# Looking for a Test of Quotients

So I have roughly data from 8.000 persons: each person's working time and the time of absence from work due to illness. What test should I use to determine whether there is a difference of the proportions $$\text{absence quote}:=\frac{\text{absence time}}{\text{working time}}$$ in different groups, say men/women?

Edit: To clarify, I may be allowed to present the problem the I think I'm facing. I'm just interested in the described quotient. I'm given 8000 person's data.

Assume in one group ($$n_0 = 100$$) the mean of absence time for a period, say a month, is 8h, in the other group ($$n_1 = 300$$) the mean is 17h. The mean of working time in the first group is 110h, in the second 150h. Hence the quotient in the first group is $$8/110$$ whereas it's in the second $$17/150$$. Now how to perform a test for the quotients? (Sorry in case the question ist silly.)

First pic shows a box plot of working time ("Arbeitszeit"), the second on a box plot of absence time ("Ausfall").

Edit_2: Following the advice I've plotted absence vs. work time an got something to think over:

But why should I take a square root scale?

Edit_3: I did a prtest for the proportion of people which absence time is more than 30% of their working time and got significant difference: 5.9% for group 0 and 12.8% for group 1. Seems to me a much better strategy since the individual absence quotes are hardly comparable.

Thanks for your help again -- and any suggestions are welcome.

• First thing I'd do is to look at histogram/boxplot of ratios for men and women. I suppose data wouldn't be normal, but if sample distributions for men and women are similar in shape, I might try a two-sample Wilcoxon rank sum test. // It would be helpful if you can show some plots of your data. I don't see how we can give a responsible answer with so little information from you. Commented Aug 28, 2021 at 17:39
• Thanks for your suggestions; I've tried to clarify my problem. Commented Aug 28, 2021 at 18:46
• Although you are interested in the proportions, the size of the denominator likely determines the uncertainty in the proportion. An effective visual representation of the data would be a scatterplot of the numerator vs. the denominator. The use of square-root scales on both axes would be a helpful beginning. When you symbolize the points by group (such as gender), look for geometric clustering of slopes of fitted lines.
– whuber
Commented Aug 28, 2021 at 19:05
• The point is that when someone spends few hours at work, the absence quotient can be a much more variable indicator of their tendency to be absent compared to someone who spends many hours. A correct statistical test therefore takes that into consideration. When you examine only the ratios, you lose that crucial information.
– whuber
Commented Aug 28, 2021 at 19:19
• IMHO, you should pay close attention to @whuber's Comments and suggestions--even (or especially) if that means re-thinking your goals. Commented Aug 30, 2021 at 16:29

In case you are interested whether a typical absence quote (i.e. one that represents your data the best) significantly differs by gender, you can use the Yuen's t-test on 20% trimmed means (provided that no more than 20% of your dataset is comprised of outliers). In R this test can be conducted easily with the yuen function from the WRS2 package. Given your large sample size, the explanatory measure of effect size $$\xi$$ reported by this function will be also of particular importance. $$\xi = 0.15, 0.30, 0.50$$ translates roughly to small, medium respectively large effect size.
library(WRS2)