1. Dealing With Statistical Uncertainty
Richard Mott
Wellcome Trust Centre for Human Genetics

2. Synopsis Hypothesis testing P-values Confidence intervals T-test
Non-parametric tests
Sample size calculations
Permutations
Bootstrap

3. Hypothesis testing Null Hypothesis H0 Alternative Hypothesis H1
Examples of H0:
Mean of a population is 3.0
In a genetic association study, there is no association between disease state and the genotypes of a particular SNP
Examples of H1:
Mean of population > 3.0
There is an association between disease and genotype

4. The Normal Distribution
N(m,s)
Mean m
Variance s2
Many traits have Normal Distributions
Height
Weight
Central Limit Theorem
average of many random quantities often has a Normal Distribution even if the individual observations do not

5. P-values
The P-value of a statistic Z is the probability of sampling a value more extreme than that observed value z if the null hypothesis is true.
For a one-sided test, the p-value would be
a = Prob( Z > z | H0)
For a two-sided test it is
a = Prob( |Z| > z | H0)
Note this is not the probability density at x, but the tails of the cumulative distribution function
Upper and lower 2.5% tails of the
N(0,1) distribution,
corresponding to |z|=1.96

6. Power
The power of a statistic Z is the probability of sampling a value more extreme than the observed value z if the alternative hypothesis is true.
For a one-sided test, the power would be
b = Prob( Z > z | H1)
For a two-sided test it is
b = Prob( |Z| > z | H1)
Note this is not the probability density at x, but the tails of the cumulative distribution function
H0: N(1,0) H1: N(1.5,1) H2: N(4,1)
0.05 = Prob( Z> 1.644|H0) (5% upper tail)
0.44 = Prob( Z> 1.644|H1)
0.99 = Prob( Z> 1.644|H2)

7. The Likelihood Likelihood = “probability of the data given the model”
Basis of parametric statistical inference
Different hypotheses can often be expressed in terms of different values of the parameters q
Hypothesis testing equivalent to comparing the likelihoods for different q

8. The Likelihood Ratio Test
General Framework for constructing hypothesis tests:
S = Likelihood( data | H1) / Likelihood( data | H0)
Reject H0 if S > s(H0, H1)
Threshold s is chosen such that there is a probability a of making a false rejection under H0.
is the size of the test (or false positive rate or Type I error) e.g. a=0.05
is the power of the test, the probability of correctly rejecting H0 when H1 is true e.g. b=0.8
1- b is the type II error, the false negative rate
Generally, for fixed sample size n, if we fix a then we can’t fix b
If we fix a and b then we must let n vary.
The Neyman Pearson Lemma states that the likelihood ratio test is the most powerful of all tests of a given size
Type III error: "correctly rejecting the null hypothesis for the wrong reason".

9. Example: The mean of a Normal Distribution
H0: mean = m0 vs H1: mean = m1 data: y1, … yn independently and identically distributed with a Normal distribution N(m,s2) Assume variance is the same and is known Therefore we base all inferences on the sample mean compared to the difference in the hypothesised means

10. The Normal and T distributions
For a sample from a Normal distribution with known mean and variance, the sample mean can be standardised to follow the standard Normal distribution
And the 95% confidence interval for the mean is given by
But we often wish to make inferences about samples from a Normal Distribution when the variance is unknown: The variance must be estimated from the data as the sample variance
Because of this uncertainty the distribution of the sample mean is broader, and follows the Tn-1 distribution

11. The T distribution
The Tn and Normal distributions are almost identical for n>20,
As a rule of thumb, an approximate 95% confidence interval for n>20 is

12. T-tests The T-test compares the means of samples
The T-test is optimal (in LR sense) if the data are sampled from a Normal distribution
Even if the data are not Normal the test is useful in large samples
There are several versions, depending on the details

13. T tests for the comparison of sample means One Sample Test
One sample y1 ….. yn
H0: the sample mean = m0
H1: the sample mean != m0
Reject H0 if
tn-1(a/2) is the quantile of the Tn-1 distribution such that the probability of exceeding tn-1(a/2) is a/2
Two sided test
in R, tn-1(0.025) = qt(0.025,n-1)

14. T tests for the comparison of sample means Paired T-test
Two samples of paired data (x1,y1), (x2,y2) …. (xn,yn)
Example:
two experimental replicates taken from n individuals, we wish to test if the replicates are statistically similar
H0: mean(x) = mean(y)
Take differences d1= x1 - y1 … dn= xn - yn
H0: mean(d) = 0
One sample T-test with m0=0

15. T tests for the comparison of sample means Two-Sample T-test
Two samples of unpaired data
x1, x2 ….. xn
y1, y2 ….. ym
H0: mean(x) = mean(y)
If we assume the variance is the same in each group then we can estimate the variance as the pooled estimator
The test statistic is
(The case with unequal variances is more complicated)

16. T tests in R
t.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), mu = 0, paired = FALSE, var.equal = FALSE, conf.level = 0.95, ...)
x a numeric vector of data values.
y an optional numeric vector data values.
alternative a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less".
mu a number indicating the true value of the mean
paired a logical indicating whether you want a paired t-test.
var.equal a logical variable indicating whether to treat the two variances as being equal.
conf.level confidence level of the interval.
formula a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs a factor with two levels giving the corresponding groups.
data an optional data frame containing the variables in the model formula.
subset an optional vector specifying a subset of observations to be used.

17. T tests in R
samp1 <- rnorm( 20, 0, 1) # sample 20 numbers from N(0,1) distribution
samp2 <- rnorm( 20, 1.5, 1) # sample from N(1.5,1)
t.test( samp1, samp2,var.equal=TRUE )
Two Sample t-test
data: samp1 and samp2
t = , df = 38, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y

18. Outliers A small number of contaminating observations with different distribution
Outliers can destroy statistical genuine significance
Reason: the estimate of the variance is inflated, reducing the T statistic
And sometimes create false significance :
Type III error: "correctly rejecting the null hypothesis for the wrong reason".
There are Robust Alternatives to the T-test: Wilcoxon tests
> samp3 <- samp2
> samp3[1] = 40 # add one outlier
> t.test( samp1, samp3,var.equal=TRUE )
Two Sample t-test
data: samp1 and samp3
t = , df = 38, p-value =
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
sample estimates:
mean of x mean of y
> var(samp2)
[1]
> var(samp3)
[1]

19. Non-parametric tests

20. Non-parametric tests: Wilcoxon signed rank test (alternative to Paired T-test)
Two samples of paired data (x1,y1), (x2-y2) …. (xn,yn)
Take differences
d1= x1 - y1 … dn= xn – yn
H0: distribution of d’s is symmetric about 0
Compute the ranks of the absolute differences, rank(|d1|) etc
Compute the signs of the differences
Compute W, the sum of the ranks with positive signs
If H0 true then W should be close to the mean rank n/2
The distribution of W is known and for n>20 a Normal approximation is used.
in R
wilcox.test(x, y, alternative = c("two.sided", "less", "greater"), mu = 0, paired = TRUE, exact = NULL, correct = TRUE, conf.int = FALSE, conf.level = 0.95, ...)

21. Wilcoxon rank sum test (alternative to unpaired T-test)
Two samples of unpaired data
x1, x2 ….. xn
y1, y2 ….. ym
Arrange all the observations into a single ranked series of length N = n+m.
R1 = sum of the ranks for the observations which came from sample 1. The sum of ranks in sample 2 follows by calculation, since the sum of all the ranks equals N(N + 1)/2
Compute U = min(R1 ,R2) - n(n + 1)/2.
z = (U –m)/s Normal approximation
m = nm/2
s2 = nm(n=m+1)/12
The Wilcoxon Rank sum test is only about 5% less efficient than the unpaired T-test (in terms of the sample size required to achieve the same power) in large samples.

22. Wilcoxon rank sum test (alternative to unpaired T-test)
>wilcox.test(samp1,samp2) Wilcoxon rank sum test data: samp1 and samp2 W = 82, p-value = alternative hypothesis: true mu is not equal to 0 > wilcox.test(samp1,samp3) # with outliers data: samp1 and samp3

23. Binomial Sign Test (T-test on one sample)
One sample y1 ….. yn
H0: the sample mean = m0
Count M, the number of y’s that are > m0
If H0 is true, this should be about n/2, and should be distributed as a Binomial random variable B(n/2,n)

24. Contingency Tables
Smoker
Non-smoker
Male 10 20
Female 5 12 17 15 22 37
Do Males and Females have different rates of smoking?

25. Contingency Tables
Data are a table of counts, with the row and column margins fixed.
Ho: the counts in each cell are consistent with the rows and columns acting independently
n11
r1c1/N
n12
r1c2/N
r1 = n11+ n12
n21
r2c1/N
n22
r2c2/N
r2 = n21+ n22 c1
= n11+ n21 c2
= n12+ n22 N

26. The Chi-Squared statistics

27. Fisher’s Exact Test for 2 x 2 Contingency tables
Similar idea to a permutation test (see below)
Given the marginal totals in a 2x2 table, we can compute the probability P of getting the observed cell counts assuming the null hypothesis of no associations between classifying factors.
hypergeometric distribution.
Find all tables with same margins and with probabilities no bigger than P
P-value of the table is the sum of these probabilities

28. Fisher’s Exact Test R implementation:
Smoker
Non-smoker
Male a b
a+b
Female c d
c+d
a+c
b+d n
Smoker
Non-smoker
Male 10 20
Female 5 12 17 15 22 37
R implementation:
m <- matrix(c(10,10,5,12),nrow=2)
fisher.test(m)
Fisher's Exact Test for Count Data
data: m
p-value =
alternative hypothesis: true odds ratio is not equal to 1
95 percent confidence interval:
sample estimates:
odds ratio
>

29. Permutation Tests Another non-parametric Alternative
Permutation is a general principle with many applications to hypothesis testing.
But it is not useful for parameter estimation
The method is best understood by an example:
Suppose a data set is classified into N groups. We are interested in whether the differences between group means (eg if N=2 then the two sample T test or Wilcoxon Rank Sum test is appropriate)
Compute a statistic S (eg the difference between the maximum and minimum group mean)
If there are no differences between groups then any permutation of the group labellings between individuals should produce much the same result for S
Repeat i = 1 to M times:
permute the data
compute the statistic on the permuted data, Si
The Permutation p-value is the fraction of permutations that exceed S
It’s that simple!

30. Bootstrapping
Given a set of n things S = {x1 …. x2}, a bootstrap sample is formed by sampling with replacement from S, to create a new set S* of size n
Original
Resample
Some elements of S will be missing in S* and some will be present multiply often (about 1/e = 63% of the elements will be present in S* on average)
S* can be used as a replacement for S in any algorithm that works on S, and we can repeatedly draw further bootstrap samples.
Any statistic Z that can be calculated on S can also be calculated on S*, and so we can construct the bootstrap sampling distribution of Z from repeated samples.
From this we can make inferences about the variation of Z, eg construct a 95% confidence interval (if Z is numeric)
Can be used for parameter estimation

31. Bootstrapping
Bootstrapping is a general way to evaluate the uncertainty in estimates of statistics whilst making few assumptions about the data
It is particularly useful for complex statistics whose distribution cannot be evaluated analytically
Example:
In a gene expression study, we wish to find a minimal set of SNPs that in combination will predict the variation in a phenotype.
Suppose we have a deterministic algorithm A that will find us a plausible set of SNPs (ie a model).
This is likely to be a complex programme and contain some arbitrary steps.
It is also unlikely to give any measure of uncertainty to the model.
There may be many possible sets of SNPs with similar claims, so we would like a way to average across different models, and evaluate the evidence for a SNP as the fraction of models in which the SNP occurs
To do this, we need a way to perturb the data, to generate new data sets on which we can apply the procedure A
Bootstrapping is one way to do this