Say we fit a regression model between the binary variables, $\mathbf{X} = (X_1, X_2, X_3) $ and a continuous response variable, $Y$ with a $$E(Y| \mathbf{X}) = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3X_3 + \beta_{2,3}X_2 X_3.$$

Does including the interaction term $\beta_{2,3}$ affect the power to reject $\beta_1$ even though $X_1$ is not included in the interaction? Does including the interaction affect the estimation of $\beta_1$ in any way?

Can we say anything about the behavior of $\hat{\beta_1}$ if we add more binary variables, $X_2, \dots, X_J$, and add interaction terms? Can this be extended to generalized linear models? See the formula below:

$$g\{ E(Y_i | \mathbf{X}) \} = \beta_0 + \beta_1 + \sum_{j=2}^{J}\beta_j X_j + \sum_{j,k \neq 1} \beta_{j,k} X_j X_k.$$

Edit: I'll be more precise about what I mean by power. Say you have a test with 80% power to reject $H_0: \beta_1 = 0$ when $\beta_1 = \beta_{1}^*$ and the true model is $E(Y|X) = \beta_0 + \beta_1 X_1$. By adding additional $X_j$ where $j\neq 1$, you are fitting a misspecified model. Adding additional interaction terms results in a "more" misspecified model. Is there any effect of adding multiple interactions that don't involve $X_1$, e.g. $\sum_{j,k \neq 1} \beta_{j,k} X_j X_k$

  • 1
    $\begingroup$ Because "power" has a specific meaning in this context, and that meaning does not seem to be intended, and various colloquial senses of "power" are plausible (ranging from magnitude to significance to sensitivity), could you please explain what you mean by "power"? $\endgroup$
    – whuber
    Oct 20 '21 at 22:26
  • $\begingroup$ No, I do intend to use the specific meaning of power. I've clarified my question. $\endgroup$
    – Eli
    Oct 21 '21 at 0:10
  • $\begingroup$ Thanks--it's clearer now in the edit. Note, though, that a phrase like "power of $\beta_1$" is unusual and ambiguous: a test has power; a parameter does not. $\endgroup$
    – whuber
    Oct 21 '21 at 15:25
  • $\begingroup$ Agreed that it's ambigious. I make this more explicit by adding "power to reject $\beta_1$. Between that addition and the previous edit, it should be clear. $\endgroup$
    – Eli
    Oct 21 '21 at 18:40

Yes, it affects the power in three ways.

First, adding $X_2X_3$ to the model changes the true value of $\beta_1$ unless $X_2X_3$ is uncorrelated with $X_1$. In some designed experiments it would be natural for these to be uncorrelated, but in other sorts of data there typically isn't a reason to expect them to be uncorrelated. The coefficient could change by a large or small amount in either direction; the power could go up or down. The power might even become not-very-well-defined if the new value of $\beta_1$ was 0.

Second, again if $X_2X_3$ is correlated with $X_1$, putting it in the model will affect the variance of $\hat\beta_1$ because the variance of $\hat\beta_1$ is inversely proportional to the variance of $X_1$ conditional on everything else in the model. This effect will tend to reduce the power; the variance conditional on $X_2X_3$ is smaller than the variance not conditional on it [*]

Third, adding $X_2X_3$ to the model will tend to reduce the residual variance (if its coefficient is not zero), and so reduce the standard error of $\hat\beta_1$ and increase the power.

[*] I'm being loose with language, 'conditional' here is about linear projections rather than true conditional expectations.

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    $\begingroup$ Fourth, while adding $X_2X_3$ reduces the residual variance it also reduces the degrees of freedom that are used to measure this residual variance. If $X_2X_3$ has nothing useful to add, then it will reduce the variance but at the cost of an extra degree of freedom and effectively power is reduced. $\endgroup$ Oct 20 '21 at 22:15
  • $\begingroup$ What about when Is $X_1$? is uncorrelated with $X_2 X_3$? Is there any impact on power. $\endgroup$
    – Eli
    Oct 21 '21 at 0:05
  • $\begingroup$ That's the third point. And the fourth in the comment. $\endgroup$ Oct 21 '21 at 1:43
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    $\begingroup$ It isn't evident that this answers the question. The problem is that including a new variable changes the very meaning of all the coefficients in the model. The symbol "$\beta_1$" means something different depending on which variables are included in the model. How, then, does it even make sense to talk about the "power" (presumably of a test of that coefficient against zero) changing?? $\endgroup$
    – whuber
    Oct 21 '21 at 15:27
  • $\begingroup$ Adding a new variable may change the meaning of coefficients, but does not change the meaning of accepting or rejecting the null hypothesis about $beta_1$, and that's what power is about here. $\endgroup$
    – TMBailey
    Oct 21 '21 at 20:14

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