conditional effects interpretations

[Adrienne Wold]

Hello,
I know there are many threads on interpreting regressions when interaction terms are significant but main effects are not (I understand that if the main effect in insignificant this means there is no sig relationship between a variable x and y conditional on the third variable being 0, and a significant interaction term would indicate a significant relationship between x and y conditional on a non-zero value of the third variable. What I'm confused about though the predicted value of y when the third variable is 1 with a non-significant main effect of X. For example, with:

y= Ax + Bz + C(x*z)

For binary variable z, if z=1, is the total effect of x on y still A+C or just C?

[Clyde Schechter]

Just do the algebra. When you set z = 1 you have:

y = Ax + Bz + C(x*z) = Ax + B + C(x) = (A+C)*x + Bz

so the effect of x on y is A+C.

and a significant interaction term would indicate a significant relationship between x and y conditional on a non-zero value of the third variable

This is not correct. A "significant" interaction term indicates that the relationship between x and y conditional on a non-zero value of the third variable is significantly different from the relationship between x and y conditional on a zero value of the third variable. However, both of those relationships could still be non-"significant."

[Adrienne Wold]

Ok, thank you. How do you then assess the significance of the effect on X when z=1 (against a null that there is no effect) rather than whether this relationship is different from the effect on x when z=0?

[Carlo Lazzaro]

Adrienne:
you can use -lincom-, like in the following toy-example that compares domestic and foreign cars which received 3 maintenance procedures:

use "C:\Program Files\Stata17\ado\base\a\auto.dta"
quietly regress price i.foreign##i.rep78
mat list e(b)
<snip>
. lincom  ([_cons]+_b[0b.foreign]+(_b[3.rep78]+(_b[0b.foreign#3o.rep78])))-([_cons]+_b[1.foreign]+(_b[3.rep78]+(_b[1.foreign#3.rep78])))

 ( 1)  0b.foreign - 1.foreign + 0b.foreign#3o.rep78 - 1.foreign#3.rep78 = 0

------------------------------------------------------------------------------
       price | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
         (1) |   1778.407    1830.91     0.97   0.335    -1882.723    5439.538
------------------------------------------------------------------------------

.

The difference does not reach ststistical signifcance.

[Adrienne Wold]

Thank you. Conceptually, is this testing b1+b3 does not equal 0 using all combinations of the variable values included?

[Adrienne Wold]

And does this change if there are other controls in the model in additional to x and z?

[Carlo Lazzaro]

Adrienne:
1) conceptually is testing if the following equation (which, in turn, is the result of some simplifications)=0:

 
 0b.foreign - 1.foreign + 0b.foreign#3o.rep78 - 1.foreign#3.rep78 = 0

2) the procedure remains the same (maybe a tad more code demanding) if you have more predictors and/or interactions to test.