# Identification of confounder in a logistic regression model example in “Applied Logistic Regression”

I am reading Hosmer's Applied Logistic Regression, and I am a bit stuck in chapter 3, when taking about interaction and confounders.

In page 77, it states the following:

Using the estimated coefficient for LWD in model 1 we estimate the odds ratio as $\exp(1.054)=2.87$. The results shown in Table 3.14 indicate that AGE is not a strong confounder, $\Delta(\hat{\beta})\%=4.2$, but it does interact with LWD, $p=0.076$.

I am OK with the $\exp(1.054)=2.87$, but I can't figure out how he calculate the $\Delta(\hat{\beta})\%=4.2$.

Can anyone help? Thanks.

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 I believe that value of 4.2 may be a misprint for -4.4, which is computed as -0.044/0.01: merely the conversion of the change in betas for AGE from model 1 to model 2 (from 0 to -0.044) to percent per year. – whuber♦ Jan 10 '12 at 15:22

This strategy for finding 'confounders' has been seriously challenged in the past few years. It is likely to result in confidence intervals for exposure effects that do not have the claimed properties. There is really no need to be parsimonious when adjusting for confounding.

The H&L strategy has also been shown to be arbitrary. For one thing it matters whether you apply the % change rule to the log odds ratio or to the odds ratio.

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Frank, I'm not sure I understand your statement "There is really no need to be parsimonious when adjusting for confounding" correctly. Do you mean that apart from the main predictor variables of interest, it's ok to include as many other variables in a regression model as seems necessary, as long as they're seen as only being there to adjust for confounding? Are confounders not included in the 10:1 or 15:1 rule of thumb for ratio of sample size to number of predictors? That doesn't seem right to me, so I have probably misinterpreted your statement. – mark999 May 1 '12 at 20:28
That's mainly correct. The 15:1 rule is for developing accurate predictions. The ratio of cases:predictors can be much less than that for adjustment. If using penalized maximum likelihood estimation (without penalizing the exposure effect) the ratio can be as low as 2:1. Going back to the original H&L strategy, the parsimony achieved is partially a mirage because of using Y to achieve that. – Frank Harrell May 2 '12 at 12:10
Thanks for your reply Frank. To be more specific, suppose OLS is to be used (as opposed to penalised MLE), and the goal is to get estimates and confidence intervals of effects, and p-values, for the predictor variables of interest. By "effects" I mean the regression coefficient if the predictor appears linearly in the model, or the change in dependent variable for some given change in predictor value, if the predictor appears non-linearly. In this situation can we ignore the 15:1 guideline and include lots of confounders in the model? – mark999 May 3 '12 at 20:55
If you are not penalizing the confounder effects, my guess is that you can get away with a 5:1 events:variables ratio. But we need more simulation studies. But the original proposal of moving variables in and out of the model doesn't help. – Frank Harrell May 4 '12 at 3:18
Thanks Frank, I appreciate your taking the time to answer these questions. When discussing an events:variables ratio, what consequences do you have in mind if that ratio is violated? My understanding is that having too many variables results in larger standard errors and hence wider confidence intervals, but the coefficient estimates are still unbiased, confidence intervals still have correct coverage, and p-values are still accurate. Are there other consequences of having too many variables? – mark999 May 4 '12 at 5:39
It is the percentage change in the coefficient for LWD when AGE is added to the model: $$\Delta\beta\%=(1.054-1.010)/1.054 \times 100\% = 4.2\%$$.