Table 1.
Left: notation used in this paper for the 2 × 2 contingency table; Right: an example data of the 2 × 2 contingency table, which are also presented in Fig. 2 and Example 2 in Fig. 5.
Table 2.
P-values obtained from the analysis of example data in Table 1 using several methods.
Fig 1.
The data are shown on the top left panel.
On top right panel, all possible configurations of tables (y1 and y2) are listed when only y+ is known. The corresponding maximum likelihood estimate of the log odds ratio ψ for each possible table, denoted as , is also shown. The nuisance parameter λ* = (n1 π1+n2 π2)/(n1 + n2) is the marginal probability of success among all treated subjects. (A) Contour plot of the likelihood L = L(ψ,λ*;y1,y2), which is the joint likelihood of different values of ψ and λ* given the observed values ofy1 and y2. Lighter colors denote higher values of L; (B) Contour plot of the marginal likelihood L2 = L(ψ,λ*;y+) given the success total y+ as a function of ψ andλ*; (C) The likelihood L given y1 and y2 plotted against ψ at five different fixed values ofλ*. The profile likelihood function is also plotted; (D) The marginal likelihood L2 given y+ plotted against ψ at fixed values of λ*. The conditional likelihood L1 = L(ψ;y1∣y+) is also plotted in red. These graphs demonstrate that for balanced sample sizes the marginal success total tells us virtually nothing about ψ, and hence should be treated as an ancillary statistic.
Fig 2.
See Fig. 1 for an explanation of these panels.
In this example, the sample sizes are the same for both treatments, but the success rates are different.
Fig 3.
See Fig. 1 for an explanation of these panels.
In this example, the treatments have unequal sample sizes. For these tables, the marginal success total still tells us very little about ψ although it is slightly more informative than in balanced tables (see also Fig. 4).
Fig 4.
See Fig. 1 for an explanation of these panels.
This example has both unequal sample sizes and unequal success rates. It is even more extreme than the example in Fig. 3.
Fig 5.
The standardized conditional, modified profile, and profile likelihood functions are depicted for the log odds ratio ψ using the data in Fig. 1, Fig. 2, Fig. 3 and Fig. 4.
The example numbers in this figure correspond to the examples described in Figs. 1–4. The profile likelihood is represented by a dashed black line, while the conditional and modified profile likelihoods are represented by thick red and black dotted lines, respectively. The horizontal lines represent 1/6.8 (upper), 1/8 (middle) and 1/32 (lower) likelihood support intervals (SIs). The maximum likelihood estimate (MLE) of each likelihood was also shown. For normally distributed data, a 1/6.8 SI and a Frequentist 95% confidence interval are identical. Note that the modified profile and conditional likelihoods are indistinguishable for all examples, while the profile and conditional likelihoods are similar for the examples of the null (i.e., ψ = 0 in Examples 1 & 3). In these two examples, the profile likelihood is not visible because it is overlain by the conditional likelihood.
Table 3.
The sample space for the data in Table 1 where n1 = n2 = 10 without conditioning: combinations of y1 and y2 yield 11 × 11 = 121 possible configurations of tables.
The sample space with conditioning on the observed success total is in bold face.
Table 4.
The sample space for the data in Table 1 where n1 = n2 = 10 with conditioning on the success total: there are only 7 possible table configurations.
The observed y1 and y2 are in bold face.