Calculating the Magnitude of Change in Hazard/Odds Ratios for Different Fractional Units of Change

Question

I have a Logistic and Cox PH model built. My covariates are Age Since Start, Incentive and Starting Balance. All covariates are numerical. The dependent variable is whether or not the account made a prepayment on their monthly bill -- A 1 for prepayment and 0 for no prepayment. After running the model and exponentiating the coefficients I have my odds ratios:

• Age: .991
• Incentive: 1.044
• Log(Starting Balance): .883

For my Cox model after exponentiating the coefficients I have hazard rates:

• Incentive: 1.09
• Log(Starting Balance): .86

The incentive covariate is referring to an account's incentive to refinance which is expressed as a percentage. So given my hazard results, which I interpret as a 1% increase in Incentive increase the chances of prepayment by 9%.

Recently I was asked to find the magnitude of change in the odds/hazard ratios given, say, a .25% increase in the Incentive or a 25% increase in Starting Balance. I've tried searching for ways to do this but only came up with one similar question. How do I go about calculating this magnitude of change?

Thanks

Answer 1

This is easy if you want to calculate a change in log-odds or log-hazard. For a logistic model, a regression coefficient is the estimated change in log-odds per unit change in the predictor. For a Cox model, it's the change in log hazard. You don't say enough about the units of your predictors to provide specifics here. For example: is Age in months or years? Is Incentive in percentage points, basis points, or something else? Is the log used for StartingBalance the natural log, base-10 log, or something else? As you know what units you modeled in, getting corresponding changes in log-odds or log-hazard for any change in predictor values is straightforward.

What's tricky is looking for changes in odds or hazard, not on the log scale. How much a change in log-odds or log-hazard means in terms of a change in odds or hazard depends on the original value of odds or hazard from which you want to calculate the difference.

To get a change in odds or hazard, you thus need to specify both a reference condition and a new condition. Those specified conditions should include values for all predictors in the model for both conditions, including the intercept. With those, you calculate the log-odds or log-hazard for each condition. Get the odds/hazard for each condition by exponentiating, and only then calculate the difference in odds or hazard between the conditions.