Interaction between a continuous variable and a 2 category variable (comparison of slopes)

Here we examine the special case where we wish to test the interaction between a continuous variable (x) and a two category variable (group). This is equivalent to comparing the slopes of x between the two groups.

This can be formulaed as a t-statistic of the regression estimate for the x by group interaction (x by group product) term (fitted in a regression also containing x and group). We usually code the group as a dichotmous 0-1 variable. The t-statistic for the interaction's regression coefficient can be used to assess if the interaction is present.

Cribari-Neto (2004) recommends using a variant of the t-statistic which he calls the HC4-based quasi-t test which adjusts the standard error of the interaction regression coefficient. This may be computed using this [attachment:HC4.xls spreadsheet.] This spreadsheet also computes an extension of the HC4-based quasi-t recommended by Ng and Wilcox (2010) which is more robust to departures from assumptions underlying the computing of the variance of the regression estimates - specifically the assumption that the regression model is equally precise in both groups for all values of x (homoscedasticity). They call their estimate the HC4-based wild bootstrap quasi-t test which additionally accounts for sampling variability by sampling continually (with replacement) from the current sample to assess the variation in the t value (the 'bootstrapping' bit!) Details of how to compute this using the spreadsheet using complete cases are given [:FAQ/HChowto: here.]

Both the asymptotic HC4 estimator and its bootstrap version may also be computed using R 2.10 and later. For the former you will need to add-in the libraries 'foreign' and 'sandwich' which contain the macros you will need to evaluate the adjusted variance of the x by group interaction term if you have not installed these already. An example SPSS file containing the data used below is [attachment:modeg.sav given here.] Note: You will need to change the file location as necessary, to the one on your PC, in the read.spss() command example given below. The data for this example may be found in this [attachment:modHC4.sav SPSS file.]

install.packages("sandwich")
install.packages("foreign")

The following syntax produces tval (t-value for the group by x interaction), its degrees of freedom (df) and its 2-sided p-value (pout).

library(foreign)
library(sandwich)
dat <- read.spss("C:\\Documents and Settings\\peterw\\Desktop\\My Documents\\My Documents2\\BARNEY MODERATION + R HC4 CODE\\HC4 test data on WIKI.sav")
attach(dat)
dat <- na.omit(dat)
vc <- vcovHC(fm,type="HC4")
tval <- fm$coeff[4]/sqrt(vc[4,4])
df <- length(dat$A)-4
pout <- 2*pt(-abs(tval),df)

The usual asymptotic results may be outputted using

vc <- vcov(fm)
tval <- fm$coeff[4]/sqrt(vc[4,4])
df <- length(dat$A)-4
pout <- 2*pt(-abs(tval),df)

The HC4 bootstrap version may be computed by installing the reg2ci macro. Details with a worked example are in the appendix of Ng and Wilcox.

References

Cribari-Neto, F. (2004). Asymptotic inference under heteroscedasticity of unknown form. Computational Statistics and Data Analysis 45 215-233.

Ng, M. and Wilcox, R.R. (2010). Comparing the regression slopes of independent groups. British Journal of Mathematical and Statistical Psychology 63 319-340.