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.

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

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

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:

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.

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.

Confidence interval for a proportion

As the measurement of risk is simply a proportion, the confidence interval for the population measurement of risk can be calculated as for any proportion. If the number of individuals in a random sample of size n who experience a particular outcome is r, then r/n is the sample proportion, p. For large samples the distribution of p can be considered to be approximately Normal, with a standard error of [2]:

The 95% confidence interval for the true population proportion, p, is given by p - 1.96 × standard error to p + 1.96 × standard error, which is:

where p is the sample proportion and n is the sample size. The sample proportion is the risk and the sample size is the total number exposed to the risk factor.

For the study conducted by Rivers and coworkers [6] the 95% confidence interval for the risk for death on early goal-directed therapy is 0.325 ± 1.96(0.325 [1-0.325]/117)^0.5 or (24.0%, 41.0%), and on the standard therapy it is (40.6%, 58.6%). The interpretation of a confidence interval is described in (see Statistics review 2 [3]) and indicates that, for those on early goal-directed therapy, the true population risk for death is likely to be between 24.0% and 41.0%, and that for the standard therapy between 40.6% and 58.6%.

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.

Difference between two proportions

Confidence interval

For the Rivers study, instead of examining the ratio of the risks (the relative risk) we can obtain a confidence interval and carry out a significance test of the difference between the risks. The proportion of those on early goal-directed therapy who died is p₁ = 38/117 = 0.325 and the proportion of those on standard therapy who died is p₂ = 59/119 = 0.496. A confidence interval for the difference between the true population proportions is given by:

(p₁ - p₂) - 1.96 × se(p₁ - p₂) to (p₁ - p₂) + 1.96 × se(p₁ - p₂)

Where se(p₁ - p₂) is the standard error of p₁ - p₂ and is calculated as:

Thus, the required confidence interval is -0.171 - 1.96 × 0.063 to -0.171 + 1.96 × 0.063; that is -0.295 to -0.047. Therefore, the difference between the true proportions is likely to be between -0.295 and -0.047, and the risk for those on early goal-directed therapy is less than the risk for those on standard therapy.

Hypothesis test

We can also carry out a hypothesis test of the null hypothesis that the difference between the proportions is 0. This follows similar lines to the calculation of the confidence interval, but under the null hypothesis the standard error of the difference in proportions is given by:

where p is a pooled estimate of the proportion obtained from both samples [5]:

So:

The test statistic is then:

Comparing this value with a standard Normal distribution gives p = 0.007, again suggesting that there is a difference between the two population proportions. In fact, the test described is equivalent to the χ²test of association on the two by two table. The χ² test gives a test statistic of 7.31, which is equal to (-2.71)² and has the same P value of 0.007. Again, this suggests that there is a difference between the risks for those receiving early goal-directed therapy and those receiving standard therapy.

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.

McNemar's test

In this situation, because the χ² test does not take pairing into consideration, a more appropriate test, attributed to McNemar, can be used when comparing these correlated proportions.

For example, in the comparison of two diagnostic tests used in the determination of Helicobacter pylori, the breath test and the Oxoid test, both tests were carried out in 84 patients and the presence or absence of H. pylori was recorded for each patient. The results are shown in Table 10, which indicates that there were 72 concordant pairs (in which the tests agree) and 12 discordant pairs (in which the tests disagree). The null hypothesis for this test is that there is no difference in the proportions showing positive by each test. If this were true then the frequencies for the two categories of discordant pairs should be equal [5]. The test involves calculating the difference between the number of discordant pairs in each category and scaling this difference by the total number of discordant pairs. The test statistic is given by:

	Breath test
Oxoid test	+	-	Total
+	40	8 (b)	48
-	4 (c)	32	36
Total	44	40	84(n)

Where b and c are the frequencies in the two categories of discordant pairs (as shown in Table 10). The calculated test statistic is compared with a χ² distribution with 1 degree of freedom to obtain a P value. For the example b = 8 and c = 4, therefore the test statistic is calculated as 1.33. Comparing this with a χ² distribution gives a P value greater than 0.10, indicating no significant difference in the proportion of positive determinations of H. pylori using the breath and the Oxoid tests.

The test can also be carried out with a continuity correction attributed to Yates [5], in a similar way to that described above for the χ²test of association. The test statistic is then given by:

and again is compared with a χ² distribution with 1 degree of freedom. For the example, the calculated test statistic including the continuity correct is 0.75, giving a P value greater than 0.25.

As with nonpaired proportions a confidence interval for the difference can be calculated. For large samples the difference between the paired proportions can be approximated to a Normal distribution. The difference between the proportions can be calculated from the discordant pairs [8], so the difference is given by (b - c)/n, where n is the total number of pairs, and the standard error of the difference by (b + c)^0.5/n.

For the example where b = 8, c = 4 and n = 84, the difference is calculated as 0.048 and the standard error as 0.041. The approximate 95% confidence interval is therefore 0.048 ± 1.96 × 0.041 giving -0.033 to 0.129. As this spans 0, it again indicates that there is no difference in the proportion of positive determinations of H. pylori using the breath and the Oxoid tests.

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.

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.

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