Solved – How to calculate probabilities for a 4 by 2 table

fishers-exact-testodds-ratior

I have collected data about test subjects which are classified into 4 categories, and I want to calculate the probabilities (odds ratios?) of developing some condition for a test subject in one category. The data are as follows: 

             Condition present    Condition not present
Category A   4                    22
Category B   7                    13
Category C   16                   12
Category D   30                   6

Fisher's exact test from R (fisher.test) gives p=3,49E-04. I guess that means that there is a correlation between categories and the increased likelihood of developing that condition? If so, how should I calculate exact probability of developing the condition for each category?

Best Answer

To calculate the probabilities, you first sum up each row and each column. For you that would be:

             present     not present     TOTAL
Category A    4          22               26
Category B    7          13               20
Category C   16          12               28
Category D   30           6               36  

TOTAL        57          53              110

Then you divide each value by the total to get the percentage. Thus, you could get probabilities for the condition being present (or not) by row, for being in a given category by column, or being in a particular combination of condition and category by the whole table; which you should use depends on which makes sense in your situation. For example, the probability of a subject in Category A having the condition present is $4/26=.154$. The odds is the ratio of present to not present, for Category A it is $4/22=.182$. Finally, the odds ratio is the ratio of the odds for some category to some other category, for instance, the odds ratio for Category A vs. Category B is $.182/.538=.338$. Since it's less than 1, the odds of the condition being present, if you're in Category A, is lower than the odds of the condition being present, if you're in Category B ($.182<.538$). The odds ratio gives you the multiplicative factor for converting one odds into the other, it is also a measure of effect size. Mathematically, you can move from probabilities to odds and back again:
$$ \text{odds}=\frac{\text{probability}}{(1-\text{probability})} $$
$$ \text{probability}=\frac{\text{odds}}{(1+\text{odds})} $$
As to the question of whether you should say that the category is correlated with the condition, I guess you could, but I would just say that the probability of the condition being present varies by category.

Related Question