Regression approach to ANOVA

#1
I am trying to fit regression for some categorical variables. I have two treatment A(3 levels) and B (2 levels) and their interaction.

The output are 6 variables (plus intercept) and their estimate. How to interpretate these estimate?
A1=-9, A2=-3, B1=-3, A1B1=5, A2B1=1


IF the interaction not included in the model, Is the interpretation same?
 

JohnM

TS Contributor
#2
ANOVA is merely a special case of regression, specifically for categorical independent variables with a continuous dependent variable. However, ANOVA and regression will produce the same results.

The estimates from ANOVA are interpreted the same way as in regression, i.e., how much will the dependent variable change with a unit change in the independent variable.
 
#3
Thank a lot

Thanks a lot JohnM. I think I am not clear enough on my question, what I am really want to know is how the indepdendent variable related to each other.

if no interaction, when A1=-9, I think it probably means A1 smaller than A3, but I am not sure about interaction involved. is that mean A1 smaller than A3 overall, or A1 smaller than A3 under B1 or some other interpretation.

THank you.
 

JohnM

TS Contributor
#4
This gets kind of tricky, and a lot of books that cover multiple regression conveniently avoid discussion of interpretation of interaction coefficients. The coefficients in front of the main effects have a "slope" interpretation, and they will change if you leave out or include interaction terms, much in the same way that effects will change in ANOVA if you do the same thing....
 

Dr.D

New Member
#5
HI lovedieer

This is how you interpret the estimates. Now you have 2 treatments: A and B.
A has three levels:A1, A2, and A3. You enter only A1 and A2 in the regression, and left out A3 (which is good protocol for regression to avoid multicollinearity). Now A3 is the referene category, meaning the estimates of A1 and A2 is compared to this category. Lets use a real-life exanple, say A1 means counselling treatment, A2 = yoga treatment, and A3 = no treatment. Say the dependent variable is levels of stress.

If A1 and A2 have estimates which are negative (like the one you found): A1 = -9, and A2 = -3. It means that those in counselling (A1) and yoga (A2) treatment groups have lower levels of stress, compared with A3 (those in the non-treatment group), or those in A3 have higher levels of stress compared with those in the other two groups - of course, the effects must be statistically significant (so one must not look at only estimates but alsp p-values).

In terms of interaction, you can conduct a two-way ANOVA between A1 and B1 (dummys), etc. Request a plot which would have the two categories on the bottom axis (for example, A1 and not A1), and two separate lines indicating B1 and not B1. Look at the relationship between one variable (the independent variable you have) for each category of the moderator (the other variable). You interpret the an interaction effect as such where the assumption is that the relationship between A1 and dependent variable (for example) is different for each level of B (under B1 - it may be positive; under not B1, it may be negative or depict no relationship). Remember the interaction term must be significant in the regression, if not..there is no sense to conduct one.
 
#6
Thanks for your detail interpretation.
I have one more question:
if there's no interaction. A3 and B2 are set as 0, did the parameter estimate for A1=-9 mean A1 has smaller stress than A3 at B2 level or it mean A1 has smaller stress than A3 across two B levels?
 

Dr.D

New Member
#7
If there is no interaction. The other effects are known as main effects: e.g.
A1 = -9. These effects are interpreted independent of each other. So A1= -9 just mean AI has lower levels of stress, compared with A3 (nothing to do with B). You just compare to the corresponding reference category.