Reporting two way interactions

An example

Here we consider reporting an analysis involving a two-way interaction, using the example of Crampton’s (1947) study of the effects of dose and method of delivery of vitamin C on the development of teeth in guinea pigs. Read the study background in the post about understanding two-way interactions.  As a reminder, the plot of means is reproduced here:

The outcome is Odontoblast length (microns), and there are two explanatory factors — dose of vitamin C and method of delivery.

The analysis of these data discussed here uses a non-additive model — a model including an interaction. An appropriate report of the analysis may include summary statistics, the estimated interaction effects with 95% confidence intervals, and P-values.  The interaction P-value is for a test of the null hypothesis of no effect moderation or no interaction: the effect of one of the factors (method of delivery) is the same at each level of the other factor (dose). The analysis for the non-additive model also will provide so-called tests of the main effects, with P-values; the corresponding null hypothesis is, for any factor, that there is no difference between true means for different levels of the factor.  In the non-additive model, these tests are carried out in the presence of the interaction; this means that they are generally not the focus of interpretation.

The analysis of these data, with the methods illustrated here, assumes that: the observations are independent, the random errors arise from a Normal distribution, and the variance of the random errors is constant.  A good analysis involves assessment of these assumptions, including examination of residual plots, and this may be discussed in a description of the statistical methods used or in the reporting of the results.

Here is an example of a summary table that provides descriptive statistics.

Table 1: Summary statistics for Odontoblast length (microns) by Method of delivery and Dose (mg/day)

The hypothesis test results can be summarised as follows:

Table 2: Statistical tests from a two-way Anova model of Odontoblast length (microns)

Consistent with the pattern of means in the plot, the P-value for the test of the interaction is small.

The next aspect of reporting focuses on interpretation of the interaction effect.  The plot of means, as provided above, is often used as an aid to understanding the interaction.  This shows little effect of method of delivery when the dose is 2 mg/day, but mean differences with higher mean odontoblast lengths for orange juice when the dose is 0.5 or 1.0 mg/day.

The interaction effects can be quantified, for example, by estimating pairwise mean differences for levels of one factor for each level of the other factor.  How best to do this can depend on the factors in the analysis and how many levels they have.  In this example, pairwise comparisons of the method of delivery for each dose are suitable; this corresponds to a natural interpretation of the plot of means given above.

The table below provides the estimates of the difference of means (Orange juice minus Ascorbic acid) for each dose; 95% confidence intervals quantify the precision of these estimates.  The P-values correspond to tests of the null hypotheses of no effect of method of delivery at each dose level.

Table 3: Pairwise comparisons of mean Odontoblast length (microns) according to Method of delivery for each level of dose

It can be helpful to plot the pairwise comparisons above, as shown in the next plot.

In some cases, it may also be useful to obtain the estimates and confidence intervals from the model, and to plot these, like the example below.  This may be the case if there are many levels of one or both of the factors involved.