Practical Considerations Surrounding Normality

Transcription

1 Practical Considerations Surrounding Normality Prof. Kevin E. Thorpe Dalla Lana School of Public Health University of Toronto KE Thorpe (U of T) Normality 1 / 16 Objectives Objectives 1. Understand the role of normality including misconceptions. 2. Appreciate the correct approach to analysis of change in the clinical trial. KE Thorpe (U of T) Normality 2 / 16

2 Role of Normality Introduction In statistics the normal distribution is fundamental to much statistical inference. Many statistical methods/models have some form of normality assumption. However, the nature and implications of this assumption are frequently misunderstood or applied incorrectly; one example being that the data should be normally distributed. KE Thorpe (U of T) Normality 3 / Female Male.1 Density Height

3 2 15 Percent of Total Height Height qnorm

4 Role of Normality What is Normal Then? If the data are not normal, then where does normality fit in? Sampling distributions via CLT. The Error distribution of certain models. Maximum likelihood theory. KE Thorpe (U of T) Normality 7 / 16 An Illustration: Analysis of Change Comparing Change Background In parallel group designs with repeated measurements on the same subjects, the implicit goal is often to determine if one group changes (e.g. improves) more than the other. For this reason, comparing change scores, as opposed to follow-up values, between groups is intuitively appealing. A second argument often put forth in favour of comparing change is the belief that doing so adjusts for baseline differences in the outcome. KE Thorpe (U of T) Normality 8 / 16

5 An Illustration: Analysis of Change Comparing Change Issues Assumes that a simple difference is a valid measurement of change. If the baseline values are the same, treatment effect estimates will be the same whatever approach is taken, while significance tests will depend on the correlation between baseline and follow-up. Low correlation will make comparison of follow-up more likely to be significant. High correlation will make comparison of change more likely to be significant. It is, of course, inappropriate to choose your method based on what is more likely to give a significant result. If baseline scores are different, because of regression to the mean, change scores do not in fact control for baseline differences. KE Thorpe (U of T) Normality 9 / 16 An Illustration: Analysis of Change Comparing Change An Appropriate Analysis A more generally appropriate analysis is Analysis of Covariance (ANCOVA), which is really a regression model. post i = β + β 1 pre i + β 2 group i + ε i In this model β 2 provides the estimate of the treatment effect. Its interpretation as the average (expected) difference in the post value (or change in value) between the two treatment groups, adjusted for the pre value. A 95% confidence interval is also readily available from the analysis. KE Thorpe (U of T) Normality 1 / 16

6 An Illustration: Analysis of Change Role of Normality Normality Assumption for ANCOVA The ANCOVA model is (more) formally: post i = β + β 1 pre i + β 2 group i + ε i where the subscript i indexes the subjects and ε i N(, σ 2 ) is called the error term. The normality assumption for ANCOVA is that the error is normal. The meaning of this is that if you were to sample from the population of subjects with a given pre value and group, the post values sampled would follow a normal distribution with a particular mean (dependent on pre and group) and standard deviation σ. This is not the same as saying that the post values are (unconditionally) normally distributed with some mean and standard deviation σ. KE Thorpe (U of T) Normality 11 / 16 An Illustration: Analysis of Change Role of Normality Normality Assumption for ANCOVA So what? An outcome measure that is not normally distributed at follow-up is also probably not normally distributed pre-randomization. In fact, their distributions likely share similarities and are in fact related to each other because of within patient correlation. Thus, by adjusting for the pre-randomization values, it is not unreasonable that approximately normal errors will be left. KE Thorpe (U of T) Normality 12 / 16

7 Post A Post B Percent of Total 5 Pre A Pre B Data Pre Post A B Data Quantiles Normal Quantiles

8 Residual Quantiles Normal Quantiles T-Test Distributional Assumption Robustness The main distributional assumption for the t-test is that the population is normally distributed. Consider by simulation Poisson and exponential population distributions. Sample size of 41 per group obtained by assuming population means of 1 and 2 and standard deviation of about 1.58 and 8% power. Type I error (assuming mean 1 in both groups) was.49 for Poisson and.47 for exponential. Power (assuming means of 1 and 2) was.948 for Poisson and.853 for exponential. KE Thorpe (U of T) Normality 16 / 16