Journal of Drug Abuse Open Access

Keywords

Multiple correlation; Regression; Effect size; Hypothesis testing; Computer program; SAS

Suppose that a researcher is interested in predicting cocaine use frequency in adults from a number of putative risk factors such as sensation seeking tendencies, level of psychosocial stress, frequency of other illicit drug use, and number of negative life events. A commonly used statistical approach to modeling such data is linear multiple regression analysis, and when using regression it is standard practice to examine whether the squared multiple correlation coefficient, R² (i.e., the proportion of variance in the outcome variable accounted for by the predictors) is statistically significant. The intent of such a test is to determine whether R² differs significantly from zero, and the null hypothesis may be stated as H_o: ρ²=0, where ρ² represents the population value (parameter) for the squared multiple correlation coefficient. Although this test is widely used, it is misleading because the expected value of R² is not zero when ρ=0 (where ρ represents the population value for the multiple correlation coefficient). Rather, as Morrison [1] pointed out, the expected value, or expected long-run mean, of R² is equal to p/n-1, where p is the number of predictor variables and n is the sample size. The implication of this equation is that R² should be examined in relation to the expected value of R², E(R²), because the latter quantity is the value of R² that can be expected simply “by chance.” In light of this realization, it seems more appropriate for researchers to test the null hypothesis, H_o: ρ²=ρ_o ², where ρ_o ²=E(R²). Unfortunately, commonly used statistical software programs such as IBM SPSS, Minitab, and SAS do not implement this adjusted hypothesis test whereby R² is tested against the expected value of R².

Huberty [2] recognized the importance of testing R² against its expected value and he proposed an adjusted R² index that takes into account the value of E(R²). This adjusted R² index “corrects” the obtained squared multiple correlation coefficient by explicitly incorporating the expected value of R². Huberty [2] also presented an effect size measure for linear multiple regression studies that is calculated by subtracting E(R²) from Huberty’s adjusted R² index. Darlington [3] gave an F statistic for testing the null hypothesis that R² equals the expected value of R² (i.e., H_o: ρ²=ρ_o ²). To calculate the aforementioned statistical quantities, I wrote a user-friendly computer program called DARHUB (short for Darlington and Huberty statistics). DARHUB calculates the expected value of R², Darlington’s F statistic for testing the null hypothesis that ρ²=ρ_o ², the observed probability value for Darlington’s F, Huberty’s adjusted R² index, and Huberty’s effect size measure. The user specifies only the squared multiple correlation coefficient, R², from a linear regression analysis (i.e., the R² obtained by performing a regression analysis using SPSS, Minitab, SAS, etc.), the number of predictor variables, p, and the sample size, n. DARHUB is written in SAS data step language [4], and SAS is compatible with PC, Mac and Linux workstations. User documentation is included in the program listing. Table 1 contains sample input, the full program listing (the SAS data step code), and output from the analysis (Table 1). The DARHUB program and accompanying ReadMe file can be downloaded from the following academic website: https://psychology.cofc. edu/personal-pages/james-hittner,-phd.---software-page.php.

Output:

Table 1: The DARHUB SAS program with sample input and output.

References

1. Morrison DF (1990) Multivariate statistical methods. McGraw-Hill, New York.
2. Huberty CJ (1994) A note on interpreting an R2 value. Journal of Educational andBehavioral Statistics19: 351-356.
3. Darlington RB (1990) Regression and linear models. McGraw-Hill, New York.
4. SAS Institute (1990) SAS/STAT User’s Guide (Release 6.04). SAS, Cary, North Carolina.