# How to convert the odds standard error to probability standard error?

I have conducted a logistic regression analysis using dummy variables in R. I would like to convert the outputted oddsratio standard error to probability (interpreted as risk) standard error. I am able to convert the outputted odds ratio to risk through:

1. ln(odds) = intercept + ln(odds ratio)
2. odds = exp(ln(odds))
3. risk = odds / (1+odds)

How should I convert the odds ratio standard error to risk standard error or confidence intervals?

Thanks!

Work on the log scale for as long as you can and then convert to the scale of odds at the last moment. So compute the confidence interval on the log scale and then convert the limits and the estimate to the odds scale as in your step 2. I suppose you could compute a standard error for the odds using the delta method but the way I suggest is simpler I believe.

You can do the following to calculate odds ratios with custom CI.

library(oddsratio)
fit.glm <- glm(admit ~ gre + gpa + rank, data = data.glm, family = "binomial")
calc.oddsratio.glm(data = data.glm, model = fit.glm,
incr = list(gre = 380, gpa = 5), CI = .70)

predictor oddsratio CI.low (15 %) CI.high (85 %)          increment
1       gre     2.364         1.540          3.647                380
2       gpa    55.712        10.084        314.933                  5
3     rank2     0.509         0.366          0.706 Indicator variable
4     rank3     0.262         0.183          0.374 Indicator variable
5     rank4     0.212         0.136          0.325 Indicator variable


And then use your formula to convert the odds ratio CIs to probability CIs.

• Thanks so much pat-s and mdewey! For the second part of my analysis, it turns out I need standard error of the ln(risk) (rather than confidence intervals). It's not clear to me how I can go from ln(std error of odds) to ln(std error of the risk). After finding the ln(risk) for each dummy variable, I will need to find the relative ln(risk) - i.e. dividing each of the ln(risk)s by the lowest ln(risk) among the dummy variables. What happens to the ln(std error) at that point? Thanks again for your help. – pau Feb 9 '17 at 2:59