When you are deciding how to display two by two (or larger) tables, you
have a variety of ways to do this. No way is correct all the time, and some
of choices reflect subjective judgment. But here are some rules I use.
1. Never display more than one type of number in a table.
Statistical software like SPSS can produce counts, row percents, column
percents, cell percents, expected counts, residuals, and/or cell contribution
to chi-squared values. At one time or another you might want to use each of
these statistics, but never all at one time. Two or more numbers in a table
causes confusion and makes your tables harder to interpret.
Present a single summary statistic in the table if at all possible. If you
need to display two summary statistics (for example, both counts and row
percentages), then place the counts in one table and the row percentages in a
different table. If you have to fit them in the same table, place the two
numbers side by side with the less important number appearing second and in
parentheses For example, 54% (257).
2. Row percentages are usually best. Row percentages are
the percentages you compute by dividing each count by the row total. Row
percentages place the comparison between two numbers within a single column,
so that one number is directly beneath the number you want to compare it to.
This is usually better than column percents, where the numbers you want to
compare are side by side. If you find that column percentages make more
sense. Consider swapping the rows and columns.
If you find that cell percentages make the most sense, consider creating
composite categories that combine the row and column categories. Cell
percentages are the percentages that you get when you divide each cell count
by the overall total. When cell percents are interesting, it usually means
that you are interested in the four distinct categories in your two by two
table. For example, you are interested in seeing what fraction of job
candidates are white males, rather than seeing how the probability of being
male influences the probability of being white. For this type of data, treat
it as a single categorical variable with four levels (white males, white
females, black males, black females) rather than two categorical variables
with each having two levels (black/white, male/female).
3. Place the treatment/exposure variable as rows and outcome
variable as columns. This relates to the above item. You usually are
interested in the probability of an outcome like death or disease, and you
are interested in how this probability changes when the treatment or exposure
changes. Arranging the table thusly and using row percents usually gets you
the comparison you are interested in.
4. If one variable has a lot more levels than the other variable, place
that variable in rows. A table that is tall and thin is usually easier to
read than a table that is short and wide. It is easier to scroll up and down
rather than left and right. For a really large number of levels, you might
have to print your table on two or more pages. Usually it is a lot easier to
align these pages if the table is tall and thin. A short wide table that is
split on two or more pages is often a disaster.
5. Whenever you report percentages, always round. A change
on the order of tenths of a percent are almost never interesting or
important. Displaying that tenth of a percent makes it harder to manipulate
the numbers to see the big picture.
6. Don't worry about whether your percentages add up to 99% or
101%. First of all, it can't happen with a two by two table unless
you round incorrectly. For a larger table, it can happen, but your audience
is sophisticated enough to understand why this is the case. No one, for
example, is going to be upset when 33% plus 33% plus 33% adds up to less than
100%.
7. When in doubt, write out your table several different ways.
Pick out the one that gives the clearest picture of what is really happening.
Don't rely on the first draft of your table, just like you would never rely
on the first draft of your writing.
A simple fictitious example will help illustrate these points.
We classify people by their income (rich/poor) and also by their attitude
(happy/miserable). There are, for example, 30 rich happy people in our
sample and 70 poor miserable people.
This figure shows column percentages. We compute this by dividing each
number by the column total.
We see for example that only 25% of all happy people are rich. This is a
conditional probability and is usually written as P[Rich | Happy]. Read the
vertical bar as "given." So this probability is read as the probability of
being rich given that you are happy.
This figure shows row percentages. We compute this by dividing each number
by the row total.
We see, for example that 75% of rich people are happy. This is a different
conditional probability, P[Happy | Rich]. Read this as the probability of
being happy given that you are rich.
Notice the distinction between the two probabilities. Only a few happy
people are rich, but most rich people are happy.
This figure shows cell percentages. We compute this by dividing each number
by the grand total. Each percentage represents the probability of having two
conditions. For example, there is a 15% chance of being rich and happy.
The table above shows a good format for combining two numbers in a single
table.
This is an alternate way of displaying cell percentages.
If we had a six categories for attitude rather than just two, we might
arrange the table differently.
Notice that this table would not require any sideways scrolling.