interpretation of interaction-term in linear regression, with and without main-effect

Question

In a case-control study including men and women of various ages, I wish to investigate if there is a difference in a measured variable (X) between cases and controls. The data are stored in a dataframe/tibble d as such:

# A tibble: 1,103 × 4
     CaCo Gender   Age        X
   <fctr> <fctr> <dbl>    <dbl>
1    Case  Woman    59 1.225700
2    Case  Woman    61 1.153512
3    Case  Woman    50 1.125951
4    Case  Woman    30 1.316410
5    Case    Man    28 1.248292
6    Case    Man    52 1.226141
7    Case  Woman    45 1.332503
8    Case    Man    31 1.272777
9    Case    Man    30 1.150000
10   Case  Woman    41 1.186069
# ... with 1,093 more rows

xtabs(~ CaCo + Gender, data = d)
         Gender
CaCo      Man Woman
  Control 401   271
  Case    256   175

The reference category for the CaCo-term is Control and for the Gender-term it is Man.

I use linear regression lm in R to apply model m1:

#-----
Call:
lm(formula = "X ~ CaCo + Age + Gender + CaCo:Gender", data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.5736 -0.1111 -0.0128  0.1007  1.1256 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           1.0924392  0.0276614  39.493  < 2e-16 ***
CaCoCase              0.0117859  0.0141087   0.835    0.404    
Age                  -0.0029474  0.0004465  -6.601 6.36e-11 ***
GenderWoman           0.0037238  0.0138262   0.269    0.788    
CaCoCase:GenderWoman  0.0325746  0.0220949   1.474    0.141    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1757 on 1098 degrees of freedom
Multiple R-squared:  0.05002,   Adjusted R-squared:  0.04655 
F-statistic: 14.45 on 4 and 1098 DF,  p-value: 1.662e-11
#----

It is my understanding that the coefficients should be interpreted as follows:

• The CaCoCase-term represents the differences between cases and controls, among males (male cases have 0.0117859 higher levels than male controls - not significant).

• The GenderWoman-term represents the difference between genders, among controls (female controls have 0.0037238 higher levels than male - not significant)

• The CaCoCase:GenderWoman-term represents how much greater the difference between cases and controls is among females than among males (i.e. female cases have 0.0117859 + 0.0325746 higher levels than male controls).

I hope I am right so far...?

Now, because I don't believe there is an effect of gender on X, but I suspect that the difference between cases and controls is mainly observed among women, I drop the main Gender-term and keep only the interaction CaCo:Gender, to get model m2:

#-----
Call:
lm(formula = "X ~ CaCo + Age + CaCo:Gender", data = d)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.5736 -0.1111 -0.0128  0.1007  1.1256 

Coefficients:
                          Estimate Std. Error t value Pr(>|t|)    
(Intercept)              1.0924392  0.0276614  39.493  < 2e-16 ***
CaCoCase                 0.0117859  0.0141087   0.835   0.4037    
Age                     -0.0029474  0.0004465  -6.601 6.36e-11 ***
CaCoControl:GenderWoman  0.0037238  0.0138262   0.269   0.7877    
CaCoCase:GenderWoman     0.0362984  0.0172355   2.106   0.0354 *  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1757 on 1098 degrees of freedom
Multiple R-squared:  0.05002,   Adjusted R-squared:  0.04655 
F-statistic: 14.45 on 4 and 1098 DF,  p-value: 1.662e-11
#-----

The model statistics are identical (as far as I can tell) between m1 and m2. It appears that the GenderWoman-term from m1 (the main effect of gender) have become CaCoControl:GenderWoman in m2, but I am assuming the interpretation is the same.

The only other difference between the models is the interaction term CaCoCase:GenderWoman, where the coefficient is slightly larger with a smaller error and consequently a much lower p-value.

The effect of the interaction term appears identical between m1 and m2 when illustrated using the effects-package:

library(effects)
plot(effect("CaCo:Gender", m1))

(As a side note, when modelling men and women separately by X ~ CaCo + Age, it appears clear that there is a difference between cases and controls among women, but not among men)

My questions are:

Are the interpretations of the coefficients the same between the models m1and m2? If so, what are the reasons they differ? If not, how should the coefficients be interpreted?

Any help is much appreciated!

Answer 1

I had forgot about this… If anyone out there is interested, here is a long explanation:

Important to remember that the interpretation of the main effects depend on which categories were set as reference during modelling (in this case men and controls).

The models are indeed identical! In m1, the interaction-term (CaCoCase:GenderWoman) represents both the difference in CaCo-effect among men and women, as well as the difference in gender-effect among controls and cases.

In m1, to calculate the gender-effect (difference between men and women) among cases, one must sum up the gender-effect among controls (GenderWoman) and the difference in gender-effect among controls and cases (i.e. the interaction term CaCoCase:GenderWoman). Thus, among cases, women have 0.0037238 + 0.0325746 = 0.0362984 higher levels than men.

Similarly, in m1, to calculate the CaCo-effect (difference between controls and cases) among women, one must sum up the CaCo-effect among men (CaCoCase) and the difference in CaCo-effect among men and women (i.e. the interaction term CaCoCase:GenderWoman). Thus, among women, cases have 0.0117859 + 0.0325746 = 0.0443605 higher levels than controls.

In both cases, it is also possible to calculate the standard error of these sums: https://stats.stackexchange.com/a/3657.

Importantly, in m2, above, I attempted to “drop” the Gender-term. The fact is that because it is included in the interaction, it will also be included as a main effect by the model (although not explicitly printed). I.e. the formula for m2 could just as well have been written as formula = "X ~ CaCo + Age + Gender + CaCo:Gender".

In m2 the size of the interaction-term is not explicitly returned by the summary function. Instead, the gender-effect is printed separately for controls (CaCoControl:GenderWoman = 0.0037238, identical to the GenderWoman-term from m1) and for cases (CaCoCase:GenderWoman = 0.0362984, same as in m1 after summation).

By rearranging the order of the model-terms specified in the formula argument of the lm function, one can refocus the returned summary of the model, as such (m3):

Call:
lm(formula = "X ~ Gender + Age + Gender:CaCo", data = d2)

Residuals:
    Min      1Q  Median      3Q     Max 
-0.5736 -0.1111 -0.0128  0.1007  1.1256 

Coefficients:
                       Estimate Std. Error t value Pr(>|t|)    
(Intercept)           1.0924392  0.0276614  39.493  < 2e-16 ***
GenderWoman           0.0037238  0.0138262   0.269  0.78772    
Age                  -0.0029474  0.0004465  -6.601 6.36e-11 ***
GenderMan:CaCoCase    0.0117859  0.0141087   0.835  0.40370    
GenderWoman:CaCoCase  0.0443605  0.0171180   2.591  0.00968 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.1757 on 1098 degrees of freedom
Multiple R-squared:  0.05002,   Adjusted R-squared:  0.04655 
F-statistic: 14.45 on 4 and 1098 DF,  p-value: 1.662e-11

In m3 the size of the interaction-term is not explicitly printed (same as for m2). Instead, the CaCo-effect is printed separately for men (GenderMan:CaCoCase = 0.0117859, identical to the CaCoCase term from m1) and for women (GenderWoman:CaCoCase = 0.0443605, same as in m1 after summation). (((This was what I was originally interested in!)))

All three models are identical, but the summary of the models are returned in slightly different ways. Notice that all the conclusions from m2 and m3 were readily available already in m1 by summing main-effects and interaction-effects appropriately.