Interaction terms in Cox model - Why does the result when I multiply variables myself?

Question

I have a data set with the following variables:

outcome: outcome (value: 0,1)

time

chemical1: exposure level of chemical 1 (value: 1,2,3)

chemical2: exposure level of chemical 2 (value: 1,2,3)

confounders: other confounding variables.

I hope to evaluate the interaction between chemical1 and chemical2 in cox regression:

coxph(Surv(time, outcome) ~ chemical1 * chemical2 + confounders, data = data)

The result is as following:

                                           coef exp(coef)  se(coef)      z Pr(>|z|) 
factor(chemical1)3:factor(chemical2)3  0.734954  2.085386  0.483811  1.519  0.12874  

                                         exp(coef) exp(-coef) lower .95 upper .95
factor(chemical1)3:factor(chemical2)3    2.0854     0.4795    0.8079     5.383

(Other rows omitted)

I created another variable interaction. This variable has value 0 when both chrmical1 and chemical2 are 1, value 1 when both are 3.

I plug this value into the regression:

coxph(Surv(time, outcome) ~ interaction + confounders, data = data)

The results:

                           coef exp(coef)  se(coef)      z Pr(>|z|)    
factor(interaction)1  1.551730  4.719627  0.194017  7.998 1.27e-15 ***

                         exp(coef) exp(-coef) lower .95 upper .95
factor(interaction)1    4.7196     0.2119    3.2267     6.903

Could somebody help explain why the values beween interaction and factor(chemical1)3:factor(chemical2)3 is different? Thank you!

Answer 1

The way you explain what you have done, the 2 models are substantially different.

The first model, with the chemical1 * chemical2 term, implies (along with effects of the confounders) 2 "main effects" for each of the chemicals, representing differences for each chemical of level 2 and level 3 from its level 1. It also produces interaction terms between the chemicals for each pair of their levels 2 and 3.

The second model only adds to the confounders a single interaction term, between the 2 chemicals each at level 3. It omits the "main effects" and the other interaction terms.

If you run 2 models with different sets of predictors, you should expect to get different results between them for any predictor that you include in both. This is an example of omitted-variable bias, which can happen even in ordinary least-squares regression and can be even more troublesome in Cox regressions.