# What kind of linear mixed model is most appropriate for this data?

I previously wrote about this data here, and was advised that a linear mixed model applied to the raw data would provide "more precision and power" than the originally suggested approach of a regression on the average scores.

Participants in a study all respond to stimuli under two conditions, and some difference in response time is expected between Condition 1 and Condition 2.

I have three continuous outcome variables:

• A participant's average response time in Condition 1
• A participant's average response time in Condition 2
• The difference between C1 and C2, i.e. C1 minus C2

I also have 7 variables (1 categorical, 6 continuous) that are believed to predict these outcome variables. In particular, they're strongly believed to predict the difference score, because the variables are believed to predict that some people will be more affected by C1 than C2, or vice-versa.

What sort of linear mixed model should I use? It would be appreciated if some advice can be provided on why the selected approach is appropriate.

• What kind of data do you have more precisely: you have N different participants and for each of them you have a response1 and response 2 (and the difference) and for each prticiapnt you also have values for 7 other variables ? Can you give an example of these 7 variables ? – user83346 Sep 10 '15 at 8:40
• That is correct. An example of one the predictor variables is a test of linguistic fluency where the scores varied from about 300 to 800. The N is 88. – user1205901 Sep 10 '15 at 8:51
• And you have all these observed at different points in time ? – user83346 Sep 10 '15 at 9:55
• We obtained all the data in a single one-hour session, in which participants completed various tests. Each session had only one participant. – user1205901 Sep 10 '15 at 9:59

I think about three main possibilities. The first one is a multiple regression model with C2 as a function of C1 and other predictor variables. The second one is the difference score (C2-C1 seems most natural to me) as a function of C1 and other predictor variables. The third is test score as the dependent variable, and using a mixed models to account for the paired nature of the data. In this form, C1 and C2 are combined in the long format, so that each individual has two rows, one with the value of C1 as the score, and one with the value of C2. They should then also have a categorical variable with the categories C1 or C2 indicating the source of the outcome variable for each row.

So the first possibility is a standard multiple regression. You will then model C2 as a function of C1:

lm (C2 ~ C1 + x1 + ... + x7)


The predictor variables will indicate how each predictor affects the mean of C2, regardless of the value of C1. You may think that certain predictors will influence the way that C1 affects C2. For example, perhaps a good C1 score is associated with an even better C2 score in women, but there is no such relationship in men. Then you need to include interaction effects between C1 and the predictor variables:

lm (C2 ~ C1 * x1 + ... + C1 * x7)


And you may of course also include interactions between predictors. You may also have a non-linear relationship between C1 and C2, and you can test this by including polynomials of C1 in the model and then comparing the models:

lm (C2 ~ C1 + I(C1*C1) + x1 + ... + x7)
lm (C2 ~ C1 + I(C1*C1) + I(C1*C1*C1) + x1 + ... + x7)


There are also other techniques for modeling non-linearity.

The second approach, with using the difference score, was not recommended in the answer to your previous question. I haven't used this approach, but I can see some problems with it and no obvious advantages.

The third approach is using mixed models. The data should be organized this way:

subject score test x1 x2 x3 ...
1       145   C1   3  2  0
1       172   C2   3  2  0
2       142   C1   0  14 2
2       154   C2   0  14 2


Etc. In this approach, you'll model score as a function of test type (C1/C2) and predictor variables, and you'll allow each individual to have their own baseline score and also, if you want to, their individual effect of test type (C1/C2) on the score. First the random effects section:

library(lme4)
library(lmerTest)
lmer (score ~ fixed effects + (1|subject))


Here we take into account that the observations are paired, and every subject will have their own baseline score level on both tests.

lmer (score ~ fixed effects + (type|subject))


Now we added a different slope per individual, so that each individual will have their own effect of type (C1 vs C2) on the score. And we can now add fixed effects:

lmer (score ~ type + x1 + ... + x7 + (type|subject))


But now we're only modeling the score, and you're mainly interested in the effect of predictor variables on the difference between C1 and C2 if I understand you correctly. We then have to model this by using interactions between type and predictor variables:

lmer (score ~ type * x1 + ... + type * x7 + (type|subject))


Interactions between other predictor variables can of course be added as well. I think this starts to look like a full model. There are many parameters that must be estimated in this model: 2*7 (for all x's and interaction with type) and 1 for type, and then intercept, standard deviation and the two random effects for subject. 20 in total if I'm not mistaken. This might be compared to 17 parameters in the multiple regression model with all interactions between x's and C1. But this model has twice the number of observations.

Which method is preferable is up to you, and you can create many models and compare them to see which seems to be the best (using anova, AIC etc).

Good luck!

• Thanks for your response! I understand AIC, but how does one use ANOVA to compare models? – user1205901 Sep 13 '15 at 14:34
• In R, you can just type anova(model1, model2) to compare them. The test will calculate an F statistic based on the difference in sum of squared residuals between the models with the difference in number of estimated parameters as the degrees of freedom for the statistic, and an F-test will be made so that you get a p-value. If the p-value is low, there is evidence that the more complex model provides a better fit, and if the p-value is high, you should keep the simpler model (that is nested within the other). – JonB Sep 14 '15 at 7:46
• When you write "The data should be organized this way", is the score variable meant to be the RT for that person on a particular item? So the data should be organized like this: i.imgur.com/jyhlkAN.png? – user1205901 Sep 16 '15 at 12:31
• Yes, precisely! – JonB Sep 16 '15 at 14:11