Cross-tabulation and the chi-square statistic
4.3 The chi-square statistical test
We have seen, then, that the sex of patients was not equally distributed across the counsellors. But we might ask was this a chance result? If we started the service again and the patients were randomly assigned to the counsellors might we have found a similar result? Perhaps even the opposite – with most male patients seeing the female counsellor. More likely of course, by chance, we would expect them to be distributed quite evenly across the sexes. The question is: how much of an unequal distribution across the sexes does there have to be for us to conclude that there was a significant bias for male patients to see a male counsellor and vice versa? Or, to put it another way: how much of an unequal distribution across the sexes does there have to be for us to reject the possibility that this has occurred by chance? This is where a statistical test comes in handy.
Chi-square (represented as χ2) applies a statistical test to cross-tabulation by comparing the actual observed frequencies in each cell of tables with expected frequencies. Expected frequencies are those we would expect if data is
‘randomly distributed’.
Table 4.3 Cross-tabulation of sex and counsellor
Cross-tabulation and the chi-square statistic 63
It may help at this point to think of the four cells in the cross-tabulation table as buckets. Now imagine a lottery machine designed to pump balls into each of the buckets through four pipes – one leading into each bucket. If the distribution of balls by the lottery machine is truly random, then we would expect a similar number of balls in each bucket. Thus, if the lottery machine was set up to distribute 200 balls we would expect roughly 50 balls in each of the four buckets.
So, using the hypothetical data in Table 4.4 we can see that there is no dif- ference across male/female patients and counsellor. The actual observed counts match the expected counts.2
In Table 4.5 we have the opposite scenario: all the males saw John, whereas all the females saw Jane. Notice that the expected frequencies remain the same – 50 in each cell. The divergence from expected frequencies would strongly suggest that there is a relationship between sex of the patient and the counsellor they saw: we have 50 more male patients than expected seeing John (and 50 less than expected seeing Jane); and 50 more female patients than expected seeing Jane (and 50 less than expected seeing John).
Table 4.4 Group/counsellor cross-tabulation with no difference across males/females (hypothetical data)
Table 4.5 Group/counsellor cross-tabulation with a large difference across males/females (hypothetical data)
What the chi-square statistic does is to calculate the odds of this distribution happening by chance.
Let us run chi-square on our counselling data and see what happens.
Running chi-square in SPSS
1 From the menu at the top of the screen click: Analyze then Descriptive Statistics then Crosstabs.
2 Move the variable sex into the rows box and counsellor into the columns box.
3 Click the Cells button. Ensure that observed and expected counts are ticked, and row percentages (since we want to know the percentages of males and females who saw each counsellor).
Screenshot 4.2
4 Click Continue to close this box.
5 Click on the Statistics tab and put a tick in the chi-square box.
Cross-tabulation and the chi-square statistic 65
Screenshot 4.3
6 Click Continue then click OK to run the analysis.
The SPSS output viewer should produce the cross-tabulation table and a table listing the chi-square and related tests.
Note: SPSS can produce some scary output tables listing all kinds of statistics with obscure and complicated names which may or may not be relevant to your analysis. The chi-square test table is a mild example. The trick is to know what you are looking for, as will be illustrated.
Table 4.6 Cross-tabulation of sex and counsellor with expected counts
The first thing to note is the footnote (b) which tells us that ‘0 cells have expected count less that 5’. This is an important assumption of chi-square.
If you are intending to cross-tabulate data and use the chi-square statistic, you should always try to ensure that you have a large enough sample size to maintain sufficient values in the cells.3
The values we are interested in are along the top Pearson chi-square row. The Pearson chi-square value is 6.467, with a significance or probability (p) value of .011. This means that, according to the chi-square calculation, the probability of this distribution of values occurring by chance is less than .01 – or 1 in 100, so probability (p) = .01. We would accept these odds as statistically significant and conclude that there is a relationship between the sex of patients and the counsellor they saw: male patients were more likely to see the male counsellor and female patients were more likely to see the female counsellor.4
So, in your report to the doctor you would first point out that cross-tabulation between the sex of patients and counsellors showed that a greater proportion tended to see the same sex counsellor: 71 per cent of male patients saw John; 75 per cent of female patients saw Jane. But then you would also add that the chi-square statistic showed this to be statistically significant: χ2= 6.467, p = .011. You would then go on to comment on this result, for example, possible explanations.