# Notation of probability matrix corresponding to a contingency table

Contingency tables tables are typically formatted as tables similar to matrices in mathematics, see this example.

Is the equation below an accepted notation of expressing the probabilities of the outcomes as a matrix? If not, what would be the accepted way? Are there any published materials using the same notation? $$\widehat{Pr_\text{outcome}} = \begin{matrix} & \begin{matrix} \text{RH} & \text{LH} \end{matrix} \\ \begin{matrix} \text{male} \\ \text{female} \end{matrix} & \begin{bmatrix} 43 & 9 \\ 44 & 4 \end{bmatrix} \end{matrix} \cdot \frac{1}{100} = \begin{matrix} & \begin{matrix} \text{RH} & \text{LH} \end{matrix} \\ \begin{matrix} \text{male} \\ \text{female} \end{matrix} & \begin{bmatrix} 0.43 & 0.09 \\ 0.44 & 0.04 \end{bmatrix} \end{matrix}$$

• Out of curiosity, what would be the rationale for displaying a table this way? – chl Apr 11 '11 at 21:29
• @chl The four frequencies (and probabilities) have a natural ordering, the 2 by 2 table. So the matrix notation is an appealing way to express the probability space. Also the determinant of the matrix is useful, see my comment below. So I speculated that although I had not met this notation, a similar notation might be in use. (I also like the concept of a Karnaugh map.) – GaBorgulya Apr 11 '11 at 22:04
• @Ga I pointed out in a comment to your deleted post that a matrix notation for bivariate probabilities is (and long has been) in use. Most statistical software will produce such tables, e.g. It makes perfect sense and is quite clear: it's difficult to imagine someone misunderstanding what is meant. – whuber Apr 13 '11 at 1:32

It looks reasonable to me. Though realize that with an N larger than say 1,000, your table of probabilities won't represent exact counts because you'll end up truncating to three or so decimal places.

• Downvoted. How is the truncation issue relevant? – charles.y.zheng Apr 13 '11 at 10:30

I'm not sure I can justify what I'm about to say, but I would be uneasy about expressing probabilities in a form like this. The structure is too reminiscent of other things that don't properly apply and suggests you should be able to do stuff like matrix multiplication that wouldn't mean anything here.

If the goal is to express a set of 4 mutually-exclusive outcomes, then your probabilities should be a single vector with 4 entries adding to 1.

OTOH, if it is to express a structure within that -- the difference in the handedness distribution by sex -- then it would make more sense if each of the rows added to 1.

And if what you actually want is a contingency table, use a contingency table. Dividing through by the total count doesn't really gain you anything.

Using a convention whereby all entries in a matrix add to 1 seems to me, well, unconventional. But I admit this objection is a bit subjective and hand-wavy.

• @walkytalky You mention “other things that don't properly apply”, but let me give a counter example. The phi coefficient and Cramér's V rely on the determinant of such a contingency table: $$\phi=\frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet0}n_{\bullet1}}}= \frac{\text{det}\begin{bmatrix} n_{11} & n_{10}\\n_{01} & n_{00} \end{bmatrix}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet0}n_{\bullet1}}}$$ – GaBorgulya Apr 11 '11 at 12:57
• @walkytalky I divided through by the total count to estimate probabilities with relative frequencies. – GaBorgulya Apr 11 '11 at 13:04
• @GaBorgulya Those stats work from count tables, though. And estimating the probabilities is not what I'm objecting to, it's presenting a single set of probabilities (summing to one) as a 2x2 matrix. These are two different kinds of information, and this presentation risks confusing them. I am not saying it is actively wrong, just that it is unclear. – walkytalky Apr 11 '11 at 13:16
• @walky "Unclear" suggests there is at least one other natural way in which the expression could be interpreted. Could you give an example of such an alternative? Given that the left hand side explicitly announces that a probability is forthcoming, I'm having a hard time imagining how the values on the right hand side would be misunderstood. – whuber Apr 11 '11 at 16:04
• If the goal is to represent a joint pmf of two discrete variables (which is how I interpreted "probabilities of the outcomes"), how is this presentation unclear? The brackets? – JMS Apr 11 '11 at 17:21