What is the meaning of the beta-coefficient for an interaction term in a crossover study?

Question

I have asked a similar question here: stackoverflow I am puzzled by the interpretation for an interaction term. In my data my Y is an interval variable with the health outcome of an experiment. I have used an interaction term in which I have interacted the condition with the predisposition of the subject considering health status at base level. They are both categorical variables (factor variables in R).

Now it gets complicated because the Condition was two treatments: in treatment A subjects got the placebo first and then the real medicine whereas in treatment B they go the real medicine first and the placebo second. All it changes is the order.

Health outcome = a + Condition * Health.Base

I have the worst state of health at the base level as my reference category I find that interaction with the Condition is statistically significant but I am not sure how to interpret this.

I use the lm() function of R (although my design looks more like an ANOVA) and in the output I get the b coefficient in an output that looks like this:

ConditionB:Health.Base.So.and.So         (Beta and p-value)
ConditionB:Health.Base.Excellent         (Beta and p-value)

A statistically significant interaction term for those in Excellent health at baseline would mean that they are affected by the Condition B more than Condition A compared to the reference category people (Poor health at baseline). Is this right? What does the beta-coefficient represent?

If I would like to examine for each Health category at the base line separately without comparing to a reference category I would have to code each category as a dummy variable. However, in this case I would compare whether membership to a specific health status at the base line significantly changes between conditions compared to those who belong to the other health statuses at the base line. Is this right? Again, what does the beta-coefficient represent?

Would it be right to assume that the choice of the interaction between the Condition and the dummy variables is easier to interpret?

--- EDIT --- R output:

Call:
lm(formula = HealthOutcome ~ Condition * HealthStatus, 
    data = datA)

Residuals:
    Min      1Q  Median      3Q     Max 
-1.5957 -0.5942 -0.2640  0.4423  2.4423 

Coefficients:
                                       Estimate Std. Error t value     Pr(>|t|)    
(Intercept)                            2.595652   0.053044  48.934  < 2e-16 ***
ConditionCondB                        -0.001449   0.077071  -0.019    0.985    
HealthStatusSo.and.So                 -0.331693   0.078094  -4.247  2.35e-05 ***
HealthStatusExcellent                 -0.836724   0.092692  -9.027  < 2e-16 ***
ConditionCondB:HealthStatusSo.and.So   0.137490   0.110612   1.243    0.214    
ConditionCondB:HealthStatusExcellent  -0.199787   0.133943  -1.492    0.136    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.8045 on 1059 degrees of freedom
  (68 observations deleted due to missingness)
Multiple R-squared:   0.16, Adjusted R-squared:  0.156 
F-statistic: 40.33 on 5 and 1059 DF,  p-value: < 2.2e-16

Answer 1

You mention that you could convert to dummy variables, but that is what R does for you automatically. The default is that the baseline value is 0 for all dummy variables and the other factor levels each have their own dummy variable that is 1 for that level and 0 otherwise.

Consider a simple case where we have 2 factors as predictors, factor A has levels No, and Yes with No as the baseline. Factor B has 3 levels, lo, med, hi with lo as the baseline. So that means that a full model would be something like:

$$ y = \beta_0 + \beta_1 A{\rm Yes} + \beta_2 B{\rm med} + \beta_3 B{\rm hi} + \beta_4 A{\rm Yes} B{\rm med} + \beta_5 A{\rm Yes} B{\rm hi} + \varepsilon$$

Where AYes is 0 if A is No and 1 if it is Yes and Bmed is 1 for B med and 0 otherwise and Bhi is 1 for B having value hi and 0 otherwise. Now just plug each of the 6 possible combinations of A and B into the equation and you can start to see the effects of each $\beta$. If A is No and B is lo then all the variables are 0 and only $\beta_0$ remains, so $\beta_0$ is the mean for the No/lo combination. Now if A is Yes and B is lo then we have $\beta_0 + \beta_1$ so $\beta_1$ is the difference between Yes and No when B is lo. Similarly $\beta_2$ is the difference between med and lo when A is NO and $\beta_3$ is the difference between B hi and baseline.

Now the interactions: If we only had an additive model, no interactions, then the mean for the combination A Yes B med would be the baseline plus the effect of A Yes plus the effect of B med which would be $\beta_0 + \beta_1 + \beta_2$, but with the interaction that mean is $\beta_0 +\beta_1+\beta_2+\beta_4$, so the $\beta_4$ term is the difference between the actual cell mean and the mean that would be predicted from an additive model. If $\beta_4=0$ then that means additive works fine, if $\beta_4<0$ then the mean is less than what the additive model predicts and if $\beta_4 > 0$ then we have a higher mean than the additive model predicts. Similar or $\beta_5$.