12.2 - Interactions12.2 - Interactions
It is important to examine treatment × covariate interactions. For example, it is possible that the responses in the treatment groups differ for low levels of the prognostic factor, but not differ for high levels of the prognostic factor. Interpretations of statistical results in the presence of treatment × covariate interactions can become complex.
It is unfortunate that the detection of interactions is model-dependent. For example, if the treatment × covariate interaction exists in an ANCOVA model of the outcome variable (additive model), it is possible that it will disappear in an ANCOVA model of the logarithm of the outcome variable (multiplicative model). Therefore, interactions can exist or fail to exist based on the selection of the statistical model and the assumptions associated with it.
Notice in the figure above that the difference between Treatments A and B is constant and does not depend on the value of the covariate. There is no treatment by covariate interaction in this scenario above.
Now, in the figure above, the difference between Treatments A and B is not constant and does depend on the value of the covariate.
Prognostic factors can be continuous measures (age, baseline cholesterol, etc.), ordinal (age categories, baseline disease severity, etc.), binary (gender, previous tobacco use, etc.), or categorical (ethnic group, geographic region, institution in multi-center trials, etc.).
Most prognostic factors are measured at baseline and do not change over time, hence they are called time-independent covariates. Time-independent covariates are easily included in many types of statistical models.
On the other hand, some prognostic factors do change over time; these are called time-dependent covariates. For example, consider a clinical trial in diabetics with selected dietary intake variables measured on a regular basis over the course of a six-month clinical trial. Dietary intake could affect the severity of disease and exacerbate problems for diabetic patients, so a statistical analysis of the trial might incorporate these prognostic factors as time-dependent covariates. We have to be very careful when using time-dependent covariates.
A statistical analysis that uses time-dependent covariates can yield misleading results if the time-dependent covariates are affected by treatment. In such a situation, removal of the effect of the time-dependent covariates can also cause removal of the treatment effect. For example, suppose that one of the treatments in the aforementioned diabetes example has an additional effect of increasing appetite. Then removing the effect of dietary intake also could remove the effect of the treatment!
The schematic below represents the situation where the treatment is affecting both the covariates and the outcome. The covariates also affect the outcome. Adjusting for the effect of the covariate over time may account for the majority of the treatment effect on the outcome.
As another example, consider a clinical trial in which an experimental therapy for decreasing diastolic blood pressure is compared to placebo. Suppose that the investigator wants to use pulse as a time-dependent covariate in an ANCOVA model because there is a positive correlation between pulse and diastolic blood pressure. Suppose that the treatment not only reduces diastolic blood pressure, but reduces pulse as well. Then an analysis that removes the effect of pulse, measured over the course of the trial, could remove the effect of treatment. The misleading results from the model would cause the treatment effect on diastolic blood pressure to remain undetected.