1a) According to critics of misconduct in mediation analyses (e.g., Klaus Fiedler, Rex B. Kline, Charlotte Tate) the mediator has to follow a temporal and causal order after the predictor. That is the the predictor has to precede (come before) the mediator in a time ordered manner. In other words, the mediator also has to follow the predictor temporally and conceptually be the pathway from predictor to the outcome (Fiedler et al., 2015; Tate, 2015). In Dana’s analysis, verbal ability and age conceptually occur at the same time, therefore the time ordered pathway does not match the mediation requirements. Additionally it proposes that the only pathway the predictor would work through is verbal ability and it would logically and empirically seem plausible that a child’s age would predict morality through other pathways than just verbal ability.
Alternatively, Dana could also consider a moderation analysis to see if the effect of age on moral performance changes at different levels of verbal ability. This could help to clarify the role that verbal ability plays in the relationship between age and moral behavior. Dana could run a multiple regression analysis with child’s age, verbal ability, and the (Age*Verbal Ability) interaction term as predictors, and morality test performance as the outcome variable.
1b) Dana can not conclude full mediation by only observing the p value being over .05. There is still a correlation between age and morality scores and the lack of significance may be due to an inadequately small sample size. She may have found partial mediation, but the non-significant direct effect could be due to lack of statistical power.
1c) From the information given, we cannot calculate the indirect effect and its confidence interval by hand, because we’re not given the coefficient for the effect of child’s age on verbal ability (path a), or the effect of verbal ability on performance controlling for child’s age (path b). These values would be necessary to calculate the indirect effect (a*b).
2a) Anel has not performed an accurate mediation analysis. In a mediation analysis, the goal is to show that the effect of the predictor variable (work satisfaction) on the outcome variable (productivity) operates through the mediator variable (task challenge). For this, you would typically need to follow these steps: a) Show that work satisfaction (the predictor) is significantly associated with productivity (the outcome variable), b) Show that work satisfaction is significantly associated with task challenge (the mediator), c) Show that task challenge is significantly associated with productivity, controlling for work satisfaction, and d) The relationship between work satisfaction and productivity should be significantly reduced when controlling for task challenge. This is typically referred to as “the indirect effect”.
2b) What you have conducted here is a sequential modeling analysis which can show if task challenge explains additional variance in productivity beyond that explained by work satisfaction. This can demonstrate whether task challenge is a significant predictor of productivity above and beyond work satisfaction, but it does not establish a mediation effect.
3a) Chandak seems to have conducted a moderation analysis rather than a mediation analysis. Which is the better approach since a mediation analysis may not be appropriate, due to the mediator not following the main predictor in a causal and temporal order. The interaction term Chandak created (stress x coping) is used to test if the effect of work stress on counterproductive work behavior changes at different levels of coping skills, which is the definition of moderation. Mediation, on the other hand, tests whether the effect of work stress on counterproductive work behavior goes through (i.e., is mediated or carried by) coping skills. Chandak correctly centered the predictor variables to reduce multicollinearity before creating the interaction term. He appropriately included all three terms (centered work stress, centered coping skills, and the centered interaction term) in a regression model predicting counterproductive work behavior. Given his findings, his conclusion that coping skills do not moderate the relationship between work stress and counterproductive work behavior is valid because the interaction term was not significant. However, if Chandak wishes to understand more about the relative contribution of work stress and coping skills in predicting counterproductive work behavior, he could conduct a sequential regression analysis. In this approach, variables are entered into the regression equation in steps or blocks, with each block representing a set of predictors that are distinct. For instance, Chandak could enter work stress in the first step of the model, then add coping skills in the second step. By examining the change in R-square between steps, he could determine whether adding coping skills to the model significantly improves the model’s prediction of counterproductive work behavior above and beyond work stress alone.
4a) In the case of many multivariate models, the high VIF values for the ‘self-esteem’ predictor and ‘attachment’ would suggest that these two predictors show an inflated and inaccurate degree of variance overlap (sometimes due to differences in their raw units). This is referred to as collinearity. Collinearity can be resolved by standardizing the variables via dividing the raw units by their standard deviation. In this particular case, the problem of essential collinearity could also be masked by what is called non-essential collinearity. Non-essential collinearity happens due to the scaling of the predictor variables in the equation when there is an interaction term. To remedy this you can center the variables, by subtracting the mean of the variable from each score, which Kelley does. There is also a separate issue of non-essential collinearity when there is a categorical by continuous interaction. That is when you center the continuous first-order predictor variables and introduce categorical variables in an interaction term, the adjusted Y-axis intercept can be correlated with the categorical predictor variables, and that by itself produces non-essential collinearity (whereby some statisticians would say you don’t have to do a collinearity test).
4b) So it’s still appropriate to move forward with your analysis even when given those high VIFs. In the categorical by continuous interaction case dummy coding or effects coding is useful. Dummy coding typically codes one category as a reference group (coded as 0) and compares other categories against this reference. On the other hand, effects coding compares each category to the overall average or grand mean, and thus, it often provides a more intuitive interpretation in some contexts as it takes into account all variables.
Assuming that adult attachment is a categorical variable in your case, with “avoidant” and “secure” as the two categories, you could use dummy coding to create a new variable where “secure” is coded as ‘1’ and “avoidant” as ‘0’. In this case the avoidant group can be the “reference group” and represents the Y intercept and that would show if the mean of the “secure” group is different from the “avoidant” group. You could then include this new variable in your regression model:
(when g = 2)
The regression equation categories (0 = “avoidant” group, 1 = “secure” group) are as follows:
“Avoidant” group: Ŷ = b\(_{0}\) + b\(_{1}\)(0), which means that b\(_{0}\)= “avoidant” group (reference group.)
“Secure” group: Ŷ = b\(_{0}\) + b\(_{1}\)(1), which means that b\(_{1}\) = “secure” group as compared to reference group.
In this equation, the reference group is the same as the regression intercept.
A one-unit change in C\(_{1}\) (C\(_{1}\) = 0 to C\(_{1}\) = 1) represents the difference in value of the C\(_{1}\) = 1 mean from the reference group mean (C\(_{1}\) = 0) on the outcome.
Your attachment variable has two categories, “avoidant” and “secure”, you could assign “avoidant” the value of -1 and “secure” the value of 1. You could then include this effects-coded variable and its interaction with self-esteem in your regression model. This interaction term will now represent the differential impact of self-esteem on relationship satisfaction between secure and avoidant attachment styles. Here are the basics for your regression equation:
Ŷ = b\(_{0}\) + b\(_{1}\), which means that b\(_{0}\) + b\(_{1}\)= M\(_{1}\) (here, M\(_{1}\) = mean of secure group). Ŷ = b\(_{0}\) + b\(_{1}\)(-1), which means that b\(_{0}\) - b\(_{1}\) = M1 (here, M\(_{1}\) = mean of avoidant group).
In this equation, the b is the (unweighted) mean of both groups A one-unit change in C\(_{1}\) (C\(_{1}\) = 0 to C\(_{1}\) = 1) represents the discrepancy of the C\(_{1}\) = +1 mean from the (unweighted) grand mean on the outcome.
Also, it was beneficial for Kelley to review the zero-order correlations between predictors and the outcome before proceeding with the regression analysis.
Zero order correlations represent the correlation of the categorical variable based on coding (1, -1).
Positive rs = group coded 1 has a higher value than group coded - 1.
Negative rs = group coded 1 has a lower value than the group coded -1.