0
$\begingroup$

Suppose we are interested in English scores ($E_{ij}$) and Math scores ($M_{ij}$) in students in various classes. The scores are in different scales from each other. We perform two linear regressions. The regression with the English scores has $X$ as the covariates. The regression with the Math scores has $Z$ and $W$ as the covariates. Is there a way to combine the two regressions into one regression? Note that we use GEE linear regression with an exchangeable working variance covariance matrix.

So we have:

$$E(E_{ij}) = \beta_{0}+ \beta_{1}X_{ij}$$ $$E(M_{ij}) = \gamma_{0}+ \gamma_{1} Z_{ij} + \gamma_{2}W_{ij}$$

Could we somehow combine the covariates? For example, would it make sense to consider:

$$E(M_{ij}-E_{ij}) = (\gamma_{0}-\beta_{0})+(\gamma_{1}Z_{ij}-\beta_{1}X_{ij})+ \gamma_{2} W_{ij}$$

$\endgroup$
1
  • $\begingroup$ What is the story you are going to tell? If you wish to explain a greater skill in Math than in English then $M-E$ looks good, but if you are interested in overall skill, why not $M+E$? $\endgroup$
    – Sergio
    May 9, 2014 at 5:47

1 Answer 1

0
$\begingroup$

$E_{ij}=\beta_0+\beta_1X_{ij}+\epsilon_E$

$M_{ij}=\gamma_0+\gamma_1Z_{ij}+\gamma_2W_{ij}+\epsilon_M$

$M_{ij}-E_{ij}=\gamma_0+\gamma_1Z_{ij}+\gamma_2W_{ij}+\epsilon_M-\beta_0-\beta_1X_{ij}-\epsilon_E$

$(M_{ij}-\gamma_0-\gamma_1Z_{ij}-\gamma_2W_{ij})-(E_{ij}-\beta_0-\beta_1X_{ij})=\epsilon_M-\epsilon_E\sim N(0,\sigma_E^2+\sigma_M^2)$

$\sum(\epsilon_M-\epsilon_E)^2=\sum[(M_{ij}-\gamma_0-\gamma_1Z_{ij}-\gamma_2W_{ij})-(E_{ij}-\beta_0-\beta_1X_{ij})]^2=\sum(M_{ij}-\gamma_0-\gamma_1Z_{ij}-\gamma_2W_{ij})^2+\sum(E_{ij}-\beta_0-\beta_1X_{ij})^2-2\sum(M_{ij}-\gamma_0-\gamma_1Z_{ij}-\gamma_2W_{ij})(E_{ij}-\beta_0-\beta_1X_{ij})$

After the least square estimation we have

$\sum(\epsilon_M-\epsilon_E)^2=\sum(M_{ij}-\hat{M_{ij}})^2+\sum(E_{ij}-\hat{E_{ij}})^2-2\sum(M_{ij}-\hat{M_{ij}})(E_{ij}-\hat{E_{ij}})$

If and only if you have $\epsilon_M$ and $\epsilon_E$ to be independent, which you have $Cov(\epsilon_M,\epsilon_E)=0$, you have what you want.

Otherwise you've got new regression model for the combined result.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.