[R-sig-ME] Dependence of difference on mean level

[Kim Colyvas]

I am analysing data from a study comparing 3 methods of assessing total daily nutritional intake for a person.
The study design called for 3 measurements (one per day), to be taken in the first week using method 1 (photographic recording of all meals with computer aided assessment), and similarly 3 measurements in the second week using method 2 (24 hr recall) and 3 measurements in the 3rd week using method 3 (where method 3 is method 1 repeated). The study design was not ideal for comparing methods but was the best that could be done given the practicalities of the situation.

I fitted a mixed model with 2 random effects, the first being participant to account for differences between people in daily intake and the second being week by participant to capture week to week variations in intake. The residual term representing day to day variation in daily intake as well as measurement error. The variable method (3 levels) was fit as a fixed effect to assess if there was a difference (bias) between methods. The analyses I have done indicate there is. I was happy with this till I began to wonder if the size of the bias varied with the level of intake. Would the size of the difference between methods be greater when the daily intake was large compared to when it was smaller?

What I would like help with is
a) whether there is a way that can I adapt the mixed model approach to test for the dependence of the difference between methods on the intake level (e.g. mean intake per person could be represented by the participant random effect) and
b) have a formal statistical test for whether the difference is constant or varies as a function of the intake level and
b) a reference to support the approach.

I chose a method that I thought would be suitable but I have no references to support it and I have become increasingly uneasy about the validity of what I did. My alternative was to average all measurements for a participant (not always 9 days of data due to various kinds of losses) and use that to represent a participant’s intake level. The mixed model was then modified by dropping the participant random effect and replacing it with a fixed effect in the model using the mean intake as a covariate. The second random effect (week by participant) was retained in the model. This model was equivalent to the parallel slopes model used when carrying out ANCOVA. Then the dependence of the size of the difference between methods was tested by adding an interaction term between method and mean intake to provide a test for differences between slopes. I found in several cases this term was significant indicating that the bias between methods was dependent on the mean level of a person’s intake. If there was a reference that validates this approach I would be happy with that as a solution.

Thanks,
Kim Colyvas

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