Assessing moderation effects with a heterogeneous moderated regression analysis

Abstract

Previous research has examined moderation effects with traditional analyses such as ANOVA, ANCOVA, moderated regression analysis (MRA), or a combination of MRA and subgroup analysis. However, there exists some confusion in such analyses, because the analyses do not separately consider two possible effects of a moderator on the form and strength of relationship between a focal predictor and a dependent variable. The effect on the form is measured with the interaction effect between the focal predictor and the moderator whereas the effect on the strength is measured with the effect of the moderator on predictability of the focal predictor on the dependent variable. This paper proposes a heterogeneous MRA that allows the moderation effect to be heterogeneous in the population, and shows that it allows one to examine the two possible moderation effects separately. Furthermore, this paper shows that previous research based on the traditional analyses might have incorrectly led to conclusions that there did not exist moderation effects even though the moderation effects were strongly supported by theories. The heterogeneous MRA can examine moderation effects with a data set collected for the traditional analyses. Thus, this paper recommends one to use the heterogeneous MRA together with the traditional analyses.

Appendices

Appendix 1

Equation 1 is equivalent to:

$$Y=\alpha +{\beta }_{x}{x}_{F}+{\beta }_{x}\left(X-{x}_{F}\right)+{\beta }_{z}Z+{\beta }_{xz}~{x}_{F}Z+{\beta }_{xz}\left(X-{x}_{F}\right)Z+e,$$

(12)

where ${x}_{F}$ indicates a specific value of $X$. Simplifying Eq. 12 leads to:

$$Y={\alpha }^{t}+{\beta }_{x}{X}^{t}+{\beta }_{z}^{t}Z+{\beta }_{xz}{X}^{t}Z+e,$$

(13)

where ${X}^{t}=X-{x}_{F}$, ${\alpha }^{t}=\alpha +{\beta }_{x}{x}_{F}$, and ${\beta }_{z}^{t}={\beta }_{z}+{\beta }_{xz}{x}_{F}$.

Thus, in Eq. 13, Z can be classified as the Pure Moderator if ${x}_{F}=-{\beta }_{z}/{\beta }_{xz}$ (implying ${\beta }_{z}^{t}=0$) when ${\beta }_{xz}\ne 0$, whereas it can be classified as the Quasi-moderator if ${x}_{F}\ne -{\beta }_{z}/{\beta }_{xz}$ (implying ${\beta }_{z}^{t}\ne 0$) when ${\beta }_{xz}\ne 0$.

Appendix 2

The variance of the error term in Eq. 6 is written as:

$$Var\left({e}_{i}\right)=\left[1+\left({\tau }_{xz}^{2}/{\sigma }^{2}\right){X}_{i}^{2}{Z}_{i}^{2}\right]{\sigma }^{2}={\sigma }^{2}+{\tau }_{xz}^{2}{X}_{i}^{2}{Z}_{i}^{2} \mathrm{~~for }~i=1, 2, \cdot \cdot \cdot , N.$$

(14)

Thus, it is possible to replace unknown squared errors ${e}_{i}^{2}$ with ${\widehat{e}}_{i}^{2}={\left({Y}_{i}-{\widehat{Y}}_{i}\right)}^{2}$ where:

${e}_{i}={Y}_{i}-\left(\alpha +{\beta }_{x}{X}_{i}+{\beta }_{z}{Z}_{i}+{\beta }_{xz}{X}_{i}{Z}_{i}\right)$ for i = 1, 2, ···, N, and then regress ${\widehat{e}}_{i}^{2}$ in Eq. 14, i.e.:

$${\widehat{e}}_{i}^{2}={\sigma }^{2}+{\tau }_{xz}^{2}{X}_{i}^{2}{Z}_{i}^{2}+{u}_{i},$$

(15)

where ${\sigma }^{2}\ge 0$ and ${\tau }_{xz}^{2}\ge 0$, of which the coefficients (${\sigma }^{2}$ and ${\tau }_{xz}^{2}$) can be estimated by non-negative least squares. Eq. 15 is the auxiliary model of the HMRM, which is used to probe whether the error term satisfies the assumption of homoscedasticity in Breusch-Pagan test (Breusch and Pagan 1979).

If ${\widehat{\tau }}_{xz}^{2}=0$ in the recovered regression model, one can conclude that the third variable (Z) is Non-Homologizer because the variance of the error term does not depend on levels of Z. If ${\widehat{\tau }}_{xz}^{2}\ne 0$ in the recovered regression model, one can conclude that the third variable is Homologizer.

Appendix 3

With the estimates in Eq. 15, one can calculate the weight used in the WHMRM, which is written as:

$${\widehat{W}}_{i}=1+\left({\widehat{\tau }}_{xz}^{2}/{\widehat{\sigma }}^{2}\right){X}_{i}^{2}{Z}_{i}^{2} \mathrm{~~for }~i=1, 2, \cdot \cdot \cdot , N.$$

(16)

Then, one can estimate the WHMRM written as:

$${Y}_{i}^{t}=\alpha {H}_{i}+{\beta }_{x}{X}_{i}^{t}+{\beta }_{z}{Z}_{i}^{t}+{\beta }_{xz}{X}_{i}^{t}{Z}_{i}^{t}+{e}_{i}^{t} \mathrm{~~for }~i=1, 2, \cdot \cdot \cdot , N,$$

(17)

where ${Y}_{i}^{t}={Y}_{i}^{t}/\sqrt{{\widehat{W}}_{i}}$, ${H}_{i}=1/\sqrt{{\widehat{W}}_{i}}$, ${X}_{i}^{t}={X}_{i}/\sqrt{{\widehat{W}}_{i}}$, ${Z}_{i}^{t}={Z}_{i}/\sqrt{{\widehat{W}}_{i}}$, ${X}_{i}^{t}{Z}_{i}^{t}={X}_{i}{Z}_{i}/\sqrt{{\widehat{W}}_{i}}$ and ${e}_{i}^{t}={e}_{i}/\sqrt{{\widehat{W}}_{i}}$.

The WHMRM satisfies the assumption of homoscedasticity.