Module

The module can estimate several linear models:

• Linear model
• Poisson model
• Poisson overdispersed
• Negative binomial model
• Logistic model
• Probit model
• Ordinal model (proportional odds)
• Multinomial model
• Custom model with combinations of distribution and link function

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

• Linear model: identity LF, gaussian distribution, yielding a general linear model for continuous 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.

• Poisson (overdispersion) model: 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.

• Negative binomial model: logarithm LF, negative binomial distribution, maximum likelihood estimation, with overdispersion, modelling count dependent variables. This model is often used for overdispersed data.

• 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.

• Ordinal model: s proportional odds logistic regression, cumlative logit LF, multinomial distribution, modelling ordered dependent variables.

• Multinomial model: logit LF, multinomial distribution, modelling categorical dependent variables.

• 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.

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. An Offset variable can be defined, which is included in the model with a fixed coefficient (set to 1) (see )

The options of this panel are:

Model

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

Interactions between continuous variables or categorical and continuous ones can be set by selecting one or more variables and 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.

Models comparison

When Activate is flagged, models comparison options become visible.

Two models will be estimated and compared. The current model defined in Model Terms and the model defined in the Nested Model field. By default, the Nested Model terms are empty, so an intercept-only model 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 w) and two interactions w*z and w*z. The nested model terms are only composed by the main effects (x, z and w). Thus, the loglikelihood ratio test that it is performed to compare the model will test the significance of the two interactions together. The output offers a Table in which each model fit indices and tests are presented, and the two models comparison test is presented.

The options are:

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.

Covariates Scaling

Continuous variables can be centered, standardized (z-scores), log-transformed (Log) 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, simply select none.

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.

Plots interpretation varies depending on the model being estimated. All plots are, however, depicting predicted values in the response original scale (usually probabilities). See details and interpretation discussion.

The options are:

Simple Effects

Simple effects can be computed for any combination of types of variables, making it easy to probe interactions, simple slopes, and combinations of them. Simple effects can estimated up to any order of interaction. If only one moderator is set in the Moderators field, the effect of the variable in the Simple effects variable is computed at different levels of the moderator. If more than one moderator is defined, the effects are estimated for all combinations of the moderator levels.

Simple effects are computed following the same logic of the plots. They correspond to the plotted effects as defined above. As for plots, the effects are estimated for different levels of the categorical moderators and for the conditioning values of the continuous moderators defined in Covariates Scaling panel.

When there is more than one moderator, one can activate Simple interactions to obtain estimation and tests for lower order interactions at different levels of a moderator. Simple interactions are computed using the last variable appearing in the Moderators field as moderator. In the case depicted in the figure above, the interaction w*x is estimated and tested at different levels of z.

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