to 0.009 (not shown). We notice that this adjustment parallels that made in SPSS by adjusting for continuity. When expected counts per cell are relatively small (a working rule is that they should be at least five in each cell), one can also request Fisher's exact test (see Fisher, 1922a), which we note also mirrors the output generated by SPSS:
> fisher.test(diag.table) Fisher's Exact Test for Count Data data: diag.table p-value = 0.008579 alternative hypothesis: true odds ratio is not equal to 1 95 percent confidence interval: 1.466377 26.597383 sample estimates: odds ratio 5.764989
Other useful statistics for contingency tables include the phi coefficient and Cramer's V. Phi, ϕ, is a measure of association for 2 × 2 contingency tables, computed as
where χ2 is the chi‐square statistic calculated on the 2 × 2 table, and n is the total sample size. The maximum ϕ can attain is 1.0, indicating maximal association. ϕ can be computed in SPSS by /statistics = phi and is available in R in the psych package (Revelle, 2015). Cramer's ϕc extends on ϕ in that it allows for contingency tables of greater than 2 × 2. It is included in the /statistics = phi command and also available in R's psych package. It is given by:
where k is the minimum of the number of rows or columns. The relationship between ϕc and ϕ is easily shown for k = 2:
2.2.1 Power for Chi‐Square Test of Independence
We can estimate power5and required sample size for the chi‐square test of independence using the package pwr in R:
> library(pwr) > pwr.chisq.test (w =, N =, df =, sig.level =, power = )
where w is the anticipated or required effect size, estimated as:
and p0i and p1i are the probabilities in a given cell i under the null and alternative hypotheses, respectively. We demonstrate by estimating power for w = 0.2:
> pwr.chisq.test(w = 0.2, N =, df = 5, sig.level = .05, power = 0.90) Chi squared power calculation w = 0.2 N = 411.7366 df = 5 sig.level = 0.05 power = 0.9 NOTE: N is the number of observations
Table 2.2 Contingency Table for 2 × 2 × 2 Design
| Exposure | Condition Absent (0) | Condition Present (1) | Total | |
|---|---|---|---|---|
| Males | Yes | 10 | 20 | 30 |
| No | 15 | 5 | 20 | |
| Females | Yes | 13 | 17 | 30 |
| No | 12 | 8 | 20 | |
| Total | 50 | 50 | 100 |
R estimates that a total of approximately 411 subjects are required to achieve power set at 0.90. Such a large sample is required because w = 0.2 constitutes a relatively small effect size (see Cohen (1988) for details).
The reader may ask at this point how one might go about analyzing data for higher‐dimensional frequency tables. The example for the chi‐square test of the data in Table 2.1 is only for that of a 2 × 2 layout. Suppose we added a third factor to our analysis, such as gender, making our contingency table appear as in Table 2.2.
For data such as that in Table 2.2 featuring higher‐dimensional frequency data, log‐linear models are a possibility (Agresti, 2002). Log‐linear models are an option in the wider class of generalized linear models, to be discussed further in Chapter 10, where we discuss in some detail a special case of the generalized linear model called the logistic regression model.
2.3 SENSITIVITY AND SPECIFICITY
Sensitivity and specificity are measures historically used in diagnostic situations but can be applied to other contexts as well. We can easily adapt the data in Table 2.1 to suit a brief discussion of these measures. We keep the same cell frequencies, but modify variable names so the data become a bit more applicable to a discussion of sensitivity and specificity (see Table 2.3).
The sensitivity of the diagnostic instrument is the probability that the test is positive given that the individual has the disease. In the margins, we see that 30 people have the disease, of which 20 were diagnosed with it. Thus, the sensitivity of the test is 20/30 = 0.66. The specificity of the diagnostic instrument is the probability that the test is negative, given that the individual does not have the disease. In the margins, we see that 20 people do not have the disease, of which 15 were diagnosed with not having the disease. Hence, the specificity of the test is 15/20 = 0.75. The overall prevalence of the disease is equal to 30/50 (i.e., 30 people have the disease out of 50). One can also compute what are known as positive and negative predictive values from such tables. For these and other measures useful for diagnostic situations, see Dawson and Trapp (2004).
2.4 SCALES OF MEASUREMENT: NOMINAL, ORDINAL, INTERVAL, RATIO
Recall that in our discussion of the so‐called “soft” versus “hard” sciences in Chapter 1, we concluded that a key principal difference between the two is not necessarily one of different statistical or analytical methods used in drawing conclusions, but rather in the actual material that is subjected to measurement. Though this book is not about measurement per se, we nonetheless wish to review the scales of measurement as first proposed by S.S. Stevens in 1946 (Stevens, 1946).
