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There are two groups:

  • treatment
  • control.

Each group has three categories of outcomes:

  • High
  • medium and
  • low blood pressure.

The number of subjects is 100 for each group.

Now, let's say, for control group, the number of observations is 40, 40, 20 for high, medium, and low blood pressure, respectively. And, for the treatment group, the number of observations is 10, 50, 40 for high, medium and low. What I want to test is whether the ratio of occurrences (High/Low) ignoring the medium is different between these two groups. 2 (=40/20) for control vs. 0.25 (=10/40) for treatment.

What statistical test should I use?

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  • $\begingroup$ The approach here seems difficult to justify. Looking at ratios of frequencies is equivalent to considering frequencies on logaritmic scale and that itself points to Poisson regression. But that aside ignoring the medium group when it's part of the measurement protocol and part of the total pattern of variation seems arbitrary. For most readerships (whether you are preparing dissertation, thesis, report, presentation, paper) that's likely to seem somewhere between puzzling and perverse and something you would need to explain and defend at length. $\endgroup$
    – Nick Cox
    Aug 19 '13 at 16:11
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    $\begingroup$ It's rather late to make this suggestion, but you might bear it in mind for your next experiment. In order to assess whether blood pressure was high, medium or low, you must have actually measured it and got a result in mmHg. You would get much better power from such an analysis using the actual blood pressures in mmHg than categorising them. Additionally, categorising the blood pressures before doing the analysis, presumes you know how to categorise appropriately. It is quite conceivable that for different treatments different categorisations would be appropriate. $\endgroup$
    – Robert Jones
    Aug 19 '13 at 16:33
  • $\begingroup$ Thanks all for your answers. Just to let you know, the thresholds for High, Medium, Low were preset before the experiment so there is no hindsight bias. As someone suggested changing these thresholds could change our inferences, but by carefully setting the thresholds that are meaningful to our experiment before the experiment, we avoid the data-snooping issues. Thanks Jean for your suggestion on using either a randomization test or a bootstrapping method. $\endgroup$
    – Jen
    Aug 21 '13 at 8:22
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I can't help but ask, why do you want to compare the high/low ratios? Is this some standard way to assess blood pressures? Why not use a chi-squared test to compare the groups? Did you know you were going to conduct this sort of analysis before you saw the data? Or did you see the data, see the differences in the high and low categories, and decide the best way to show a difference between the groups was to look at the high/low ratio?

If you knew in advance that you wanted to analyze the high/low ratios, then you could use a randomization test or bootstrapping.

For a randomization test, you would randomly assign the 200 subjects to treatment and control groups of 100 subjects each and calculate the difference between the trt hi/lo and the ctl hi/lo. This random assignment and difference calculation is repeated 1000 times or more, so that you have a distribution of randomly generated differences. Then you compare the observed difference (1.75) to that distribution and see if it is in the extreme 5% (or whatever alpha) tails.

For bootstrapping, you would randomly select (with replacement) 200 subjects from the full sample of 200 subjects and calculate the difference between the trt hi/lo and the ctl hi/lo. This random selection and difference calculation is repeated 1000 times or more ... and the rest is similar to the randomization test described above. (You may want to use a bias-corrected bootstrap.)

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