Patients were treated with a drug at four dose levels (100mg, 200mg, 300mg, 400mg) and then monitored for toxicity. The data are tabulated in Figure 12.1.
Notice that there is a natural ordering across both the rows and the columns of the above contingency table. There is also the suggestion that progressively increasing drug i = 1 2 ..., , ,r
Figure 12.1 Crosstabulation of dose-response data
Count
Mild Moderate Severe Death TOXICITY
Drug Dose * TOXICITY Crosstabulation
4 4×
Jonckheere-Terpstra Test
Figure 12.1 shows the data in crosstabulated format to illustrate the concept of applying the Jonckheere-Terpstra test to doubly ordered tables, however this test is obtained from the Nonparametric Tests procedure, and your data must be structured appropriately for Nonparametric Tests. Figure 12.2 shows a portion of these data displayed in the Data Editor. The data consist of two variables. Dose is an ordered grouping variable that indicates dose level, and toxicity is an ordered categorical variable with four values, where 1=Mild, 2=Moderate, 3=Severe, and 4=Death. Note that although value labels are displayed, these variables must be numeric. This is a large data set, with 227 cases, and therefore Figure 12.2 shows only a small subset of these data in order to illustrate the necessary data structure for the Jonckheere-Terpstra test. The full data set was used in the following example.
You can run the Jonckheere-Terpstra test on the dose-response data shown in Figure 12.2. The results are shown in Figure 12.3.
Figure 12.2 Dose-response data, displayed in the Data Editor
Doubly Ordered R x C Contingency Tables 163
The value of the observed test statistic, defined by Equation 8.38, is , the mean is , the standard deviation is 181.8, and the standardized test statistic, cal-culated by Equation 8.41, is . The standardized statistic is normally distributed with a mean of 0 and a variance of 1, while its square is chi-square distributed with one degree of freedom.
The asymptotic two-sided p values are evaluated as the tail areas under a standard normal distribution. In calculating the one-sided p value, which is not displayed in the output, a choice must be made as to whether to select the left tail or the right tail at the observed value . In Exact Tests, this decision is made by selecting the tail area with the smaller probability. Thus, the asymptotic one-sided p value is calculated as
Equation 12.5 where is the tail area from to z under a standard normal distribution. In the present example, it is the right tail area that is the smaller of the two, so that the asymp-totic one-sided p value is evaluated as the normal approximation to , which works out to 0.0490. The asymptotic two-sided p value is defined as double the one-sided:
Equation 12.6 Since the square of a standard normal variate is a chi-square variate with one degree of freedom, an equivalent alternative way to compute the asymptotic two-sided p value is to evaluate the tail area to the right of from a chi-square distribution with one degree of freedom. It is easy to verify that this too will yield 0.099 as the asymptotic two-sided p value.
The exact one-sided p value is computed as the smaller of two permutational probabilities:
Equation 12.7
4 227 9127.000 8827.500 181.760 1.648 .099 .100 .049 .000
TOXICITY
Grouping Variable: Drug Dose 1.
Figure 12.3 Results of Jonckheere-Terpstra test for dose-response data
t = 9127
In the present example, the smaller permutational probability is the one that evaluates the right tail. It is displayed on the screen as . The exact one-sided p value is the point probability . This probability, 0.000, is a natural measure of the discreteness of the test statistic. Some statisticians advocate subtracting half its value from the exact p value, thereby yielding a less conservative mid-p value.
(See Lancaster, 1961; Pratt and Gibbons, 1981; and Miettinen, 1985 for more information on the role of the mid-p value in exact inference.) Equation 12.8 defines the exact two-sided p value
Equation 12.8 Notice that this definition will produce the same answer as Equation 9.4, with
for all .
Sometimes the data set is too large for an exact analysis, and the Monte Carlo method must be used instead. Figure 12.4 displays an unbiased estimate of the exact one- and two-sided p value for the Jonckheere-Terpstra test based on a crude Monte Carlo sample of 10,000 tables from the reference set.
The Monte Carlo point estimate of the exact one-sided p value is 0.051, which is very close to the exact one-sided p value of 0.049. Moreover, the Monte Carlo method also produces a confidence interval for the exact p value. Thus, although this point estimate might change slightly if you resample with a different starting seed or a different random number generator, you can be 99% confident that the exact p value is contained in the interval 0.045 to 0.057. The Monte Carlo point estimate of the exact two-sided p value is 0.101, and the corresponding 99% confidence interval is 0.093 to 0.109. More tables could be sampled from the reference set to further narrow the widths of these intervals.
Pr(T*≥1.65) = 0.049 Pr(T*= 1.65)
p2 = Pr(T* ≥1.648) = 0.100
D y( ) = (T∗( )y )2 y∈Γ
Figure 12.4 Monte Carlo results for Jonckheere-Terpstra test for dose-response data
4 227 9127.000 8827.500 181.760 1.648 .099 .1012 .093 .109 .0512 .045 .057
TOXICITY Monte Carlo Sig. (2-tailed)
Sig. Monte Carlo Sig. (1-tailed) Jonckheere-Terpstra Test1
Grouping Variable: Drug Dose 1.
Based on 10000 sampled tables with starting seed 2000000.
2.
1. Grouping Variable: Drug Dose
2. Based on 10000 sampled tables with starting seed 2000000.
Jonckheere-Terpstra Test1
Doubly Ordered R x C Contingency Tables 165