# Beta coefficients from stratified analysis when there are covariates?

Suppose I have a regression model shown below

Model 1: $$Y = \beta_0^\ + \beta_1SEX\ + \beta_2ALCOHOL\ + \beta_3SEX*ALCOHOL\$$

The predictors I am interested in are SEX (binary: 0 female, 1 male) and Alcohol consumption (binary: drinker, non-drinker). Suppose that I found a significant interaction between SEX and ALCOHOL and decided to stratify the data by sex. So I would have two new models:

Model 2a: $$\text{Female: } Y_F = \beta_0^\ + \beta_2ALCOHOL\$$

So for the female subset, the intercept is still $\beta_0$ and the slope for ALCOHOL is $\beta_2$

Model 2b: $$\text{Male: } Y_M = (\beta_0^\ + \beta_1) + (\beta_2\ +\beta_3) ALCOHOL$$

For the male subset, the intercept is now $\beta_0^\ + \beta_1$ and the slope for ALCOHOL is $\beta_2^\ + \beta_3$

This is pretty straightforward. If you fit a model like this in any statistical package, you would get this kind of result. However, if say, in the model, I actually included an additional variable AGE, which is a covariate (assuming that it does not interact with either SEX or ALCOHOL), the original model would be the one below:

Model 3: $$Y = \beta_0^\ + \beta_1SEX\ + \beta_2ALCOHOL\ + \beta_3SEX*ALCOHOL\ + \beta_4AGE\$$

Further suppose that we still have a significant interaction between SEX and ALCOHOL and I would like to stratify the data again. I would get two models below if I followed the logic above:

Model 4a: $$\text{Female: } Y_F = \beta_0^\ + \beta_2ALCOHOL\ + \beta_4AGE\$$

Model 4b: $$\text{Male: } Y_M = (\beta_0^\ + \beta_1) + (\beta_2\ +\beta_3)ALCOHOL + \beta_4AGE\$$

However, the actual beta coefficients obtained using a computer program can be very different from what you obtain using the updated equations above. The difference in $\beta_0$ makes sense, because the stratified models in 4a and 4b still assume the pooled mean for age; namely, it estimates $\bar{Y}$ at the mean of all subjects' age, whereas in the stratified analyses done by a computer program, the intercepts of the models estimate $\bar{Y}$ at the mean age of a sub-group.

However, I wonder why the slopes are different. In other words, why are the slopes in Models 4a and 4b different from those produced by a statistical package.

• I'm not sure this is what the OP is asking. I think s/he wants to know why $\beta_4$, $\beta_2$ and $\beta_3$ are different in models 4a, 4b and 3. – Peter Ellis Dec 16 '12 at 19:50
When you fit models 4a and 4b you are allowing the estimates of $\beta_4$ to be different for the two different sex groups. In effect, you are not fitting model 3 in two steps, but a variant on model 3 that has an interaction between sex and age. So there is no real reason the coefficients in 4a and 4b should match those in 3.
Once $\beta_4$ is different, the other estimates are bound to be different in 4a and 4b from their estimates in 3, unless age is perfectly orthogonal to sex and alcohol (which is vanishingly unlikely).