Sensitivity and specificity of multiple parallel diagnostic tests. What is the net sensitivity and the net specificity when tests that may have different sensitivities and specificities are combined to diagnose a particular disease or condition? If any test is positive, then the disease or condition is considered present. The tests for this evaluation are completely independent of each other. In short, overall sensitivity is greater than any single test but overall specificity is less than any single test.

‘The premise of independence underlies the entire approach to the use of multiple tests. However, it seems unlikely that tests for most diseases are fully independent of one another. If the assumption that the tests are completely independent is wrong, calculation of the probability of disease from several tests would tend to over estimate the test’s value.’ Clinical Epidemiology: The Essentials By Robert H. Fletcher, Suzanne W. Fletcher .

Sensitivity is the proportion of true positives that are correctly identified by the test.

Specificity is the proportion of true negatives that are correctly identified by the test.

Investigate net sensitivity using Monte Carlo simulation

  set.seed(123)                                      #allow reproducible results
  sim<-10e6                                          #Monte Carlo simulations
  sens<-c(0.6, 0.8)                                  #two diagnostic test sensitivities
  no.of.diag.test<-length(sens)
 
  x  = matrix(NA, nrow=sim, ncol=(no.of.diag.test))  #empty matrix to collect results
        
#Independent Bernoulli random generation. The probability is the sensitivity, so if a 1 is generated this is considered a success. Do this for the battery of tests. Repeat a large number of times.

  for(i in 1: no.of.diag.test ){                     #perform simulations
    x[,i] = rbinom(sim, size = 1 , prob = sens[i])   
  }

  head(x)                                            #examine first rows of simulations
     [,1] [,2]
[1,]    1    1
[2,]    0    1
[3,]    1    1
[4,]    0    1
[5,]    0    1
[6,]    1    1
  y<-rowSums(x)            #how many of the parallel tests were positive/negative
  1-length(y[y==0])/sim    #1 - number of times all tests are negative, sensitivity
[1] 0.9200041

Net sensitivity analytic solution

  1- prod(1-sens)  
[1] 0.92

Investigate net specificity using Monte Carlo simulation

  spec<-c(0.99,0.94,0.9,0.87,0.68)                      #mixture of specificities
  no.of.diag.test<-length(spec)
   
  x  = matrix(NA, nrow=sim, ncol=(no.of.diag.test))     #empty matrix to collect results
        
#The probability is 1-specificity, so if a 1 is generated this is considered a failure. Do this for the battery of tests. Repeat a large number of times.
  
  for(i in 1: no.of.diag.test ){                        #perform simulations
      x[,i] = rbinom(sim, size = 1 , prob = 1-spec[i]) 
  }

  head(x)                                               #examine first rows of simulations
     [,1] [,2] [,3] [,4] [,5]
[1,]    0    0    0    0    1
[2,]    0    0    0    0    0
[3,]    0    0    0    0    1
[4,]    0    0    1    0    1
[5,]    0    0    0    0    0
[6,]    0    0    0    0    1
  y<-rowSums(x)         #how many of the parallel tests were positive/negative
  length(y[y==0])/sim   #proportion negative
[1] 0.4954899

Net specificity analytic solution

  prod(spec)      
[1] 0.4954887

Reference and useful information

https://books.google.co.uk/books?id=NJ0uu9xTiIIC&pg=PA54&lpg=PA54&dq=sensitivity+of+3+parallel+tests&source=bl&ots=kSMLwgAsxW&sig=C-PXbjRGiPRCT8_8qMHXxYhZauM&hl=en&sa=X&ved=0CE4Q6AEwB2oVChMIl_DB5ZD_xwIVhEIUCh1bowBU#v=onepage&q=sensitivity%20of%203%20parallel%20tests&f=false Clinical Epidemiology: The Essentials By Robert H. Fletcher, Suzanne W. Fletcher

http://ocw.jhsph.edu/courses/FundEpi/PDFs/Lecture11.pdf

http://www.ajronline.org/doi/abs/10.2214/ajr.184.1.01840014

http://radiopaedia.org/articles/sensitivity-and-specificity-of-multiple-tests

Computing Environment

R version 3.2.2 (2015-08-14)
Platform: x86_64-w64-mingw32/x64 (64-bit)
Running under: Windows 8 x64 (build 9200)

locale:
[1] LC_COLLATE=English_United Kingdom.1252  LC_CTYPE=English_United Kingdom.1252   
[3] LC_MONETARY=English_United Kingdom.1252 LC_NUMERIC=C                           
[5] LC_TIME=English_United Kingdom.1252    

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] MKmisc_0.99 knitr_1.11 

loaded via a namespace (and not attached):
 [1] DEoptimR_1.0-3     magrittr_1.5       formatR_1.2.1      tools_3.2.2        htmltools_0.2.6    RColorBrewer_1.1-2
 [7] yaml_2.1.13        stringi_0.5-5      rmarkdown_0.8      robustbase_0.92-5  stringr_1.0.0      digest_0.6.8      
[13] evaluate_0.8      
[1] "~/X/DEVELOPMENT\\AGREEMENT SENS SPEC SAMPLE SIZE"

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