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Maintenance and sample solutions

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hyginn 2020-09-23 06:07:40 +10:00
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FND-STA-Significance.R
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@ -1,20 +1,16 @@
# tocID <- "FND-STA-Significance.R" # tocID <- "FND-STA-Significance.R"
# #
# ---------------------------------------------------------------------------- #
# PATIENCE ... #
# Do not yet work wih this code. Updates in progress. Thank you. #
# boris.steipe@utoronto.ca #
# ---------------------------------------------------------------------------- #
# #
# 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: 1.2 # Version: 1.3
# #
# Date: 2017 09 - 2019 01 # Date: 2017-09 - 2020-09
# Author: Boris Steipe (boris.steipe@utoronto.ca) # Author: Boris Steipe (boris.steipe@utoronto.ca)
# #
# Versions: # Versions:
# 1.3 2020 Maintenance. Add sample solution.
# 1.2 Update set.seed() usage # 1.2 Update set.seed() usage
# 1.1 Corrected treatment of empirical p-value # 1.1 Corrected treatment of empirical p-value
# 1.0 First contents # 1.0 First contents
@ -31,18 +27,22 @@
#TOC> ========================================================================== #TOC> ==========================================================================
#TOC> #TOC>
#TOC> Section Title Line #TOC> Section Title Line
#TOC> ------------------------------------------------------------------ #TOC> ------------------------------------------------------------------
#TOC> 1 Significance and p-value 43 #TOC> 1 Significance and p-value 49
#TOC> 1.1 Significance levels 54 #TOC> 1.1 Significance levels 60
#TOC> 1.2 probability and p-value 71 #TOC> 1.2 probability and p-value 77
#TOC> 1.2.1 p-value illustrated 103 #TOC> 1.2.1 p-value illustrated 109
#TOC> 2 One- or two-sided 158 #TOC> 2 One- or two-sided 165
#TOC> 3 Significance by integration 198 #TOC> 3 Significance by integration 209
#TOC> 4 Significance by simulation or permutation 204 #TOC> 4 Significance by simulation or permutation 215
#TOC> 5 Final tasks 312 #TOC> 5 Final tasks 327
#TOC> #TOC> 6 Sample solutions 336
#TOC> 6.1 338
#TOC> 6.2 342
#TOC> 6.3 346
#TOC>
#TOC> ========================================================================== #TOC> ==========================================================================
@ -106,7 +106,7 @@ print(x, digits = 22)
# curve, as a fraction of the whole. # curve, as a fraction of the whole.
# === 1.2.1 p-value illustrated # === 1.2.1 p-value illustrated
# 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:
@ -146,19 +146,20 @@ rect(x, 0, xMax, y, col = "#00EE0044", border = TRUE)
# ... and a red line for our observation. # ... and a red line for our observation.
abline(v = x, col = "#EE0000", lwd = 2) abline(v = x, col = "#EE0000", lwd = 2)
# The p-value of our observation is the area colored red as a fraction of the # The p-value of our observation is the red area as a fraction of the
# whole histogram (red + green). # whole histogram (red + green).
# Task: # Task:
# Explain how the expression sum(r < x) works to give us a count of values # Explain how the expression sum(r < x) works to give us a count of values
# with the property we are looking for. # with the property we are looking for. E.g., examine -4:4 < x
# Task: # Task:
# Write an expression to estimate the probability that a value # Write an expression to estimate the probability that a value
# drawn from the vector r is less-or-equal to x. The result you get # drawn from the vector r is less-or-equal to x. The result you get
# will depend on the exact values that went into the vector r but it should # will depend on the exact values that went into the vector r but it should
# be close to 0.185 That expression is the p-value associated with x. # be close to 0.185 That expression is the p-value associated with x.
# (Sample solution 6.1)
# = 2 One- or two-sided =================================================== # = 2 One- or two-sided ===================================================
@ -173,7 +174,7 @@ abline(v = x, col = "#EE0000", lwd = 2)
sum(r > 1.96) sum(r > 1.96)
# [1] 24589 # [1] 24589
# Wait - that's about 2.5% , not 5% as expected. Why? # Wait - that's about 2.5% of 1,000,000, not 5% as expected. Why?
# The answer is: we have to be careful with two-sided distributions. 2 standard # The answer is: we have to be careful with two-sided distributions. 2 standard
# deviations away from the mean means either larger or smaller than 1.96 . This # deviations away from the mean means either larger or smaller than 1.96 . This
@ -193,6 +194,9 @@ sum(r > quantile(r, probs = 0.95))
# [1] 50000 # [1] 50000
# which is 5%, as expected. # which is 5%, as expected.
# Task:
# Use abline() to add the p = 0.05 boundary for smaller values to the histogram.
# (Sample solution 6.2)
# To summarize: when we evaluate the significance of an event, we divide a # To summarize: when we evaluate the significance of an event, we divide a
# probability distribution into two parts at the point where the event was # probability distribution into two parts at the point where the event was
@ -201,6 +205,7 @@ sum(r > quantile(r, probs = 0.95))
# significant. # significant.
# #
# = 3 Significance by integration ========================================= # = 3 Significance by integration =========================================
# If the underlying probability distribution can be analytically or numerically # If the underlying probability distribution can be analytically or numerically
@ -307,13 +312,17 @@ for (i in 1:N) {
chs[i] <- chSep(sample(v, length(v))) # charge chs[i] <- chSep(sample(v, length(v))) # charge
} }
hist(chs) hist(chs, breaks = 50)
abline(v = chSep(v), col = "#EE0000") abline(v = chSep(v), col = "#EE0000")
# Contrary to our expectations, the actual observed mean minimum charge # Contrary to our expectations, the actual observed mean minimum charge
# separation seems to be larger than what we observe in randomly permuted # separation seems to be larger than what we observe in randomly permuted
# sequences. But is this significant? Your task to find out. # sequences. But is this significant? Your task to find out.
# Task:
# Calculate the empirical p-value for chsep(v)
# (Sample solution 6.3)
# = 5 Final tasks ========================================================= # = 5 Final tasks =========================================================
@ -321,7 +330,22 @@ abline(v = chSep(v), col = "#EE0000")
# be 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)) seal(sum(chs))
# = 6 Sample solutions ====================================================
# == 6.1 ==================================================================
#
sum(r <= x) / length(r)
# == 6.2 ==================================================================
#
abline(v = quantile(r, probs = c(0.05)))
# == 6.3 ==================================================================
#
( x <- (sum(chs >= chSep(v)) + 1) / (length(chs) + 1) )
# [END] # [END]