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Correct calculation of empirical p-vale

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hyginn 2017-11-01 23:29:18 -04:00
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42
FND-STA-Probability_distribution.R
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@ -3,12 +3,13 @@
# Purpose: A Bioinformatics Course: # Purpose: A Bioinformatics Course:
# R code accompanying the FND-STA-Probability_distribution unit. # R code accompanying the FND-STA-Probability_distribution unit.
# #
# Version: 1.0 # Version: 1.1
# #
# Date: 2017 10 10 # Date: 2017 10
# Author: Boris Steipe (boris.steipe@utoronto.ca) # Author: Boris Steipe (boris.steipe@utoronto.ca)
# #
# Versions: # Versions:
# 1.1 Corrected empirical p-value
# 1.0 First code live version # 1.0 First code live version
# #
# TODO: # TODO:
@ -27,20 +28,20 @@
#TOC> #TOC>
#TOC> Section Title Line #TOC> Section Title Line
#TOC> ----------------------------------------------------------------------- #TOC> -----------------------------------------------------------------------
#TOC> 1 Introduction 54 #TOC> 1 Introduction 49
#TOC> 2 Three fundamental distributions 117 #TOC> 2 Three fundamental distributions 112
#TOC> 2.1 The Poisson Distribution 120 #TOC> 2.1 The Poisson Distribution 115
#TOC> 2.2 The uniform distribution 173 #TOC> 2.2 The uniform distribution 168
#TOC> 2.3 The Normal Distribution 193 #TOC> 2.3 The Normal Distribution 188
#TOC> 3 quantile-quantile comparison 234 #TOC> 3 quantile-quantile comparison 229
#TOC> 3.1 qqnorm() 244 #TOC> 3.1 qqnorm() 239
#TOC> 3.2 qqplot() 304 #TOC> 3.2 qqplot() 299
#TOC> 4 Quantifying the difference 321 #TOC> 4 Quantifying the difference 316
#TOC> 4.1 Chi2 test for discrete distributions 355 #TOC> 4.1 Chi2 test for discrete distributions 350
#TOC> 4.2 Kullback-Leibler divergence 446 #TOC> 4.2 Kullback-Leibler divergence 441
#TOC> 4.2.1 An example from tossing dice 457 #TOC> 4.2.1 An example from tossing dice 452
#TOC> 4.2.2 An example from lognormal distributions 579 #TOC> 4.2.2 An example from lognormal distributions 574
#TOC> 4.3 Kolmogorov-Smirnov test for continuous distributions 620 #TOC> 4.3 Kolmogorov-Smirnov test for continuous distributions 616
#TOC> #TOC>
#TOC> ========================================================================== #TOC> ==========================================================================
@ -606,9 +607,10 @@ abline(v = KLdiv(pmfL1, pmfL2), col="firebrick")
# How many KL-divergences were less than the difference we observed? # How many KL-divergences were less than the difference we observed?
sum(divs < KLdiv(pmfL1, pmfL2)) #933 sum(divs < KLdiv(pmfL1, pmfL2)) #933
# Therefore the probability that the samples came from the same distribution # Therefore the empirical p-value that the samples came from the same
# is only 100 * (N - 933) / N (%) ... 6.7%. You see that this gives a much more # distribution is only 100 * ((N - 933) + 1) / (N + 1) (%) ... 6.8%. You see
# powerful statistical approach than the chi2 test we used above. # that this gives a much more powerful statistical approach than the chi2 test
# we used above.
# == 4.3 Kolmogorov-Smirnov test for continuous distributions ============== # == 4.3 Kolmogorov-Smirnov test for continuous distributions ==============
@ -628,7 +630,7 @@ ks.test(dl1, dg1.5)
ks.test(dl1, dg1.9) ks.test(dl1, dg1.9)
ks.test(dg1.5, dg1.9) # the two last gammas against each other ks.test(dg1.5, dg1.9) # the two last gammas against each other
# (The warnings about the presence of ties comes from the 0s in our function # (The warnings about the presence of ties comes from the 0's in our function
# values). The p-value that these distributions are samples of the same # values). The p-value that these distributions are samples of the same
# probability distribution gets progressively smaller. # probability distribution gets progressively smaller.

52
FND-STA-Significance.R
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@ -3,12 +3,13 @@
# Purpose: A Bioinformatics Course: # Purpose: A Bioinformatics Course:
# R code accompanying the FND-STA-Significance unit. # R code accompanying the FND-STA-Significance unit.
# #
# Version: 1.0 # Version: 1.1
# #
# Date: 2017 09 05 # Date: 2017 09 - 2017 10
# Author: Boris Steipe (boris.steipe@utoronto.ca) # Author: Boris Steipe (boris.steipe@utoronto.ca)
# #
# Versions: # Versions:
# 1.1 Corrected treatment of empirical p-value
# 1.0 First contents # 1.0 First contents
# #
# TODO: # TODO:
@ -20,27 +21,22 @@
# answer, or ask your instructor. Don't continue if you don't understand what's # answer, or ask your instructor. Don't continue if you don't understand what's
# going on. That's not how it works ... # going on. That's not how it works ...
# ============================================================================== # ==============================================================================
#TOC> ========================================================================== #TOC> ==========================================================================
#TOC> #TOC>
#TOC> Section Title Line #TOC> Section Title Line
#TOC> ------------------------------------------------------------ #TOC> ------------------------------------------------------------
#TOC> 1 Significance and p-value 46 #TOC> 1 Significance and p-value 42
#TOC> 1.1 Significance levels 57 #TOC> 1.1 Significance levels 53
#TOC> 1.2 probability and p-value 74 #TOC> 1.2 probability and p-value 70
#TOC> 1.2.1 p-value illustrated 100 #TOC> 1.2.1 p-value illustrated 100
#TOC> 2 One- or two-sided 152 #TOC> 2 One- or two-sided 153
#TOC> 3 Significance by integration 192 #TOC> 3 Significance by integration 193
#TOC> 4 Significance by simulation or permutation 198 #TOC> 4 Significance by simulation or permutation 199
#TOC> 5 Final tasks 298 #TOC> 5 Final tasks 302
#TOC> #TOC>
#TOC> ========================================================================== #TOC> ==========================================================================
#TOC>
#TOC>
# = 1 Significance and p-value ============================================ # = 1 Significance and p-value ============================================
@ -56,7 +52,7 @@
# == 1.1 Significance levels =============================================== # == 1.1 Significance levels ===============================================
# A "very small" probability is purely a matter of convention, a cultural # A "very small" probability is purely a matter of convention - a cultural
# convention. In the biomedical field we usually call probabilities of less then # convention. In the biomedical field we usually call probabilities of less then
# 0.05 (5%) small enough to reject the null-hypothesis. Thus we call # 0.05 (5%) small enough to reject the null-hypothesis. Thus we call
# observations with a probability of less than 0.05 "significant" and if we want # observations with a probability of less than 0.05 "significant" and if we want
@ -67,7 +63,7 @@
# also be 0.0498631, or 0.03, or 1/(pi^3). 0.05 is just the value that the # also be 0.0498631, or 0.03, or 1/(pi^3). 0.05 is just the value that the
# British statistician Ronald Fisher happened to propose for this purpose in # British statistician Ronald Fisher happened to propose for this purpose in
# 1925. Incidentally, Fisher later recommended to use different cutoffs for # 1925. Incidentally, Fisher later recommended to use different cutoffs for
# diffrent purposes (cf. # different purposes (cf.
# https://en.wikipedia.org/wiki/Statistical_significance). # https://en.wikipedia.org/wiki/Statistical_significance).
@ -93,8 +89,12 @@ print(x, digits = 22)
# 2: that distribution can be integrated between any specific value # 2: that distribution can be integrated between any specific value
# and its upper and lower bounds (or +- infinity). # and its upper and lower bounds (or +- infinity).
# Then what we really mean by the probability of an observation in the context of that distribution is: the # Then what we really mean by the probability of an observation in the context
# probability of observing that value, or a value more extreme than the one we have. We call this the p-value. Note that we are not talking about an individual number anymore, we are talking about the area under the curve between our observation and the upper (or lower) bound of the curve, as a fraction of the whole. # of that distribution is: the probability of observing that value, or a value
# more extreme than the one we have. We call this the p-value. Note that we are
# not talking about an individual number anymore, we are talking about the area
# under the curve between our observation and the upper (or lower) bound of the
# curve, as a fraction of the whole.
# === 1.2.1 p-value illustrated # === 1.2.1 p-value illustrated
@ -102,6 +102,7 @@ print(x, digits = 22)
# Let's illustrate. First we draw a million random values from our # Let's illustrate. First we draw a million random values from our
# standard, normal distribution: # standard, normal distribution:
set.seed(112358)
r <- rnorm(1000000) r <- rnorm(1000000)
# Let's see what the distribution looks like: # Let's see what the distribution looks like:
@ -201,8 +202,11 @@ sum(r > quantile(r, probs = 0.95))
# be warranted for biological data, can be a highly technical question. # be warranted for biological data, can be a highly technical question.
# Fortunately, we can often simply run a simulation, a random resampling, or a # Fortunately, we can often simply run a simulation, a random resampling, or a
# permutation and then count the number of outcomes, just as we did with our # permutation and then count the number of outcomes, just as we did with our
# rnorm() samples. Here is an example. Assume you have a protein sequence and # rnorm() samples. We call this an empirical p-value. (Actually, the "empirical
# you speculate that postively charged residues are close to negatively charged # p-value" is defined as (Nobs + 1) / (N + 1). )
# Here is an example. Assume you have a protein sequence and
# you speculate that positively charged residues are close to negatively charged
# residues to balance charge locally. A statistic that would capture this is the # residues to balance charge locally. A statistic that would capture this is the
# mean minimum distance between all D,E residues and the closest R,K,H # mean minimum distance between all D,E residues and the closest R,K,H
# residue. Let's compute this for the sequence of yeast Mbp1. # residue. Let's compute this for the sequence of yeast Mbp1.
@ -297,8 +301,8 @@ abline(v = chSep(v), col = "#EE0000")
# = 5 Final tasks ========================================================= # = 5 Final tasks =========================================================
# From chs, compute the probability of a mean minimum charge separation to be # From chs, compute the empirical p-value of a mean minimum charge separation to
# larger or equal to the value observed for the yeast MBP1 sequence. Note # be larger or equal to the value observed for the yeast MBP1 sequence. Note
# the result in your journal. Is it significant? Also note the result of # the result in your journal. Is it significant? Also note the result of
# the following expression for validation: # the following expression for validation:
writeBin(sum(chs), raw(8)) writeBin(sum(chs), raw(8))