Why not alter the specificity?
Revised: 2001-10-28

Question: How is positive etiologic predictive value different from altering the specificity to predict post-test probability for disease? Could it be that altering the specificity makes Etiologic Predictive Value unnecessary?

Consider the following example: A conventional throat culture has been obtained during a summer period from 36 children of age 3-15 years having a sore throat possibly caused by group A beta-hemolytic streptococci (GABHS)¹. Among those 36 cultures a throat culture indicated presence of the bacterium GABHS in 11 (31%).

However, some of the children might be ill due to a virus as well as carrying GABHS. To investigate this phenomenon we may collect data from healthy children. During the same period of time throat cultures were obtained from 290 healthy children of age 3-15 living in the same geographical area, and showing no signs indicating possible GABHS-caused tonsillopharyngitis¹. Among those 290 cultures a throat culture indicated presence of GABHS in 37 (13%).

How well does the outcome of the throat culture predict whether the sore throat is caused by the bacterium GABHS? Let us compare three different approaches to this question:

1. Ignore the importance of carriers and calculate a prediction (Predictive Values) of presence of GABHS in the throat.
    
2. Compensate for carriers by altering the specificity.
    
3. Use Etiologic Predictive Values (EPV) to compensate for carriers and calculate a prediction of the disease a sore throat caused by GABHS.

1. Ignoring carriers

If the sensitivity of throat culture to find presence of GABHS in the throat is assumed to be 90%^{2, 3, 4, 5, 6, 7} and the specificity to be 97%^{5, 6} then the predictive values in respect of presence of GABHS in the throat are estimated so (see also Appendix 1):

2. Altering the specificity

There are two alternatives to alter specificity to compensate for presence of carriers (we will later discuss if it is appropriate to do this).

Compensating for carriers - lowering the specificity
In our example 30.6% of the children with a sore throat had growth of GABHS. At the same time 12.8% of healthy children had growth of GABHS. Let us compensate for carriers by lowering the specificity (97%) with 12.8% (We will later discuss if it is appropriate to do this). The predictive values in respect of presence of GABHS in the throat are estimated so (see also Appendix 2):

Compensating for carriers - specificity is 1 minus carriers
In our example 30.6% of the children with a sore throat had growth of GABHS. At the same time 12.8% of healthy children had growth of GABHS. Let us, as some authors do⁸, compensate for carriers by letting the specificity to find absence of disease be 100% minus 12.8% (We will later discuss if it is appropriate to do this). The predictive values in respect of presence of GABHS in the throat are estimated so (see also Appendix 3):

3. Compensate for carriers using EPV

As mentioned on other pages EPV will compensate for carriers. In the above mentioned example positive EPV is 67.9% and negative EPV is 96.7%.

Comparison between approaches

As we can see (Table IV) the different approaches will yield quite different post-test probability for disease.

It is easy to understand that ignoring carriers will yield a higher positive predictive value than if we consider carriers. However, it is interesting to see that compensating for carriers by altering the specificity might yield another estimate of predictive values than using EPV (Table IV). To understand this we must clarify the situation.

We can look at the relation between test outcome and presence or absence of the etiologic agent (Table V).

Since the disease is caused by M then M must be present in the patient at some stage of the disease. Thus, the second column will have no patients (se also the page defining markers).

We can also look at the relation between test outcome and the presence or absence of disease (Table VI).

From Table V and VI we can understand that the specificity for the test to correctly identify absence of the marker (GABHS) and the specificity for the test to correctly identify absence of the disease a sore throat caused by GABHS are not the same. The first may be denoted as P(T^-|M^-) and the second as P(T^-|S⁺D^-) where S⁺ denotes patients with a sore throat. The second specificity is only of interest in patients having a sore throat. The first specificity may be of interest in both patients having a sore throat and in healthy individuals.

Is it appropriate to lower the specificity?
Using altered specificity, as in the first alternative in the second approach, will mix the specificity for the test to correctly identify absence of the marker (GABHS) and the specificity for the test to correctly identify absence of the disease a sore throat caused by GABHS. This may be described as

where S⁺ denotes patients with a sore throat and S^- denotes healthy individuals (not having a sore throat). Since the disease a sore throat caused by GABHS is impossible to have if the individual has no sore throat then P(T⁺|S^-) is the same as P(T⁺|S^-D^-). Furthermore, if theta is 1 we will find that P(T⁺|S^-D^-) is the same as P(T⁺|S⁺D^-). P(T⁺|S⁺D^-) is the same as 1-P(T^-|S⁺D^-). Thus, the above expression may be rewritten as

and this expression can be simplified to

The conclusion is that this procedure is correct only if both theta and the specificity for the test to correctly identify absence of the marker (GABHS) are 1. In all other circumstances this procedure will result in a false estimate of the specificity to predict absence of disease. Thus, our estimate of 58.5% is not correct.

Is it appropriate to assume specificity=1-carriers?
This approach is similar to the above and may be described as

Simplifying this formula in a similar way as described above will result in that 1=1, which of course is true. The conclusion is that this procedure is correct if theta is 1. The specificity for the test to correctly identify absence of the marker (GABHS) is no longer needed. Thus, our estimate of 67.7% will be correct under the assumption that theta is 1. If we want to include the possibility of altering theta and have confidence intervals we will end up with EPV. Thus, this is an alternative deduction that also will lead to EPV.

The small difference seen in Table IV between this approach and EPV is due to that we have calculated our estimation in Table IV with a limited number of decimals. If we had used more decimals in every step (also when we calculate P(T⁺|S^-)) then our estimate of the probability for the disease a sore throat caused by GABHS would be 67.8652% which is exactly the same as the prediction provided by positive EPV.

Other WebPages of interest

Other pages with subjects that might be of interest is:

• Defining disease
• Defining carriers
• Defining a marker
• Defining populations
• Why not use likelihood ratio?

(You can click on these links to quickly see the pages)

References

1. Gunnarsson, R.K., Holm, S.E. and Söderström, M. `The prevalence of beta-haemolytic streptococci in throat specimens from healthy children and adults. Implications for the clinical value of throat cultures' , Scand J Prim Health Care, 15, 149-155 (1997).
    
2. Centor, R.M., Meier, F.A. and Dalton, H.P. `Throat cultures and rapid tests for diagnosis of group A streptococcal pharyngitis' , Ann Intern Med, 105, 892-899 (1986).
    
3. Kaplan, E.L. Unresolved problems in diagnosis and epidemiology of streptococcal infection. Streptococci and Streptococcal diseases, Academic Press, New York. 1972. pp 557-570.
    
4. Stillström, J., Schwan, A. and Björklind, A. `Streptococcal throat infection: calculation of test standards and a comparison between an antigen detection test and culture' , Scand J Prim Health Care, 9, 149-154 (1991).
    
5. White, C.B., Harris, R., Weir, M.R., Gonzales, I. and Bass, J.W. `Streptococcal pharyngitis. Comparison of latex agglutination and throat culture' , Clin Pediatr (Phila), 27, 431-434 (1988).
    
6. Lewey, S., White, C.B., Lieberman, M.M. and Morales, E. `Evaluation of the throat culture as a follow-up for an initially negative enzyme immunosorbent assay rapid streptococcal antigen detection test' , Pediatr Infect Dis J, 7, 765-769 (1988).
    
7. Christensen, P., Danielsson, D., Hovelius, B. and Kjellander, J. `Preliminary identification of beta-hemolytic streptococci in throat swab cultures with a commercial blood agar slide (streptocult)' , J Clin Microbiol, 15, 981-983 (1982).
    
8. O'Marcaigh AS, Jacobson RM. `Estimating the predictive value of a diagnostic test. How to prevent misleading or confusing results´ , Clin Pediatr (Phila), 32, 485-491 (1993).

Appendix 1

Assuming that a throat culture has a sensitivity of 0.9 and a specificity of 0.97 then our example with children having a sore throat may be described (Table VII).

We can see that

Formula 1

and also that

Formula 2

If we replace A in formula 2 with formula 1 we get

As we know B then the remaining cells in Table VII is easily calculated and we end up with Table I.

Appendix 2

If we lower the specificity to predict absence of disease to adjust for carriers we may for example have a specificity of 97%-12.8% = 84.2%. Note that column four and five in Table VI has similar denotations in the rows for T+ and T-. Thus, these columns can be merged. Our example with children having a sore throat may then be described as (Table VIII)

In a similar way as presented in Appendix 1 we would find that

and thus B=28.84. As we know B then the remaining cells in Table VIII is easily calculated and we end up with Table II.

Appendix 3

If we use the second alternative to lower the specificity to adjust for carriers we will in our example have a specificity of 100%-12.8% = 87.2%. As mentioned in Appendix 2 the fourth and fifth column will be merged. Our example with children having a sore throat may then be described as (Table IX)

In a similar way as presented in Appendix 1 we would find that

and thus B=27.72. As we know B then the remaining cells in Table IX is easily calculated and we end up with Table III.

Ronny Gunnarsson MD PhD
Department of Primary Health Care
Göteborg University
SWEDEN

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