Adding coefficients to obtain interaction effects - what to do with SEs?

Question

I have a multivariate regression, which includes interactions. For example, to get the estimate of the treatment effect for the poorest quintile I need to add the coefficients from the treatment regressor to the coefficient from the interaction variable (which interacts treatment and quintile 1). When adding two coefficients from a regression, how does one obtain standard errors? Is it possible to add the standard errors from the two coefficients? What about the t-stats? Is it possible to add these as well? I'm guessing not but I can't find any guidance on this.

Thanks so much in advance for your help!

Answer 1

I think this the expression for $SE_{b_{new}}$:

$$\sqrt{SE_1^2 + SE_2^2+2Cov(b_1,b_2)}$$

You can work with this new standard error to find your new test statistic for testing $H_o: \beta=0$

Answer 2

I assume you mean 'multivariable' regression, not 'multivariate'. 'Multivariate' refers to having multiple dependent variables.

It is not considered to be acceptable statistical practice to take a continuous predictor and to chop it up into intervals. This will result in residual confounding and will make interactions misleadingly significant as some interactions can just reflect lack of fit (here, underfitting) of some of the main effects. There is a lot of unexplained variation within the outer quintiles. Plus, it is actually impossible to precisely interpret the "quintile effects."

For comparisons of interest, it is easiest to envision them as differences in predicted values. Here is an example using the R rms package.

require(rms)
f <- ols(y ~ x1 + rcs(x2,3)*treat)  # or lrm, cph, psm, Rq, Gls, Glm, ...
# This model allows nonlinearity in x2 and interaction between x2 and treat.
# x2 is modeled as two separate restricted cubic spline functions with 3
# knots or join points in common (one function for the reference treatment
# and one function for the difference in curves between the 2 treatments)
contrast(f, list(treat='B', x2=c(.2, .4)),
            list(treat='A', x2=c(.2, .4)))
# Provides a comparison of treatments at 2 values of x2
anova(f) # provides 2 d.f. interaction test and test of whether treatment
# is effective at ANY value of x2 (combined treat main effect + treat x x2
# interaction - this has 3 d.f. here)

Answer 3

To be more general, if you create a (row) vector for the estimate that you care about $R$ such that your estimator is equal to $R\beta$, then the variance of that estimator is $R\hat{V}R^\prime$, where $\hat{V}$ is the estimated variance-covariance matrix of your regression. Your estimate is distributed Normal or t, depending upon the assumption that you are making (Law of Large Numbers v. assuming normal errors in your regression model). Alternatively, you can test multiple estimates if you let $R$ be a matrix. This is known as a Wald test. The distribution in this case is a $\chi^2_r$, where $r$ is the number of rows in your matrix (assuming that the rows are linearly independent).