Moderation Analysis Using a Two-Level Regression Model

Abstract

Moderation analysis is widely used in social and behavioral research. The most commonly used model for moderation analysis is moderated multiple regression (MMR) in which the explanatory variables of the regression model include product terms, and the model is typically estimated by least squares (LS). This paper argues for a two-level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables. An algorithm for estimating the parameters of the two-level model by normal-distribution-based maximum likelihood (NML) is developed. Formulas for the standard errors (SEs) of the parameter estimates are provided and studied. Results indicate that, when heteroscedasticity exists, NML with the two-level model gives more efficient and more accurate parameter estimates than the LS analysis of the MMR model. When error variances are homoscedastic, NML with the two-level model leads to essentially the same results as LS with the MMR model. Most importantly, the two-level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables. When applied to data from General Social Surveys 1991, NML with the two-level model identified a significant moderation effect of race on the regression of job prestige on years of education while LS with the MMR model did not. An R package is also developed and documented to facilitate the application of the two-level model.

Appendices

Appendix A. Iteratively Reweighted Least Squares (IRLS) Algorithm for NMLEs

This Appendix contains the development of the IRLS algorithm that maximizes the likelihood function l(θ) in (8). Setting the partial derivatives of l(θ) with respect to γ and σ at zero yields the normal estimating equations

$$ \sum_{i=1}^n \frac{1}{\tau_i^2}\bigl(y_i-\mathbf {c}_i'\boldsymbol{\gamma}\bigr)\mathbf {c}_i=\mathbf {0}, $$

(A.1)

and

$$ \sum_{i=1}^n \frac{1}{2\tau_i^4}\bigl[\bigl(y_i-\mathbf {c}_i' \boldsymbol{\gamma}\bigr)^2-{\bf h}_i' \boldsymbol{\sigma}\bigr]{\bf h}_i=\mathbf {0}. $$

(A.2)

We need to solve (A.1) and (A.2) for NMLEs of γ and σ. If the $\tau_{i}^{2}$s are known, then the solution to (A.1) is

$$ \hat{\boldsymbol{\gamma}}=\Biggl(\sum _{i=1}^n\frac{1}{\tau _i^2}\mathbf {c}_i \mathbf {c}_i'\Biggr)^{-1} \Biggl(\sum _{i=1}^n\frac{1}{\tau_i^2}\mathbf {c}_iy_i \Biggr), $$

(A.3)

and that to (A.2) is

$$ \hat{\boldsymbol{\sigma}}=\Biggl(\sum _{i=1}^n \frac{1}{\tau_i^4}{\bf h}_i{ \bf h}_i'\Biggr)^{-1}\Biggl[ \sum _{i=1}^n \frac{1}{\tau_i^4}{\bf h}_i \bigl(y_i-\mathbf {c}_i'\hat{\boldsymbol{\gamma}}\bigr)^2\Biggr]. $$

(A.4)

Although the $\tau_{i}^{2}$s are unknown in practice, they can be estimated as ${\bf h}_{i}'\hat{\boldsymbol{\sigma}}$ once a $\hat{\boldsymbol{\sigma}}$ is available. Thus, Equations (A.1) and (A.2) can be solved by the following IRLS algorithm:

• (S1) With an initial value of σ, obtain each $\tau_{i}^{2}={\bf h}_{i}'\boldsymbol{\sigma}$, i=1,2,…,n.

• (S2) Obtain $\hat{\boldsymbol{\gamma}}$ by (A.3) and $\hat {\boldsymbol{\sigma}}$ by (A.4).

• (S3) Update the initial σ by $\hat{\boldsymbol{\sigma}}$ in (S2) and go back to (S1).

• (S4) Continue (S1) to (S3) until $\hat{\boldsymbol{\gamma}}$ and $\hat{\boldsymbol{\sigma}}$ stabilize.

The converged solutions are the NMLEs of γ and σ.

Appendix B. Two-Level Regression with p Predictors and m Moderators

This Appendix contains the details of extending the simple two-level regression model in (4) to cases with p predictors and m moderators. With p predictors, the counterpart of the first regression equation in (4) is

$$ y_i=\beta_{i0}+\beta_{i1}x_{i1}+ \beta_{i2}x_{i2}+\cdots+\beta _{ip}x_{ip}+e_i= \mathbf {c}'_i\boldsymbol {\beta }_i+e_i, \quad i=1,2,\ldots,n, $$

(B.1)

where $\mathbf {c}_{i}'=(1,x_{i1},x_{i2},\ldots,x_{ip})$ and β _i =(β _i0,β _i1,β _i2,…,β _ip )′. The counterpart of the 2nd and 3rd regression equations in (4) are

$$ \beta_{ij}=\gamma_{j0}+\gamma_{j1}u_{i1}+ \gamma_{j2}u_{i2}+\cdots +\gamma _{jm}u_{im}+ \varepsilon_{ij},\quad j=0,1,2,\ldots, p;\ i=1,2,\ldots ,n. $$

(B.2)

Let u _i =(1,u _i1,u _i2,…,u _im )′, ε _i =(ε _i0,ε _i1,ε _i2,…,ε _ip )′,

$$\boldsymbol {\Gamma }=\left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \gamma_{00}&\gamma_{01}&\gamma_{02}&\cdots&\gamma_{0m}\\ \gamma_{10}&\gamma_{11}&\gamma_{12}&\cdots&\gamma_{1m}\\ \gamma_{20}&\gamma_{21}&\gamma_{22}&\cdots&\gamma_{2m}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \gamma_{p0}&\gamma_{p1}&\gamma_{p2}&\cdots&\gamma_{pm} \end{array} \right ). $$

We can rewrite (B.2) in matrix form as

$$ \boldsymbol {\beta }_i=\boldsymbol {\Gamma }\mathbf {u}_i+ \boldsymbol {\varepsilon }_i,\quad i=1,2,\ldots, n. $$

(B.3)

The counterpart of Equation (5) is obtained by putting (B.3) into (B.1),

$$ y_i=\mathbf {c}_i'\boldsymbol {\Gamma }\mathbf {u}_i+\delta_i, $$

(B.4)

where $\delta_{i}=\mathbf {c}_{i}'\boldsymbol {\varepsilon }_{i}+e_{i}$. Let

$$\boldsymbol {\Sigma }=\mathrm {Var}(\boldsymbol {\varepsilon }_i)= \left ( \begin{array}{c@{\quad}@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \sigma_{00}&\sigma_{01}&\sigma_{02}&\cdots&\sigma_{0p}\\ \sigma_{10}&\sigma_{11}&\sigma_{12}&\cdots&\sigma_{1p}\\ \sigma_{20}&\sigma_{21}&\sigma_{22}&\cdots&\sigma_{2p}\\ \cdots&\cdots&\cdots&\cdots&\cdots\\ \sigma_{p0}&\sigma_{p1}&\sigma_{p2}&\cdots&\sigma_{pp} \end{array} \right ). $$

Then

$$ \tau_i^2=\mathrm {Var}(\delta_i)= \mathbf {c}_i'\boldsymbol {\Sigma }\mathbf {c}_i+\sigma_e^2. $$

(B.5)

Similarly, the parameters σ ₀₀ and $\sigma_{e}^{2}$ are not distinguishable in (B.5) due to not having a nested data structure.

As the counterpart of (16),

$$R_j^2=\frac{\hat {\boldsymbol {\gamma }}_j'\mathbf {S}_{uu}\hat {\boldsymbol {\gamma }}_j}{\hat {\boldsymbol {\gamma }}_j'\mathbf {S}_{uu}\hat {\boldsymbol {\gamma }}_j+\hat{\sigma}_j^2} $$

is the estimate of the percentage of variance of β _ij accounted for by the m moderators, where $\hat{\boldsymbol{\gamma}}_{j}=(\hat{\gamma}_{j1},\hat{\gamma}_{j2},\ldots, \hat{\gamma}_{jm})'$, j=1,2,…,p; and S _uu is the sample covariance matrix of u _i =(u _i1,u _i2,…,u _im )′, i=1,2,…,n.

Appendix C. R Package

This Appendix introduces an R package to perform NML estimation of the two-level regression model. The package can be downloaded at http://www3.nd.edu/~kyuan/moderation/NML.R. A simulated data set with 6 variables (y,x ₁,x ₂,u ₁,u ₂,u ₃) and 500 cases is used to illustrate the use of the package, and the data set can be downloaded at http://www3.nd.edu/~kyuan/moderation/simudata.dat. Both of these files are saved in the folder d:/moderation/ in this illustration with names NML.R and simudata.dat, respectively.

The code for running the package and its utilities are documented in Appendix D. The first three lines of the code are to change the working directory, to load the package into the R Console, and to read the data, respectively. Lines 4 to 10 of Appendix D are to identify the number of cases, the dependent variable (y), possible level-1 predictors (x1, x2), and possible level-2 predictors or moderators (u1, u2, u3). In this package, we regard the level-1 and level-2 intercepts as the regression coefficients corresponding to the predictor x _i0=1 and u _i0=1, respectively. These are fulfilled by lines 11 and 12, where the labels or column names are for identifying parameter estimates corresponding to intercepts. Lines 16 to 20 are to specify the level-1 and level-2 regression models. In the example, the level-1 model specified by L1=cbind(x0,x1,x2) has three coefficients, β _i0, β _i1, and β _i2, corresponding to the coefficients of x _i0=1, x _i1 and x _i2, respectively. Lines 17 to 19 are to specify the level-2 models corresponding to each of the level-1 coefficients. L20=cbind(u0,u1,u2) assigns three predictors to β _i0: u _i0=1, u _i1 and u _i2; L21=cbind(u0,u2,u3) assigns three predictors to β _i1: u _i0=1, u _i2 and u _i3; and L22=cbind(u0,u1,u2,u3) assigns four predictors to β _i2: u _i0=1, u _i1, u _i2 and u _i3. The 20th line in Appendix D puts all the level-2 predictors together to pass to the package. Notice that the specification of level-2 predictors must correspond to the level-1 predictors. For example, if you decide to leave out the intercept in level-1, then the specification for level-1 and level-2 predictors become

L1=cbind(x1,x2);#level-1 predictors;

L21=cbind(u0,u2,u3);#level-2 predictors for beta_i1;

L22=cbind(u0,u1,u2,u3);#level-2 predictors for beta_i2;

L2=list(L21,L22); #all level-2 predictors;

Lines 24 to 26 in Appendix D are to setup a H matrix corresponding to ${\bf h}_{i}$ in Equation (7) that contains the predictors for variance parameters in σ. Line 24 requests six variance parameters be estimated in the order of σ ₁₁=Var(ε _i1), σ ₂₂=Var(ε _i2), σ ₁₂=Cov(ε _i1,ε _i2), σ ₀₁=Cov(ε _i0,ε _i1), σ ₀₂=Cov(ε _i0,ε _i2), and $\sigma_{0e}^{2}=\mathrm {Var}(\varepsilon_{i0})+\mathrm {Var}(e)$. If one chooses not to let the prediction errors at level-2 covary, then line 24 needs to be set as H_mat=cbind(x1*x1, x2*x2, 1), which typically results in smaller SEs for the variance estimates. Lines 25 and 26 are to label the estimates for the variance estimates to be correctly identified in the output. For example, if not allowing the level-2 prediction errors to covary, then line 25 should be set as H_name=cbind("x1x1", "x2x2", "x0e"). In particular, the label x1x1 and x2x2 are used to identify the proper variance estimates to calculate R-squares in the package. We do not encourage users to change the notation used in labeling.

Line 28 is to run the package, and the output of running Appendix D is in Appendix E. In addition to the results of NML for the two-level model, the default output also contains the results of LS analysis for the corresponding MMR model, where xjuk corresponds to the level-1 coefficient of xj predicted by the level-2 predictor uk, e.g., x0u0 corresponds to the intercept γ ₀₀ or the level-1 coefficient of x0 predicted by the level-2 predictor u0. For the LS estimates, the default output contains SE_ls, SE_sw0, SE_sw3, SE_sw4, and the corresponding z-scores. The results corresponding to SE_sw0 are not in Appendix E because there is not enough horizontal space. We choose to output these four sets of SEs because SE_lss are the default for LS analysis; SE_sw0s are the most commonly used consistent SEs in software, SE_sw3s perform the best with smaller sample sizes according to Long and Ervin (2000), and SE_sw4s are most reliable when having high-leverage observations (Cribari-Neto 2004).

The labels for the results of NML are the same as those for the results of LS. The default output of NML also contains SE_sw0s and the corresponding z-scores, which are not included in Appendix E due to space limitation.

Appendix D. R Code for NML Estimation for the Model Described in Appendix B

Appendix E. The Output of Running the Example in Appendix C