# Direction of relationship in 2x2 contingency tables

I've been studying contingency table analysis and got to experiment a little bit with Fisher's exact test and Fisher's power test. Now I want to be able to determine the direction of the relationships.

Can anyone tell me what are the best methods to determine the direction of the relationship in 2x2 contingency tables? Is it just by using the odds? Or are there any standard, formal methods to calculate it?

The easiest way to describe the direction is with a sentence like "The proportion of pilots among men was 0.13 higher than among women". You can do this descriptively even if you use Fisher or the odds ratio for the formal test. Note that Fisher is exact for the very rare situation in which both row and column totals are fixed in advance (before the data are gathered). It can also be used for experimental data in which either the row or column totals are fixed. It is not appropriate for any sampling situation. For those it is extremely conservative. Chi-squared will then be closer to the truth, though not real close. I suspect Fisher is commonly (mis)used because of the word "exact" which people read too much into. It's an exact answer to (in most cases) another question.

By way of clarification, I think the earlier posts address the issue of choosing the direction of the alternative in a hypothesis test rather than just describing the observed direction (which is what I addressed).

Welcome to the site.

2x2 tables do not have a direction of relationships.

In any case, statistics do not reveal the direction of relationships, thought and logic and substantive knowledge does. That is, you (the researcher) figures out which direction (if any) the relationship might go in and then you use statistics to examine that relationship.

Since you have 2x2 tables, you would probably want logistic regression as a tool to do that examination. You would pick a dependent variable and an independent variable and then do the regression.

If you tell us more about your particular case, we may be able to suggest more precise solutions.

To expand a little on Peter Flom's answer, you need to know the science and what question you are trying to answer to interpret any output. For a 2x2 table there are several possible ways to look at direction, some simple approaches include calculating a confidence interval on the odds ratio, calculating a confidence interval on the difference of 2 proportions, comparing the observed values in the table to the expected values (under a null of no relationship).

Any of the above will give a feel for direction and strength of difference, but it is meaningless to say that the odds ratio is significantly greater than 1 without understanding the odds of what given what. I remember a consulting session that started off a little confusing because the direction of the odds ratio was opposite what was expected, until we realized that one of us was talking about the odds of death and the other was talking about the odds of surviving, once we were looking at the same concept then everything made sense.

Why not try a Bayesian route? For example, suppose that you want to estimate the posterior distribution of the difference or ratio of a proportion between two groups.

Step 1: Open R.

Step 2: Simulate the posterior distributions of the proportion in each group with two calls of the rbeta() function, parameterized with the success and failure counts in the focal category (i.e., the "success") of the appropriate group. You could do 1,000,000 draws from each posterior in very little time. If you have at least one observation in each category, you can use an improper beta prior with shape and scale parameters equal to 0. If you don't, you can use a flat and still somewhat uninformative prior by adding a one to the counts in each category.

Step 3: Use the - operator to subtract one empirical distribution from the other to obtain the empirical distribution of the difference. Or use the / operator if you are interested in a ratio.

Step 4: Use the quantile() function to find the 2.5th, 50th, and 97.5th pecentile. Use the hist() or plot(density()) functions to visualize the distribution.

Asymptotically, the answer you get will be the same as with a frequentist method, except this way is easier. You can do virtually the same thing with any other exponential family distribution by looking up the conjugate prior of the sampling distribution, and how to parameterize the posterior.

Disclaimer This isn't a critique of frequentist methods, just a proposal of a simpler way to solve this particular problem using Bayesian inference.