Introduction
Rashid, my opposite number in Penang Medical College, has the bad habit of asking statistical questions when I am trying to concentrate on pulling and crawling my way up steeply-sloping rainforest. And it was in such a situation that he came up with a rather good question:
Just what is Chi-squared anyway?
And I said
Chi-squared is a measure of surprise.
Think of it this way. Here was have a table of data, and we are wondering if there is a pattern in there. The data show the relationship between loneliness and prevalence of depressed mood in people aged 65 and over in a community survey.
. tab depressed loneliness, col
+-------------------+
| Key |
|-------------------|
| frequency |
| column percentage |
+-------------------+
| Feels lonely
Depressed mood (Q24) | Not lonely Lonely | Total
----------------------+----------------------+----------
No depression last mo | 1,310 621 | 1,931
| 95.48 81.28 | 90.40
----------------------+----------------------+----------
Depressed last month | 62 143 | 205
| 4.52 18.72 | 9.60
----------------------+----------------------+----------
Total | 1,372 764 | 2,136
| 100.00 100.00 | 100.00
Is there anything surprising here? Well we seem to have more people who are depressed among the lonely (19% depression prevalence) than the non-lonely (5% prevalence). But how surprising is that difference? Well, that depends on what we were expecting.
If loneliness had no impact on depression, then we would expect to see more or less the same prevalence of depression in the lonely and the non-lonely. Of course, the prevalence probably wouldn’t be exactly the same, but a few people one way or another wouldn’t cause much surprise. What we are looking for is a surprising number of people too many or too few.
So how do we judge when the number of people too many or too few is surprising? Well, that depends on how many people we were expecting, doesn’t it? If ten people more than I expected turn up, I am not surprised if I am running a conference, but I am pretty taken aback if I had thought I was going out on an intimate dinner for two.
So the amount of surprise depends on the number of people I saw, but also on the number I was expecting.
What are we expecting?
Let’s have a look at that table again. Notice that 9·6% of people were depressed. So if loneliness has nothing to do with depression, we expect 9·6% of the lonely and 9·6% of the non-lonely to be depressed.
We have 1,372 non-lonely people; 9·6% of that is
. di 1372*0.096
131.712
I let Stata’s display command do the work for me. Essentially, we are expecting 132 of the non-lonely people to be depressed, and so we are expecting
. di 764*0.096
73.344
73 of the lonely people to be depressed.
We can work out the remainder of the numbers, but Stata will display them for us. Now we know how Stata expected those frequencies, that is.
. tab depressed loneliness, exp
+--------------------+
| Key |
|--------------------|
| frequency |
| expected frequency |
+--------------------+
| Feels lonely
Depressed mood (Q24) | Not lonely Lonely | Total
----------------------+----------------------+----------
No depression last mo | 1,310 621 | 1,931
| 1,240.3 690.7 | 1,931.0
----------------------+----------------------+----------
Depressed last month | 62 143 | 205
| 131.7 73.3 | 205.0
----------------------+----------------------+----------
Total | 1,372 764 | 2,136
| 1,372.0 764.0 | 2,136.0
We have four cells in the table, and in each cell we can compare the number we saw with the number we expected. And remember, the expectation is based on the idea that loneliness has nothing to do with depression. From that idea follows the reasoning that if 9·6% of the whole sample is depressed, then we are expecting 9·6% of the lonely, and of the non-lonely to be depressed. And, of course, 90·4% of the whole sample is not depressed, so we are expecting 90·4% of the lonely, and 90·4% of the non-lonely to be depressed.
The Chi-squared test visits each cell of the table, calculating how far apart the expected and observed frequencies are, as a measure of how surprising that cell is, and totalling up all those surprises as a total surprise score.
Of course, we need to relate that total surprise score to the number of cells in the table. A big surprise score could be the result of adding together a lot of small surprises coming from the cells of a large table – in which case we aren’t really surprised – or it could come from a very small number of cells in a small table, in which case we are surprised.
Degrees of freedom: the capacity for surprise
So a Chi-squared value has to be interpreted in the light of the number of cells in the table. Which is where the degrees of freedom come in. Degrees of freedom measures the capacity of a table to generate surprising results. Looking at our table, once we had worked out one expected frequency, we could have filled in the other three by simple subtraction. If we expect 131·7 people to be depressed but not lonely, then the rest of the depressed people must be depressed and lonely. And the rest of the non-lonely people must be non-depressed. See – once I know one number in that table, I can work out the rest by subtraction.
So that 2 x 2 table has only one potential for surprise.
And by extension, once I know the all but one of the numbers in the row of a table, I can fill in the last one by subtraction. Same goes for the columns. So the capacity of a table for surprises is one less than the number of rows multiplied by one less than the number of columns.
And what is the Chi-squared value for the table anyway?
. tab depressed loneliness, col chi
+-------------------+
| Key |
|-------------------|
| frequency |
| column percentage |
+-------------------+
| Feels lonely
Depressed mood (Q24) | Not lonel Lonely | Total
----------------------+----------------------+----------
No depression last mo | 1,310 621 | 1,931
| 95.48 81.28 | 90.40
----------------------+----------------------+----------
Depressed last month | 62 143 | 205
| 4.52 18.72 | 9.60
----------------------+----------------------+----------
Total | 1,372 764 | 2,136
| 100.00 100.00 | 100.00
Pearson chi2(1) = 114.0215 Pr = 0.000
A very large index of surprise: a Chi-squared of 114, with one degree of freedom. And a P-value that is so small that the first three digits are all zero. We would write P<0·001.
So that’s it . Chi-squared is a measure of surprise, and degrees of freedom measure the capacity of a table to come up with surprises.
Those aren’t mathematical definitions, but they are useful ways of thinking about statistical testing, and the difficult idea of degrees of freedom.
