This page provides supplementary material to the paper

DOI: 10.1017/S0950268816002028

**Abstract:**
The risk ratio or relative risk quantifies the risk of disease in a study population relative to a reference population. Standard methods of estimation and testing assume a perfect diagnostic test having sensitivity and specificity of 100%. However, this assumption typically does not hold, and this may invalidate naive estimation and testing for the risk ratio. We propose procedures that control for sensitivity and specificity of the diagnostic test, given the risks are measured by proportions, as it is in cross-sectional studies or studies with fixed follow-up time (in this case the risk ratio is also called prevalence risk ratio or simply prevalence ratio). The proposed procedures provide an exact unconditional test and an exact two-sided confidence interval for the true risk ratio. The methods also cover the case when sensitivity and specificity differ in the two groups (differential misclassification). The resulting test and confidence interval may be useful in epidemiological studies as well as in clinical and vaccine trials. We illustrate the method with real life examples which demonstrate that ignoring sensitivity and specificity of the diagnostic test may lead to considerable bias in the estimated risk ratio.

Full text from the publisher

Author's original version of the manuscript

R functions (If you use them in a publication, please cite our paper.)

On-line calculator (test version)

**Two further real life applications**

to illustrate why correcting the risk ratio for Se and Sp is important

King et al. (2012, Veterinary Parasitology, 187, 85-92) studied the seroprevalence of*Neospora caninum* in dogs in different areas in Australia. They used an ELISA
test with an optimized cut-off value resulting in sensitivity = specificity = 67.8%.

For the comparison of infection rates in different areas, one usually selects a reference region and calculates the risk ratios relative to that for all other regions. For illustration, let us select now the region with the lowest prevalence as a reference, which is Yuendumu, Northern Territory, with an observed (apparent) seroprevalence of 27% (41/151), and calculate the RR for Goodooga, New South Wales, where 10 out of the 16 tested dogs (63%) were found to be positive.

The ratio of the apparent prevalences is 10/16 / (41/151) = 2.30 (95% CI: 1.27 to 3.43). The adjustment for sensitivity and specificity results in dramatic changes in both the point estimate and the confidence interval. Since the true prevalence in the reference area is estimated as 0, the estimated true RR is infinite, and the 95% CI for the true RR expands from 6.26 to infinity. Notice that the CIs for the apparent and true RR do not even overlap.

The following computer output shows the results by the R function RRSeSp.CI06. The confidence interval for the apparent prevalence is obtained by setting sensitivity and specificity equal to 1. The function calls are displayed in red and the results in blue.

Wu et al. (2014, BMC Veterinary Research, 10:100) studied the prevalence of*Lawsonia intracellularis* infection in pig farms in China.
They used an ELISA test with sensitivity and specificity of 72% and 93%, respectively.

Let us compare the infection status of two provinces by the prevalence risk ratio, one province is selected from the Southern part of the country, Guanxi (35%, 43/124) and another from the central part, Henan (63%, 118/187).

The apparent RR is 1.82 (95% CI: 1.41 to 2.41). Correction for sensitivity and specificity results in a true RR of 2.03 (95 % CI: 1.43 to 3.06). Although the difference is not as dramatic as in the previous example, it is not at all negligible.

The computer output with the results:

Reiczigel J, Singer J, Lang Zs (2017)
Exact inference for the risk ratio with an

imperfect diagnostic test

Epidemiology and Infection, Volume 145, Issue 1, 187-193.
imperfect diagnostic test

DOI: 10.1017/S0950268816002028

Full text from the publisher

Author's original version of the manuscript

R functions (If you use them in a publication, please cite our paper.)

On-line calculator (test version)

to illustrate why correcting the risk ratio for Se and Sp is important

King et al. (2012, Veterinary Parasitology, 187, 85-92) studied the seroprevalence of

For the comparison of infection rates in different areas, one usually selects a reference region and calculates the risk ratios relative to that for all other regions. For illustration, let us select now the region with the lowest prevalence as a reference, which is Yuendumu, Northern Territory, with an observed (apparent) seroprevalence of 27% (41/151), and calculate the RR for Goodooga, New South Wales, where 10 out of the 16 tested dogs (63%) were found to be positive.

The ratio of the apparent prevalences is 10/16 / (41/151) = 2.30 (95% CI: 1.27 to 3.43). The adjustment for sensitivity and specificity results in dramatic changes in both the point estimate and the confidence interval. Since the true prevalence in the reference area is estimated as 0, the estimated true RR is infinite, and the 95% CI for the true RR expands from 6.26 to infinity. Notice that the CIs for the apparent and true RR do not even overlap.

The following computer output shows the results by the R function RRSeSp.CI06. The confidence interval for the apparent prevalence is obtained by setting sensitivity and specificity equal to 1. The function calls are displayed in red and the results in blue.

> RRSeSp.CI06(k1=41,n1=151,k2=10,n2=16,Se1=1,Sp1=1,

plot=F,print=T,level=.95)

Observed (apparent) p1 = 0.272 , p2 = 0.625 , RR(=p2/p1) = 2.3

Estimated true p1 = 0.272 , p2 = 0.625 , RR(=p2/p1) = 2.3

95 % CI: 1.266 3.428

> RRSeSp.CI06(k1=41,n1=151,k2=10,n2=16,Se1=.678,Sp1=.678,

plot=F,print=T,level=.95)

Observed (apparent) p1 = 0.272 , p2 = 0.625 , RR(=p2/p1) = 2.3

Estimated true p1 = 0 , p2 = 0.851 , RR(=p2/p1) = Inf

95 % CI: 6.258 Inf

Wu et al. (2014, BMC Veterinary Research, 10:100) studied the prevalence of

Let us compare the infection status of two provinces by the prevalence risk ratio, one province is selected from the Southern part of the country, Guanxi (35%, 43/124) and another from the central part, Henan (63%, 118/187).

The apparent RR is 1.82 (95% CI: 1.41 to 2.41). Correction for sensitivity and specificity results in a true RR of 2.03 (95 % CI: 1.43 to 3.06). Although the difference is not as dramatic as in the previous example, it is not at all negligible.

The computer output with the results:

> RRSeSp.CI06(k1=43,n1=124,k2=118,n2=187,Se1=1,Sp1=1,

plot=F,print=T,level=.95)

Observed (apparent) p1 = 0.347 , p2 = 0.631 , RR(=p2/p1) = 1.82
Estimated true p1 = 0.347 , p2 = 0.631 , RR(=p2/p1) = 1.82

95 % CI: 1.408 2.411

> RRSeSp.CI06(k1=43,n1=124,k2=118,n2=187,Se1=.72,Sp1=.93,

plot=F,print=T,level=.95)

Observed (apparent) p1 = 0.347 , p2 = 0.631 , RR(=p2/p1) = 1.82

Estimated true p1 = 0.426 , p2 = 0.863 , RR(=p2/p1) = 2.03

95 % CI: 1.431 3.059