82 lines
2.5 KiB
TeX
82 lines
2.5 KiB
TeX
\documentclass[addpoints,10pt]{exam}
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\usepackage{url}
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\usepackage{color}
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\usepackage{hyperref}
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\pagestyle{headandfoot}
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\runningheadrule
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\firstpageheadrule
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\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
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-- 11/06/2014}
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%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\firstpagefooter{}{}{}
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\runningfooter{}{}{}
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\pointsinmargin
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\bracketedpoints
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%\printanswers
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%\shadedsolutions
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%% Submission instructions %%%%%%%%%%%%%%%%%%%%%%%%%
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\sffamily
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% \begin{flushright}
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% \gradetable[h][questions]
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% \end{flushright}
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\begin{center}
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\input{../disclaimer.tex}
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\end{center}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
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\begin{questions}
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\question The p-value corresponds to the probability
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$$P(\mbox{result seems significant}| H_0 \mbox{is true}).$$
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This means that if your significance threshold is $\alpha=0.05$ and
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you accept all test with $p \le \alpha$ as significant, then $5\%$
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of all cases in which $H_0$ was true (there was no effect) your test
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will appear significant (false positive).
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The problem with that is that you do not know for how many of the
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tests $H_0$ is actually true. What you really would like to know is:
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From all those tests that came out significant ($p\le\alpha$) how
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many of them are false positives? This probability corresponds to
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$$P(H_0 \mbox{is true}|\mbox{result seems significant})$$ and is
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called {\em false discovery rate}. In general you cannot compute
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it. However, if you have many p-values, then you can actually
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estimate it. The corresponding ``p-value'' for the false discovery
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rate is called ``q-value''.
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In the paper
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{\em Storey, J. D., \& Tibshirani, R. (2003). Statistical
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significance for genomewide studies. Proceedings of the National
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Academy of Sciences of the United States of America, 100,
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9440–9445. doi:10.1073/pnas.1530509100}
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you can find an algorithm how to compute q-values from p-values.
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The attached data file {\tt p\_values.dat} contains p-values from
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test of several neurons whether they respond to a certain stimulus
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condition or not.
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\begin{parts}
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\part Plot a histogram of the p-values.
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\part Read and understand the paper by Storey and
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Tibshirani. Visualize their method at your histogram.
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\part Implement their method and convert each p-value to a
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q-value.
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\part From running the script, estimate the proportion of neurons
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that show a true effect (i.e. $P(H_A)$).
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\end{parts}
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\end{questions}
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\end{document}
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