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How would one compare the odds ratio of a sample to that of the odds ratio of its source population? For instance:

I have child murder records/data from mortuaries for a geographical area (my "population"). Let's say I use a regression model (e.g. Poisson) with a regressor variable: race of child (black vs white), and I calculate that odds of child being black is 1.14 vs white.

Now I do the same exercise but for newspaper coverage in the same geographical area and time period, and end up with odds of black child murder being reported is 0.74.

If I theoretically subtract the crime coverage OR (0.74) from the epidemiological OR (1.14),I should get an estimate of media undereporting. But given that my two samples are dependant and non-parametric, how would I go about doing this?

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  • $\begingroup$ Hmmm, so should I follow Frank's or Peter's advice? @Frank would you explain a bit further as to how one would go about calculating the ratio of the odds ratio please? $\endgroup$
    – The_Dude
    Commented Oct 20, 2014 at 22:00

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Depending on whether your two data sources are linked, you have 2 options to model this.

  1. When your 2 data sources are linked. This means for every child murder cases, you know whether it was reported or not. In this case, create a response variable $y_i$ and let $y_i =1$ if the $i$th murdered child was reported in the news and $y_i =0$ if unreported. You then use logistic regression with $y$ as response variable and race as your covariate. The odds ratio of race would be what you need.

  2. When your 2 data sources are NOT linked. This means you only know the total number of child murders cases and child murder reports by race. You then calculate 2 proportions: probability of a murder case being reported for black and white children. You can then do a 2-sample binomial test i.e test a difference between 2 proportions.

Hope this helps.

Peter

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I don't know how you get odds ratios out of Poisson regression so I respond as if this were logistic regression. You would describe this not as a difference in odds ratios but rather as a ratio of odds ratios. To get the proper confidence intervals with overlapping samples, I think you would need to have all the raw data so that you could use bootstrapping to estimate the variance of the log of the ratio of odds ratios or more directly estimate quantiles. You may be able to approximate this knowing the variance of your original log odds ratio (standard output of logistic regression) plus the same for the larger region, and you could multiply the sum of these variances by a fudge factor (say 1.5) to account for the overlap, giving a conservative analysis.

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