1
$\begingroup$

Suppose I have an experiment where I have three groups $A, B , C$, such that groups $A$ and $B$ have specific (but different) characteristics and group $C$ is just a control group, with no special characteristics. We also have that $A, B$ and $C$ are disjoint groups.

Now, I want to find the effect of a treatment on group $A$ and on group $B$ (specifically, I want to find how the treatment affects a person, given that they have characteristic $A$ vs characteristic $B$).

If I specify the model as follows, I have multicollinearity:

$$Y_{it} = \alpha_0 + \beta_0 T_t + \beta_1 A_i + \beta_2 B_i + \beta_3 C_i + \gamma_1 (A_i\cdot T_t)+ \gamma_2 (B_i\cdot T_t)+ \gamma_3 (C_i\cdot T_t) + \epsilon_{it},$$

If I omit the $C_i$ dummy, I still have a multi collinearity if I include $C_i\cdot T_t$. How can I avoid such multicollinearity, and still specify a model that I can interpret meaningfully?

$\endgroup$

1 Answer 1

1
$\begingroup$

Since each case belongs to exactly one of group A, B, or C, if the case doesn't belong to either A or B then it must belong to group C. So there are only 2 independent predictors among the three of A, B, and C. Choose 2 for your dummy variables; then your intercept will represent the value for the third group. Similar considerations for the interaction terms.

In response to comment: Note that if the values of $T$ start at 0 then all of the interaction terms have values of 0 at time $T = 0$, as they are products of group-membership dummies with time. Say that you choose to set A and B as your dummies. With the way you formulate the model, the intercept will be the value for Group C at time 0; it includes no interactions. Coefficients for A and B will represent differences from the value of Group C at time 0. The coefficient for $T$ will be the slope in time for Group C. The interaction terms with $T$ for Groups A and B will represent the differences in slope from that of Group C.

As $T$ represents time, make sure that you are handling the time series and potential repeated measures on the same individuals appropriately. Those are not always trivial tasks.

$\endgroup$
2
  • $\begingroup$ I’m not sure I follow how to interpret the intercept if I exclude both $C_i$ and $C_i\cdot T_t$, since when $T_t$ is 1, the intercept will need to capture both $C_i$ and $C_i \cdot T_t$, but when $T_t$ is 0, it only captures $C_i$. And $t$ does represent time, yes, since the treatment is time dependent. $\endgroup$
    – jackson5
    Commented Mar 5, 2018 at 16:17
  • $\begingroup$ @jackson5 : I've added some elaboration to the answer to address your comment. $\endgroup$
    – EdM
    Commented Mar 5, 2018 at 18:21

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.