Computing the confidence intervals for predictive values

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Dadong Zhang, Jingye Wang, Suqin Cai, and Johan Surtihadi

2024-04-12

Introduction

CIfinder is an R package intended to provide functions to compute confidence intervals for the positive predictive value (PPV) and negative predictive value (NPV) based on varied scenarios. In situations where the proportion of diseased subjects does not correspond to the disease prevalence (e.g. case-control studies), this package provides two types of solutions: I) five methods to estimate confidence intervals for PPV and NPV via ratio of two binomial proportions, including Gart & Nam (1988) https://doi.org/10.2307/2531848, Walter (1975) https://doi.org/10.1093/biomet/62.2.371, MOVER-J (Laud, 2017) https://doi.org/10.1002/pst.1813, Fieller (1954) https://www.jstor.org/stable/2984043, and Bootstrap (Efron, 1979) https://doi.org/10.1201/9780429246593; II) three direct methods to compute the confidence intervals, including Pepe (2003) https://doi.org/10.1002/sim.2185, Zhou (2007) doi:10.1002/sim.2677, and Delta https://doi.org/10.1002/sim.2677. In prospective studies where the proportion of diseased subjects provides an unbiased estimate of the disease prevalence, the confidence intervals for PPV and NPV can be estimated by methods that are applicable to single proportions. In this case, the package provides six methods for cocmputing the confidence intervals: “clopper.pearson”, “wald”, “wilson”, “wilson.correct”, “agresti”, and “beta”.For more information, please see the Details and References sections in the user’s manual.

Installation

You can install the latest version of CIfinder by:

install.packages("CIfinder")

Example use

To demonstrate the utility of the CIfinder::ppv_npv_ci() function for calculating the confidence intervals for PPV and NPV, we will use a case-control study published by van de Vijver et al. (2002) https://doi.org/10.1056/NEJMoa021967. In the study, Cases were defined as those that have metastasis within 5 years of tumour excision, while controls were those that did not. Each tumor was classified as having a good or poor gene signature which was defined as having a signature correlation coefficient above or below the ‘optimized sensitivity’ threshold, respectively. This ‘optimized sensitivity’ threshold is defined as the correlation value that would result in a misclassification of at most 10 per cent of the cases. The performance of their 70-gene signature as prognosticator for metastasis is summarized in Table 1 below:

Table 1. Breast cancer data from a molecular signature classifier
	Case	Control
Poor_Signature	31	12
Good_Signature	3	32
Total	34	44

Generate the confidence interval for \(PPV\) or \(NPV\)

Since this was a case-control study, the proportion of cases (34/(34+44)=43.6%) is not an unbiased estimate of its prevalence (assumed 7%). PPV and NPV and their confidence intervals can’t be estimated directly. To calculate the confidence interval based “gart and nam” method:

library(CIfinder)
ppv_npv_ci(x1 = 31, n1 = 34, x0 = 32, n0 = 44, prevalence = 0.07,
           method = "gart and nam")
#> $method
#> [1] "gart and nam"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 0.7272727
#> 
#> $phi_ppv
#> phi_ppv_est   phi_ppv_l   phi_ppv_u phi_ppv_mle 
#>   0.2991202   0.1715910   0.4656310   0.3009710 
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u   ppv_mle 
#> 0.2010444 0.1391548 0.3049051 0.2000554 
#> 
#> $phi_npv
#> phi_npv_est   phi_npv_l   phi_npv_u phi_npv_mle 
#>   0.1213235   0.0323270   0.3090900   0.1266740 
#> 
#> $npv
#>   npv_est     npv_l     npv_u   npv_mle 
#> 0.9909508 0.9772641 0.9975727 0.9905554

In this output, phi_ppv denotes \(\phi_{PPV}=\frac{1-specificity}{sensitivity}\) in the function to estimate \(PPV=\frac{\rho}{\rho+(1-\rho)\phi_{PPV}}\), where \(\rho\) is the prevalence. Similarly, phi_npv denotes \(\phi_{NPV}=\frac{1-sensitivity}{specificity}\) in the function to estimate \(NPV=\frac{1-\rho}{(1-\rho)+\rho\phi_{NPV}}\). ppv and npv provide the results for the estimates, lower confidence limit, upper confidence limit and the maximum likelihood estimate based on score method described in the Gart and Nam paper.

To calculate the confidence interval based “zhou’s method”

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 32, n0 = 44, prevalence = 0.07,
           method = "zhou")
#> $method
#> [1] "zhou"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 0.7272727
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 0.2010444 0.1217419 0.2803468 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9909508 0.9811265 1.0007750 
#> 
#> $ppv_logit_transformed
#>   ppv_est     ppv_l     ppv_u 
#> 0.2010444 0.1331384 0.2919216 
#> 
#> $npv_logit_transformed
#>   npv_est     npv_l     npv_u 
#> 0.9909508 0.9734141 0.9969560

In this output, since continuity correction isn’t specified, the estimates and confidence intervals in ppv and npv are same as the standard delta method where the ppv_logit_transformed and npv_logit_transformed refer the standard logit method described in the paper. If continuity.correction is being specified:

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 32, n0 = 44, prevalence = 0.07,
           method = "zhou",
           continuity.correction = TRUE)
#> $method
#> [1] "zhou"
#> 
#> $sensitivity
#> [1] 0.8699646
#> 
#> $specificity
#> [1] 0.7090237
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 0.1836999 0.1148465 0.2525533 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9863836 0.9750497 0.9977175 
#> 
#> $ppv_logit_transformed
#>   ppv_est     ppv_l     ppv_u 
#> 0.1836999 0.1244835 0.2626354 
#> 
#> $npv_logit_transformed
#>   npv_est     npv_l     npv_u 
#> 0.9863836 0.9688986 0.9940985

In this case, a continuity correction value \(\frac{z_{\alpha/2}^2}{2}\) is applied. The ppv and npv outputs refer the “Adjusted” in the paper and ppv_logit_transformed and npv_logit_transformed denote the “Adjusted logit” method described in the paper.

Also comparing to the Bootstrap methods:

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 32, n0 = 44, prevalence = 0.07,
           method = "boot")
#> $method
#> [1] "boot"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 0.7272727
#> 
#> $phi_ppv
#> phi_ppv_est   phi_ppv_l   phi_ppv_u 
#>   0.2991202   0.1680917   0.4713499 
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 0.2010444 0.1376989 0.3092894 
#> 
#> $phi_npv
#> phi_npv_est   phi_npv_l   phi_npv_u 
#>  0.12132353  0.03216665  0.31249927 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9909508 0.9770191 0.9975847

And Fieller method

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 32, n0 = 44, prevalence = 0.07,
           method = "fieller")
#> $method
#> [1] "fieller"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 0.7272727
#> 
#> $phi_ppv
#> phi_ppv_est   phi_ppv_l   phi_ppv_u 
#>   0.2991202   0.1538974   0.4509565 
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 0.2010444 0.1430353 0.3284463 
#> 
#> $phi_npv
#> phi_npv_est   phi_npv_l   phi_npv_u 
#>   0.1213235   0.0000000   0.2606399 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9909508 0.9807594 1.0000000

Confidence interval for special cases

Assume we have a special case data where \(x_0=n_0\):

Table 1. Example of a special testing data
	Case	Control
Poor_Signature	31	0
Good_Signature	3	44
Total	34	44

In this situation, Pepe, Delta,Zhou (standard logit), and boot methods can not be used without continuity correction. Walter method can be used, but there may have skewness concerns. gart and nam and mover-j could be considered.

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 44, n0 = 44, prevalence = 0.07,
           method = "gart and nam")
#> $method
#> [1] "gart and nam"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 1
#> 
#> $phi_ppv
#> phi_ppv_est   phi_ppv_l   phi_ppv_u phi_ppv_mle 
#>    0.000000    0.000000    0.067785    0.004121 
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u   ppv_mle 
#> 1.0000000 0.5261573 1.0000000 0.9480916 
#> 
#> $phi_npv
#> phi_npv_est   phi_npv_l   phi_npv_u phi_npv_mle 
#>  0.08823529  0.02376800  0.21956500  0.09223300 
#> 
#> $npv
#>   npv_est     npv_l     npv_u   npv_mle 
#> 0.9934025 0.9837423 0.9982142 0.9931056

Comparing to the Walter output:

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 44, n0 = 44, prevalence = 0.07,
           method = "walter")
#> $method
#> [1] "walter"
#> 
#> $sensitivity
#> [1] 0.9117647
#> 
#> $specificity
#> [1] 1
#> 
#> $phi_ppv
#>          phi        lower        upper 
#> 0.0000000000 0.0007803411 0.1940673904 
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 1.0000000 0.2794604 0.9897390 
#> 
#> $phi_npv
#>        phi      lower      upper 
#> 0.08823529 0.03758016 0.27386671 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9934025 0.9798027 0.9971794

Note, for walter, no continuity.correction should be used as 0.5 has been used as described by the original paper.

Also comparing to the Zhou’s adjusted methods:

ppv_npv_ci(x1 = 31, n1 = 34, x0 = 44, n0 = 44, prevalence = 0.07,
           method = "zhou",
           continuity.correction = TRUE)
#> $method
#> [1] "zhou"
#> 
#> $sensitivity
#> [1] 0.8699646
#> 
#> $specificity
#> [1] 0.9598522
#> 
#> $ppv
#>   ppv_est     ppv_l     ppv_u 
#> 0.6199169 0.2921698 0.9476640 
#> 
#> $npv
#>   npv_est     npv_l     npv_u 
#> 0.9899059 0.9816510 0.9981609 
#> 
#> $ppv_logit_transformed
#>   ppv_est     ppv_l     ppv_u 
#> 0.6199169 0.2886800 0.8676335 
#> 
#> $npv_logit_transformed
#>   npv_est     npv_l     npv_u 
#> 0.9899059 0.9772354 0.9955563

Feedback and Report issues

We appreciate any feedback, comments and suggestions. If you have any questions or issues to use the package, please reach out to the developers.

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.