Estimates

The module provides ANOVA tables and parameters estimates for any estimated model. Variance-based effect size indices (eta, partial eta, partial omega, partial epsilon, and beta) and mean comparisons (Cohen’s d) are optionally computed.

Variables definition follows jamovi standards, with categorical independent variables defined in Factors and continuous independent variables in Covariates.

Effect size indices are optionally computed by selecting the following options (see Details: GLM effect size indices):

• \(\beta\) : standardized regression coefficients
• \(\eta^2\): (semi-partial) eta-squared
• \(\eta^2\)p : partial eta-squared
• \(\omega^2\) : omega-squared
• \(\omega^2\)p : partial omega-squared
• \(\epsilon^2\) : epsilon-squared
• \(\epsilon^2\)p : partial epsilon-squared

Confidence intervals of the parameters can be also selected in Options tab (see below).

Please check the details in Details: GLM effect size indices

The complete set of options for this panel is:

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 variables 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.

The option Intercept includes an intercept in the model. Unflagging it estimates zero-intercept models (Regression through the origin, but see here before you do it ).

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 other 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.

More details regarding custom contrasts can be found in Contrasts analysis. More details and examples for contrasts in general can be found in Rosetta store: contrasts.

Covariates Scaling

Continuous variables can be Centered, standardized (z-scores), log-transformed (Log) 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, simply select Original.

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.

The option Dependent variable scale allows to transform the dependent variable. The dependent variable can be centered, standardized (z-scores), log-transformed (Log) or used as it is (none). The default is none, so no transformation is applied.

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

Along with the means comparisons, one can obtain also the Cohen’s \(d\) effect size indices. Different formulation of the Cohen’s \(d\) are available, and they differ in the way the pooled standard error is computed.

• Cohen’s d (model SD) \(d_{mod}\): the means difference is divided by the estimated standard deviation computed based on the model residual variance.

• Cohen’s d (sample SD) \(d_{sample}\): the means difference is divided by the pooled standard deviation computed within each group.

• Hedge’s g \(g_{sample}\): the means difference is divided by the pooled standard deviation computed within each group, corrected for sample bias. The correction is the one describe by Hedges and Olkin (2014) based on the Gamma function.

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.

The remaining options are defined as follows:

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.

Assumptions checks

Options