17.2  Describing Diagnostic Tests
17.2  Describing Diagnostic TestsThe following concepts have been developed to describe the performance of a diagnostic test relative to the gold standard; these concepts are measures of the validity of a diagnostic test.
 Sensitivity
 is the probability that an individual with the disease of interest has a positive test. It is estimated from the sample as a/(a+c).
 Specificity
 is the probability that an individual without the disease of interest has a negative test. It is estimated from the sample as d/(b+d).
 Accuracy
 is the probability that the diagnostic test yields the correct determination. It is estimated from the sample as (a+d)/(a+b+c+d).
Tests with high sensitivity are useful clinically to rule out a disease. A negative result for a very sensitive test virtually would exclude the possibility that the individual has the disease of interest. If a test has high sensitivity, it also results in a low proportion of falsenegatives. Sensitivity also is referred to as "positive in disease" or "sensitive to disease".
Tests with high specificity are useful clinically to confirm the presence of a disease. A positive result for a very specific test would give strong evidence in favor of diagnosing the disease of interest. If a test has high specificity, it also results in a low proportion of falsepositives. Specificity also is referred to as "negative in health" or "specific to health".
Sensitivity and specificity are, in theory, stable for all groups of patients.
In a study comparing FNA to the gold standard (excisional biopsy), 114 women with normal physical examinations (nonpalpable masses) and abnormal mammograms received a FNA followed by surgical excisional biopsy of the same breast (Bibbo M, et al: Stereotaxic fine needle aspiration cytology of clinically occult malignant and premalignant breast lesions. Acta Cytol 1988; 32:193201.)
Cancer

No Cancer


FNA Positive 
14

8

FNA Negative 
1

91

Sensitivity = 14/15 = 0.93 or 93%
Specificity = 91/99 = 0.92 or 92%
Accuracy = 105/114 = 0.92 or 92%
SASĀ® Example
Using PROC FREQ in SAS for determining an exact confidence interval for sensitivity and specificity
Point estimates for sensitivity and specificity are based on proportions. Therefore, we can compute confidence intervals using binomial theory. See SAS Example (18.1_sensitivity_specifi.sas) below for a SAS program that calculates exact and asymptotic confidence intervals for sensitivity and specificity.
***********************************************************************
* This is a program that illustrates the use of PROC FREQ in SAS for *
* determining an exact confidence interval for sensitivity and *
* specificity. *
***********************************************************************;
proc format;
value yesnofmt 1='yes' 2='no';
run;
data sensitivity;
input positive count;
format positive yesnofmt.;
cards;
1 14
2 01
;
run;
proc freq data=sensitivity;
tables positive/binomial alpha=0.05;
weight count;
title "Exact and Asymptotic 95% Confidence Intervals for Sensitivity";
run;
data specificity;
input negative count;
format negative yesnofmt.;
cards;
1 91
2 08
;
run;
proc freq data=specificity;
tables negative/binomial alpha=0.05;
weight count;
title "Exact and Asymptotic 95% Confidence Intervals for Specificity";
run;
For the FNA study, only 15 women with cancer, as diagnosed by the gold standard, were studied. The rule for using the asymptotic confidence interval fails for sensitivity because np(1  p) = 0.9765 < 5 (the rule does hold for specificity).
As the output shows below, the exact 95% confidence intervals for sensitivity and specificity are (0.680, 0.998) and (0.847, 0.965), respectively.
Exact and Asymptotic 95% Confidence Intervals for Sensitivity  

The FREQ Procedure  
Positive  Frequency  Percent  Cumulative Frequency  Cumulative Percent 
yes  14  93.33  14  93.33 
no  1  6.67  15  100.00 
Binomial Proportion 


Proportion  0.9333 
ASE  0.0644 
95% Lower Conf Limit  0.8071 
95% Upper Conf Limit  1.0000 
Exact Conf Limits  
95% Lower Conf Limit  0.6805 
95% Upper Conf Limit  0.9983 
Test of H0: Proportion = 0.5 


ASE Under H0  0.1291 
Z  3.3566 
OneSided Pr > Z  0.0004 
TwoSided Pr > Z  0.0008 
Sample Size = 15
Exact and Asymptotic 95% Confidence Intervals for Specificity  

The FREQ Procedure  
Positive  Frequency  Percent  Cumulative Frequency  Cumulative Percent 
yes  91  91.92  91  91.92 
no  8  8.08  99  100.00 
Binomial Proportion 


Proportion  0.9192 
ASE  0.0274 
95% Lower Conf Limit  0.8655 
95% Upper Conf Limit  0.9729 
Exact Conf Limits  
95% Lower Conf Limit  0.8470 
95% Upper Conf Limit  0.9645 
Test of H0: Proportion = 0.5 


ASE Under H0  0.0503 
Z  8.3418 
OneSided Pr > Z  <.0001 
TwoSided Pr > Z  <.0001 
Sample Size = 99