How do I interpret the coefficients of the reference group for a linear regression with two dummy variables as regressors in R?

Question

I'm using R to fit the following linear regression models with the popular "mtcars" dataset:

data(mtcars)
fit1 <- lm(mpg ~ factor(am) + factor(cyl), mtcars)
fit2 <- lm(mpg ~ factor(am) * factor(cyl), mtcars)

When I take the summary of each model, I get:

For the model without interaction:

Call:
lm(formula = mpg ~ factor(am) + factor(cyl), data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-5.9618 -1.4971 -0.2057  1.8907  6.5382 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept)    24.802      1.323  18.752  < 2e-16 ***
factor(am)1     2.560      1.298   1.973 0.058457 .  
factor(cyl)6   -6.156      1.536  -4.009 0.000411 ***
factor(cyl)8  -10.068      1.452  -6.933 1.55e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.073 on 28 degrees of freedom
Multiple R-squared:  0.7651,    Adjusted R-squared:  0.7399 
F-statistic:  30.4 on 3 and 28 DF,  p-value: 5.959e-09

For the model with interaction:

Call:
lm(formula = mpg ~ factor(am) * factor(cyl), data = mtcars)

Residuals:
    Min      1Q  Median      3Q     Max 
-6.6750 -1.1000  0.1125  1.6875  5.8250 

Coefficients:
                         Estimate Std. Error t value Pr(>|t|)    
(Intercept)                22.900      1.751  13.081 6.06e-13 ***
factor(am)1                 5.175      2.053   2.521 0.018176 *  
factor(cyl)6               -3.775      2.316  -1.630 0.115155    
factor(cyl)8               -7.850      1.957  -4.011 0.000455 ***
factor(am)1:factor(cyl)6   -3.733      3.095  -1.206 0.238553    
factor(am)1:factor(cyl)8   -4.825      3.095  -1.559 0.131069    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 3.032 on 26 degrees of freedom
Multiple R-squared:  0.7877,    Adjusted R-squared:  0.7469 
F-statistic: 19.29 on 5 and 26 DF,  p-value: 5.179e-08

Using these models as examples, I would like to know:

1. How do you interpret the coefficient of the reference group in each case? (I know how to interpret these and the coefficients of each variable in the case when I fit mpg with just one dummy variable. But I can't find information on how to interpret these with two different dummy variables). Additionally, how do you interpret the other coefficients? As difference in the means with respect to what?

2. How do you interpret the coefficients in the interaction terms?

I'm not interested in performing a two-way ANOVA in this case, I would like to perform this with linear regression.

Thanks.

Answer 1

The intercept is the estimated value for when both factors take on their reference level, presumably in this case am==0 and cyl==4. The coefficient for am==1 has to be added to the intercept to get the estimated value when am==1. Similarly for cyl. If you have am==1 and cyl==6 then you add both coefficients.

In the case of the interaction everything proceeds in the same way except for the two cells where am==1 and cyl == 6 | 8. For those you also have to add in the coefficient for the corresponding interaction term.