I received an email from India (isn't the Internet wonderful?) that asked me to comment on
the differences between a Chi-square test, Fisher's Exact test, and logistic regression.
Let's take each of these in sequence.
A Chi-square test can mean a variety of different things depending on the context of the
problem. I outline some of these in
For the record, there is no standard spelling for this. The terms "Chi square" and "Chi
squared" are both acceptable, and you can also hyphenate the two words (Chi-square) or make
them a single word (Chisquare). You can also use an initial capital letter or lower case for
"Chi" and/or "Square". I use different versions throughout my web pages. Two quotes about
consistency (found at en.thinkexist.com) capture my
feelings on the topic:
-
Consistency is the last refuge of the unimaginative. Oscar Wilde, and
-
A foolish consistency is the hobgoblin of little minds, adored by little statesmen and
philosophers and divines. Ralph Waldo Emerson.
In the context of the original email, the Chi-square test was intended to compare the
proportion of patients who die under two separate surgical procedures. This produces a
two-by-two table. I outline some of the data entry issues associated with two-by-two tables
at
and discuss some of the alternative statistical summaries (odds ratio, relative risk,
number needed to treat) at
The Chi-square test has some flexibility in that if there are more than two rows or more
than two columns in your table, you can still use this test. Beware, though, that the
Chi-square test can be inefficient if the variable representing multiple rows or multiple
columns is ordinal rather than
nominal.
Fisher's Exact test is an alternative to the Chi-square test for two-by-two tables. The
adjective "exact" stresses that this test does not rely on an approximation. Although some
have cautioned about the conservative nature of Fisher's Exact test, there is general
consensus in the research community that this test is preferred to the Chi-square test for
small data sets, though there are varying rules for deciding whether a data set is small
enough to warrant the use of Fisher's Exact test.
There used to be some reluctance to use Fisher's Exact test for moderate size data sets
because the calculations needed for this test increase rapidly as the sample size increases.
But with recent improvements in the computational algorithms, and with the ever increasing
speed and power of computers, there is no reason to avoid this test, except, perhaps, for the
very largest of data sets.
There are extensions of Fisher's Exact test to situations where there are more than two
rows/columns, but these extensions are not available in most statistical software packages.
I discuss the use of Fisher's Exact test including some examples of how to report the
results in a publication at
A simple logistic regression model will produce results that are very similar to the
Chi-square test and Fisher's Exact test. A more complex logistic model, though, such as one
that includes additional covariates, may produce radically different results than either of
these tests. The reason for this is that Fisher's Exact test cannot adjust for possible
confounding variables.
Logistic regression is also preferred when you are trying to predict a binary outcome
using a continuous predictor variable. Logistic regression uses odds ratios as a measure of
the relationship between the outcome variable and the predictor variable. I discuss what this
means and how to interpret results from a logistic regression model at
I also have a cautionary tale about overfitting in the context of a logistic regression
model:
07/08/2008.
