In the previous statistics reviews most of the procedures discussed are appropriate for quantitative measurements. However, qualitative, or categorical, data are frequently collected in medical investigations. For example, variables assessed might include sex, blood group, classification of disease, or whether the patient survived. Categorical variables may also comprise grouped quantitative variables, for example age could be grouped into 'under 20 years', '20–50 years' and 'over 50 years'. Some categorical variables may be ordinal, that is the data arising can be ordered. Age group is an example of an ordinal categorical variable.

When using categorical variables in an investigation, the data can be summarized in the form of frequencies, or counts, of patients in each category. If we are interested in the relationship between two variables, then the frequencies can be presented in a two-way, or contingency, table. For example, Table 1 comprises the numbers of patients in a two-way classification according to site of central venous cannula and infectious complications. Interest here is in whether there is any relationship, or association, between the site of cannulation and the incidence of infectious complications. The question could also be phrased in terms of proportions, for example whether the proportions of patients in the three groups determined by site of central venous cannula differ according to type of infectious complication.

In order to test whether there is an association between two categorical variables, we calculate the number of individuals we would get in each cell of the contingency table if the proportions in each category of one variable remained the same regardless of the categories of the other variable. These values are the frequencies we would expect under the null hypothesis that there is no association between the variables, and they are called the expected frequencies. For the data in Table 1, the proportions of patients in the sample with cannulae sited at the internal jugular, subclavian and femoral veins are 934/1706, 524/1706, 248/1706, respectively. There are 1305 patients with no infectious complications. So the frequency we would expect in the internal jugular site category is 1305 × (934/1706) = 714.5. Similarly for the subclavian and femoral sites we would expect frequencies of 1305 × (524/1706) = 400.8 and 1305 × (248/1706) = 189.7.

We repeat these calculations for the patients with infections at the exit site and with bacteraemia/septicaemia to obtain the following:

Exit site: 245 × (934/1706) = 134.1, 245 × (524/1706) = 75.3, 245 × 248/1706 = 35.6

Bacteraemia/septicaemia: 156 × (934/1706) = 85.4, 156 × (524/1706) = 47.9, 156 × (248/1706) = 22.7

We thus obtain a table of expected frequencies (Table 2). Note that 1305 × (934/1706) is the same as 934 × (1305/8766), and so equally we could have worded the argument in terms of proportions of patients in each of the infectious complications categories remaining constant for each central line site. In each case, the calculation is conditional on the sizes of the row and column totals and on the total sample size.

The test of association involves calculating the differences between the observed and expected frequencies. If the differences are large, then this suggests that there is an association between one variable and the other. The difference for each cell of the table is scaled according to the expected frequency in the cell. The calculated test statistic for a table with r rows and c columns is given by:

where O_ij is the observed frequency and E_ij is the expectedfrequency in the cell in row i and column j. If the null hypothesis of no association is true, then the calculated test statistic approximately follows a χ² distribution with (r - 1) × (c - 1) degrees of freedom (where r is the number of rows and c the number of columns). This approximation can be used to obtain a P value.

For the data in Table 1, the test statistic is:

1.134 + 2.380 + 1.314 + 6.279 + 21.531 + 2.052 + 2.484 + 14.069 + 0.020 = 51.26

Comparing this value with a χ² distribution with (3 - 1) × (3 - 1) = 4 degrees of freedom, a P value of less than 0.001 is obtained either by using a statistical package or referring to a χ² table (such as Table 3), in which 51.26 being greater than 18.47 leads to the conclusion that P < 0.001. Thus, there is a probability of less than 0.001 of obtaining frequencies like the ones observed if there were no association between site of central venous line and infectious complication. This suggests that there is an association between site of central venous line and infectious complication.

The χ² test indicates whether there is an association between two categorical variables. However, unlike the correlation coefficient between two quantitative variables (see Statistics review 7 1), it does not in itself give an indication of the strength of the association. In order to describe the association more fully, it is necessary to identify the cells that have large differences between the observed and expected frequencies. These differences are referred to as residuals, and they can be standardized and adjusted to follow a Normal distribution with mean 0 and standard deviation 1 2. The adjusted standardized residuals, d_ij, are given by:

Where n_i. is the total frequency for row i, n._j is the total frequency for column j, and N is the overall total frequency. In the example, the adjusted standardized residual for those with cannulae sited at the internal jugular and no infectious complications is calculated as:

Table 4 shows the adjusted standardized residuals for each cell. The larger the absolute value of the residual, the larger the difference between the observed and expected frequencies, and therefore the more significant the association between the two variables. Subclavian site/no infectious complication has the largest residual, being 6.2. Because it is positive there are more individuals than expected with no infectious complications where the subclavian central line site was used. As these residuals follow a Normal distribution with mean 0 and standard deviation 1, all absolute values over 2 are significant (see Statistics review 2 3). The association between femoral site/no infectious complication is also significant, but because the residual is negative there are fewer individuals than expected in this cell. When the subclavian central line site was used infectious complications appear to be less likely than when the other two sites were used.

The use of the χ² distribution in tests of association is an approximation that depends on the expected frequencies being reasonably large. When the relationship between two categorical variables, each with only two categories, is being investigated, variations on the χ²test of association are often calculated as well as, or instead of, the usual test in order to improve the approximation. Table 5 comprises data on patients with acute myocardial infarction who took part in a trial of intravenous nitrate (see Statistics review 3 4). A total of 50 patients were randomly allocated to the treatment group and 45 to the control group. The table shows the numbers of patients who died and survived in each group. The χ² test gives a test statistic of 3.209 with 1 degree of freedom and a P value of 0.073. This suggests there is not enough evidence to indicate an association between treatment and survival.

Table 8 comprises the numbers of patients in a two-way classification according to AVPU classification (voice and pain responsive categories combined) and subsequent survival or death of 1306 patients attending an accident and emergency unit. (AVPU is a system for assessing level of consciousness: A = alert, V = voice responsiveness, P = pain responsive and U = unresponsive.) The χ² test of association gives a test statistic of 19.38 with 2 degrees of freedom and a P value of less than 0.001, suggesting that there is an association between survival and AVPU classification.

Because the categories of AVPU have a natural ordering, it is appropriate to ask whether there is a trend in the proportion dying over the levels of AVPU. This can be tested by carrying out similar calculations to those used in regression for testing the gradient of a line (see Statistics review 7 1). Suppose the variable 'survival' is regarded as the y variable taking two values, 1 and 2 (survived and died), and AVPU as the x variable taking three values, 1, 2 and 3. We then have six pairs of x, y values, each occurring the number of times equal to the frequency in the table; for example, we have 1110 occurrences of the point (1,1).

Following the lines of the test of the gradient in regression, with some fairly minor modifications and using large sample approximations, we obtain a χ² statistic with 1 degree of freedom given by 5:

For the data in Table 8, we obtain a test statistic of 19.33 with 1 degree of freedom and a P value of less than 0.001. Therefore, the trend is highly significant. The difference between the χ² test statistic for trend and the χ² test statistic in the original test is 19.38 - 19.33 = 0.05 with 2 - 1 = 1 degree of freedom, which provides a test of the departure from the trend. This departure is very insignificant and suggests that the association between survival and AVPU classification can be explained almost entirely by the trend.

Some computer packages give the trend test, or a variation. The trend test described above is sometimes called the Cochran–Armitage test, and a common variation is the Mantel–Haentzel trend test.

Another application of a two by two contingency table is to examine the association between a disease and a possible risk factor. The risk for developing the disease if exposed to the risk factor can be calculated from the table. A basic measurement of risk is the probability of an individual developing a disease if they have been exposed to a risk factor (i.e. the relative frequency or proportion of those exposed to the risk factor that develop the disease). For example, in the study into early goal-directed therapy in the treatment of severe sepsis and septic shock conducted by Rivers and coworkers 6, one of the outcomes measured was in-hospital mortality. Of the 263 patients who were randomly allocated either to early goal-directed therapy or to standard therapy, 236 completed the therapy period with the outcomes shown in Table 9.

From the table it can be seen that the proportion of patients receiving early goal-directed therapy who died is 38/117 = 32.5%, and so this is the risk for death with early goal-directed therapy. The risk for death on the standard therapy is 59/119 = 49.6%.

Another measurement of the association between a disease and possible risk factor is the odds. This is the ratio of those exposed to the risk factor who develop the disease compared with those exposed to the risk factor who do not develop the disease. This is best illustrated by a simple example. If a bag contains 8 red balls and 2 green balls, then the probability (risk) of drawing a red ball is 8/10 whereas the odds of drawing a red ball is 8/2. As can be seen, the measurement of odds, unlike risk, is not confined to the range 0–1. In the study conducted by Rivers and coworkers 6 the odds of death with early goal-directed therapy is 38/79 = 0.48, and on the standard therapy it is 59/60 = 0.98.

To assess the importance of the risk factor, it is necessary to compare the risk for developing a disease in the exposed group with the risk in the nonexposed group. In the study by Rivers and coworkers 6 the risk for death on the early goal-directed therapy is 32.5%, whereas on the standard therapy it is 49.6%. A comparison between the two risks can be made by examining either their ratio or the difference between them.

Matched pair designs, as discussed in Statistics review 5 9, can also be used when the outcome is categorical. For example, when comparing two tests to determine a particular condition, the same individuals can be used for each test.

For a χ² test of association, a recommendation on sample size that is commonly used and attributed to Cochran 5 is that no cell in the table should have an expected frequency of less than one, and no more than 20% of the cells should have an expected frequency of less than five. If the expected frequencies are too small then it may be possible to combine categories where it makes sense to do so.

For two by two tables, Yates' correction or Fisher's exact test can be used when the samples are small. Fisher's exact test can also be used for larger tables but the computation can become impossibly lengthy.

In the trend test the individual cell sizes are not important but the overall sample size should be at least 30.

The analyses of proportions and risks described above assume large samples with similar requirement to the χ² test of association 8.

The sample size requirement often specified for McNemar's test and confidence interval is that the number of discordant pairs should be at least 10 8.

The χ² test of association and other related tests can be used in the analysis of the relationship between categorical variables. Care needs to be taken to ensure that the sample size is adequate.

None declared.

This article is the eighth in an ongoing, educational review series on medical statistics in critical care.

Previous articles have covered 'presenting and summarizing data', 'samples and populations', 'hypothesestesting and P values', 'sample size calculations', 'comparison of means', 'nonparametric means' and 'correlation and regression'.

Future topics to be covered include:

Chi-squared and Fishers exact tests

Analysis of variance

Further non-parametric tests: Kruskal–Wallis and Friedman

Measures of disease: PR/OR

Survival data: Kaplan–Meier curves and log rank tests

ROC curves

Multiple logistic regression.

If there is a medical statistics topic you would like explained, contact us at editorial@ccforum.com.

AVPU: A = alert, V = voice responsiveness, P = pain responsive and U = unresponsive