Bootstrapping without raw data

[Tom Yates]

Hopefully a simple question, but I can't seem to find an answer online.

I have estimates of two proportions, each with a confidence interval. I am interested in obtaining the product of these proportions, with an appropriate confidence interval. I believe taking multiple random draws from each of the two proportion distributions could produce a bootstrapped confidence interval for the product.

The STATA instructions for the bootstrap command seem to start with a dataset - see here for example using ratio of two means. I am starting with summary statistics rather than data.

Is it possible to obtain bootstrapped confidence intervals from summary statistics? Alternatively, I suppose I could simulate data.

If anyone has example code for either of those options, I would be keen to see it!

Thanks,
Tom

[Daniel Feenberg]

The proportions would each be gaussian, and the formula for the distribution of the product of gaussians is known, but wouldn't you need to have the covariance of the two proportions to calculate the variance of the product? That is what the original data would provide to the bootstrap. Maybe this will give you some ideas.

[Tom Yates]

Thanks, I hadn't considered whether it might be possible to calculate this directly. Are the two distributions not binomial (Bernoulli) rather than Gaussian?

Covariance will be hard to quantify, as the two proportions are (in most settings) taken from different studies. I am seeking to estimate the proportion of tuberculosis disease attributable to household transmission by multipling the proportion of cases with a history of household TB exposure by the proportion of household case pairs with the same strain of TB.

The alternative here is to use the ends of the two confidence intervals, a conservative approach. That feels a bit ugly and doesn't get you a 95% CI, but is transparent and simple.

[Leonardo Guizzetti]

What you are proposing to do is a simulation, not a bootstrap. Explicit in that process is the assumption of some sampling distribution for each parameter.

Since you have the estimates and their CIs, I would use the MOVER-R method [1], which extends the broad set of previously derived MOVER estimators. Note that this would assume independence of proportions.

I haven't implemented this in Stata (nor ready the article in much depth -- though am familiar with other MOVER estimators). You should be able to follow it easily enough.

1. Newcombe RG. MOVER-R confidence intervals for ratios and products of two independently estimated quantities. Statistical Methods in Medical Research. 2016;25(5):1774-1778. doi:10.1177/0962280213502144

[Tom Yates]

Many thanks Leonardo - the MOVER method looks useful. I see Robert Newcombe has Excel sheets implementing this approach on his website [1].

1. https://profrobertnewcomberesources.yolasite.com/