bch441-work-abc-units/FND-STA-Significance.R

# FND-STA-Significance.R
#
# Purpose:  A Bioinformatics Course:
#              R code accompanying the FND-STA-Significance unit.
#
# Version:  1.0
#
# Date:     2017  09  05
# Author:   Boris Steipe (boris.steipe@utoronto.ca)
#
# Versions:
#           1.0    First contents
#
# TODO:
#
#
# == DO NOT SIMPLY  source()  THIS FILE! =======================================
#
# If there are portions you don't understand, use R's help system, Google for an
# answer, or ask your instructor. Don't continue if you don't understand what's
# going on. That's not how it works ...
# ==============================================================================
 
#TOC> ==========================================================================
#TOC> 
#TOC>   Section  Title                                        Line
#TOC> ------------------------------------------------------------
#TOC>   1        Significance and p-value                       46
#TOC>   1.1      Significance levels                            57
#TOC>   1.2      probability and p-value                        74
#TOC>   1.2.1    p-value illustrated                           100
#TOC>   2        One- or two-sided                             152
#TOC>   3        Significance by integration                   192
#TOC>   4        Significance by simulation or permutation     198
#TOC>   5        Final tasks                                   298
#TOC> 
#TOC> ==========================================================================
 

#TOC>
#TOC>




# =    1  Significance and p-value  ============================================

# The idea of the probability of an event has a precise mathematical
# interpretation, but how is it useful to know the probability? Usually we are
# interested in whether we should accept or reject a hypothesis based on the
# observations we have. A rational way to do this is to say: if the probability
# of observing the data is very small under the null-hypothesis, then we will
# assume the observation is due to something other than the null-hypothesis. But
# what do we mean by the "probability of our observation"? And what is "very
# small"?

# ==   1.1  Significance levels  ===============================================

# A "very small" probability is purely a matter of convention, a cultural
# 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
# observations with a probability of less than 0.05 "significant" and if we want
# to highlight this in text or in a graph, we often mark them with an asterisk
# (*). Also we often call observations with a probability of less than 0.01
# "highly significant" and mark them with two asterisks (**). But there is no
# special significance in these numbers, the cutoff point for significance could
# 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
# 1925. Incidentally, Fisher later recommended to use different cutoffs for
# diffrent purposes (cf.
# https://en.wikipedia.org/wiki/Statistical_significance).


# ==   1.2  probability and p-value  ===========================================

# But what do we even mean by the probability of an observation?
# Assume I am drawing samples from a normal distribution with a mean of 0 and a
# standard deviation of 1. The sample I get is ...

set.seed(sqrt(5))
x <- rnorm(1)
print(x, digits = 22)
# [1] -0.8969145466249813791748

# So what's the probability of that number? Obviously, the probability of
# getting exactly this number is very, very, very small. But also obviously,
# this does not mean that observing this number is in any way significant - we
# always observe some number. That's not what we mean in this case. There are
# several implicit assumptions when we speak of the probability of an
# observation:

# 1: the observation can be compared to a probability distribution;
# 2: that distribution can be integrated between any specific value
#      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
# 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                      

# Let's illustrate. First we draw a million random values from our
# standard, normal distribution:

r <- rnorm(1000000)

# Let's see what the distribution looks like:

(h <- hist(r))

# The histogram details are now available in the list h -  e.g. h$counts

# Where is the value we have drawn previously?
abline(v = x, col = "#EE0000")

# How many values are smaller?
sum(r < x)

# Let's color the bars:
#    first, make a vector of red and green colors for the bars with breaks
#    smaller and larger then x, white for the bar that contains x ...
hCol <- rep("#EE000044", sum(h$breaks < x) - 1)
hCol <- c(hCol, "#FFFFFFFF")
hCol <- c(hCol, rep("#00EE0044", sum(h$breaks > x) - 1))
# ... then plot the histogram, with colored bars ...
hist(r, col = hCol)
# ... add two colored rectangles into the white bar ...
idx <- sum(h$breaks < x)
xMin <- h$breaks[idx]
xMax <- h$breaks[idx + 1]
y <- h$counts[idx]
rect(xMin, 0, x, y, col = "#EE000044", border = TRUE)
rect(x, 0, xMax, y, col = "#00EE0044", border = TRUE)
# ... and a red line for our observation.
abline(v = x, col = "#EE0000", lwd = 2)

# The p-value of our observation is the area colored red as a fraction of the
# whole histogram (red + green).


# Task:
#    Explain how the expression sum(r < x) works to give us a count of values
#    with the property we are looking for.

# Task:
#    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
#    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.


# =    2  One- or two-sided  ===================================================

# The shape of our histogram confirms that the rnorm() function has returned
# values that appear distributed according to a normal distribution. In a normal
# distribution, readily available tables tell us that 5% of the values (i.e. our
# significance level) lie 1.96 (or approximately 2) standard deviations away
# from the mean. Is this the case here? How many values in our vector r are
# larger than 1.96?

sum(r > 1.96)
# [1] 24589

# Wait - that's about 2.5% , not 5% as expected. Why?

# 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
# can give rise to errors. If we are simply are interested in outliers, no
# matter larger or smaller, then the 1.96 SD cutoff for significance is correct.
# But if we are specifically interested in, say, larger values, because a
# smaller value is not meaningful, then the significance cutoff, expressed as
# standard deviations, is relaxed. We can use the quantile function to see what
# the cutoff values are:

quantile(r)
quantile(r, probs = c(0.025, 0.975)) # for the symmetric 2.5% boundaries
# close to ± 1.96, as expected
quantile(r, probs = 0.95) # for the single 5% boundary
# close to 1.64 . Check counts to confirm:
sum(r > quantile(r, probs = 0.95))
# [1] 50000
# which is 5%, as expected.


# 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
# observed. We then ask whether the integral over the more extreme part is less
# or more than 5% of the whole. If it is less, we deem the event to be
# significant.
#

# =    3  Significance by integration  =========================================

# If the underlying probability distribution can be analytically or numerically
# integrated, the siginificance of an observation can be directly computed.


# =    4  Significance by simulation or permutation  ===========================

# But whether the integration is correct, or relies on assumptions that may not
# be warranted for biological data, can be a highly technical question.
# 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
# rnorm() samples. Here is an example. Assume you have a protein sequence and
# you speculate that postively charged residues are close to negatively charged
# 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
# residue. Let's compute this for the sequence of yeast Mbp1.

MBP1 <- paste0("MSNQIYSARYSGVDVYEFIHSTGSIMKRKKDDWVNATHILKAANFAKAKRTRILEKEVLK",
               "ETHEKVQGGFGKYQGTWVPLNIAKQLAEKFSVYDQLKPLFDFTQTDGSASPPPAPKHHHA",
               "SKVDRKKAIRSASTSAIMETKRNNKKAEENQFQSSKILGNPTAAPRKRGRPVGSTRGSRR",
               "KLGVNLQRSQSDMGFPRPAIPNSSISTTQLPSIRSTMGPQSPTLGILEEERHDSRQQQPQ",
               "QNNSAQFKEIDLEDGLSSDVEPSQQLQQVFNQNTGFVPQQQSSLIQTQQTESMATSVSSS",
               "PSLPTSPGDFADSNPFEERFPGGGTSPIISMIPRYPVTSRPQTSDINDKVNKYLSKLVDY",
               "FISNEMKSNKSLPQVLLHPPPHSAPYIDAPIDPELHTAFHWACSMGNLPIAEALYEAGTS",
               "IRSTNSQGQTPLMRSSLFHNSYTRRTFPRIFQLLHETVFDIDSQSQTVIHHIVKRKSTTP",
               "SAVYYLDVVLSKIKDFSPQYRIELLLNTQDKNGDTALHIASKNGDVVFFNTLVKMGALTT",
               "ISNKEGLTANEIMNQQYEQMMIQNGTNQHVNSSNTDLNIHVNTNNIETKNDVNSMVIMSP",
               "VSPSDYITYPSQIATNISRNIPNVVNSMKQMASIYNDLHEQHDNEIKSLQKTLKSISKTK",
               "IQVSLKTLEVLKESSKDENGEAQTNDDFEILSRLQEQNTKKLRKRLIRYKRLIKQKLEYR",
               "QTVLLNKLIEDETQATTNNTVEKDNNTLERLELAQELTMLQLQRKNKLSSLVKKFEDNAK",
               "IHKYRRIIREGTEMNIEEVDSSLDVILQTLIANNNKNKGAEQIITISNANSHA")

# first we split this string into individual characters:
v <- unlist(strsplit(MBP1, ""))

# and find the positions of our charged residues

ED  <- grep("[ED]", v)
RKH <- grep("[RKH]", v)

sep <- numeric(length(ED)) # this vector will hold the distances
for (i in seq_along(ED)) {
  sep[i] <- min(abs(RKH - ED[i]))
}

# Task: read and explain this bit of code

# Now that sep is computed, what does it look like?

table(sep)  # these are the minimum distances
# 24 of D,E residues are adjacent to R,K,H;
# the longest separation is 28 residues.

# What is the mean separation?
mean(sep)

# The value is 4.1 . Is this significant? Honestly, I would be hard pressed
# to solve this analytically. But by permutation it's soooo easy.

# First, we combine what we have done above into a function:

chSep <- function(v) {
  # computes the mean minimum separation of oppositely charged residues
  # Parameter: v (char) a vector of amino acids in the one-letter code
  # Value: msep (numeric) mean minimum separation

  ED  <- grep("[EDed]", v)
  RKH <- grep("[RKHrkh]", v)

  sep <- numeric(length(ED))
  for (i in seq_along(ED)) {
    sep[i] <- min(abs(RKH - ED[i]))
  }
  return(mean(sep))
}

# Execute the function to define it.

# Confirm that the function gives the same result as the number we
# calculated above:
chSep(v)

# Now we can produce a random permutation of v, and recalculate
set.seed(pi)
w <- sample(v, length(v)) # This shuffles the vector v. Memorize this
                          # code paradigm. It is very useful.
chSep(w)
# 3.273 ... that's actually less than what we had before.

# Let's do this 10000 times and record the results (takes a few seconds):

N <- 10000
chs <- numeric(N)
for (i in 1:N) {
  chs[i] <- chSep(sample(v, length(v))) # charge
}

hist(chs)
abline(v = chSep(v), col = "#EE0000")

# Contrary to our expectations, the actual observed mean minimum charge
# separation seems to be larger than what we observe in randomly permuted
# sequences. But is this significant? Your task to find out.


# =    5  Final tasks  =========================================================

# From chs, compute the probability of a mean minimum charge separation to 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 following expression for validation:
writeBin(sum(chs), raw(8))


# [END]