Distribution of Predictor Variables
The mean scores on the IIP-C variables were: agency − 0.31 (SD 0.63, range − 2.25; 1.36), communion 0.23 (SD 0.54, range − 1.89; 1.88) and elevation 0.22 (SD 0.69, range − 1.37; 1.66), showing a considerable variation in the degree and kind of interpersonal problems reported by the parents in this sample. On the SCL-90-R revised depression scale (items rated 1–5), the mean item score was 1.94 (SD 0.78, range 1; 4). Some parents reported clearly clinical levels of depressive symptoms: 39 (40.2%) were at or above the mean raw score of a clinical outpatient sample [
41]. Posterior estimates of the predictor variable correlation matrix are displayed in Table
2.
Table 2
Estimated predictor correlation coefficients (posterior means and 93% CI)
Agency | − .42 (− .55; − .27) | | |
Communion | − .06 (− .26; .15) | − .22 (− .45; .05) | |
Elevation | .58 (.45; .68) | − .54 (− .67; − .36) | − 0.18 (− .43; .12) |
Evaluating Models by Leave-one-out Cross-validation and Model Stacking
The four models and the differences in expected log posterior density are displayed in Table
3, along with the stacking weights obtained from the stacking_weights() function from the R package loo [
57].
Table 3
Results of leave-one-out crossvalidation and model stacking
Parental interpersonal problems | 0 | 0 | 0.75 |
Parental interpersonal problems with parent gender interaction | 3.89 | 1.58 | 0 |
Parental depressive symptoms | 8.35 | 6.21 | 0 |
Parental depressive symptoms with parent gender interaction | 10.07 | 7.28 | 0.25 |
Observing the expected log posterior densities and their standard errors, several conclusions may be drawn. Firstly, the data do not support modelling an interaction between parent gender and interpersonal problems, given the difference in expected log posterior density and the stacking weights. Secondly, the data supports both parent interpersonal problems and parental depressive symptoms as predictors of parent-adolescent conflict. Though the difference between the model with parental interpersonal problems and the models with parental depression is larger than the standard deviation, it is not by much. The stacking weights imply that a combination of the model with parental interpersonal problems and the model with parental depression and an interaction with parent gender gives the highest expected predictive accuracy, but with most weight given to parental interpersonal problems.
Regression Parameter Estimates
The regression model parameter estimates from the two models given a positive stacking weight are displayed in Table
4. Both models also have a large number of hierarchical parameters (such as IRT item parameters and hyperparameters, latent trait estimates, and random intercepts per adolescent). These parameters are summarised in the Supplementary material.
Table 4
Regression model parameter estimates
Interpersonal problems model |
Agency | 0.19 | 0.07 | 0.07; 0.31 | 3768 | 1 |
Communion | 0.02 | 0.06 | − 0.08; 0.12 | 5126 | 1 |
Elevation | 0.08 | 0.07 | − 0.04; 0.21 | 6509 | 1 |
Adolescent age | − 0.03 | 0.03 | − 0.09; 0.04 | 7081 | 1 |
Variance of errors | 0.27 | 0.06 | 0.18; 0.39 | 1016 | 1 |
Variance of random effects | 0.21 | 0.07 | 0.05; 0.34 | 1214 | 1 |
Degrees of freedom in t-likelihood | 21.88 | 14.15 | 4.84; 53.70 | 13,611 | 1 |
Parental depressive symptoms model | | | | | |
Intercept | − 0.03 | 0.12 | − 0.24; 0.18 | 840 | 1 |
Depressive symptoms | − 0.16 | 0.07 | − 0.30; − 0.04 | 4719 | 1 |
Depressive symptoms × mother | 0.16 | 0.09 | 0; 0.33 | 5495 | 1 |
Mother | 0.02 | 0.08 | − 0.12; 0.17 | 9331 | 1 |
Adolescent age | − 0.03 | 0.03 | − 0.10; 0.03 | 8449 | 1 |
Variance of errors | 0.30 | 0.06 | 0.20; 0.41 | 1639 | 1 |
Variance of random effects | 0.16 | 0.08 | 0.02; 0.30 | 1610 | 1 |
Degrees of freedom in t-likelihood | 20.8 | 13.85 | 4.57; 51.59 | 14,430 | 1 |
The regression parameter estimates show a positive association between parental agency-related interpersonal problems and parent-adolescent conflict. The positive sign of the coefficient implies that as parents report more problems related to being too interpersonally domineering, their adolescents will tend to report more conflict. The posterior distribution of regression coefficient values (summarised in the table by its mean, standard deviation and the 3.5th and 96.5th percentiles) shows that the data are not at all consistent with a negative association under this model. The data are also not very consistent with a near-zero association, with only a 0.08 probability of a standardised regression coefficient smaller than 0.1.
The posterior distribution for Communion is symmetric around 0, which means the data are most consistent with no strong association between parent-adolescent conflict and parents reporting difficulties either being too cold and distant or overly concerned with maintaining relationships. It is worth noting that in a Bayesian data analysis, an estimate of 0 is no less certain than any other estimate, unlike in classical hypothesis testing, where failure to reject the null hypothesis cannot be interpreted as evidence for the null hypothesis being true [
58].
The posterior mean estimate for Elevation, the interpersonal problem variable measuring general interpersonal distress, is weakly positive, but there is considerable uncertainty in this estimate. An association near zero (between − 0.1 and 0.1) is quite consistent with the data, with a probability of 0.60, but any association is probably positive, with a 0.89 probability of a regression coefficient larger than 0. This means there may be an association between parental general interpersonal distress and parent-adolescent conflict, and that any association is probably positive and of small magnitude, but that the data does not provide conclusive evidence.
For parental depression, the coefficients show a negative association for fathers only, as the positive coefficient for the interaction with dummy-coded parent gender is of similar magnitude as the coefficient for parental depression. The coefficient for paternal depressive symptoms is below − 0.1 with a 0.81 probability. The posterior distribution of the total coefficient for maternal depressive symptoms (obtained by elementwise addition of the posterior samples for the two coefficients) shows evidence for no strong association between maternal depressive symptoms and parent-adolescent conflict, with a 0.92 probability of a coefficient between − 0.1 and 0.1. Both the regression coefficient for parent gender and the intercept (necessary in a model with a dummy-code, to estimate the effect of belonging to the reference category, in this case a father-adolescent relationship), is estimated very close to 0, implying that there are probably no large differences in reported conflict level between mother-adolescent dyads and father-adolescent dyads as groups. The coefficient for adolescent age is also very close to 0 in both models.
In summary, there are two main findings: Adolescent reported conflict is predicted to be higher when parents report more problems than average related to being too interpersonally domineering, and lower when parents report more problems than average being interpersonally submissive, and this applies regardless of parent gender. Given the model weighting, this interpersonal tendency has higher predictive utility than parental depressive symptoms. Conflict is also predicted to be higher with fathers who report less than average depressive symptoms, and lower when fathers report more depressive symptoms, while the depressive symptoms of mothers do not appear to be strongly associated with parent-adolescent conflict.