Statistical Methods in Clinical Trials Ziad Taib Biostatistics AstraZeneca February 22, 2012
Types of Data Continuous Blood pressure Time to event Categorical sex quantitative qualitative Discrete No of relapses Ordered Categorical Pain level
Types of data analysis (Inference) Parametric Vs Non parametric Model based Vs Data driven Frequentist Vs Bayesian
Steps in data analysis and presentation of results • Data cleaning. • Descriptive analyses. • Statistical Inference. • Reporting and presenting the results.
Continuous Data Part I Classical InferencePart II Bayesian Inference
Part IClassical Inference • One sample • Paired data • Two samples • Many samples • One way anova • Two way anova • Ancova
One sample From a population of patients we draw a sample of size n of patients and measure some response on each of these. We want to test whether the average response in this population is equal to (or is larger or smaller than) some predefined value μ0 (e.g. the corresponding value in the healthy population).
Estimation • Assume is a sample (i.i.d) of size n from some distribution F with mean μ and standard deviationσ. Then and • Are (i) unbiased and (ii) consistent estimators of μ and σ² i.e. (i) (LLN) (ii)
Estimation in general • The maximum likelihood method • The moment method • Uniformly Minimum Variance Unbiased estimators • Least squares • etc
Hypothesis testing We can e.g. test the null hypothesis against using the test statistic when n is large, T follows, under the null hypothesis, the standard normal distribution (CLT).
Hypothesis testing For moderate values of n assuming follow a normal distribution, then follows, under the null hypothesis, the t (student) distribution with (n-1) degrees of freedom.
The distribution of the average when n is large The true distribution among the patients (exponential with mean 0.5).
Confidence intervals • A (1-α)100% confidence interval for μ has the form when n is large (i) (ii) when n moderate but sample normally distributed. A hypothesis (or a confidence interval) can be one sided or two sided
95% confidence interval The normal distribution 1-α=0.95 The t distribution 1-α=0.95
Correspondence Theorem Hypothesis testing and confidence intervals are two sides of the same coin! Reject null hypothesis (α) Parameter value according to null hypothesis not included in confidence interval (α)
Paired data • Sometimes we measure a response at baseline and at the end of the treatment. A suitable analysis can be to consider the endpoint changes from baseline. • and to use the (one sample) test statistic
Two independent samples From two populations/distributions we draw two samples of sizes n1 and n2 of patients and measure some response on each of these. We want to test whether the average responses, μ1 and μ2, in these two populations are equal. The population variances are denoted by σ1² and σ2² respectively.
Two samples (continued) • When both sample sizes are large we can use the central limit theorem according to which the test statistic below follows the standard normal distribution • When the sample sizes are moderate but normal with equal variances we can use the t-statistic (having n1+n2-2 d.f.) and the pooled variance. • The remaining case can be handled using a method known as Satterwaite’s confidence intervals.
Many samplesOne way analysis of variance • Here we want to compare several (k) groups (treatments) with respect to the average response in each group. The following model is assumed
Anova1 • Main point: individuals only differ with respect to one criterion: group belonging. • Intuitively: Looking at the pictures we see that the variation says something about the difference between the cases where the groups are different and when they are similar.
Anova1 continued • The null hypothesis and the alternative hypothesis can be formulated as • The deviation of an individual observation from the overall mean can be described by SSE/SSW SSA/SSB SST/SST
Anova 1 continued • Manipulating these gives: MSE MSA MSA • When the null hypothesis is false we expect the ratio MSA/MSE to be large. To calculate exact significance levels we use the fact that it F-distributed with (k-1,N-k) d.f.
Example: 3 groups with the same mean 20 and the same standard deviation 2. We reject for large values of F0. F is F-distributed and F0 the value of MSA/MSE We cannot reject
Example: 3 groups with means 20, 20 and 20.5 and the same standard deviation 3. We can reject
Multiple comparisons • When we reject the null hypothesis we only know that the groups are not equal but some of them might still be. To find out more we have to consider all the pairwise comparisons between the groups. With k groups this gives m=k(k-1)/2 such comparisons. How do we do this and still have a reasonable overall significance level? The simplest way to deal with this is using Bonferroni’s inequality. This implies that when performing m tests if each test is at the 1-α/m level then the tests taken simultaneously will be on the 1- α level. We will deal with problem in detail later on.
Two way anova • Here we assume that individuals can differ with respect to two factors (two drugs, treatment and centre). The following model is usually suitable for this situation • As before the deviation of an individual observation from the overall average can be decomposed into terms related to the effects of treatment, center, treatment by center as well as to a pure random component.
Anova2 • We use a similar notation as before
Tests of treatment effect, center effect or treatment by center interaction can be performed by using the appropriate ratio between mean square values.
Anova2 • As an example assume we want to test if there is a treatment effect: • This can be tested using the test statistic • Which has under the null hypothesis an F distribution with (a-1) and ab(n-1) degrees of freedom. • The various tests are summarized in the following table
Analysis of covariance • Here we assume that individuals can differ with respect baseline values. It is sometimes desirable to adjust the model for the endpoint measures Y so baseline values X are taken into account. The following model can then be useful The analysis uses an F distribution based on Mean sums of squares (cf. p.319)
Various forms of models and relation between them Classical statistics (Observations are random, parameters are unknown constants) Mixed models, both random and constant parameters • LM: Assumptions: • independence, • normality, • constant parameters Repeated measures: Assumptions 1) and 3) are modified LMM: Assumptions 1) and 3) are modified GLMM: Assumption 2) Exponential family and assumptions 1) and 3) are modified GLM: assumption 2) Exponential family Longitudinal data Maximum likelihood Non-linear models LM - Linear model GLM - Generalised linear model LMM - Linear mixed model GLMM - Generalised linear mixed model Bayesian statistics, parameters are random
The Linear Mixed-effects Model • The linear mixed effects model is quite flexible and does not need balance, independence etc. Usually some version of maximum l likelihood is used for the inference Average evolution Subject specific
What is a probability? 1. the limit of a relative frequency Problem: Not all events all repeatable! 2. the degree of plausibility Or of belief Problem: subjective? P=? P=1/6
Bayes in brief • Bayes’ conclusions were accepted by Laplace in 1781 • Rediscovered by Condorcet, and remained unchallenged until • Boole questioned them. • Since then Bayes' techniques have been subject to controversy.
Is Classical Inference Really Classical? • Only in the sense that it started a realtively long time ago with Fisher around 1920. • Bayesian Statistics is even more classical since it was dominating until then. • Classical inference was introduced as a way to introduce objectivity in the scientific process.
Simplified scientific Process • A theory or a hypothesis needs to be tested • We perform an experience and obtain some data • Are our data in agreement with our theory? • If the answer to the above question is no, we reject the theory. Otherwise we cannot reject it!
Classical ParadigmvsThe Bayesian paradigm • The classical paradigm is based on the consideration of P[ Data | Theory ] (1) • How likely is the data if the theory was to be true?
Classical vs Bayesian (cont’d) • The Bayesian paradigm is based on the consideration of P[ Theory | Data] (2) • How much support or belief is there in the theory given the data?
Bayes’ Formula Where T = Theory and D = Data Simple formula with many interesting implications. States that (3)