Interpreting 0-1 dummy variables

Let us suppose we have a group with two levels and a standardized covariate, x. Then the prediction of outcome, y, may be written using regression estimates, B, as below.

$$ y = B_text{constant} + B_text{group} \mbox{I(group=1)} + B_text{cov} (\mbox{covariate value not equal to zero}) $$ $$ + B_text{int} (\mbox{group = 1 and cov not equal to zero}) $$

For predictions, P, of y using the above 0-1 coding it follows:

[P(increase of 1 sd from mean in x in group 1) - P(mean of x in group 1)] - [P(increase of 1 sd from mean in group 2) - P(mean of x in group 2)]

= $$(B_text{constant} + B_text{group} + B_text{cov} + B_text{int} - B_text{constant} - B_text{group}) - (B_text{constant} + B_text{cov} - B_text{constant})$$

= $$ B_text{int} $$

So the covariate by group interaction term measures the difference in covariate slopes between the two groups.

These dummy codings are extended for groups with K or more levels by using K-1 0-1 dummy codings representing linearly independent pairs of dummy codings that span the space (ie can be combined to give information on all possible pairwise group differences) as above to represent the difference in covariate slopes between the i-th group and reference group, K, i $$ne$$ K.

[P(increase of 1 sd from mean in x in group i) - P(mean of x in group i)] - [P(increase of 1 sd from mean in group K) - P(mean of x in group K)] (for an i $$ne$$ K)

= $$(B_text{constant} + B_text{group} + B_text{cov} + B_text{int(i)} - B_text{constant} - B_text{group}) - (B_text{constant} + B_text{cov} - B_text{constant})$$

= $$ B_text{int(i)} $$