Question

Variable Description id unique identifier perio 1 = periodontitis, 0 = no periodontitis smoke 1 = current or ex-smoker, 0 = non-smoker oralhyg 1 = daily or more frequent brushing of teeth, 0 = less than daily brushing sepo 1 = high SEP in childhood, 0 = low SEP in childhood sep1 1= high SEP in adult life, 0 = low SEP in adult life

Fit a logistic regression model with perio as the outcome and exposures sep0, sep1, smoke and oralhyg.

The result is following:

Provide an interpretation of the estimated parameters smoke and oralhyg - what are the implications for someone who smokes but who also has good oral hygiene

habits?

I know this question is a bit tricky, so if you get some idea, I will regard it as helpful!

Answer 1

For the interpretation, we would need to only look at the table Odds Ratio column, and the smoke and oralhyg rows. Notice that we can write a multiple logistic regression as estimating $\hat{p}$ as $\ln{\frac{\hat{p}}{1-\hat{p}}}=\beta_0+\sum\beta_ix_i$ and thus my odds ratio comes out to be $OR=e^{\beta_0+\sum\beta_ix_i}$ . Now this is when all binary variables take value 0. Since we want to see the implications for a smoking person with good oral hygiene, i.e. smoke=1 and oralhyg=1, we get that $OR_{req}=e^{\beta_0+\sum\beta_ix_i}\times e^{\beta_1}\times e^{\beta_2}=OR \times e^{\beta_1}\times e^{\beta_2}$ . Now, we have those specific values from the tables, and thus finally we get:

$\frac{OR_{req}}{OR}=5.079021\times 0.2668406 = 1.35528901105$

Thus we can infer that people who smoke and have good oral hygiene has approximately 1.355 times more chance of getting periodontitis than people who neither smokes nor has good oral hygiene. The confidence intervals are not too wide, and hence there is a high chance that this inference is good enough for a large population.