Hierarchical Regression

Hierarchical regression (HR) is one of several regression methods subsumed under multiple regression. HR is primarily focused on explaining how effects are manifested by examining variance accounted for in the dependent variable. The aim of HR is typically to determine whether an independent variable explains variance in a dependent variable beyond that already explained by some other independent variable(s).

It is typical that the additional amount of explained variance is evaluated for statistical significance based on change in R2 (ΔR2). R2 represents the amount of variance in a dependent variable that is explained by an optimal linear combination of independent variables. Thus, ΔR2 represents the change in variance explained in the dependent variable by including an additional independent variable.

For example, say a researcher is interested in studying influences on math achievement. Specifically, the researcher is interested in whether math self-concept explains variance in math achievement. [Page 779]But, the researcher knows, based on a literature review, that socioeconomic status (SES) and intelligence also explain variance in math achievement and math self-concept (and importantly, a possible relation between math self-concept and math achievement). Therefore, the researcher is interested in whether math self-concept explains variance in math achievement beyond that of SES and intelligence.

The researcher collects data on the three independent variables (SES, intelligence, and math self-concept), along with data on the dependent variable—math achievement scores. HR is used to analyze the data. The two so-called blocks of entry are used. In the first block, SES and intelligence are included, simultaneously, to explain variance in math achievement scores. In the second block, math self-concept is included to explain variance in math achievement, beyond that explained by the variables in the first block. Because SES and intelligence are already included in the regression, the math self-concept variable explains variance beyond that already explained by the combination of SES and intelligence.

In this example, if the ΔR2 is statistically significant, then math self-concept explains unique variance in math achievement. To determine practical significance, the researcher may interpret the ΔR2. A better estimate of the practical significance may be obtained by taking the square root of ΔR2. This estimate is called the semipartial correlation. Using the aforementioned example, the semipartial correlation represents the relation between math self-concept and math achievement, after removing the influences of SES and intelligence from math self-concept.

The example we used seems to be the most common of HR. HR has several other uses including estimating total, direct and indirect effects, providing a standardized measure of effect size in the form of proportion of variance explained or semipartial correlation, and performing moderator analyses. Although ΔR2 is often the focal point in HR, the regression coefficients can still be interpreted individually for their statistical significance, sign, and magnitude. One issue with interpreting the regression coefficients is how to interpret them in each block because they change when additional independent variables are included in subsequent blocks. Ultimately, the rationale for order of entry in HR should be based on theory and be logically defensible.

See also Multicollinearity; Multiple Linear Regression; Partial Correlations; Residuals; Simple Linear Regression; Stepwise

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