# Multivariate Repeated Measures Analysis with Multiple Conditions

I'm running a repeated measures experiment where each participant performs tasks under 3 different conditions (A, B, C). Under each condition, each participant performs tasks at 10 difficulty levels (i.e. each participant performs 30 trials, difficulty level 1 to 10 under condition A, difficulty level 1 to 10 under condition B, etc.).

Each task has 3 measurements, one is a score on a scale and two are dichotomous outcomes (correct or incorrect).

How can I analyse the data? Is there an equivalent to a two-way MANOVA that also works for dichotomous outcomes?

• If you expect a monotonic relationship between difficulty and your DV's, make sure you take that into account by testing the linear component of trend possibly pooled with the quadratic. Don't average these with the cubic, quartic, etc. as you would be doing in the full ANOVA. Believe it or not simulations have shown ANOVA works pretty well with dichotomous variables even though normality is violated as long as the proportions aren't too close to 0 or 1. I'm not sure about MANOVA. – David Lane May 29 '17 at 2:24
• Thanks for your comment! That's really interesting. Since my dataset is split up into 3x10 categories, I don't think I have quite enough data for each point to be far enough from 0 or 1. Good to know though! – hkflyer May 29 '17 at 3:59

You have 3 response variables: one is a score and two are dichotomous. Assume you want to analyze them separately (fit a model for each of them).

Conditions A, B and C will be treated as categorical covariate by using dummy variables $X_{ij1}$ and $X_{ij2}$, and difficulty levels as continue covariate $X_{ij3}$, where $i$ indicates the participant and $j = 1,\dots, 30$ means trials.

Considering the score as continuous variable $Y_{ij}$. The linear mixed model can be:

$$Y_{ij} = \beta_0+\beta_1X_{ij1}+\beta_2X_{ij2}+\beta_3X_{ij3} + \gamma_{i}+\epsilon_{ij}$$ and $\gamma_{i} \sim N(0,\sigma_\gamma^2)$, $\epsilon_{ij} \sim N(0,\sigma^2)$ and all of the random terms are independent.

For dichotomous response variables, the mixed effect logistic regression maybe works.

Same as above, Let $$Y=\begin{cases} 1 & \text{if correct}\\ 0 & \text{if incorrect} \end{cases}$$ and $$\mu_{ij} = \beta_0+\beta_1X_{ij1}+\beta_2X_{ij2}+\beta_3X_{ij3} + \gamma_{i}$$

Fit the model $$\Pr(Y=1)=\frac{\exp(\mu_{ij})}{1+\exp(\mu_{ij})}$$

When data are available, need to check the assumptions based on which the models are established. It is possible all of these models are not suitable for your data.

• Thanks for your reply! That makes sense. I suspect that the 3 measures are somewhat correlated as they measure a similar construct. Would adding the 2 dichotomies outcomes as a random effect into the mixed model account for that? Or what is the best way to deal with that issue? Thanks for your help! – hkflyer May 29 '17 at 4:04
• Fake example: A program to help decrease the weight of the people. The change of weight is measured and change of BMI also recorded. When modeling weight change without BMI change in model, the result indicate the program is useful; but after adding BMI change as a covariate, the program has no effect on weight change. Need to think hard on your purpose. Need to consider the relationship among the variables, – user158565 May 29 '17 at 18:04