Comparison of the accuracy of two tests using relative true positive and false positive fraction

Calculates two-sided Wald confidence intervals and performs a Wald test for the relative true positive fraction (TPF) (sensitivity) and false positive fraction (FPF) (i.e., one minus specificity) of two binary diagnostic tests in a paired study design.

This function is primarily intended for the analysis of paired screen positive studies, i.e. those paired studies where the disease (outcome) is ascertained using the gold standard test only in subjects who screen positive to either or both diagnostic tests. However, this function can also be used with data from standard paired studies, i.e. where the gold standard test is applied to all subjects.

Usage

tpffpf.rel(tab, alpha)

Arguments

tab: an object of class tab.paired.
alpha: significance level alpha used to compute two-sided 100(1-alpha)%-confidence intervals, the default is 0.05.

Value

A list containing:

tpf: a named vector containing rel.tpf (the relative true positive fraction, Test2 vs. Test1), se.log.rel.tpf (the standard error for log(rel.tpf)), lcl.rel.tpf (the lower confidence limit for rel.tpf), ucl.rel.tpf (the upper confidence limit for rel.tpf), and pval.rel.tpf (the p-value from the test for the null hypothesis: relative true positive fraction=1).
fpf: a named vector containing rel.fpf (the relative false positive fraction, Test2 vs. Test1), se.log.rel.fpf (the standard error for log(rel.fpf)), lcl.rel.fpf (the lower confidence limit for rel.fpf), ucl.rel.fpf (the upper confidence limit for rel.fpf), and pval.rel.fpf (the p-value from the test for the null hypothesis: relative false positive fraction=1).
alpha: significance level alpha for 100(1-alpha)%-confidence intervals for rel.tpf and rel.fpf.

Details

If relative true positive fraction>1, the percentage increase in true positive fraction for Test2 relative to Test1 is computed as 100(relative true positive fraction-1)%. If relative true positive fraction<1 the percentage decrease in true positive fraction for Test2 relative to Test1 is computed as 100(1-relative true positive fraction)%. Percentage increase/decrease in false positive fraction is computed in an analogous fashion.

Given the independence of relative TPR and relative TNR, a possible joint 100(1-alpha)% confidence region for {relative TPF, relative FPF} is formed by the rectangle {lcl.rel.tpf, ucl.rel.tpf} x {lcl.rel.fpf, ucl.rel.fpf}, where {lcl.rel.tpf, ucl.rel.tpf} and {lcl.rel.fpf, ucl.rel.fpf} are 100(1-alpha*)% confidence intervals for relative TPF and relative FPF, respectively, and alpha*=1-sqrt(1-alpha).

In screen positive studies, only relative TPF and relative FPF can be estimated from the data. Their constituents, i.e. TPF and FPF for the two tests, are not estimable. Relative specificity is not estimable either. Therefore, sesp.rel should not be used to attempt to estimate those quantities from studies with a paired screen positive design. McNemar's test (sesp.mcnemar) can, however, be used to test the null hypothesis of equality in specificities in paired screen positive studies (Schatzkin et al., 1987).

References

Schatzkin, A., Connor, R. J., Taylor, P. R., & Bunnag, B. (1987). Comparing new and old screening tests when a reference procedure cannot be performed on all screenees: example of automated cytometry for early detection of cervical cancer. American Journal of Epidemiology, 125(4), 672-678.

Cheng, H., & Macaluso, M. (1997). Comparison of the accuracy of two tests with a confirmatory procedure limited to positive results. Epidemiology, 104-106.

Alonzo, T. A., Pepe, M. S., & Moskowitz, C. S. (2002). Sample size calculations for comparative studies of medical tests for detecting presence of disease. Statistics in medicine, 21(6), 835-852.

Examples

# Data from Cheng and Macaluso (Table 2)
breast.cancer.data <- read.tab.paired(
 10, 24, 21, NA,
 13, 144, 95, NA,
 testnames=c("Mammography", "Physical examination")
)
breast.cancer.data
#> Two binary diagnostic tests (paired design)
#> 
#> Test1: 'Mammography'
#> Test2: 'Physical examination'
#> 
#> Diseased:
#>            Test1 pos. Test1 neg. Total
#> Test2 pos.         10         24    34
#> Test2 neg.         21         NA    NA
#> Total              31         NA    NA
#> 
#> Non-diseased:
#>            Test1 pos. Test1 neg. Total
#> Test2 pos.         13        144   157
#> Test2 neg.         95         NA    NA
#> Total             108         NA    NA
#> 
tpffpf.rel.results <- tpffpf.rel(breast.cancer.data)
str(tpffpf.rel.results)
#> List of 3
#>  $ tpf  : Named num [1:5] 1.097 0.207 0.732 1.644 0.655
#>   ..- attr(*, "names")= chr [1:5] "rel.tpf" "se.log.rel.tpf" "lcl.rel.tpf" "ucl.rel.tpf" ...
#>  $ fpf  : Named num [1:5] 1.4537 0.11872 1.15191 1.83457 0.00163
#>   ..- attr(*, "names")= chr [1:5] "rel.fpf" "se.log.rel.fpf" "lcl.rel.fpf" "ucl.rel.fpf" ...
#>  $ alpha: num 0.05
tpffpf.rel.results
#> $tpf
#>        rel.tpf se.log.rel.tpf    lcl.rel.tpf    ucl.rel.tpf   pval.rel.tpf 
#>      1.0967742      0.2066265      0.7315373      1.6443641      0.6548356 
#> 
#> $fpf
#>        rel.fpf se.log.rel.fpf    lcl.rel.fpf    ucl.rel.fpf   pval.rel.fpf 
#>    1.453703704    0.118723652    1.151908920    1.834567318    0.001626358 
#> 
#> $alpha
#> [1] 0.05
#>