3
$\begingroup$

Suppose we have the following Poisson regression model:

$\log(y) = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2$

Where, for example, $y$ is the number of children in a family, $x_1$ is the years that the mother spent in formal education, and $x_2$ is the years that the father spent in formal education.

Suppose we estimate

$\hat{\beta}_0 = +0.702 $
$\hat{\beta}_1 = -0.001 $
$\hat{\beta}_2 = -0.002 $
$\hat{\beta}_3 = -0.002 $

So ignorant parents get 2 children: $ \log(y) = 0.702 \Rightarrow y = e^{0.702} \approx 2$

Parents with 18 years of formal education each gets 1 children: $\log(y) = 0.702 − (0.001 × 18) − (0.002 × 18) − (0.002 × 18 × 18) = 0 \Rightarrow y = \exp(0) = 1$

and increasing formal education decreases the expected number of children, because the coefficients are negative.

Now I want to find out the effect of an additional year of father formal education on the number of children.

We have proven that for an additional year of father formal education, the number of children gets multiplied by $e^{\beta_2} (e^{\beta_3})^{\bar{x}_1}$.

So, starting from a situation where both of them have 10 years of formal education, an additional year of father's formal education multiply the number of children by: $ e^{−0.002} (e^{−0.002})^{10} = 0.978 $.

My questions are:

  1. What is the effect of 1 additional year of father education if the estimated $\beta_2$ is not statistically significant?

    In this case, we have no evidence to say that the coefficient is different than 0 in the population.

    So to find out the effect in the population, I should use 0, not the estimated $-0.002$ in the formula above, right?

    So the effect becomes $ e^{0} (e^{−0.002})^{10} = 0.98 $.

    Right?

  2. What if it's $\beta_3$ not being statistically significant?

$\endgroup$
7
  • $\begingroup$ What is exactly your question ? $\endgroup$
    – CaroZ
    Commented Mar 4 at 19:20
  • $\begingroup$ @CaroZ how to compute the effect of one additional unit of a covariate, when there is an interaction term between that covariate and another covariate, and the estimated coefficient of the interaction term is statistically significant but the estimated coefficient on the other covariate is not. $\endgroup$ Commented Mar 4 at 19:53
  • 1
    $\begingroup$ oh when there's an interaction, we can only compute the effect of one additional unit conditional on the value of the other covariate forming the interaction. So in your example, a 1 unit increase in $x_1$ results in a $-0.001$ unit change in the response when $x_2=0$, and it results in a $-0.003$ unit change if $x_2=1$. This is independent of the significance of anything in your model. $\endgroup$ Commented Mar 4 at 20:02
  • 1
    $\begingroup$ Ask yourself: when you are making predictions from a linear model, at what point in the process do you look at p-values associated with the coefficients? At no point, right? You just take the X variable and multiply it by the regression coefficients. These are independent things. Seeing a big p-value might prompt you to create a new model which has fewer variables in it, at which point it would behave like you expect it to. (Whether or not it's a good idea to drop non-significant variables from a model is another question). $\endgroup$ Commented Mar 4 at 20:12
  • 1
    $\begingroup$ @JohnMadden uhm I understand what you are saying but I'm still not persuaded $\endgroup$ Commented Mar 4 at 20:50

2 Answers 2

4
$\begingroup$

You shouldn't put any credence in the p-values for the individual regression coefficients of predictors involved in interactions. Centering one such predictor will change the value (and thus the apparent "significance") of the individual coefficient for any predictor with which it is interacting. This page has a worked-through example in linear regression, but the same phenomenon will occur in any regression model with interactions.

For example: if you had centered the $x_2$ years of education values for the fathers, the value of the $\hat \beta_1$ coefficient for the $x_1$ mothers' education, and thus its apparent "significance" (whether it's different from 0), would be altered. Yet there's nothing fundamentally different in the model. You centered the values of one predictor before you fit the model, yet changed the "significance" of the coefficient of its interacting predictor.

For that reason alone you shouldn't omit any individual coefficients for predictors involved in interactions, even if their values aren't significantly different from 0.

Beyond that, John Madden is correct in general. If you are going to make a prediction from a predictive model, you need to include all the predictors whether they pass any arbitrary "statistical significance" threshold. The values for the coefficients of the "significant" predictors are predicated on having those "insignificant" predictors also in the model. Furthermore, there's a risk of omitted-variable bias when you omit outcome-associated predictors from a model, even if they don't appear to be "significant" based on a small sample.

I'd recommend that you look at Chapter 4 of Frank Harrell's Regression Modeling Strategies for an outline of how to approach this type of modeling.

$\endgroup$
2
$\begingroup$

Think about it in this way : there is an interaction between 2 variables, which means that the effect of variable 1 differs according to the value of variable 2. Therefore the effect of variable 1 alone is irrelevant, even if significant.

$\endgroup$

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.