0
$\begingroup$

$Y$ (scores among black students) $\sim X_1 + X_1^2 + X_2 + (X_1 * X_2) + (X_1^2 * X_2)$

X1           0.089626*** (e.g. same ethnic teacher)
X1^2         -0.008001***
X1*X2        0.003887*** (e.g. same ethnic teacher * principals' leadership
X1^2*X2      -0.000231***

In this case, How can I interpret (X1^2*X2)?

$\endgroup$
  • $\begingroup$ Without the coefficients this is unlikely to get a sensible answer. What is your scientific question which caused you to fit such a model? $\endgroup$ – mdewey Dec 24 '18 at 15:48
  • $\begingroup$ I put the coefficients and it would be two-level hierarchical linear modeling (students within schools). I would like to examine the effects of the size of black teachers on black students' score and how it varies by principals' multicultural leadership. And I assume that the relationship between the size and students' score may be non-linear. $\endgroup$ – Jay Dec 24 '18 at 16:11
  • 1
    $\begingroup$ What sort of values do X1 and X2 take? From your very brief post they sound like binary (0 or 1) variables, in which case I don't understand why you would square one of them. That question aside, in a test-score context it would be very rare to have a productive use for all 3 types of effects (main effect, interaction and squared term) involving the same variable. $\endgroup$ – rolando2 Dec 24 '18 at 19:59
  • $\begingroup$ Sorry for confusing. X1 is the percentage of black teachers and X2 is Likert scale. $\endgroup$ – Jay Dec 24 '18 at 21:24
1
$\begingroup$

This regression can be written more simply as:

$$Y \sim (X_1 + X_1^2)*X_2.$$

This model involves main effect terms plus interaction for the variable $X_2$ and a second-order polynomial in the first variable $X_1$. In such a model, the main effects and interactions are:

$$\begin{matrix} \text{Main effect of variable } X_1 & & & & X_1+X_1^2 \\[6pt] \text{Main effect of variable } X_2 & & & & X_2 \\[6pt] \text{Interaction effect of variables } X_1 \text{ and } X_2 & & & & (X_1+X_1^2):X_2 \\[6pt] \end{matrix}$$

The individual term $X_1^2:X_2$ is not really meaningful in itself, since it is an interaction with only one of the terms in the second-order polynomial for your variable $X_1$. When interpreting the variables you should keep all the parts of your polynomial variable together.

$\endgroup$
  • $\begingroup$ I don't follow. The original regression explicitly references five functionally independent variables and (perhaps, depending on what "$\sim$" is intended to mean), implicitly references a constant and therefore estimates either five or six independent parameters. Your model estimates only three or four (including the constant). In what sense is it equivalent, then? $\endgroup$ – whuber Dec 26 '18 at 20:30
  • $\begingroup$ The symbol $\sim$ is used in the same way the OP uses it in his question ---i.e., as part of R syntax for a regression equation. (The symbol $*$ is product interaction and the symbol $:$ is single interaction.) In this syntax, the regression $Y \sim (X_1 + X_1^2) * X_2$ is just a shorthand for the equivalent regression $Y \sim X_1 + X_1^2 + X_2 + X_1:X_2 + X_1^2:X_2$ (both of which implicitly include constants, since there is no $-1$ term in either expression). $\endgroup$ – Ben Dec 26 '18 at 23:05
  • $\begingroup$ The model matrix for the formula y ~ (x1 + x1^2)*x2 has four columns while the model matrix for the formula y ~ x1 + I(x1^2) + x2 + I(x1*x2) + I(x1^2*x2) corresponding to (perhaps one, but a natural one) interpretation of the OP's model has six columns. The models are not the same, so I'm struggling to see how your answer addresses the question. $\endgroup$ – whuber Dec 26 '18 at 23:35

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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