Maintenance and sample solutions
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# tocID <- "FND-STA-Significance.R"
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#
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# ---------------------------------------------------------------------------- #
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# PATIENCE ... #
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# Do not yet work wih this code. Updates in progress. Thank you. #
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# boris.steipe@utoronto.ca #
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# ---------------------------------------------------------------------------- #
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#
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# Purpose: A Bioinformatics Course:
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# R code accompanying the FND-STA-Significance unit.
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#
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# Version: 1.2
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# Version: 1.3
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#
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# Date: 2017 09 - 2019 01
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# Date: 2017-09 - 2020-09
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# Author: Boris Steipe (boris.steipe@utoronto.ca)
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#
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# Versions:
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# 1.3 2020 Maintenance. Add sample solution.
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# 1.2 Update set.seed() usage
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# 1.1 Corrected treatment of empirical p-value
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# 1.0 First contents
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@ -31,18 +27,22 @@
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#TOC> ==========================================================================
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#TOC>
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#TOC>
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#TOC> Section Title Line
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#TOC> ------------------------------------------------------------------
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#TOC> 1 Significance and p-value 43
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#TOC> 1.1 Significance levels 54
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#TOC> 1.2 probability and p-value 71
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#TOC> 1.2.1 p-value illustrated 103
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#TOC> 2 One- or two-sided 158
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#TOC> 3 Significance by integration 198
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#TOC> 4 Significance by simulation or permutation 204
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#TOC> 5 Final tasks 312
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#TOC>
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#TOC> 1 Significance and p-value 49
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#TOC> 1.1 Significance levels 60
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#TOC> 1.2 probability and p-value 77
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#TOC> 1.2.1 p-value illustrated 109
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#TOC> 2 One- or two-sided 165
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#TOC> 3 Significance by integration 209
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#TOC> 4 Significance by simulation or permutation 215
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#TOC> 5 Final tasks 327
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#TOC> 6 Sample solutions 336
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#TOC> 6.1 338
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#TOC> 6.2 342
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#TOC> 6.3 346
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#TOC>
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#TOC> ==========================================================================
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@ -106,7 +106,7 @@ print(x, digits = 22)
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# curve, as a fraction of the whole.
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# === 1.2.1 p-value illustrated
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# === 1.2.1 p-value illustrated
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# Let's illustrate. First we draw a million random values from our
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# standard, normal distribution:
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@ -146,19 +146,20 @@ rect(x, 0, xMax, y, col = "#00EE0044", border = TRUE)
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# ... and a red line for our observation.
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abline(v = x, col = "#EE0000", lwd = 2)
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# The p-value of our observation is the area colored red as a fraction of the
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# The p-value of our observation is the red area as a fraction of the
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# whole histogram (red + green).
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# Task:
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# Explain how the expression sum(r < x) works to give us a count of values
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# with the property we are looking for.
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# with the property we are looking for. E.g., examine -4:4 < x
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# Task:
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# Write an expression to estimate the probability that a value
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# drawn from the vector r is less-or-equal to x. The result you get
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# will depend on the exact values that went into the vector r but it should
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# be close to 0.185 That expression is the p-value associated with x.
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# (Sample solution 6.1)
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# = 2 One- or two-sided ===================================================
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@ -173,7 +174,7 @@ abline(v = x, col = "#EE0000", lwd = 2)
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sum(r > 1.96)
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# [1] 24589
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# Wait - that's about 2.5% , not 5% as expected. Why?
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# Wait - that's about 2.5% of 1,000,000, not 5% as expected. Why?
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# The answer is: we have to be careful with two-sided distributions. 2 standard
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# deviations away from the mean means either larger or smaller than 1.96 . This
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@ -193,6 +194,9 @@ sum(r > quantile(r, probs = 0.95))
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# [1] 50000
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# which is 5%, as expected.
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# Task:
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# Use abline() to add the p = 0.05 boundary for smaller values to the histogram.
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# (Sample solution 6.2)
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# To summarize: when we evaluate the significance of an event, we divide a
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# probability distribution into two parts at the point where the event was
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@ -201,6 +205,7 @@ sum(r > quantile(r, probs = 0.95))
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# significant.
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#
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# = 3 Significance by integration =========================================
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# If the underlying probability distribution can be analytically or numerically
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@ -307,13 +312,17 @@ for (i in 1:N) {
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chs[i] <- chSep(sample(v, length(v))) # charge
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}
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hist(chs)
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hist(chs, breaks = 50)
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abline(v = chSep(v), col = "#EE0000")
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# Contrary to our expectations, the actual observed mean minimum charge
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# separation seems to be larger than what we observe in randomly permuted
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# sequences. But is this significant? Your task to find out.
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# Task:
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# Calculate the empirical p-value for chsep(v)
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# (Sample solution 6.3)
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# = 5 Final tasks =========================================================
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@ -321,7 +330,22 @@ abline(v = chSep(v), col = "#EE0000")
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# be larger or equal to the value observed for the yeast MBP1 sequence. Note
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# the result in your journal. Is it significant? Also note the result of
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# the following expression for validation:
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writeBin(sum(chs), raw(8))
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seal(sum(chs))
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# = 6 Sample solutions ====================================================
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# == 6.1 ==================================================================
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#
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sum(r <= x) / length(r)
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# == 6.2 ==================================================================
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#
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abline(v = quantile(r, probs = c(0.05)))
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# == 6.3 ==================================================================
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#
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( x <- (sum(chs >= chSep(v)) + 1) / (length(chs) + 1) )
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# [END]
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