GALMj version ≥ 2.0.0

**Generalized Mixed Linear Models** module of the GAMLj suite for jamovi

The module estimates generalized mixed linear models with categorial and/or continuous variables, with options to facilitate estimation of interactions, simple slopes, simple effects, post-hoc, etc. In this page you can find some hint to get started with the mixed models module.

The module can estimate generalized 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 and a plethora of dependent variable types.

The module can estimate several mixed linear models:

- Poisson model
- Poisson overdispersed
- Logistic model
- Probit model
- Custom model with combinations of distribution and link function

For each model, any combination of categorical and continuous variables can be set as independent variables, thus providing an easy way for multiple regression, ANOVA-like, ANCOVA-like and moderation analysis for categorical and count dependent variables.

Models are defined by a link function (LF) and the dependent variable distribution, thus allowing to model different types of dependent variables:

- Logistic model: logit LF, binomial distribution, modelling dichotomous dependent variables.
Probit model: inverse of the cumulative normal distribution link function, binomial distribution, modelling dichotomous dependent variables.

- Poisson model: logarithm LF, Poisson distribution, modelling count dependent variables. This model is often called log-linear model when the independent variables are all categorical.
Overdispersed Poisson model: logarithm LF, Poisson distribution, quasi-maximum likelihood estimation, with overdispersion, modelling count dependent variables. This model is often used for overdispersed data.

Custom model: combination of distribution family and link function.

The available distributions are:

- Gaussian
- Binomial
- Gamma
- Inverse Gaussian

- The available link functions are:
- Identity
- Log
- Inverse
- Inverse squared

The plausibility of the distribution/link function combination is no checked, but errors are issued if the data do not conform to the chosen custom model.

The module provides 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. That can be easily solved by recoding the clustering variables ( see Technical Details ).

Confidence intervals for the parameters estimates and for the exponential of them (expB).

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

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 by 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.

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.

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 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 toghether”. 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.

More details and examples Rosetta store: contrasts.

Continuous variables can be centered, standardized, cluster-based centered, cluster-based standardized, log-transformed or used as they are (`none`

). 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 none.

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.

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. 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

The offset should be within 5 and 50.

Note that with either of these two options, one can estimate simple effects and plots for any value of the continuous IV.

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

Labels+Values produces labels of the form \(Mean \pm SD=XXXX\), where

`XXXX`

is the actual value.

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

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 three-way 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.

Flagging the Display options add Confidence intervals (or confidence bands) or Standard errors to the plots.

Plot options allows to add observed data to the plot (Observed scores) or to fix the range of the plot to the actual range of the dependent variable (Y-axis observed range), without the need to plot the actual predicted values. When `Separate plots`

are required, in each plot are showed only the observed scores of the moderator level for which the plot is done when the moderator is a categorical one, otherwise all data are plot in each plot.

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.

Print the estimate expected means, SE, df and confidence intervals of the precicted dependent variable by factors in the model. When `Include covariates is selected, factors levels are crossed also with the conditiong levels of the continuous variables (if any). The conditioning values are selected in Covariates Scaling panel.

The precision/speed parameter applies to mixed models with one random effect. Higher numbers increase precision in the estimation with a speed cost. It corresponds to the `nAGQ`

parameter of `glmer`

command in R package lme4. For models with more than

The CI method decides which method is used for computing confidence intervals. Wald uses Wald statistics and it is less precise than the other methods, but it is significantly faster. Profile is more precise but slower. Boot, bootstrap method, is the slowest. Details of the implementation can be found in lme4 manual

Some worked out examples of the analyses carried out with jamovi GAMLj Mixed models are posted here (more to come)

Some more information about the module specs can be found here