Module

The module can estimates REML and ML linear mixed models for any combination of categorical and continuous variables, thus providing an easy way of obtaining multilevel or hierarchical linear models for any combination of independent variables types.

Estimates

The module provides a parameter estimates of the fixed effects, the random variances and correlation among random coefficients.

Variables definition follows jamovi standards, with categorical independent variables defined in “fixed factors” and continuous independent variables in “covariates”.

The grouping variable is simply set by putting the corresponding variable(s) into cluster. In this version, multiple clustering variables are possible, but not combinations of classifications ( see Technical Details ).

The actual estimation occurs when the dependent variable, the clustering variable and at least one random coefficient (random effect) has been selected.

Fixed effects Model

By default, the model fixed effects terms are filled in automatically for main effects and for interactions with categorical variables.

Interactions between continuous variables or categorical and continuous can be set by clicking the second arrow icon.

Polinomial effects for continuous variables can be added to the model. When a variable is selected in the Components field, a little number appears on the right side of the selection. The number indicates the order of the effect.

By increasing that number before dragging the term into the Model Terms field, one can include any high order effect. Increasing the order number and combining the selection with other variables allows including interactions involving higher order effects of a variable.

Random effects

Random effects across clustering variables are automatically prepared by the module following R lmer() standards: term | cluster indicates that the coefficient associated with term is random across cluster.

By default the module assumes correlated random effects. All the effects varying across the same cluster variable appearing in the Random coefficients will be correlated. To obtain a variance component model, select Not correlated. A custom pattern of correlation can be obtained by selecting Correlated block. For instance, in Fig. below, a custom structure has been defined by allowing the intercept and the effect of x to be correlated, whereas the effect of wfac is independent from the others.

The option LRT for random effects produces a table of Likelihood Ratio Tests for the random effects. The table is estimated with lmerTest::ranova command, documented here. The test basically compares the likelihood of a model with the effect included versus a model with the effect excluded. For example, x in (1+x|cluster) means that the model with (1+x|cluster) random structure is compared with a model with 1|cluster) random structure. If significant, the model with random effect x is significantly better (in terms of likelihood) than the model with (1|cluster) structure.

Models comparison

When Model Comparison Activate is flagged, model comparison options become visible. Both for fixed effects

and for random effects.

Two models will be estimated and compared. The current model defined in by the Model Terms and the Random Coefficients is compared with the the model defined in the Nested Model and Nested Model Random Coefficients. By default, the Nested Model terms are empty and the Nested Model Random Coefficients are filled with the full model random terms, so a model without fixed effects is compared with the current. When the user defines nested terms, the comparison is updated.

Consider the following example:

The current model is composed by three main effects (x, z) and the random coefficients for (Intercept) and x. The nested model terms are have the same fixed effects, but it has only the random intercept. Thus, the loglikelihood ratio test that is performed to compare the models will test the significance of the random effect of x. The output offers a Table in which each model fit indices and tests are presented, and the two models comparison test is presented.

Factors coding

It allows to code the categorical variables according to different coding schemas. The coding schema applies to all parameters estimates. The default coding schema is simple, which is centered to zero and compares each means with the reference category mean. The reference category is the first appearing in the variable levels.

Note that all contrasts but dummy (and custom) guarantee to be centered to zero (intercept being the grand mean), so when involved in interactions the other variables coefficients can be interpret as (main) average effects. If contrast dummy is set, the intercept and the effects of other variables in interactions are estimated for the first group of the categorical IV.

Contrasts definitions are provided in the estimates table. More detailed definitions of the comparisons operated by the contrasts can be obtained by selecting Show contrast definition table.

Differently to standard R naming system, contrasts variables are always named with the name of the factor and progressive numbers from 1 to K-1, where K is the number of levels of the factor.

In reading the contrast labels, one should interpret the (1,2,3) code as meaning “the mean of the levels 1,2, and 3 pooled together”. If factor levels 1,2 and 3 are all levels of the factor in the samples, (1,2,3) is equivalent to “the mean of the sample”. For example, for a three levels factor, a contrast labeled 1-(1,2,3) means that the contrast is comparing the mean of level 1 against the mean of the sample. For the same factor, a contrast labeled 1-(2,3) indicates a comparison between level 1 mean and the subsequent levels means pooled together.

Custom contrasts weights can be defined by first selecting custom for the variable of interest. Upon choosing custom for a variable, a new field appears and we can input the contrast weights we wish to test. Only one contrast per variable can be defined, but if more contrasts are required one can always run different analyses, one for each contrast. The coding weights are input with the simple syntax w1,w2,w3. The other of the weights follow the other of the factor levels in the datasheet.

More details and examples Rosetta store: contrasts.

Covariates Scaling

Continuous variables can be centered, standardized, cluster-based centered, cluster-based standardized, log-transformed or used as they are (Original). The default is centered because it makes our lives much easier when there are interactions in the model, and do not affect the B coefficients when there are none. Thus, if one is comparing results with other software that does not center the continuous variables, without interactions in the model one would find only a discrepancy in the intercept, because in GAMLj the intercept represents the expected value of the dependent variable for the average value of the independent variable. If one needs to unscale the variable, simple select Original.

Centered clusterwise and z-scores clusterwise center each score using the mean of the cluster in which the score belongs. For z-scores clusterwise the score is also divided by the cluster standard deviation. Log applies a simple natural logarithm transformation to the variable.

The same transformations can be applied to the dependent variable by selecting an option in Dependent variable scale.

Covariates conditioning rules how the model is conditioned to different values of the continuous independent variables in the simple effects estimation and in the plots when there is an interaction in the model.

• Mean+SD: means that the IV is conditioned to the \(mean\), to \(mean+k \cdot sd\), and to \(mean-k\cdot sd\), where \(k\) is ruled by the white field below the option. Default is 1 SD.

• Percentile 50 +offset: means that the IV is conditioned to the \(median\), the \(median+k P\), and the \(median-k\cdot P\), where \(P\) is the offset of percentile one needs. Again, the \(P\) is ruled by the white field below the option. The offset should be within 5 and 50, default is 25%. The default conditions the model to:

  • \(50^{th}-25^{th}=25^{th}\) percentile

  • \(50^{th}\) percentile

  • \(50^{th}+25^{th}=75^{th}\) percentile

• Min to Max: The IV is conditioned to its \(min\), \(max\) and a number of values in between, ruled by Steps. For Steps=1 only \(min\) and \(max\) are used. For Steps=2, one value in the middle is also used, and so on.

Covariates labeling decides which label should be associated with the estimates and plots of simple effects as follows:

• Labels produces strings of the form \(Mean \pm SD\).

• Values uses the actual values of the variables, after scaling.

• Labels+Values produces labels of the form \(Mean \pm SD=XXXX\), where XXXX is the actual value.

• Unscaled Values produces labels indicating the actual value (of the mean and sd) of the original variable scale. This can be useful, for instance, when the user needs the estimates to be obtained with centered variables (because there are interactions, for instance), but the plot of the effects is preferred in the original scales of the moderators.

• Unscaled Values + Labels as the previous option, but add also the label “Mean” and “SD” to the original values.

The Scaling on option decides how the scaling of the variables handle missing values: First, keep in mind that the model will be estimated on complete cases, no matter how this option is set. When there are missing values, however, one can scale each variable only on the complete cases (the default), or scale columnwise. If columnwise is selected, the mean and standard deviation of each variable used to scale the scores are computed with the available data of the variable, independently of possible missing values in other variables.

Post-hocs

Post-hoc tests can be accomplished for the categorical variables groups by selecting the appropriated factor and flag the required tests

Post-hoc tests are implemented based on R package emmeans. All tecnical info can be found here

Plots

The “plots” menu allows for plotting main effects and interactions for any combination of types of variables, making it easy to plot interaction means plots, simple slopes, and combinations of them. The best plot is chosen automatically.

By filling in Horizontal axis one obtains the group means of the selected factor or the regression line for the selected covariate.

By filling in Horizontal axis and Separated lines one obtains a different plot depending on the type of variables selected:

• Horizontal axis and Separated lines are both factors, one obtains the interaction plot of group means.
• Horizontal axis is a factor and Separated lines is a covariate. One obtains the plot of group means of the factor estimated at three different levels of the covariate. The levels are decided by the Covariates conditioning options above.
• Horizontal axis and Separated lines are covariates. One obtains the simple slopes graph of the simple slopes of the variable in horizontal axis estimated at three different levels of the covariate.

By filling in Separate plots one can probe higher-order interactions. If the selected variable is a factor, one obtains a two-way graph (as previously defined) for each level of the “Separate plots” variable. If the selected variable is a covariate, one obtains a two-way graph (as previously defined) for the Separate plots variable centered to conditioning values selected in the Covariates conditioning options. Any number of plots can be obtained depending on the order of the interaction.

By flagging Random effects one obtains the random effects estimated values in the plot along with the fixed effects. In case of multiple cluster variables, the first cluster variable in the cluster field of “variable role” panel is used (if it is included in the model). To change the cluster variable used to plot the random effects, change the order of the variables in the “variable role” definition.

Estimated marginal means

Print the estimate expected means, SE, df and confidence intervals of the predicted dependent variable by factors in the model. Any combination available in the model (main effects, interactions, non-linear terms), can be requested. If the term involves categorical independent variables, means of each level of the variable are presented. If the term involves continuous variables, expected means computed at the levels defined in Covariate Scaling are presented.

Options