# Testing if coefficients are statistically significantly different across models

I will be building two zero-inflated negative binomial (ZINB) regression models, where each model is aiming to predict different disease count outcomes based on the exact same independent variables and data. For example:

$$Y_A =\beta_1^A X_1 + \beta_2^A X_2 + \beta_i^A X_i... + \beta_n^A X_n \tag*{Eq.(A)}$$ $$Y_B = \beta_1^B X_1 + \beta_2^B X_2 + \beta_i^B X_i... + \beta_n^B X_n \tag*{Eq.(B)}$$

What would be the appropriate method to test if the $$\beta$$ coefficients are statistically significantly across models when the data from the independent variables are the same across the models? Such that:

$$H_0: \beta_1^A ~\text{in Eq.(A)} = \beta_1^B ~\text{in Eq.(B)} \\ H_A: \beta_1^A ~\text{in Eq.(A)} \ne \beta_1^B ~\text{in Eq.(B)}$$

I have investigated the use of Seemingly-Unrelated Regression (SUR) methods, but I am unsure if there is relevant literature to warrant their use in the form of a ZINB model. Is there another more simple means of testing if these $$\beta$$ coefficients are statistically significantly different from each other?

So if you have something like:

Scenario 1:

Data =
$$\begin{bmatrix} y_{1} & b_{1} & b_{2} & \dots & b_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ y_{1_k} & b_{1_k} & b_{2_k} & \dots & b_{n_k} \end{bmatrix}$$

model 1: Y ~ B1 + B2 + . . . Bn

AND

Scenario 2:

Data =
$$\begin{bmatrix} y_{1} & b_{1} & b_{2} & \dots & b_{n} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ y_{1_j} & b_{1_j} & b_{2_j} & \dots & b_{n_j} \end{bmatrix}$$

model 1: Y ~ B1 + B2 + . . . Bn

Why not stack both data sets and add a dummy variable (G below) to denote each group? Where G = 'one' is group 1 and G = 'two' is group 2.

Combined scenario:

Data =
$$\begin{bmatrix} y_{1} & b_{1} & b_{2} & \dots & b_{n} & G\\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ y_{1_{k+j}} & b_{1_{k+j}} & b_{2_{k+j}} & \dots & b_{n_{k+j}} & G_{k+j} \end{bmatrix}$$

Now your model can test the importance of the group variable.

model 3: Y ~ B1 + B2 + . . . Bn + G + B1:G + B2:G + . . . Bn:G

You could also have interaction terms like B1:G in the model to test the importance of the interaction on that particular beta.

• You should probably expand on this answer, since it may not be clear to the OP what "stack both data sets" means and how the dummy predictor variable would be set up / work. I see what you're doing, but an actual example would help less-experienced people a great deal! – jbowman Feb 25 '20 at 19:34
• I have thought about this, but what this will get you is if each predictor is significant or not in each group. I am interested in quantifying the difference in coefficients when they are statistically significantly associated with the outcome in both groups. In other words, Group 1 might have a significant coefficient of 1.5 for beta 1 and group 2 might have a significant coefficient of 1.8 for beta 1. Thus, can I test if these two coefficients are significantly different? – coconn41 Feb 25 '20 at 20:02
• My thinking is that the 'combined' version will get at that. It sounds like you want to test for a group effect which is kind of what I was getting at. The simple version (with an intercept and 1 beta for each group and normal data) would be an ANCOVA analysis, right? – JebLuvsStats Feb 25 '20 at 20:17
• This is normally dist. data but I hope that the idea behind the model equation is useful: stats.stackexchange.com/a/280732/252122 – JebLuvsStats Feb 25 '20 at 20:29