diff --git a/statistics/assignments/day2.tex b/statistics/assignments/day2.tex new file mode 100644 index 0000000..504a3d7 --- /dev/null +++ b/statistics/assignments/day2.tex @@ -0,0 +1,46 @@ +\documentclass[addpoints,10pt]{exam} +\usepackage{url} +\usepackage{color} +\usepackage{hyperref} + +\pagestyle{headandfoot} +\runningheadrule +\firstpageheadrule + +\firstpageheader{Scientific Computing}{afternoon assignment day 02}{10/21/2014} +%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014} +\firstpagefooter{}{}{} +\runningfooter{}{}{} +\pointsinmargin +\bracketedpoints + +%\printanswers +\shadedsolutions + + +\begin{document} +%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%% +\sffamily +%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% + +\begin{questions} + \question Download example002 from yesterday (brain weights). + \begin{parts} + \part Simulate a null distribution via permutation. + \part Determine whether you can reject ``means are equal'' on a + 5\% significance level using the simulated null distribution. + \part Check whether the means are different with a two sample + t-test in matlab ({\tt ttest2}). + \part Plot the data appropriately and generate a single slide that + contains the plot and short discussion of the test that respects + the nature statistical checklist (ignore all question whether the + assumptions of the test are satisfied). + \part Upload the slide and the code to Ilias. Deadline is 19h00. + \end{parts} +\end{questions} + + + + + +\end{document} diff --git a/statistics/examples/example002.tex b/statistics/examples/example002.tex index 3aa5596..4748dee 100644 --- a/statistics/examples/example002.tex +++ b/statistics/examples/example002.tex @@ -1,3 +1,3 @@ -Brain Weight In 1888, P. Topinard published data on the brain weights +Brain Weight: In 1888, P. Topinard published data on the brain weights of hundreds of French men and women. 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100755 index 0000000..2efd4bb Binary files /dev/null and b/statistics/figs/statistic3.png differ diff --git a/statistics/figs/statistic4.png b/statistics/figs/statistic4.png new file mode 100755 index 0000000..f3a1885 Binary files /dev/null and b/statistics/figs/statistic4.png differ diff --git a/statistics/lecture_statistics01.tex b/statistics/lecture_statistics01.tex index ac11763..771bebf 100644 --- a/statistics/lecture_statistics01.tex +++ b/statistics/lecture_statistics01.tex @@ -22,14 +22,19 @@ % \useoutertheme{miniframes} } -\AtBeginSection[] +\AtBeginSubsection[] { \begin{frame} \begin{center} \Huge \insertsectionhead \end{center} + \tableofcontents[ + currentsubsection, + hideothersubsections, + sectionstyle=show/hide, + subsectionstyle=show/shaded, +] % \frametitle{\insertsectionhead} - % \tableofcontents[currentsection,hideothersubsections] \end{frame} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 @@ -84,25 +89,7 @@ Bernstein Center T\"ubingen} \end{frame} -\begin{frame} - \frametitle{plan} - \setcounter{tocdepth}{1} - \tableofcontents -\end{frame} -\begin{frame} - \frametitle{information} - \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} - %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % errorbars (error bar paper) % confidence intervals (sources of error) @@ -170,8 +157,7 @@ Bernstein Center T\"ubingen} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section[descriptive statistics, errorbars, and plots]{Day 1 -- - descriptive statistics and plots} +\section{Day 1 -- descriptive statistics and plots} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{types of data} @@ -523,7 +509,7 @@ correlation coefficient does not have that property. \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{nominal scale} +\subsection{plotting data} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %------------------------------------------------------------- diff --git a/statistics/lecture_statistics02.tex b/statistics/lecture_statistics02.tex index ac11763..1ea8246 100644 --- a/statistics/lecture_statistics02.tex +++ b/statistics/lecture_statistics02.tex @@ -22,14 +22,19 @@ % \useoutertheme{miniframes} } -\AtBeginSection[] +\AtBeginSubsection[] { \begin{frame} \begin{center} \Huge \insertsectionhead \end{center} + \tableofcontents[ + currentsubsection, + hideothersubsections, + sectionstyle=show/hide, + subsectionstyle=show/shaded, +] % \frametitle{\insertsectionhead} - % \tableofcontents[currentsection,hideothersubsections] \end{frame} } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%5 @@ -84,12 +89,7 @@ Bernstein Center T\"ubingen} \end{frame} -\begin{frame} - \frametitle{plan} - \setcounter{tocdepth}{1} - \tableofcontents -\end{frame} \begin{frame} \frametitle{information} \begin{itemize} @@ -102,722 +102,671 @@ Bernstein Center T\"ubingen} \item \url{http://stats.stackexchange.com} \end{itemize} \end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -% errorbars (error bar paper) -% confidence intervals (sources of error) -% plotting (the right plot for the right data, Dan plotting paper) -% statistical test structure (bootstrapping, resampling, permutation) -% Don'ts: repeated testing, exclude data points -% study design -% PCA + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section[Prelude]{Prelude} +\section{Day 2 -- errorbars, confidence intervals, and tests} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{Types of evidence} +\begin{frame} + \scriptsize + \frametitle{Examples} + \begin{itemize} + \item Before new drugs are given to human subjects, it is common + practice to first test them in dogs or other animals. In part of + one study, a new investigational drug was given to eight male and + eight female dogs at doses of 8 mg/kg and 25 mg/kg. Within each + sex, the two doses were assigned at random to the eight dogs. Many + ``endpoints'' were measured, such as cholesterol, sodium, glucose, + and so on, from blood samples, in order to screen for toxicity + problems in the dogs before starting studies on humans. One + endpoint was alkaline phosphatase level (or APL, measured in U/l). + For females, the effect of increasing the dose from 8 to 25 mg/kg + was positive, although small (the average APL increased from 133.5 + to 143 U/l), but for males the effect of increasing the dose from + 8 to 25 mg/kg was negative.\pause + \item On 15 July 1911, 65-year-old Mrs. Jane Decker was struck by + lightning while in her house. She had been deaf since birth, but + after being struck, she recovered her hearing, which led to a + headline in the New York Times, ``Lightning Cures Deafness.'' + \pause + \item Some research has suggested that there is a genetic basis for + sexual orientation. One such study involved measuring the + midsagittal area of the anterior commissure (AC) of the brain for + 30 homosexual men, 30 heterosexual men, and 30 heterosexual + women. The researchers found that the AC tends to be larger in + heterosexual women than in heterosexual men and that it is even + larger in homosexual men. + \end{itemize} + \mycite{Samuels, Wittmer, Schaffner 2010} +\end{frame} -% ---------------------------------------------------------- -\begin{frame} -\frametitle{my expectations to this course} -\begin{itemize} -\item interest and participation -\item motivation to understand and question concepts -\item high scientific standard -\item intellectual honesty -\item sincere cooperation -\end{itemize} + +\begin{frame} + \scriptsize + \frametitle{types of evidence} + \begin{center} + \Large + {\em experiment} \\ is better than\\ {\em observational study}\\ is + better than\\ {\em anecdotal evidence} + \end{center} \end{frame} -% ---------------------------------------------------------- -\begin{frame} -\frametitle{this week will be ...} -\only<1>{ -\framesubtitle{... no \sout{fun} piece of cake} -\begin{center} - \includegraphics[height=0.7\textheight]{figs/feeding.jpg} -\end{center} -} +\subsection{What is inferential statistics?} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{sources of error in an experiment} + \begin{task}{Think about it for 2 min} + If you repeat a scientific experiment, why do you not get the same + result every time you repeat it? + \end{task} + \pause + \begin{itemize} + \item sampling error (a finite subset of the population of interest + is selected in each experiment) + \item nonsampling errors (e.g. noise, uncontrolled factors) + \end{itemize} +\end{frame} +% ---------------------------------------------------------- +\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>{ -\framesubtitle{... no \sout{fun} piece of cake} -\begin{center} - \includegraphics[height=0.7\textheight]{figs/nacho-trainer.jpg} -\end{center} -} - + \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>{ -\framesubtitle{... no lecture (please!)} -\begin{center} - \includegraphics[height=0.7\textheight]{figs/soccer.jpg} -\end{center} + \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{What you should learn this week} -\begin{itemize} -\item What makes good plots? -\item What is descriptive/inferential statistics? -\item What is the general structure of a statistical test? -\item What does a p-value mean? -\item How can I build my own tests? -\item How large should my $n$ be? -\item What is {\em maximum likelihood} and why is it important? -\end{itemize} +\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} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section[descriptive statistics, errorbars, and plots]{Day 1 -- - descriptive statistics and plots} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{types of data} +\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, (ii) old people, (iii) all people? + \begin{center} + \includegraphics[width=.6\linewidth]{figs/example02.png} + \end{center} +\end{question} +\end{frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\begin{frame} - \frametitle{data scales} - \framesubtitle{What data types are distinguished in statistics?} - \Large - {\bf Why are data types important?} - \pause - \begin{itemize} - \item selection of statistics - \item selection of plots - \item selection of correct tests - \end{itemize} +\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{data scales} - \framesubtitle{nominal/categorial scale} - \begin{itemize} - \item properties like cell type, experimental group (i.e. treatment - 1, treatment 2, control) - \item each observation/sample is put into one category - \item there is no reasonable order among the categories - \item example: [rods, cones] vs. [cones, rods] - \end{itemize} +\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{data scales} - \framesubtitle{ordinal scale} - \begin{itemize} - \item like nominal scale, but there is an order - \item {\bf but:} there is no reasonable measure of {\em distance} - between the classes - \item examples: ranks, ratings - \end{itemize} -\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} + } -\begin{frame} - \frametitle{data scales} - \framesubtitle{interval scale} - \begin{itemize} - \item quantitative/metric values - \item reasonable measure of distance between values but no absolute zero - \item examples: temperature in $^\circ$C - \end{itemize} +\end{center} \end{frame} -%------------------------------------------------------------- -\begin{frame} - \frametitle{data scales} - \framesubtitle{absolut/ratio scale} - \begin{itemize} - \item like interval scale but with absolute zero - \item example: temperature in $^\circ$K - \end{itemize} - \pause - \begin{emphasize}{relationsships between scales} - \begin{itemize} - \item scales exhibit increasing information content from nominal - to absolute - \item conversion ,,downwards'' always possible - \end{itemize} - \end{emphasize} -\end{frame} -%------------------------------------------------------------- -\begin{frame} - \frametitle{examples from neuroscience and psychology} - \begin{itemize} - \item {\bf nominal:}\pause - \begin{itemize} - \item treatment group - \item stimulus class - \item cell type - \end{itemize} - \item {\bf ordinal:} \pause - \begin{itemize} - \item ratings - \item clinical stages of a disease - \item states of an ion channel - \end{itemize} - \item {\bf Absolut-/Ratioskala:}\pause - \begin{itemize} - \item firing rate - \item membrane potential - \item ion concentration - \end{itemize} - \end{itemize} -\end{frame} -%------------------------------------------------------------- -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{statistics} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%------------------------------------------------------------- -\begin{frame} - \frametitle{What is "a statistic"?} - \begin{definition}{statistic} - A statistic (singular) is a single measure of some attribute of a - sample (e.g., its arithmetic mean value). It is calculated by - applying a function (statistical algorithm) to the values of the - items of the sample, which are known together as a set of data. - - \source{http://en.wikipedia.org/wiki/Statistic} - \end{definition} +\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{Beispiele f\"ur Teststatistiken} - \begin{itemize} - \item {\bf nominal:}\pause +\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 count - \item relative frequency/proportion + \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}{standard error vs. standard deviation} - \item {\bf ordinal:} \pause - \begin{itemize} - \item median - \item quantile/percentile - \item rank correlation - \end{itemize} - \item {\bf absolute/ratio:}\pause \begin{itemize} - \item mean - \item variance/ standard deviation - \item Pearson correlation + \item Download the dataset {\tt thymusglandweights.dat} from Ilias + \item Write a program that loads the data into matlab, extracts + the the first $80$ datapoints, and repeat the following steps + $m=500$ times: + \begin{enumerate} + \item draw $50$ data points from $x$ with replacement + \item compute their mean and store it + \end{enumerate} + Look at the standard deviation of the computed means. + \item Compare the result to the standard deviation of the original + $50$ data points and the standard error. \end{itemize} - \end{itemize} -\end{frame} - -%------------------------------------------------------------- -\begin{frame} - \frametitle{exercise} - \begin{task}{Spearman rank correlation} - \begin{enumerate} - \item Use {\tt randi} to generate two vectors - {\tt x,y} with $100$ random integers between $0$ and $10$ each. - \item Find out how to compute the Spearman - rank correlation $$\rho = 1- {\frac {6 \sum - d_i^2}{n(n^2 - 1)}}$$ with Matlab. $d_i = x_i - y_i$ is the - difference in the rank between the single data points. - \item Compute $\rho$ between $x$ and $y$, between $x$ and - $y^2$, between $\log(x+1)$ and $y^2$. - \item Compute the "standard" (Pearson) correlation coefficient - between these values. - \item What can you observe and why does it make sense? - \end{enumerate} \end{task} \end{frame} -%------------------------------------------------------------- \begin{frame}[fragile] - \frametitle{solution} - \begin{solution}{Spearman rank correlation } -\scriptsize + \frametitle{standard error} \begin{lstlisting} ->>> x = randi(10, 100, 1); ->>> y = randi(10, 100, 1); ->>> corr(x,y,'type','Spearman') -ans = - 0.1220 ->>> corr(x,y.^2,'type','Spearman') -ans = - 0.1220 ->>> corr(x,y,'type','Pearson') -ans = - 0.1074 ->>> corr(x,y.^2,'type','Pearson') -ans = - 0.0551 -\end{lstlisting} -The rank correlation does not change under a monotone transformation -of the data. Therefore, it can be used for ordinal data. The Pearson -correlation coefficient does not have that property. - \end{solution} -\end{frame} - -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{what makes a good plot} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{} - \begin{center} - \Huge What makes a good plot? - \end{center} -\end{frame} +load thymusglandweights.dat -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{features of a good plot} - A good plot - \begin{itemize} - \item helps the reader to clearly understand your point.\pause - \item is not misleading and let's the reader judge the information - on her own (different y-axis/length scales in two related plots, - "squeezing" via log-plots). \pause - \item contains information about the data (a comic might be - illustrative, but does not contain information about the - data).\pause - \item adheres to the principle of {\em ink minimization}. - \end{itemize} -\end{frame} +n = 80; +m = 500; +x = thymusglandweights(1:n); -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{features of a good plot} - \framesubtitle{design/organization} - \begin{itemize} - \item Is the display consistent with the model or hypothesis - being tested?\pause - \item Are there "empty dimensions" in the display that could be - removed (A 3D pie chart for 2D categorical data, extraneous colors - that do not encode meaningful information)?\pause - \item Does the display provide an honest and transparent portrayal - of the data (hiding, smoothing, modifying data points should be - avoided or explicitly mentioned)? - \end{itemize} - \mycite{Allen et al. 2012, Neuron} -\end{frame} - -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{features of a good plot} - \framesubtitle{axes} - \begin{itemize} - \item Are axes scales defined as linear, log, or radial?\pause - \item Does each axis label describe the variable and its units (use - "a.u." for arbitrary units)?\pause - \item Are axes limits appropriate for the data (The graphic should - not be bounded at zero if the data can take on both positive and - negative values.)?\pause - \item Is the aspect ratio appropriate for the data (When x and y - axes contrast the same variable under different conditions the - graphic should be square.)? - \end{itemize} - \mycite{Allen et al. 2012, Neuron} +mu = zeros(m,1); +for i = 1:m + mu(i) = mean(x(randi(n,n,1))); +end +disp(['bootstrap standard error: ', num2str(std(mu))]); +disp(['standard error: ', num2str(std(x)/sqrt(n))]); +\end{lstlisting} \end{frame} -%------------------------------------------------------------- +%------------------------------------------------------------------------------ \begin{frame}[fragile] - \frametitle{features of a good plot} - \framesubtitle{color mapping} + \frametitle{standard error} + \framesubtitle{bootstrapping} \begin{itemize} - \item Is a color bar provided?\pause - \item Is the color map sensible for the data type (does the data - extend to both $\pm$, does it live in an interval, is it - circular)?\pause - \item Are contrasting colors consistent with a natural interpretation? - \item Can features be discriminated when printed in grayscale? - \item Has red/green contrast been avoided to accommodate common - forms of colorblindness? + \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} - \mycite{Allen et al. 2012, Neuron} \end{frame} -%------------------------------------------------------------- + +%------------------------------------------------------------------------------ \begin{frame}[fragile] - \frametitle{features of a good plot} - \framesubtitle{uncertainty} + \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 Does the display indicate the uncertainty of estimated parameters?\pause - \item Is the type of error surface appropriate for the data? - \begin{itemize} - \item Use standard deviations to describe variability in the population.\pause - \item Use standard errors or confidence intervals to make inferences - about parameters estimated from a sample.\pause - \item Parametric confidence intervals should only be used if data - meet the assumptions of the underlying model.\pause - \end{itemize} - \item Are the units of uncertainty defined (is it standard error, is - it $95\%$ confidence interval)? + \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 error of the + median. \end{itemize} - \mycite{Allen et al. 2012, Neuron} -\end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{features of a good plot} - \framesubtitle{annotation} + \pause + {\bf Inferential statistics} \begin{itemize} - \item Are all symbols defined, preferably by directly labeling objects?\pause - \item Is the directionality of a contrast between conditions obvious?\pause - \item Is the number of samples or independent experiments indicated?\pause - \item Are statistical procedures and criteria for significance described?\pause - \item Are uncommon abbreviations avoided or clearly defined?\pause - \item Are abbreviations consistent with those used in the text? + \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} - \mycite{Allen et al. 2012, Neuron} -\end{frame} - -\subsection{bad examples} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{suboptimal example} - \begin{center} - \includegraphics[width=.5\linewidth]{figs/nobelbad} - \end{center} - \mycite{Hafting et al. 2005, nature} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{suboptimal example} - \begin{center} - \includegraphics[width=.5\linewidth]{figs/badbarright.png} - \end{center} - \source{http://en.wikipedia.org/wiki/Misleading\_graph} +% ---------------------------------------------------------- +\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}[fragile] - \frametitle{suboptimal example} - \begin{center} - \includegraphics[width=.4\linewidth]{figs/yaxisscalingleft.png} - \hspace{.5cm} - \includegraphics[width=.4\linewidth]{figs/yaxisscalingright.png} - \end{center} - \source{http://en.wikipedia.org/wiki/Misleading\_graph} -\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{frame}[fragile] - \frametitle{suboptimal example} - \begin{center} - \includegraphics[width=.4\linewidth]{figs/badscatterleft.png} - \hspace{.5cm} - \includegraphics[width=.4\linewidth]{figs/badscatterright.png} - \end{center} - \source{http://en.wikipedia.org/wiki/Misleading\_graph} -\end{frame} +\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{frame} - \frametitle{suboptimal example} - \begin{center} - \includegraphics[width=.8\linewidth]{figs/badbarplot} - \end{center} - \source{www.enfovis.com} -\end{frame} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{nominal scale} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting nominal data} - \framesubtitle{bar plot for count and relative frequency} - \begin{center} - \includegraphics[width=.8\linewidth]{figs/nominaldataplot} - \end{center} -\end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting nominal data} - \framesubtitle{bar plot for count and relative frequency} - \scriptsize -\begin{lstlisting} -% plot -bar([1,2], [50, 90], 'facecolor', 'k') - -% labels axes -ylabel('cell count') -xlabel('cell type') - -% cosmetics -xlim([0.5,2.5]) -ylim([0, 100]) -box('off') -set(gca,'XTick',1:2,'XTickLabel',{'pyramidal','interneuron'},'FontSize',20) - -% settings for saving the figure -set(gcf, 'PaperUnits', 'centimeters'); -set(gcf, 'PaperSize', [11.7 9.0]); -set(gcf, 'PaperPosition',[0.0 0.0 11.7 9.0]); -\end{lstlisting} \end{frame} -%---------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting nominal data} - \framesubtitle{pie chart for count and relative frequency} - \begin{center} - \includegraphics[width=.8\linewidth]{figs/nominaldataplot2} - \end{center} -\end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting nominal data} - \framesubtitle{exercise} - \begin{task}{pie chart} - Plot the same data ($n_{py}=50$, $n_{in}=90$) as a pie chart in Matlab. +% ---------------------------------------------------------- +\begin{frame} + \frametitle{confidence intervals for the mean via bootstrapping} + \framesubtitle{how to get the sampling distribution} + + \begin{task}{bootstrapping a confidence interval for the mean} + \begin{itemize} + \item Use the same dataset as before. + \item Bootstrap $500$ means. + \item Plot their distribution. + \item Compute the $2.5\%$ and the $97.5\%$ percentile of the + $500$ means. + \item Mark them in the plot. + \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}[fragile] - \frametitle{plotting nominal data} - \framesubtitle{pie chart for relative frequency} - \scriptsize + \frametitle{confidence intervals for the median} +\scriptsize \begin{lstlisting} -data = [50, 90]; -h = pie(data, [1,0], {'pyramidal (n=50)', 'interneuron (n=90)'}) -hText = findobj(h,'Type','text') % text object handles - -set(h(1), 'FaceColor', [.2,.2,.2]); -set(h(2), 'Rotation', 45); -set(h(3), 'FaceColor', [.8,.8,.8]); -set(h(4), 'Rotation', 45); - -title('cell count') -set(gca,'XTick',1:2,'XTickLabel',{'pyramidal', 'interneuron'}) -box('off') -set(gcf, 'PaperUnits', 'centimeters'); -set(gcf, 'PaperSize', [11.7 9.0]); -set(gcf, 'PaperPosition',[0.0 0.0 11.7 9.0]); +load thymusglandweights.dat +n = 80; +x = thymusglandweights(1:n); + +m = 500; +me = zeros(m,1); +for i = 1:m + me(i) = mean(x(randi(n,n,1))); +end + +disp(['bootstrap quantiles: ' , num2str(quantile(me,0.025)), ' ' ,num2str(quantile(me,1-0.025))]); + \end{lstlisting} \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} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{histogram} - \begin{center} - \includegraphics[width=.8\linewidth]{figs/histogram} - \end{center} -\end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{bad choice of bins} - \begin{center} - \includegraphics[width=.4\linewidth]{figs/histogrambad} - \includegraphics[width=.4\linewidth]{figs/histogrambad2} - \end{center} - \begin{summary}{Rule of thumb} - Choose the bins $b\approx n/20$. - \end{summary} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{how to do in Matlab} - \scriptsize -\begin{lstlisting} -x = randn(2000,1); % generate Gaussian data -hist(x, 50); % generate histogram -% set facecolor to gray -h = findobj(gca, 'Type','patch'); -set(h(1), 'FaceColor',[.2,.2,.2], 'EdgeColor','w', 'linewidth',2) -% plot a white grid over it -h = gridxy([],get(gca,'ytick'),'color','w','linewidth',2) -uistack(h, 'top') - -% cosmetics -box('off'); -xlabel('Data') -ylabel('Count') -\end{lstlisting} +% ---------------------------------------------------------- +\begin{frame} +\frametitle{confidence interval for the mean} +\framesubtitle{Let's search the pattern in the normal way of computing +a confidence interval for the mean} +\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}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{bar plot} - There are several ways to plot a sample $x_1, ..., x_n$ of interval/ratio/absolute - scale with a bar plot - \begin{center} - \includegraphics[width=.6\linewidth]{figs/barplots.png} - \end{center} +% ---------------------------------------------------------- +\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*} +Hint: Use the function {\tt tinv(0.025, n-1)} to get the value of +$t_{2.5\%}$ and similar for $t_{97.5\%}$. +\end{task} + + \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile,fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{bar plot} + +\begin{frame}[fragile] +\frametitle{solution} \scriptsize \begin{lstlisting} -% bar plot -x = rand(10,1); -gray = [.5,.5,.5]; - -bar(1, mean(x), 'EdgeColor','w','FaceColor', gray); -hold on - -bar(2, mean(x), 'EdgeColor','w','FaceColor', gray); -plot(0*x + 2, x, 'ok'); - -bar(3, mean(x), 'EdgeColor','w','FaceColor', gray); -errorbar(3, mean(x), std(x), 'ok'); - -bar(4, mean(x), 'EdgeColor','w','FaceColor', gray); -errorbar(4, mean(x), std(x)/sqrt(length(x)), 'ok'); -set(gca, 'xtick',[]) -ylabel('uniformly distributed random data in [0,1]') -box('off') -title('different forms of bar plots') -hold off +load thymusglandweights.dat +n = 80; +x = thymusglandweights(1:n); + +m = 500; +me = zeros(m,1); +for i = 1:m + me(i) = mean(x(randi(n,n,1))); +end + +t025 = tinv(0.025, n-1); +t975 = tinv(0.975, n-1); + +se = std(x)/sqrt(n); + +disp(['bootstrap quantiles: ' , num2str(quantile(me,0.025)), ' ' ,num2str(quantile(me,1-0.025))]); +disp(['analytical CI: ' , num2str(mean(x)+t025*se), ' ' , num2str(mean(x)+t975*se)]); + \end{lstlisting} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile,fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{bar plot and measure of central tendency and spread} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\subsection{statistical tests} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{frame} + \frametitle{ingredients into a test} + \begin{itemize} - \item A bar plot collapses real data onto a single number and some - measure of spread. This number is usually a {\em measure of central - tendency}, i.e. a typical/central value for the probability - distribution of the data.\pause - \item What measures of central tendency can you think of?\pause - \begin{itemize} - \item mean - \item median - \item geometric mean (the nth root of the product of the data values) - \item weighted mean - \item midrange (mean of the maximum and minimum values of a data set) - \end{itemize}\pause - \item Additionally, the bar plot is equipped with a measure of {\em - spread} or {\em dispersion}. What measure of spread can you think of?\pause - \begin{itemize} - \item standard deviation - \item range (maximum minus minimum of a dataset) - \item inter-quartile range - \end{itemize} + \item {\bf What is the goal of a test?}\pause + \item[] Check whether a measured + statistic looks different from what you would expect if there was no + effect.\pause + \item {\bf What are the ingredients into a test?}\pause + \item[] a test statistic (e.g. the mean, the median, ...) and a null + distribution\pause + \item {\bf What is a null distribution?}\pause + \item[] The sampling distribution of the statistic in case there is + no effect (i.e. the Null hypothesis is true). \end{itemize} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile,fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{measure of central tendency and spread} - \Large - \begin{center} - \bf The part of statistics that summarizes data in a small number - of values is called {\em descriptive statistics}. - \end{center} +\begin{frame} + \frametitle{how tests work} + \begin{enumerate} + \item Choose a statistic. + \item Get a null distribution. + \item Compare your actually measure value with the Null + distribution. + \end{enumerate} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile,fragile] - \frametitle{robust statistics} - \begin{task}{When is statistic called robust (leave-one-out)?} +\begin{frame} + \frametitle{Example: one sample test} + \framesubtitle{step 2: get a Null distribution} + \scriptsize + Assume that the expected weight of a thymus gland from the + literature is 34.3g. We want to test whether the mean of our + thymus gland dataset is different from the expectation in the + literature. Comparing a statistic of a dataset against a fixed value + is called {\em one sample test}. + \pause + + \begin{itemize} + \item {\bf How could we simulate the distribution of the data if the + mean was really 30g?}\pause + \item[] Bootstrapping. + \end{itemize} + + \begin{task}{generating a null distribution} \begin{itemize} - \item Generate an array with $20$ random numbers using {\tt - randn}. - \item Compute $20$ means: the $i^{th}$ mean is computed from the - data set {\em without} the $i^{th}$ example. - \item Repeat this with the median. - \item Make a bar plot that depicts the means of the computed means - and medians along with an appropriate measure of dispersion. - \item What can you observe? Do you understand why? + \item Write a matlab program that bootstraps 2000 means from the + thymus gland dataset. + \item How can we adjust the data that it has mean 34.3g (remember, + we want to simulate the null distribution)? + \item Plot a histogram of these 2000 means. + \item Also indicate the actual mean of the data. \end{itemize} \end{task} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{boxplot} +\begin{frame} + \frametitle{Example: one sample test} + \framesubtitle{step 3: compare the actual value to the Null distribution} \begin{minipage}{1.0\linewidth} \begin{minipage}{0.5\linewidth} + The question we want to answer in this step is: \begin{center} - \includegraphics[width=\linewidth]{figs/boxplot.png} + \color{blue} Does the actually measure value look like it came + from the Null distribution? \end{center} \end{minipage} \begin{minipage}{0.5\linewidth} - Who knows what the elements mean?\pause - \begin{itemize} - \item the box depicts the inter-quartile range - \item the line denotes the median - \item the whiskers denote the extreme value of the data not - considered outliers - \item outliers are plotted separately - \end{itemize} - \begin{task}{Outliers} - \begin{itemize} - \item Find out how an outlier is defined in a matlab boxplot. - \item Can you remove an outlier from the dataset? - \end{itemize} - \end{task} + \includegraphics[width=\linewidth]{figs/bootstraptest.png} \end{minipage} \end{minipage} + {\bf How could we do this in our bootstrapping example?}\pause + \begin{itemize} + \item Set a threshold. \pause How do we choose the threshold? \pause Via type I error.\pause + \item Specify the type I error if we used the actual measured value + as threshold (p-value). Why is that a reasonable strategy? + \end{itemize} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting interval/ratio/absolute data} - \framesubtitle{violinplot} - \begin{center} - \includegraphics[width=.8\linewidth]{figs/violinplots.png} - \end{center} - \begin{itemize} - \item Violinplots depict the distribution of the data by a - smoothed histogram. - \item Additional information (data points, median, - inter-quartile range) are plotted inside. - \end{itemize} +\begin{frame} + \frametitle{Example: one sample test} + \framesubtitle{step 3: compare the actual value to the Null distribution} + \begin{task}{type I error and p-value} + Extend the script such that it + \begin{itemize} + \item computes the $5\%$ significance boundaries from the + distribution and plot it into the histogram. + \item computes a p-value. + \end{itemize} + \end{task} \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting combinations of scales} - What could we use for a combination of categorial/nominal and - interval/ratio/absolute? - \pause - \begin{center} - \includegraphics[width=.5\linewidth]{figs/factorplot.png} - \end{center} - Each category is a single bar. +\begin{frame} + \frametitle{two sample test} + \framesubtitle{permutation test} + Brain Weight In 1888, P. Topinard published data on the brain + weights of hundreds of French men and women. Brain weights are given + in gram. The data can be downloaded from Ilias (example 002 from + yesterday). + + \vspace{.5cm} + {\bf How could we determine (similar to bootstrapping) whether the + mean brain weight of males and females are different?} + \begin{itemize} + \item What do we use as a statistic? + \item[]<2-> The difference of the means of the two groups. + \item How do we simulate the null distribution? + \item[]<3-> Shuffle the labels ``male'' and ``female'', compute + difference in means of two groups, and repeat. + \end{itemize} + \end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] - \frametitle{plotting combinations of scales} - What could we use for a combination of interval/ratio/absolute and - interval/ratio/absolute, e.g. $(x_1, y_1), ..., (x_n,y_n)$? \pause - \begin{center} - \includegraphics[width=.8\linewidth]{figs/paireddata.png} - \end{center} - Scatter plot or paired bar chart. Scatter plot can also be used for - ordinal vs. ordinal data (why not the bar chart?). -\end{frame} -%------------------------------------------------------------- -\begin{frame}[fragile] +\begin{frame} \begin{center} - \Huge - That's it. + \Huge That's it. \end{center} \end{frame} - \end{document} diff --git a/statistics/matlab/bootstrap_mean.m b/statistics/matlab/bootstrap_mean.m new file mode 100644 index 0000000..0def8fb --- /dev/null +++ b/statistics/matlab/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/matlab/ci_mean.m b/statistics/matlab/ci_mean.m new file mode 100644 index 0000000..0439252 --- /dev/null +++ b/statistics/matlab/ci_mean.m @@ -0,0 +1,19 @@ +load thymusglandweights.dat + +n = 80; +x = thymusglandweights(1:n); + +m = 5000; +me = zeros(m,1); +for i = 1:m + me(i) = median(x(randi(n,n,1))); +end + +t025 = tinv(0.025, n-1); +t975 = tinv(0.975, n-1); + +se = std(x)/sqrt(n); + +fprintf('bootstrap quantiles: %.4f, %.4f \n', quantile(me,0.025), quantile(me,0.975)); +fprintf('analytical quantile: %.4f, %.4f \n', mean(x)+t025*se, mean(x)+t975*se); + diff --git a/statistics/matlab/ci_media.m b/statistics/matlab/ci_media.m new file mode 100644 index 0000000..ad20428 --- /dev/null +++ b/statistics/matlab/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/matlab/tests.m b/statistics/matlab/tests.m new file mode 100644 index 0000000..9c16a29 --- /dev/null +++ b/statistics/matlab/tests.m @@ -0,0 +1,38 @@ +close all +clear all +load thymusglandweights.dat + +literature_mean = 34.3; + +x = thymusglandweights; +n = length(x); +y = x - mean(x) + literature_mean; + +m = 2000; +me = zeros(m,1); +for i = 1:m + me(i) = median(y(randi(n,n,1))); +end + +hist(me, 50); +hold on +mu = mean(x); +plot([mu,mu],[0,200],'--r','LineWidth',3); +xlabel('thymus gland weights [g]'); +ylabel('frequency'); +title('bootstrapped null distribution'); +hold off + +% 5% significance boundaries +low = quantile(me,0.025); +high = quantile(me,0.975); +disp(['the 5% boundaries are: ', num2str(low), ' ', num2str(high)]); + +hold on +plot([low,low],[0,200],'--g','LineWidth',3); +plot([high,high],[0,200],'--g','LineWidth',3); +hold off + +pval = mean(abs(me-literature_mean) > abs(mu - literature_mean)) + +legend('Null distribution','measured mean','5% significance boundaries') diff --git 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