Rejection Region for Paired Binary Data

Provides the rejection region of a test comparing two treatments on a binary endpoint in a matched-pairs design, such as a cross-over trial or a comparison of two diagnostic procedures on the same patients. Six tests are supported. Each can be applied with a one-sided or a two-sided alternative, and the score test also tests non-inferiority and equivalence with a margin on the scale of the difference of the response probabilities.

Usage

PairedRR(
  N,
  alpha,
  Test,
  alternative = c("greater", "less", "two.sided", "equivalence"),
  n.grid = 100,
  bb.gamma = 0,
  margin = 0,
  ref.pvalue = FALSE
)

Arguments

N: Number of pairs
alpha: Level of significance for the alternative specified by alternative
Test: Type of statistical test. Options: 'McNemar', 'McNemar-cc', 'McNemar-exact', 'McNemar-midP', 'McNemar-uncond' or 'Paired-Score'
alternative: Direction of the alternative hypothesis. Options: 'greater' (default) for p10 - p01 > -margin, 'less' for p10 - p01 < margin, 'two.sided' for p10 != p01, or 'equivalence' for -lower < p10 - p01 < upper
n.grid: Number of grid points used to search over the probability of a discordant pair in the test 'McNemar-uncond'. Default is 100
bb.gamma: Confidence level parameter of the Berger-Boos procedure for the test 'McNemar-uncond'. The default of 0 disables the procedure
margin: Margin on the scale of the difference p10 - p01. The default of 0 gives a test of superiority, and a value other than 0 tests the null hypothesis p10 - p01 <= -margin against p10 - p01 > -margin when alternative is 'greater', and p10 - p01 >= margin against p10 - p01 < margin when it is 'less'. A negative value tests for superiority by more than its absolute value. A margin other than 0 requires Test = 'Paired-Score' and a one-sided alternative. For alternative = 'equivalence' the margin is the pair c(lower, upper), both in (0, 1), or a single value used for both, and the null hypothesis p10 - p01 <= -lower or p10 - p01 >= upper is tested
ref.pvalue: Logical. If TRUE, the maximization over the probability of a discordant pair in the test 'McNemar-uncond' is refined between the grid points, see Details. Default is FALSE

Value

An object of class bbssr_prr, a logical matrix of dimension (N + 1) by (N + 1) whose entry [a + 1, b + 1] is TRUE when the null hypothesis is rejected with a pairs in which only treatment 1 responds and b pairs in which only treatment 2 responds. Entries with a + b > N are impossible outcomes and are FALSE. The p-values are stored in the attribute p.value, with NA at the impossible outcomes, and the design settings as further attributes

Details

A pair contributes one of four outcomes: both treatments respond, only treatment 1 responds (probability p10), only treatment 2 responds (probability p01), or neither responds. The difference of the response probabilities of the two treatments is p10 - p01, and only the numbers a and b of the two kinds of discordant pairs enter the tests. The function supports the following tests:

The asymptotic McNemar test (McNemar), which refers Z = (a - b) / sqrt(a + b) to the standard normal distribution without a continuity correction, with Z = 0 when a + b = 0
The asymptotic McNemar test with a continuity correction (McNemar-cc), formula (2) of Fagerland et al. (2013), which refers (|a - b| - 1) / sqrt(a + b) to the standard normal distribution, and (a - b - 1) / sqrt(a + b) for the alternative 'greater', with the p-value 1 when a + b = 0
The exact conditional McNemar test (McNemar-exact), which refers a to the binomial distribution with a + b trials and probability 1/2
The mid-p version of the exact conditional test (McNemar-midP), formulae (5) and (6) of Fagerland et al. (2013)
The exact unconditional test of Suissa and Shuster (1991) (McNemar-uncond), which orders the outcomes by Z and maximizes the null tail probability over the probability psi = p10 + p01 of a discordant pair, the test Z of Berger and Sidik (2003)
The score test of Nam (1997) and Tango (1998) for the null hypothesis p10 - p01 = -margin (Paired-Score), which refers (a - b + N margin) / sqrt(N (2 q01 - margin (margin + 1))) to the standard normal distribution, where q01 is the maximum likelihood estimate of p01 under the null hypothesis, formulae (24) to (26) of Tango (1998). With margin = 0 it is the asymptotic McNemar test

The conditional distribution of a given a + b is symmetric under the null hypothesis, so the usual conventions for the two-sided p-value of a conditional test agree. The two-sided p-value of the exact conditional test is twice the smaller tail probability, at most 1, and the two-sided mid-p value follows formulae (5) and (6) of Fagerland et al. (2013). The two-sided versions of 'McNemar', 'McNemar-uncond' and 'Paired-Score' order the outcomes by the absolute value of the statistic. The two-sided p-value of 'McNemar-cc' is twice the smaller of its two one-sided p-values, at most 1, which is the p-value of stats::mcnemar.test with correct = TRUE. Its one-sided p-value is the normal approximation with a continuity correction to the tail probability of the exact conditional test, and Fagerland et al. (2013) set the p-value to 1 when a + b = 0, where the statistic is not defined.

The test 'McNemar-uncond' maximizes the tail probability of each outcome, a polynomial of degree N in psi, over a grid of n.grid points. The maximum on the grid can fall below the maximum over the unit interval. With ref.pvalue = TRUE the grid is extended by points equally spaced on the arcsine square-root scale and every local maximum on the grid is refined by a golden-section search between its two neighbouring grid points, as in BinaryRR. With a positive bb.gamma the maximization is restricted to the Clopper-Pearson interval of level 1 - bb.gamma for psi computed from a + b, and bb.gamma is added, the confidence interval p-value of Berger and Sidik (2003).

The alternative 'less' is handled by exchanging a and b. With alternative = 'equivalence' the null hypothesis is tested by two one-sided score tests, against 'greater' with the margin lower and against 'less' with the margin upper, each at the level alpha, and the p-value of an outcome is the larger of the two one-sided p-values.

The p-values are kept for the rest of the session and reused by later calls with the same number of pairs and test, see bbssr-package. They form a matrix of (N + 1)^2 values, which takes about 2 GB of memory for 16,000 pairs, so the memory and the time of the exact calculations grow with the square of N.

References

Berger RL, Sidik K (2003). Exact unconditional tests for a 2 x 2 matched-pairs design. Statistical Methods in Medical Research, 12(2), 91-108.

Fagerland MW, Lydersen S, Laake P (2013). The McNemar test for binary matched-pairs data: mid-p and asymptotic are better than exact conditional. BMC Medical Research Methodology, 13, 91.

Nam J (1997). Establishing equivalence of two treatments and sample size requirements in matched-pairs design. Biometrics, 53(4), 1422-1430.

Suissa S, Shuster JJ (1991). The 2 x 2 matched-pairs trial: exact unconditional design and analysis. Biometrics, 47(2), 361-372.

Tango T (1998). Equivalence test and confidence interval for the difference in proportions for the paired-sample design. Statistics in Medicine, 17(8), 891-908.

Examples

RR <- PairedRR(N = 20, alpha = 0.025, Test = 'McNemar-exact')
print(RR)
#> Rejection region for paired binary data
#> 
#>   Test          : McNemar-exact
#>   Alternative   : greater
#>   Pairs         : N = 20
#>   Alpha         : 0.025
#> 
#>   Rejected outcomes: 47 of 231
#> 
#>      b=0 b=1 b=2 b=3 b=4 b=5 b=6 b=7 b=8 b=9 b=10 b=11 b=12 b=13 b=14 b=15 b=16
#> a=0  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .    .   
#> a=1  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .    .   
#> a=2  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .    .   
#> a=3  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .    .   
#> a=4  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .    .   
#> a=5  .   .   .   .   .   .   .   .   .   .   .    .    .    .    .    .        
#> a=6  X   .   .   .   .   .   .   .   .   .   .    .    .    .    .             
#> a=7  X   .   .   .   .   .   .   .   .   .   .    .    .    .                  
#> a=8  X   X   .   .   .   .   .   .   .   .   .    .    .                       
#> a=9  X   X   .   .   .   .   .   .   .   .   .    .                            
#> a=10 X   X   X   .   .   .   .   .   .   .   .                                 
#> a=11 X   X   X   .   .   .   .   .   .   .                                     
#> a=12 X   X   X   X   .   .   .   .   .                                         
#> a=13 X   X   X   X   X   .   .   .                                             
#> a=14 X   X   X   X   X   .   .                                                 
#> a=15 X   X   X   X   X   X                                                     
#> a=16 X   X   X   X   X                                                         
#> a=17 X   X   X   X                                                             
#> a=18 X   X   X                                                                 
#> a=19 X   X                                                                     
#> a=20 X                                                                         
#>      b=17 b=18 b=19 b=20
#> a=0  .    .    .    .   
#> a=1  .    .    .        
#> a=2  .    .             
#> a=3  .                  
#> a=4                     
#> a=5                     
#> a=6                     
#> a=7                     
#> a=8                     
#> a=9                     
#> a=10                    
#> a=11                    
#> a=12                    
#> a=13                    
#> a=14                    
#> a=15                    
#> a=16                    
#> a=17                    
#> a=18                    
#> a=19                    
#> a=20                    
#> 
#>   a: pairs in which only treatment 1 responds
#>   b: pairs in which only treatment 2 responds
#>   X marks rejection of the null hypothesis

# Non-inferiority with a margin of 0.1 by the score test
RR <- PairedRR(N = 44, alpha = 0.05, Test = 'Paired-Score', margin = 0.1)
sum(RR)
#> [1] 448

# \donttest{
# Exact unconditional test with the Berger-Boos procedure
RR <- PairedRR(N = 30, alpha = 0.05, Test = 'McNemar-uncond', bb.gamma = 0.0005)
plot(RR)

# }