23. Interaction, Random Effect Analysis of Variance (40 Patients)

In pharmaceutical research and development, multiple factors like age, gender, comorbidity, concomitant medication, genetic and environmental factors co-determine the efficacy of the new treatment. In statistical terms we say, they interact with the treatment efficacy.

Interaction is different from confounding. In a trial with interaction effects the parallel groups have similar characteristics. However, there are subsets of patients that have an unusually high or low response.

The above figure shows the essence of interaction: the males perform better than the females with the new medicine, with the control treatment the opposite (or no difference between males and females) is true.

Are there not only independent effects of two predictors on the outcome, but also interaction effects between two predictors on the outcome.

In a 40 patient parallel-group study of the effect of verapamil and metoprolol on paroxysmal atrial fibrillation (PAF) the possibility of interaction between gender and treatment on the outcome was assessed. The numbers of episodes of paroxysmal atrial tachycardias per patient, are the outcome variable. The entire data file is in extras.springer.com, and is entitled “chapter23interaction”. The first ten patients of the data file is given below.

Overall, metoprolol seems to perform better. However, this is only true for one subgroup (males). The presence of interaction between gender and treatment modality can be assessed several ways: (1) t-tests (see Chapter 18, Statistics on a pocket calculator part one, Springer New York, 2011, from the same authors), (2) analysis of variance, and (3) regression analysis. The data file is given underneath.

We will first perform an analysis of variance. Open the data file in SPSS.

For analysis the General Linear Model is required. It consists of four statistical models:

We will use here Univariate.

The above table shows that there is a significant interaction between gender and treatment at p = 0,0001 (* is sign of multiplication). In spite of this, the treatment modality is a significant predictor of the outcome. In situations like this it is often better to use a socalled random effect model. The “sum of squares treatment” is, then, compared to the “sum of squares interaction” instead of the “sum of squares error”. This is a good idea, since the interaction was unexpected, and is a major contributor to the error, otherwise called spread, in the data. This would mean, that we have much more spread in the data than expected, and we will lose a lot of power to prove whether or not the treatment is a significant predictor of the outcome, episodes of PAF. Random effect analysis of variance requires the following commands:

The underneath table shows the results. As expected the interaction effect remained statistically significant, but the treatment effect has now lost its significance. This is realistic, since in a trial with major interactions, an overall treatment effect analysis is not relevant anymore. A better approach will be a separate analysis of the treatment effect in the subgroups that caused the interaction.

As a contrast test we may use regression analysis for these data. For that purpose we first have to add an interaction variable:

Underneath the first 10 patients of the above data example is given, now including the interaction variable.

The interaction variable will be used together with treatment modality and gender as independent variables in a multiple linear regression model. For analysis the statistical model Linear in the module Regression is required.

The above table shows the results of the multiple linear regression. Like with fixed effect analysis of variance, both treatment modality and interaction are statistically significant. The t-value-interaction of the regression = 7,932. The F-value-interaction of the fixed effect analysis of variance = 62,916 and this equals 7,932². Obviously, the two approaches make use of a very similar arithmetic.

Unfortunately, for random effect regression SPSS has limited possibilities.

Interaction is different from confounding (Chap. 22). In a trial with interaction effects the parallel group characteristics are equally distributed between the groups. However, there are subsets of patients that have an unusually high or low response to one of the treatments. Assessments are reviewed.

More background, theoretical, and mathematical information of interaction assessments is given in Statistics applied to clinical studies 5th edition, Chap. 30, Springer Heidelberg Germany, 2012, from the same authors.