird changes its chirping + behavior when music is playing (data set {\tt + chirping.dat}. columns: day, with, without). + + Questions: + \begin{itemize} + \item What is $H_0$? What is $H_A$? \pause + $d_i=x_{\mbox{with}}-x_{\mbox{without}}$. $H_0: \hat\mu_d=0$, + $H_0: \hat\mu_d\not=0$ \pause + \item What is the test statistic? \pause $t=\frac{\hat\mu_d - + 0}{\hat\sigma_d/\sqrt{n}}$ \pause + \item Which test should did you use and why? \pause {\em Paired t-test: data + sufficiently normal, measurements are paired by day.} + \end{itemize} + \end{task} + +\end{frame} + +%----------------------------------------------------------------- + +\begin{frame} +\frametitle{} +\begin{center} + \includegraphics[height=.7\linewidth]{figs/fig5.pdf} +\end{center} + +\end{frame} + +%----------------------------------------------------------------- + + +\begin{frame} +\frametitle{} +\begin{center} + \includegraphics[width=.8\linewidth]{figs/fig6.pdf} +\end{center} + +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{two indepedendent sample test} +\begin{task}{Brain Weights (permutation test)} + The dataset {\tt brainweight.dat} contains brain weights of males + and females. It consists of {\bf (i) two samples (male/female)} + which are {\bf (ii) not paired}. We want to test whether the mean + brain weights of males and females are different. + \begin{itemize} + \item What could we use as statistic?\pause {\em~the difference in the + means} \pause + \item What would be $H_0$?\pause {\em~the difference is zero} \pause + \item Think about a way to generate an estimate of the Null + distribution with Matlab? \pause {\em~Permutation test: Shuffle the + labels, compute difference in means, repeat ...}. \pause + \end{itemize} + +\end{task} +\begin{itemize} +\item There is {\color{blue}two-sample independent t-test} is the parametric test + for this dataset. +\item If normality does not hold, you can use the + {\color{blue}Wilcoxon-Mann-Whitney test} +\end{itemize} +\end{frame} + +\begin{frame} +\frametitle{one- and two-sample t-test and sign test} +\begin{center} + \tiny +\bgroup +\def\arraystretch{2} +\begin{tabular}{|l|c|c|c|} + \hline + \textbf{name} & \textbf{statistic} & $\boldsymbol{H_{0}}$ & \textbf{Null distribution}\tabularnewline + \hline + \hline + one sample t-test & $t=\frac{\overline{x}-0}{\mbox{SE}_x}$ & mean of $t$ is zero & t-distr. with $n-1$ DoF\tabularnewline + \hline + paired sample t-test & $t=\frac{\overline{d}-0}{\mbox{SE}_d},\, d=x_{i}-y_{i}$ & mean of $t$ is zero & t-distr. with $n-1$ DoF\tabularnewline + \hline + sign test & $t=\#\left[x_{i}{ + So far, we chose a particular threshold $b$ by fixing the type I error + rate $\alpha$. + \begin{center} + \includegraphics[width=.7\linewidth]{figs/pval0.png} + \end{center} +} +\only<2>{ + \begin{itemize} + \item The {\color{blue}p-value} is the type I error rate if you use + your {\color{blue} actually measured statistic} as threshold. + \item In other words: The p-value is the minimal type I error rate + you have to accept if you call your result significant. + \end{itemize} + \begin{center} + \includegraphics[width=.7\linewidth]{figs/pval1.png} + \end{center} +} +\end{frame} +%--------------------------------------------------------- + +\begin{frame} +\frametitle{the mother of all statistics: the p-value} +\framesubtitle{Why is it a universal measure?} + +The p-value is the minimal type I error rate you have to accept if you +call your result significant. + +\begin{itemize} +\item If you have a personal $\alpha$-level that is larger than the + p-value, you automatically know that the decision threshold lies + ``further inside'' +\item This means you {\color{blue}can simply compare your $\alpha$-level with the + p-value}: if the p-value is smaller, then you call that result + significant, otherwise you don't. +\end{itemize} + +\begin{center} + \includegraphics[width=.45\linewidth]{figs/pval0.png} + \includegraphics[width=.45\linewidth]{figs/pval1.png} +\end{center} +\end{frame} +%--------------------------------------------------------- + +\begin{frame} +\frametitle{the mother of all statistics: the p-value} + +\begin{task}{p-values if $H_0$ is true} + Is the following procedure correct? + + \vspace{.5cm} + + In order to show that a sample $x_1,...,x_n$ follows a Normal + distribution with mean zero, you perform a t-test. If the p-value is + large, you conclude that there is evidence for $H_0$, i.e. accept + that $x_1,...,x_n$ has mean zero and is normally distributed. + + \vspace{.5cm} + To find the answer, simulate normally distributed random variables + with {\tt randn} in Matlab and compute the p-value with a one-sample + t-test. Repeat that several times and plot a histogram of the p-value. + +\end{task} +\pause +\begin{itemize} +\item If $H_0$ is true, the p-value is uniformly distributed between 0 + and 1. Why?\pause +\pause +\item Think about the beginning of this lecture + $$p=P(|x| > |t|) = 1 - P(|x| \le |t|) = 1 - \mbox{c.d.f.}(|t|) \sim U([0,1])$$ +\end{itemize} +\end{frame} + +%-------------------------------------------------- +\begin{frame} +\frametitle{the mother of all statistics: the p-value} + +\begin{task}{Study design} + Is the following procedure statistically sound? + + \vspace{.5cm} + + Psychophysical experiments with human subjects can be time-consuming + and costly. In order to get a significant effect with minimal effort + you use the following procedure: You start with a few subjects. If + your statistical test for the effect returns a p-value smaller than + $0.05$ you stop and publish. Otherwise you repeat adding subjects + and computing p-values until you get a significant results (or run + out of time and money). + +\end{task} +\pause + +\begin{solution}{Answer} + No, the procedure is not sound. Even if $H_0$ is true, you will + eventually get a p-value smaller than $0.05$ since it is uniformly + distributed between $0$ and $1$ in this case. +\end{solution} +\end{frame} + +%-------------------------------------------------- +\begin{frame} +\frametitle{the mother of all statistics: the p-value} + +\begin{task}{p-values over studies} + If there is no effect, how many studies would yield a significant + p-value (for $\alpha=0.05$)? +\end{task} +\pause +\begin{solution}{Answer} + $5\%$ +\end{solution} +\pause +\begin{task}{p-values in publications} + Do you think that only publishing positive findings poses a problem? +\end{task} +\pause +\begin{solution}{Answer} +Yes. If I only publish significant positive findings, then I can +publish anything if I just repeat the study long enough. +\end{solution} + +\end{frame} + +%--------------------------------------------------------- +\begin{frame} + \frametitle{the mother of all statistics: the p-value} + \begin{task}{true or false?} + \begin{itemize} + \item From $p<0.01$ you can deduce that your result is of + biological importance.\pause + + + \item {\color{gray} False. A small p-value doesn't say anything + about biological importance. It just indicates that the data + and $H_0$ are not very compatible.} \pause + + \item The p-value is the probability of observing a dataset + resulting in a test-statistic more extreme than the one at hand, + assuming the null hypothesis is true.\pause + + \item {\color{gray} True.} \pause + \item $1-p$ is the probability of the alternative hypothesis being + true. \pause + + \item {\color{gray} False. The p-value cannot tell us anything + about whether one of the hypotheses are true or not.} + \end{itemize} + \end{task} +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{multiple hypothesis testing} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%--------------------------------------------------------------- +\begin{frame} +\frametitle{two tests} +\begin{task}{Correct or not?} + You have two independent samples from a treatment group and a + control group. You are not sure whether your data meets the + requirement of a t-test. Therefore, you carry out a t-test and a + ranksum test. If one of them rejects $H_0$ you use this one to + report your findings in a paper. + +\vspace{.5cm} +\footnotesize + +To approach an answer, use Matlab and +\begin{itemize} +\item repeatedly sample two datasets from the same Normal distribution + $\mathcal N(0,1)$. +\item for each pair of datasets compute the test statistic of a + ranksum test (use {\tt ranksum}) and a t-test (use {\tt ttest2}) +\item Plot the values of the statistics against each other (using {\tt + plot(T, R, 'k.')}). What can you observe? +\item Count the number of times at least one of the tests gives a + p-value smaller than $0.05$. What can you observe? +\end{itemize} +\end{task} + + + +\end{frame} + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{two tests} + +\begin{minipage}{1.\linewidth} + \begin{minipage}{0.6\linewidth} + \begin{center} + \includegraphics[width=1.\linewidth]{figs/multipletesting.pdf} + \end{center} + \end{minipage} + \begin{minipage}{0.39\linewidth} + \small + \only<1-4>{ + \begin{itemize} + \item the two statistics are clearly correlated\pause + \item What is the type I error rate for each single test?\pause + \item Where is the type I error area in the combined plot? \pause + \item Is the type I error rate in the combined strategy lower or + larger compared to using just a single test?\pause + \end{itemize} + } + \only<5>{ + \small + \color{blue} The combined strategy has a higher error rate! This gets + worse for more tests. For that reason we have to account for multiple + testing! + } + + \end{minipage} +\end{minipage} +\end{frame} + + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{two tests} + +\begin{minipage}{1.\linewidth} + \begin{minipage}{0.49\linewidth} + \begin{center} + \includegraphics[width=1.\linewidth]{figs/multipletesting.pdf} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \small + \begin{itemize} + \item When is something called multiple testing?\pause + \item[]{\color{gray} If a hypothesis is a compound of single + hypotheses.}\pause + \item If I test $\mu_1 = \mu_2 = \mu_3$ by testing $\mu_i = \mu_j$ + for all $i\not= j$ and reject as soon as one of the test rejects, + does the type I error increase or decrease?\pause + \item[]{\color{gray} It increases, because a have the chance to make + an error in all conditions.}\pause + \item Can the type I error also go in the other direction?\pause + \item[]{\color{gray} Yes, it could. For example if the single + hypotheses are combined with ``and''.} + \end{itemize} + + \end{minipage} +\end{minipage} +\end{frame} +%--------------------------------------------------------------- + +\begin{frame} + \frametitle{summary} + \begin{itemize} + \item Multiple testing tests a {\color{blue}compound hypothesis} by + testing several single hypotheses.\pause + \item {\color{blue}Multiple testing can decrease or increase type I/II error} + dependening on how the single hypothese are combined (``or'' type + I up, ``and'' type I down).\pause + \item This can be accounted for (e.g. by {\em Bonferroni correction: + divide $\alpha$ by number of tests}). However, better is to have + a test that directly tests the compound hypothesis. ANOVA is a + typical example for that. + \end{itemize} + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{study design} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{general theme} + \begin{enumerate} + \item make an educated guess about the true parameters + \item state how accurate/powerful you want to be + \item select $n$ based on that + \end{enumerate} +\end{frame} + +\begin{frame} + \frametitle{estimating a single mean} + \framesubtitle{standard error and $\alpha$} + \begin{itemize} + \item Assume you want to make estimate the mean of some quantity.\pause + \item From a pilot study or the literature, you have an estimate $s$ + of the standard deviation and $\tilde\mu$ of the mean of that + quantity.\pause + \item $\tilde \mu$ could also be chosen to set a minimal detectable difference.\pause + \item In order to test whether your mean $\hat\mu$ is different from + a fixed mean $\mu_0$ on an $\alpha$-level of $5\%$ you know that + the $95\%$ confidence interval around $\tilde\mu$ should not + contain $\mu_0$: $$\underbrace{|\tilde\mu - \mu_0|}_{=:\delta} \ge + t_{0.025, \nu}\frac{s}{\sqrt{n}}$$ +\pause +\item This mean you should set $n$ to be +$$n \ge \left(\frac{t_{0.025, \nu}\cdot s}{\delta}\right)^2 $$ + \end{itemize} + +\end{frame} + +\begin{frame} + \frametitle{estimating means} + \framesubtitle{type I and type II error} + {\bf one can also take the desired power $1-\beta$ into account} + $$n \ge \frac{s^2}{\delta^2}\left(t_{\alpha,\nu}, + t_{\beta(1),\nu}\right)^2$$ + \only<1>{ + \includegraphics[width=.5\linewidth]{figs/experimentalDesign00.pdf} + \includegraphics[width=.5\linewidth]{figs/experimentalDesign01.pdf} + } + \pause + + {\bf rearranging the formula yields an estimate for minimal + detectable difference} + $$\delta \ge \sqrt{\frac{s^2}{n}}\left(t_{\alpha,\nu}, + t_{\beta(1),\nu}\right)$$ + \pause + + {\bf for two means, this formula becomes} + $$n \ge \frac{2s^2}{\delta^2}\left(t_{\alpha,\nu}, + t_{\beta(1),\nu}\right)^2$$ + + \pause + + \begin{emphasize}{iterative estimation} + Since $\nu$ depends on $n$ (i.e. $\nu=n-1$), we need to estimate + $n$ iteratively. + \end{emphasize} + + \mycite{Zar, J. H. (1999). Biostatistical Analysis. (D. Lynch, + Ed.)Prentice Hall New Jersey (4th ed., Vol. 4th, p. 663). Prentice + Hall. doi:10.1037/0012764} + + +\end{frame} + +\begin{frame} + \frametitle{example} + \framesubtitle{Zar, example 7.2} + \small + Researches observed the weight changes in twelve rats after being + subjected to forced exercise. The mean difference is + $\hat\mu=-0.65g$, the sample variance is $\hat\sigma^2=1.5682 + g^2$. We wish to test the difference to $\mu=0$ with $\alpha=0.05$ + and a $1-\beta=0.9\cdot 100\%$ chance of detecting a population mean + different from $\mu_0=0$ by as little as $1.0g$. + +\pause + + Let's guess that a sample size of $n=20$ would be required. Then + $\nu=19$, $t_{0.025,19}=2.093$, $\beta=1-0.9=0.1$, and + $t_{0.1,19}=1.328$. This means + $$n=\frac{1.5682}{1^2}(2.093+1.3828)^2 = 18.4.$$ + +\pause + +Now let's us $n=19$ as an estimate, in which case $\nu=18$, +$t_{0.025,18}=2.101$, $t_{0.1,18}=1.330$, +and $$n=\frac{1.5682}{1^2}(2.101+1.330)^2=18.5.$$ +Thus we need a sample size of at least $19$. +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{ANOVA} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{from linear regression to ANOVA} +\begin{frame} +\frametitle{from linear regression to ANOVA} + +\small The following table contains the impulse frequency of the +electric field from electric fish measured at several temperatures +(data for project 03). + +\begin{center} + \tiny +\begin{tabular}{lccccccc} +{\bf temperature C${}^\circ$} & \multicolumn{3}{c}{\bf impulse frequency [number/sec]} \\ \hline\\ +20.00 & 225.00 & 230.00 & 239.00 \\ +22.00 & 251.00 & 259.00 & 265.00 \\ +23.00 & 266.00 & 273.00 & 280.00 \\ +25.00 & 287.00 & 295.00 & 302.00 \\ +27.00 & 301.00 & 310.00 & 317.00 \\ +28.00 & 307.00 & 313.00 & 325.00 \\ +30.00 & 324.00 & 330.00 & 338.00 +\end{tabular} + +\end{center} + +\begin{itemize} +\item Our goal will be to test whether $\mu_{20}=...=\mu_{30}$. +\item Note that ANOVA is not the method to analyze this + dataset. Linear regression is because temperature is on an interval + scale. We will just use the ideas here for illustration. +\end{itemize} +\end{frame} + + + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{from linear regression to ANOVA} +\begin{center} + \includegraphics[width=.8\linewidth]{figs/regression01.pdf} +\end{center} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\frametitle{from linear regression to ANOVA} +\begin{center} + \includegraphics[width=.7\linewidth]{figs/regression02.pdf} +\end{center} +What kind of regression line would we expect if the means were equal? +\pause {\em One with slope $\alpha=0$.} +\end{frame} + +% ---------------------------------------------------------- +\begin{frame} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \includegraphics[width=1.\linewidth]{figs/regression02.pdf} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \begin{itemize} + \item For linear regression data, we would test whether + $\alpha=0$. + \item For categorial inputs (x-axis), we cannot compute a + regression line. Therefore, we need a different approach. + \end{itemize} + \end{minipage} +\end{minipage} +\end{frame} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{law of total variance} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\frametitle{law of total variance} +\only<1>{ + Approach law of total variance + $$\color{red} \mathbb V[f] = \color{mygreen} \mathbb V[\mu] + + \color{lightblue}\mathbb E[\mathbb V[f|\mu_i]]$$ + \begin{center} + \includegraphics[width=.7\linewidth]{figs/regression02.pdf} + \end{center} +}\pause +\only<2>{ + Approach law of total variance + $$\color{red} \mathbb V[f] = \color{mygreen} \mathbb V[\mu] + + \color{lightblue}\mathbb E[\mathbb V[f|\mu_i]]$$ + \begin{center} + \includegraphics[width=.7\linewidth]{figs/regression03.pdf} + \end{center} +}\pause +\only<3>{ + Approach law of total variance + $$\color{red} \mathbb V[f] = \color{mygreen} \mathbb V[\mu] + + \color{lightblue}\mathbb E[\mathbb V[f|\mu_i]]$$ + Data generation model for regression $f_{ij} = {\color{mygreen} \alpha t_i} + \beta + {\color{lightblue}\varepsilon_{ij}}$ + \begin{center} + \includegraphics[width=.6\linewidth]{figs/regression04.pdf} + \end{center} +}\pause +\only<4>{ + Approach law of total variance + $$\color{red} \mathbb V[f] = \color{mygreen} \mathbb V[\mu] + + \color{lightblue}\mathbb E[\mathbb V[f|\mu_i]]$$ + Data generation model for regression + $f_{ij} = {\color{mygreen} \alpha t_i} + \beta + + {\color{lightblue}\varepsilon_{ij}}: $ $${\color{mygreen} \alpha=0} + \Rightarrow {\color{mygreen} \mathbb V[\mu] = 0} \Rightarrow \mu_{20} = \mu_{22} = ... = \mu_{30}$$ + \begin{center} + \includegraphics[width=.6\linewidth]{figs/regression04.pdf} + \end{center} +} +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{single factor ANOVA} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%--------------------------------------------------------------- +\begin{frame} +\frametitle{data model for single factor ANOVA} + Approach law of total variance + $$\color{red} \mathbb V[f] = \color{mygreen} \mathbb V[\mu] + + \color{lightblue}\mathbb E[\mathbb V[f|\mu_i]]$$ + Data generation model for single factor ANOVA + $f_{ij} = \overline{\mu} + {\color{mygreen} \tau_{i}} + + {\color{lightblue}\varepsilon_{ij}}$: + $${\color{mygreen} \tau_i=\tau_j=0} + \Rightarrow {\color{mygreen} \mathbb V[\mu] = 0} \Rightarrow \mu_{20} = \mu_{22} = ... = \mu_{30}$$ + \begin{center} + \includegraphics[width=.6\linewidth]{figs/regression05.pdf} + \end{center} +\end{frame} +%--------------------------------------------------------------- +\begin{frame} +\frametitle{statistic of ANOVA} +\begin{columns} + \begin{column}{0.43\linewidth} + \begin{center} + \includegraphics[width=1.\linewidth]{figs/regression02.pdf} + + \vspace{-.2cm} + + \includegraphics[width=1.\linewidth]{figs/Fdistribution00.pdf} + \end{center} + \end{column} + \begin{column}{0.55\linewidth} + \begin{align*} + \:&\mbox{\color{lightblue} error SS}&=\color{lightblue}\sum_{ij}\left(x_{ij}-\mu_{i}\right)^{2}\\ + +\:&\mbox{\color{mygreen} group SS}&=\color{mygreen}\sum_{i}n_{i}\left(\hat{\mu}_{i}-\mu\right)^{2}\\\hline + \:&\mbox{\color{red} total SS}&=\color{red}\sum_{ij}\left(x_{ij}-\mu\right)^{2} + \end{align*} + \pause + \begin{align*} + \mbox{\color{mygreen}groups MS}=\frac{\mbox{\color{mygreen}group SS}}{\mbox{\color{mygreen}groups DF}}&=\color{mygreen}\frac{\sum_{i}n_{i}\left(\hat{\mu}_{i}-\mu\right)^{2}}{k-1}\\\mbox{\color{lightblue}error MS}=\frac{\mbox{\color{lightblue}error SS}}{\mbox{\color{lightblue}error DF}}&=\color{lightblue}\frac{\sum_{ij}\left(x_{ij}-\hat{\mu_{i}}\right)^{2}}{N-k}\\\color{dodgerblue}F&=\frac{\mbox{\color{mygreen}group MS}}{\mbox{\color{lightblue}error MS}} + \end{align*} + \end{column} +\end{columns} +\end{frame} + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{summary single factor ANOVA} +\begin{itemize} +\item {\bf Goal:} Test whether several means are equal or not.\pause +\item {\bf Strategy:} Use law of total variance to explain the overall + variance with the {\em variance of the means} and the {\em variance + within groups}\pause +\item If the total variance can be solely explained from {\em variance + within groups}, then the means do not vary and must be the same. \pause +\item Since a statistic should be large if the data does not fit to + $H_0$, we use $\frac{MS(between)}{MS(within)}$ which can be shown to + have an F-distribution under certain ...\pause +\item {\bf Assumptions:} + \begin{itemize} + \item The groups must be independent of each other. + \item In each group, the specimen must be i.i.d. from the particular + population distribution $f_{ij} \sim p(f|\mu_i) $. + \item The standard deviations of the groups are equal + ($\sigma_\varepsilon$ is the same for all groups). + \item The residuals $\varepsilon$ must be Normally distributed + \end{itemize} +\end{itemize} +\end{frame} + +\subsection{study design for ANOVA} +\begin{frame} + \frametitle{study design for ANOVA} + \begin{itemize} + \item If the means are different (but all other assumptions are + satisfied), then $F$ follows a non-central F-distribution. + \item Like in the case of one- and two-sample t-tests, this can be + used to adjust $n$ for the desired power. + \item Alternatively, one can estimate the minimal detectable + difference $\delta$ from estimates of the {\em error MS} $s^2$ + and $n$, or $n$ from $\delta$ and $s^2$, respectively. + \end{itemize} + \mycite{Zar, J. H. (1999). Biostatistical Analysis. (D. Lynch, + Ed.)Prentice Hall New Jersey (4th ed., Vol. 4th, p. 663). Prentice + Hall. doi:10.1037/0012764} + +\end{frame} + +\subsection{non-parametric ANOVA} +\begin{frame} + \frametitle{Kruskal-Wallis test} + \begin{itemize} + \item Can be applied if the data is not normally distributed. + \item Is equivalent to Mann-Whitney/Wilcoxon rank sum test for two + factor levels. + \item Needs the variances to be equal as well. + \item Instead of testing equality of means/medians it tests for + equality of distributions. + \item For more details see {\em Biostatistical Analysis}. + \end{itemize} +\end{frame} + +\begin{frame} + \frametitle{Testing the difference among several medians} + \begin{itemize} + \item Can be applied if the data is not normally distributed. + \item Does not need the variances to be equal. + \item For more details see {\em Biostatistical Analysis}. + \end{itemize} +\end{frame} + +\section{more complex ANOVAs} +\subsection{blocking} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{blocking} +\footnotesize +{\bf Blocking} +How does experience affect the anatomy of the brain? In a typical +experiment to study this question, young rats are placed in one of +three environments for 80 days: + +\begin{itemize} +\item[T1] Standard environment.The rat is housed with a single + companion in a standard lab cage. +\item[T2] Enriched environment. The rat is housed with several + companions in a large cage, furnished with various playthings. +\item[T3] Impoverished environment.The rat lives alone in a standard + lab cage. +\end{itemize} + +At the end of the 80-day experience, various anatomical measurements +are made on the rats' brains. Suppose a researcher plans to conduct +the above experiment using 30 rats. To minimize variation in response, +all 30 animals will be male, of the same age and strain. To reduce +variation even further, the researcher can take advantage of the +similarity of animals from the same litter. In this approach, the +researcher would obtain three male rats from each of 10 litters. The +three littermates from each litter would be assigned at random: one to +T1, one to T2, and one to T3. +\end{frame} + +%--------------------------------------------------------------- +\begin{frame} + \frametitle{How to create blocks} + + Try to create blocks that are as homogeneous within themselves as + possible, so that the inherent variation between experimental units + becomes, as far as possible, variation between blocks rather than + within blocks (see SWS chapter 11.6). + + {\bf Fish data:} + \begin{itemize} + \item each fish is a block + \item the different categories are the factor of interest + \item note that we have one measurement per block and factor, but + there could be more + \end{itemize} + +\end{frame} + + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{data model for block randomized ANOVA} + + Data generation model for randomized block factor ANOVA + $f_{ijk} = \overline{\mu} + \tau_{i} + \beta_j + \varepsilon_{ijk}$: + + \vspace{.5cm} + + How do we know that there is no interaction $\gamma_{ij}$ between + the blocks and the factors? + \begin{itemize} + \item {\bf a priori knowledge:} why should temperature be dependent on + fish identity + \item {\bf additivity:} for each factor $i$, the values differ by + the {\em same} amount $\beta_j$. \pause + \end{itemize} + + \begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \begin{center} + \includegraphics[width=1.\linewidth]{figs/regression06.pdf} + \end{center} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \only<2>{\color{red} Would that also be the case if the values cross at the point?} + \end{minipage} + \end{minipage} +\end{frame} + +\subsection{two factor ANOVA} +%--------------------------------------------------------------- +\begin{frame} +\frametitle{What's the funny way to write down the data model in ANOVA?} + + Data generation model for a two factor ANOVA with interaction + $$f_{ijk} = \overline{\mu} + \tau_{i} + \beta_j + \gamma_{ij} + \varepsilon_{ijk}$$ + + {\bf Note that:} + \begin{itemize} + \item The sum over the $\tau_i$, $\beta_j$, $\gamma_{ij}$, and + $\varepsilon_{ijk}$ terms are always zero. They model the {\em deviation} + from the grand mean. \pause + \item The directly correspond to the available SS/ MS terms. For + example, in the block randomized ANOVA + \begin{itemize} + \item $f_{ijk} = \overline{\mu} + \tau_{i} + \beta_j + \varepsilon_{ijk}$ + \item $SS(total) = SS(temperature) + SS(blocks) + SS(within)$ + \end{itemize} + \end{itemize} + +\end{frame} + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{different hypotheses from a 2-factor ANOVA} +\small + Data generation model for a two factor ANOVA with interaction + $$f_{ijk} = \overline{\mu} + \tau_{i} + \beta_j + \gamma_{ij} + \varepsilon_{ijk}$$ + + \begin{itemize} + \item {\bf Blocking: } Assume $\gamma_{ij}=0$. Test + $$F=\frac{\mbox{temperature MS} (\tau_i)}{\mbox{error MS} + (\varepsilon_{ijk})}$$\pause + \item {\bf Repeated Measures: } Assume $\gamma_{ij}=0$. Entity + which was repeatedly measured becomes block.\pause + \item {\bf Two factor testing factor influence: } Assume $\gamma_{ij}\not=0$. Test + $$F = \frac{\mbox{temperature MS} (\tau_i)}{\mbox{error MS} + (\varepsilon_{ijk})}$$\pause + \item {\bf Two factor testing interaction: } Assume $\gamma_{ij}\not=0$. Test + $$F=\frac{\mbox{interaction MS}(\gamma_{ij})} {\mbox{error MS} + (\varepsilon_{ijk})}$$ + + \end{itemize} + +\end{frame} + + +%--------------------------------------------------------------- +\begin{frame} +\frametitle{summary} +\begin{itemize} +\small +\item ANOVA is a very flexible method to study the interactions of + categorial variables (factors) and ratio/ interval data \pause +\item Works by checking whether a certain factor/ interaction between + factors, ... is needed to explain the variability in the data \pause +\item Relies on assumptions that need to be checked + \begin{itemize} + \item equal variance for each factor level + \item the residuals are Normally distributed + \item number of points $n_i$ should be the same + \end{itemize}\pause +\item There is a whole zoo of ANOVA techniques, for all kinds of + situations. This is just the tip of the iceberg. +\item One can often get away with violating some of the + assumptions. For more details on that check {\em Biostatistical Analysis} +\end{itemize} + +\end{frame} + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{detour I: One-tailed vs. two-tailed} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%--------------------------------------------------------------- +\begin{frame} +\hypertarget{sec:twotailed}{} + +\frametitle{one-tailed tests} +\begin{task}{Correct or not?} + Imagine a pharmaceutical company runs clinical trials for a drug + that enhances the ability to focus. To that end they apply the drug + to a treatment and measure scores in a standardized test. From the + literature it is known that normal subjects have a score of about 0. + + Since the company want to test whether the drug {\em enhances (score + > 0)} the ability to focus, they choose a one-tailed test ($H_A:$ + treatment group performs better than the performance from the + literature). +\end{task} +\end{frame} + + +%------------------------------------------------------------- + +\begin{frame} +\frametitle{one tailed test} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + {\bf two tailed test} + + \includegraphics[width=\linewidth]{figs/twotailed.png} + \footnotesize + \vspace{-1cm} + + e.g. + + \begin{itemize} + \item $H_0: \mu = 0$ + \item $H_A: \mu \not= 0$ + \vspace{1.8cm} + \end{itemize} + \end{minipage} + \begin{minipage}{0.5\linewidth} + {\bf one tailed test} + + \includegraphics[width=\linewidth]{figs/onetailed.png} + \footnotesize + \vspace{-1cm} + e.g. + + \begin{itemize} + \item $H_0: \mu = 0$ + \item $H_A: \mu > 0$ + \item $\hat\mu < 0$ must directly imply $\hat\mu$ came from + $P(\hat\mu|H_0)$ + \item if that is not the case, using one-tailed is cheating + \end{itemize} + \end{minipage} +\end{minipage} +\hyperlink{back:twotailed}{\color{gray}go back} +\end{frame} + + +% ---------------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{detour II: Statistical Power} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} +\frametitle{Why is it hard to assess the power of a test?} +\begin{minipage}{1.\linewidth} + \begin{minipage}{.5\linewidth} + \includegraphics[width=.8\linewidth]{figs/power.pdf} + \end{minipage} + \begin{minipage}{.5\linewidth} + \begin{itemize} + \item Power = 1 - P(type II error)\\ + = P(reject $H_0$| $H_A$ is true)\pause + \item in general the distribution + \begin{center} + P(test statistic|$H_A$ is true) + \end{center} + is not available to us. + \pause + \item Therefore, the power can often only be specified for a + specific $H_A$. + + \end{itemize} + + \end{minipage} +\end{minipage} +\mycite{J. H. Zar, Biostatistical Analysis} +\hypertarget{sec:power}{} +\hyperlink{back:power}{\color{gray}go back} + +\end{frame} + + +% ---------------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{detour III: Bayes rule and statistical tests} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +% ---------------------------------------------------------- +\begin{frame} +\hypertarget{sec:bayesian}{} + +\frametitle{Why is this funny (or sad)?} +\begin{center} + \includegraphics[width=.4\linewidth]{figs/frequentistsvsbayesians.png} +\end{center} +\mycite{http://xkcd.com/1132/} +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{Why is this funny (or sad)?} +\begin{minipage}{1.\linewidth} + \begin{minipage}{.5\linewidth} + \includegraphics[width=.7\linewidth]{figs/frequentistsvsbayesians.png} + \mycite{http://xkcd.com/1132/} + \end{minipage} + \begin{minipage}{.5\linewidth} + \begin{itemize} + \item $H_0:$ the sun has not gone nova + \item $H_A:$ the sun has gone nova \pause + \item test procedure: we believe the detector \pause + \item Null distribution: multinomial $n=2, p_1 = \frac{1}{6}, ..., p_6 = \frac{1}{6}$ \pause + \item the probability of making a type I error is $p(2\times + 6)=\frac{1}{6}\cdot \frac{1}{6} \approx 0.028$ + \end{itemize} + \pause + So ... what is wrong? + \end{minipage} +\end{minipage} +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{A similar example} +\begin{minipage}{1.\linewidth} + \begin{minipage}{.5\linewidth} + {\bf sensitivity \& specificity of a HIV test} + + \begin{tabular}{ccc} + & HIV & no HIV\tabularnewline + test + & 99.7\% & 1.5\%\tabularnewline + test - & 0.03\% & 98.5\%\tabularnewline + \end{tabular} + + \vspace{1cm} + + {\bf HIV prevalence (Germany)} + + \begin{tabular}{cc} + HIV & no HIV\tabularnewline + 0.1\% & 99.9\%\tabularnewline + \end{tabular} + + + \end{minipage} + \begin{minipage}{.5\linewidth} + \begin{task}{} + What is the probability that you are HIV+ if you test positive? + \end{task}\pause + In order to answer that question, you need two rules for + probability.\pause + + \vspace{1cm} + + What is the power, what is the type I error of the test? + \end{minipage} +\end{minipage} +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{Bayes rule and marginalization} +{\bf Bayes rule} +$$p(A|B)p(B) = p(B|A)p(A)$$ + +{\bf joint probability} +$$p(A,B) = p(A|B)p(B) = p(B|A)p(A)$$ + +{\bf marginalization} +$$p(B) = \sum_{\mbox{possible values a of }A}p(a,B)$$ +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{probability/Bayesian nomenclature} +\frametitle{repetition} +Let $T\in \{+, -\}$ be the test result and $H\in \{+,-\}$ whether you +are HIV positive or not. +\begin{itemize} +\item $p(T|H)$ is the {\em likelihood} \pause +\item $p(H)$ is the {\em prior} \pause +\item $p(H|T)$ is the {\em posterior} +\end{itemize} +\pause +Given the prior and the likelihood, we can compute the posterior. +\begin{align*} + p(H|T) &= \frac{P(T|H)P(H)}{P(T)} &\mbox{Bayes rule}\\ + &= \frac{P(T|H)P(H)}{\sum_h P(T,h)} &\mbox{marginalization}\\ + &= \frac{P(T|H)P(H)}{\sum_h P(T|h)p(h)} &\mbox{joint + probability} +\end{align*} + +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{HIV test} +\begin{minipage}{1.\linewidth} + \begin{minipage}{.5\linewidth} +\begin{tabular}{ccc} + & HIV & no HIV\tabularnewline + test + & 99.7\% & 1.5\%\tabularnewline + test - & 0.03\% & 98.5\%\tabularnewline +\end{tabular} + \end{minipage} + \begin{minipage}{.5\linewidth} +\begin{tabular}{cc} + HIV & no HIV\tabularnewline + 0.1\% & 99.9\%\tabularnewline +\end{tabular} + \end{minipage} +\end{minipage} + + +\begin{align*} + p(H=+|T=+)&= \frac{P(T=+|H=+)P(H=+)}{\sum_{h\in\{+,-\}} P(T=+|H=h)p(H=h)} \\ + p(H=+|T=+)&= \frac{0.997 \cdot 0.001}{0.997 \cdot 0.001 + 0.015 + \cdot 0.999} \\ + &\approx 0.062 +\end{align*} +\pause +This means with a positive HIV test, you have about $6.2$\% chance of +being HIV positive. Why is this number so low? \pause + +\only<3>{Because a lot of the people for which the test is positives + are false positives from the HIV- group. This is because HIV+ is + relatively rare.} +\end{frame} + +%----------------------------------------------------------------- +\begin{frame} +\frametitle{Why is this funny (or sad)?} +\begin{minipage}{1.\linewidth} + \begin{minipage}{.5\linewidth} + \includegraphics[width=.7\linewidth]{figs/frequentistsvsbayesians.png} + \mycite{http://xkcd.com/1132/} + \end{minipage} + \begin{minipage}{.5\linewidth} + {\bf Why is it funny:} Because it points at the fact that + statistical tests usually look at the likelihood only and ignore + the prior. + + \vspace{1cm} + + {\bf Why is it sad?} Because statistical tests usually look at + the likelihood and ignore the prior. + \end{minipage} +\end{minipage} +\hyperlink{back:bayesian}{\color{gray}go back} + +\end{frame} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{detour IV: Assessing normality with QQ plots} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{frame} +\hypertarget{sec:qqplots}{} +\frametitle{histogram equalization} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.5\linewidth} + \begin{task}{histogram equalization} + Which function $y = f(x)$ transforms $x$ such that it has the + distribution of $p(y)$? + \end{task} + \end{minipage} + \begin{minipage}{0.5\linewidth} + \only<1>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE0.png} + \end{center} + }\pause + \only<2>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE0Solution.png} + \end{center} + } + \end{minipage} +\end{minipage} +\end{frame} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{histogram equalization} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.4\linewidth} + \begin{task}{histogram equalization} + How would the function look like if the target was a Normal + distribution? + \end{task} + \end{minipage} + \begin{minipage}{0.6\linewidth} + \only<1>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE1.png} + \end{center} + }\pause + \only<2>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE1Solution.png} + \end{center} + } + \end{minipage} +\end{minipage} +\end{frame} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{histogram equalization} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.4\linewidth} + \begin{task}{histogram equalization} + Is the target distribution a Normal distribution? + \end{task} + \end{minipage} + \begin{minipage}{0.6\linewidth} + \only<1>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE2.png} + \end{center} + }\pause + \only<2>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE2Solution.png} + \end{center} + } + \end{minipage} +\end{minipage} +\end{frame} +% ---------------------------------------------------------- + +\begin{frame} +\frametitle{QQ-plots} + \begin{itemize} + \item QQ-plots can be used to visually assess whether a set of data + points might follow a certain distribution. \pause + \item A QQ-plot is constructed by + \begin{enumerate} + \item computing the fraction of data points $q_1,...,q_n$ that are + lower or equal than a given $x_1,...,x_n$ (Where do you know + that function from?)\pause + \item and plotting it against the value $y_j$ of the other + distribution which has the same $q_i$ + \end{enumerate}\pause + + \item If the two distributions are equal the QQ-plot shows a straight line.\pause + \item How would you assess the normality of data $x_1,...,x_n$ with + a QQ-plot? \pause {\em make the target distribution a Gaussian} + \end{itemize} + +\end{frame} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{histogram equalization} +\begin{minipage}{1.0\linewidth} + \begin{minipage}{0.4\linewidth} + \begin{task}{special transform} + Which function $y = f(x)$ transforms $x$ such that it has the + distribution of $p(y)$? + + + Do you know that function? + + \end{task} + + \only<2>{{\bf Answer:} The cumulative distribution function $f(x) = F(x)$.} + \end{minipage} + \begin{minipage}{0.6\linewidth} + \only<1>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE3.png} + \end{center} + }\pause + \only<2>{ + \begin{center} + \includegraphics[width=1.\linewidth]{figs/HE3Solution.png} + \end{center} + } + \end{minipage} +\end{minipage} +\hyperlink{back:detourIV}{\color{gray} back to statistical tests} +\end{frame} + +\end{document} + +