Appendix 1. Statistical methods
A short summary of some less common statistical methods used in this review is given below. A more extensive description of the model used to correct for imperfect reference standards can be found in Menten 2013.
1) Latent class analysis (LCA)
LCA is a modelling technique that can be used in situations in which there is no good reference standard. It assumes that the true disease status in a study population is unknown (or latent). The LCA model estimates the sensitivity and specificity of a set of diagnostic tests (A, B, C, …) on the basis of observed frequencies in test patterns (ABC+++, ABC++-, ABC+-+,…). As such, the LCA model provides a model-based estimate of the gold standard classification; ie the best way to group study participants in diseased or non-diseased.
The basic latent class model assumes that the observed variables are conditionally independent. This means that there should be no associations between the results of the diagnostic tests within each category of the latent variable (disease status). If this assumption does not hold, more advanced techniques (eg based on Bayesian statistical methodology) can be used. To be selected for this review, studies using LCA had to assess the conditional independence assumption between the diagnostic tests, and if conditional dependence was expected, they had to use appropriate statistical methods to take this into account. If a study was selected, the sensitivity and specificity estimates derived from the final LCA model were included in this review.
More information can be found in Hui 1980, Black 2002, Branscum 2005, and Baughman 2008 among other references.
2) The complementary Log-Log function
In the bivariate model a "link" function g(y) is used to allow the use of the Normal (Gaussian) distribution to model the underlying study-specific sensitivity (Se) and specificity (Sp) of each study included in the meta-analysis. The standard link function used in the bivariate model is the logit link:
g(y) = log (y/(1-y))
An alternative link function is the complementary log log (cloglog) link:
g(y) = log(-log(1-y))
Both link functions transform Se and Sp, which are in the interval [0,1], to any real number between minus infinity and plus infinity.
The advantage of the cloglog link with our data is that it approaches infinity less quickly when y approaches 1 and consequently it mitigates the influence of studies that report 100% Se or Sp. This also reduces the inflation of the random-effects standard deviations as is apparent from comparison of Figure 6 and Figure 10. With the logit link, the prediction region extends to below the line of no diagnostic value (Se + Sp < 1), while the study which reports the lowest diagnostic value has Se = 0.75 and Sp = 0.70 (Figure 10). On the other hand, some observed data with high Se and Sp are not contained within the prediction region. The prediction region of the model with the cloglog link contains all observed data points, while not extending far beyond the studies with lowest observed Se and Sp (Figure 6). This is reflected in a lower deviance information criterion (DIC), a measure of model fit, for the cloglog model formulation compared to the logit formulation. The model with the lowest DIC shows the best fit to the data.
3) WinBUGS code for the primary model
WinBUGS is a statistical software for Bayesian analysis using Markov chain Monte Carlo methods. WinBUGS (or its recent open-source version OpenBUGS) provides a flexible Bayesian framework for model fitting. It can be used to fit both the bivariate and HSROC models.
Below is the code to fit the basic bivariate model, allowing for data from studies that use a reference standard and for studies that use latent class analysis.
Assuming there are N=N1+N2 studies:
The code is as follows:
model{
# Binomials for Studies that Use a Reference Standard
# (where the reference standard is presumed to be perfect)
for(i in 1:N1){
TP[i] ˜ dbin(SE[i],NDiseased[i])
TN[i] ˜ dbin(SP[i],NNotDiseased[i])
}
# Normals for Studies that Use LCA
for(i in 1:N2){
Y1[i] ˜ dnorm(alpha[i+N1,1],W1[i])
Y2[i] ˜ dnorm(alpha[i+N1,2],W2[i])
}
# Bivariate normals for g(Se) and g(Sp)
for(i in 1:N){
# Implement the link function g()
# If logit is used
logit(SE[i]) <- alpha[i,1]
logit(SP[i]) <- alpha[i,2]
OR
# If cloglog is used
SE[i] <- 1-exp(-exp(alpha[i,1]))
SP[i] <- 1-exp(-exp(alpha[i,2]))
# Specify the bivariate normal
alpha[i,1:2] ˜ dmnorm(mu[],R[,])
}
# Specify Non-Informative Priors
# for means:
mu[1] ˜ dnorm(0.0,.37)
mu[2] ˜ dnorm(0.0,.37)
# for variance-covariance matrix
R[1:2,1:2] ˜ dwish(Omega[,], 2) }
Note: Prior for Omega are provided as data:
Omega = matrix(c(.001,0,0,.001),nrow=2,byrow=T)
Appendix 3. Interpretation of the clinical relevance of the findings of this review - predictive values and likelihood ratios
What follows is an interpretation of the clinical relevance of the findings of this review regarding the rK39 immunochromatographic test (ICT) when it is used to diagnose visceral leishmaniasis (VL) among patients with febrile splenomegaly and no previous history of VL.
Indian subcontinent
When the rK39 ICT is used in the Indian subcontinent, in a setting where the prior probability of VL among clinical suspects is 40%, which is typically seen in a peripheral health centre in an endemic area, the positive predictive value of the test is 87%. This means that out of 100 patients with a positive rK39 result, 87 would have VL (true positive result) and 13 would have another disease (false positive). The negative predictive value is 98%, meaning that out of 100 patients with a negative rK39 ICT result, 98 would have another disease (true negative) and 2 would have VL (false negative).
When the same test is used in a setting with a prior probability of VL of 60%, which is more typical for a referral centre in an endemic area, the positive predictive value is 94% and the negative predictive value is 95%.
A likelihood ratio is another way of expressing how informative a diagnostic test is: it indicates to what extent the rK39 ICT result changes the odds that a patient has VL. The likelihood ratio of a positive rK39 ICT result is 9.90, and the likelihood ratio of a negative test result is 0.03. This means that in the Indian subcontinent, a positive rK39 ICT result is a strong argument in favour of VL (ruling in) and that a negative rK39 ICT result is a strong argument against VL (ruling out).
East Africa
When the rK39 ICT is used in east Africa, in a setting where the prior probability of VL is 40%, which is typically seen in a peripheral health centre in an endemic area, the positive predictive value of the test is 86%. This means that out of 100 patients with a positive rK39 ICT result, 86 would have VL (true positive result) and 14 would have another disease (false positive). The negative predictive value is 90%, meaning that out of 100 patients with a negative rK39 ICT result, 90 would have another disease (true negative) and 10 would have VL (false negative).
When the same test is used in a setting with a prior probability of VL of 60%, which is more typical for a referral centre in an endemic area, the positive predictive value is 93% and the negative predictive value is 81%.
In east Africa, the likelihood ratio of a positive rK39 ICT result is 9.58, and the likelihood of a negative rk39 ICT result is 0.16. This means that a positive rK39 ICT result is strong argument in favour of VL (ruling in), and that a negative rK39 ICT result is not an absolute argument against VL (does not allow to rule out VL completely).