I’d like to test for equality between two regression coefficients, one of which is an interaction term.

I’ve been referencing Andrew P. Wheelers statistics blog:

https://andrewpwheeler.com/2016/10/19/testing-the-equality-of-two-regression-coefficients/From what I understand, I can calculate the difference in the regression coefficients and the standard error of that difference using the variance-covariance matrix. Then, apply the SE to the difference estimate to see if it is greater than zero. If it is, then the coefficients are significantly different. However, I’m stuck on the equation used to calculate the standard error of the difference because one of the regression coefficients is an interaction term.

I’m interested in calculating the difference between the regression coefficients

dayandwolf:dayfrom the model output below.Time is a 3-level categorical variable (Day, Crepuscular and Night, Night is the reference level);

Wolf is a 2 level categorical variable (a=wolves absent, b= wolves present)

Is the below equation for the standard error of the difference between the regression coefficientsThe variance-covariance matrix is below the model output.dayandwolf:daycorrect?SE_Diff = sqrt(Variance(wolf:day) + Variance(day)) – 2*Covariance(wolf:day, day))

Do I also add in Variances and Covariances forwolfsince there is an interaction term?I’m running a zero-inflated Poisson generalized linear mixed model with the R package

glmmtmb.`> summary(cougar_temporal_3_cat_time) Family: poisson ( log ) Formula: CougarActivity ~ (1 | location_id) + wolf_presence * time + offset(log(day)) Zero inflation: ~1 Data: data AIC BIC logLik deviance df.resid 4295.9 4380.3 -2140.0 4279.9 280568 Random effects: Conditional model: Groups Name Variance Std.Dev. location_id (Intercept) 0.4079 0.6387 Number of obs: 280576, groups: location_id, 64 Conditional model: Estimate Std. Error z value Pr(>|z|) (Intercept) -3.72379 0.72801 -5.115 3.14e-07 *** wolf -0.05832 0.26123 -0.223 0.82333 timecrepus 0.59630 0.21126 2.823 0.00476 ** timeday -0.40524 0.21662 -1.871 0.06138 . wolf:timecrepus 0.19332 0.29788 0.649 0.51636 wolf:timeday 0.53635 0.29287 1.831 0.06705 . ##Variance-covariance matrix vcov(cougar_temporal_3_cat_time, full=FALSE) Conditional model: (Intercept) wolf_presenceb time_3_catcrepus time_3_catday wolf_presenceb:time_3_catcrepus (Intercept) 0.52999833 -0.03298061 -0.01836126 -0.01862785 0.01793521 wolf_presenceb -0.03298061 0.06824002 0.01881935 0.01882303 -0.03951061 time_3_catcrepus -0.01836126 0.01881935 0.04463025 0.01883163 -0.04463059 time_3_catday -0.01862785 0.01882303 0.01883163 0.04692288 -0.01883176 wolf_presenceb:time_3_catcrepus 0.01793521 -0.03951061 -0.04463059 -0.01883176 0.08873388 wolf_presenceb:time_3_catday 0.01723512 -0.03951691 -0.01883286 -0.04692339 0.03953483 wolf_presenceb:time_3_catday (Intercept) 0.01723512 wolf_presenceb -0.03951691 time_3_catcrepus -0.01883286 time_3_catday -0.04692339 wolf_presenceb:time_3_catcrepus 0.03953483 wolf_presenceb:time_3_catday 0.08577567 Zero-inflation model: zi~(Intercept) zi~(Intercept) 0.6410403 ````

**Answer**

**Is the below equation for the standard error of the difference between the regression coefficients day and wolf:day correct?**

You have correctly calculated determined the formula for the difference in the coefficients since the variance of a sum is equal to the sum of the variances plus two times the sum of the covariances:

Var(ˆβwolf,dat−ˆβday)=Var(ˆβwolf,day)+Var(−ˆβday)+2Cov(ˆβwolf,dat,−ˆβday)=Var(ˆβwolf,day)+Var(ˆβday)−2Cov(ˆβwolf,day,ˆβday)

You simply take the square root of the variance to find the standard error of the differences here.

There is no need to add in any variance terms for the wolf terms since your interest is only in the estimated coefficients that you describe. So your final standard error would be:

=√Var(ˆβwolf,dat)+Var(ˆβwolf,dat)−2Cov(ˆβwolf,dat,ˆβday)=√0.08577567+0.04692288−2(−0.04692339)=√0.08577567+0.04692288−2(−0.04692339)=√0.2265453=0.4759678

**Attribution***Source : Link , Question Author : Caam , Answer Author : StatsStudent*