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

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

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

The Likelihood Ratio Test General Framework for constructing hypothesis tests: S = Likelihood( data | H0) / Likelihood( data | H1) 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".

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

Power The power of a statistic Z is the probability of sampling a value more extreme than that 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(0,1) 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)

Multiple Testing Conventionally, the significance threshold a for a test is a = 0.05, or 0.01 However, Statistical Genetics problems frequently involve multiple testing – eg a genome scan may test over one million SNPs for association, a gene expression microarray will screen tens of thousands of transcripts. On the one hand, significance thresholds should be adjusted to be more conservative, in order to prevent too many false positives ocurring. In N independent tests, the expected P-value of the most significant result to occur by chance is 1/(N+1) [because P-values are uniformly distributed on [0,1]] Whilst it is not true that tests are independent, it is a surprisingly good guide to what will be observed in reality. Often, at these scales, P-values are reported as –log10(P-value), so that N=106 implies the most significant P-value is about 10-6 , ie logP = 6. On the other hand, raising the significance threshold lowers the power to detect associations, so larger sample sizes are needed to compensate or a different approach, based around False Discovery Rates (FDR) is employed The idea behind FDR is to choose a laxer threshold so that many false positives may occur, but to limit the proportion of false positives to say 10%. The FDR is useful when the purpose of the screen is to identify items (SNPs, genes, etc) for further experimental investigation, and which will provide more information.

Example: The mean of a Normal Distribution H0: mean = m0 vs H1: mean = m1 data: y1, … yn iid 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

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

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

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

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) = pt(0.025,n-1)

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

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)

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.

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 = -3.5892, df = 38, p-value = 0.000935 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -1.9272037 -0.5372251 sample estimates: mean of x mean of y 0.4040969 1.6363113

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 = -1.5911, df = 38, p-value = 0.1199 alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -7.0519594 0.8450837 sample estimates: mean of x mean of y 0.4040969 3.5075347 > var(samp2) [1] 1.160251 > var(samp3) [1] 74.88977

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, ...)

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.

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 = 0.001041 alternative hypothesis: true mu is not equal to 0 > wilcox.test(samp1,samp3) # with outliers data: samp1 and samp3

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(1/2,n) The probability of observing more than r y’s > m0 is 0.5n Si>r (n choose i)

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 permutations that exceed S It’s that simple!

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

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 This means some elements of S will be missing in S* and some will be present multiply often (about 1/e = 63% of 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)

Degrees of Freedom Suppose x1,…x2 are iid N(0,1) Then Si xi2 is distributed as C2n And Si (xi2-0)2/n is an estimate of the sample variance. Although the E(xi)=0, knowing the values of x1….xn-1 tells us nothing extra about xn, because the observations are independent But what about Si (xi-mean)2/ ? Here, knowing x1-mean, ..xn-1 –mean forces the value of xn to be n.mean-sum(x1..xn-1) Therefore the sum is distributed as if it comprised one fewer observation, hence it has n-1 df (for example, its expectation is n-1) In general, if there are p constraints placed on a set of n numbers, then the number of degrees of freedom is n-p In particular, if p parameters are estimated from a data set, then the residuals y-x1b1 +… have p constraints on them, so they behave like n-p independent variables