Set up

In order to estimate the model with jamovi, we first need to set each variable in the rigth field.

First, we define subj as the clustering (grouping) variable, by putting it in the Cluster field. Then we put dv in the Dependent Variable field and time in the Factors field. We should push also group in the Factors but this is not allowed because the file loading module sees the variable as a continuous variable (see the scale little icon in the picture). At the moment (this may change in future releases), GAMLj requires the Factors to be nominal, nominal text, or ordinal. To get around this nuissance, we go to the data tab and define group as a nominal variable.

• Select the Data tab and click Setup

• Navigate to the group variable definition and select nominal

• Go back to the Mixed Model analysis and push group into Factors.

If we now look at the results panel, we see that the model definition is not completed yet.

We need to specify the random component by selecting what coefficients are random. We do that by expanding the Random Effects tab.

On the left side, under Components we find all possible random effects allowed in the model already prepared by jamovi. They are prepared based on the variables previously selected, so not all of them may be interesting. In our example, we need the intercept to vary randomly across participants, thus we simply select Intercept|subj. Jamovi uses the R formulation of random effects as implemented by the lme4 R package. The bar | means random across, thus we can read the “components” as Intercept random across subj, group effects random across subj, time effects random across subj, and so on.

At this point, the model is estimated and the results appear in the results panel. Before inspecting the results, we have a look at the fixed effects definition, by expanding the `Fixed Effects’ tab.

As we mentioned above, jamovi automatically includes all independent variables defined in Covariates or in Factor in the fixed effects model. Obviously, when the models are complex, one can tweak the model terms to suits the analysis aim.

Results

The first table in the output contains info about the model and the estimation.

• The Call row display the model in lme4 R package formulation. This can be useful to re-run the same analysis in R (not using GAMLj module).
• The AIC row display the Aikeke Information Criterion, which can be useful to evaluate the model, especially in comparison with other models. Details can be found in GAMLj docs and in [Zuur et. al , 2009] al.](http://www.springer.com/la/book/9780387874579)
• R-marginal and R-conditional are proportions of reduced error, or pseudo-\(R^2\). They are described in Johnson (2014) and implemented in piecewiseSEM. For our purposes, we can interpret them as follows: R-marginal is the variance explained by the fixed effects over the total (expected) variance of the dependent variable. The R-conditional is the variance explained by the fixed and the random effects together over the total (expected) variance of the dependent variable. In our example, the fixed effects explain a good share of the variance (\({R_m}^2\)=.554), but the overall model (fixed+random) captures an even bigger share of the variance (\({R_c}^2\)=.768).

Fixed effects ANOVA gives the F-tests associated with the model fixed effects. Here we see that time has a statistical significant effect (on average), group has a statistically significant effect, and there’s also an interaction between the factors.

As regards the degrees of freedom (nobody cares about them, I’ve been told before), jamovi mixed model tries to use Satterthwaite approximation as much as possible, but for complex models it may fail. When that happens, Kenword-Roger approximation is used and, if the latter does not fail, F-tests are computed. A note signals which approximation is used.

In the output there’s also a Fixed effects Parameters Estimates table that gives the fixed B coefficients. For the moment the B coefficients are not interesting, so we skip the interpretation of this table (see further down this document what we do with it).

The Random Component display the variances and SD of the random coefficients, in this case of the random intercepts. From the table we can see that there is variance in the intercepts (\({\sigma_a}^2\)=2539), thus we did well in letting the intercepts vary from cluster to cluster. \({\sigma_a}^2\) can be reported as an intra-class correlation by dividing it by the sum of itself and the residual variace (\(\sigma^2\)), that is \(v_{ic}={{\sigma_a}^2 \over {{\sigma_a}^2+{\sigma}^2}}\) (future releases will compute that for you).

Mixed vs RM Anova

If one looks at the results discussed in David C. Howell website, one can appreciate that our results are almost perfectly in line with the ones obtained with SPSS, SAS, and with a repeated measures ANOVA. The latter it is not always true, meaning that depending on the data and model charateristics, RM ANOVA and the Mixed model results may differ. RM ANOVA and the Mixed model are different strategies to estimates effects in RM designs, thus they are not always overlapping.

Estimated Means Plots

To interpret the results, we look at the means of the groups resulting from the combinations of the factors levels. We can go to Fixed Effects Plots and select time for the X-axis and group as the factor across lines.

Results show that for group 1 there is a slow decay of the dependent variable, whereas for group 2 there is a fast decay from time 0 to time 1, and a flat trend from time 1 on. On average, group 1 shows higher levels of the dv, and, on average, the dv decreases over time.

A clearer and more honest account of the model fit can be seen by looking at the estimated means over the actual observed scores (GALMj v. >= 0.9.3), option Plot -> Observed scores.

Probing the effects

We can now ask several questions to the model. There are different strategies to probe the interaction, and usually one picks the one that better suits the researcher’s hypotheses and interests. Here we use them all, mainly for showing how they can pursued in jamovi GAMLj.

Coding scheme

In case one has doubts about the meaning of the chosen coding scheme for the factors, one can ask for the table of contrasts weights. This is done in Factor coding by selecting Contrast coefficients tables.

This produces a table in the results panel as shown here.

More info about coding schemes Rosetta store: contrasts