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Estimates of sensitivity, specificity, false rates and expected proportion of population testing positive in screening tests Authors: ABSTRACT: Oyeka Ikewelugo Cyprian Anaene 1, Okeh Uchechukwu Marius 2, Igwebuike Victor Onyiaorah3, Adaora Amaoge Onyiaorah4 and Chilota Chibuife Efobi 5 This paper proposes and presents indices used as measures to evaluate or

assess results obtained from diagnostic screening tests. These indices include sensitivity, specificity, prevalence rates and false rates. We here present statistical methods for estimating these rates and for testing hypotheses concerning them. An estimate of the proportion of a population expected to test positive in a diagnostic screening test is also provided. Further interest is also to estimate the sensitivity and 2. Department of Industrial Mathematics and Applied specificity of the test and then the false rates as functions of sensitivity and specificity Statistics, Ebonyi State given knowledge or availability of an estimate of the prevalence rate of a condition in University Abakaliki, Nigeria. a population. The indices proposed ranges from -1 to 1 inclusively and therefore 3. Department of enables the researcher to determine if an association exists and if it exists between Histopathology, Nnamdi test results and condition as well as whether it is positive and direct or negative and Azikiwe University Teaching indirect which will serve as an advantage over the traditional methods. The proposed Hospital Nnewi Anambra State, indices provide estimates of the test statistic. When the proposed measures are Nigeria. applied, results indicate that it is easier to interpret and understand more than those 4. Department of obtained using the traditional approaches. In addition, the proposed measure is Opthalmology, Enugu State shown to be at least as efficient and hence as powerful as the traditional methods University Teaching Hospital Park lane Enugu State, Nigeria. when applied to sample data. Institution: 1. Department of Applied Statistics, Nnamdi Azikiwe University, Awka Nigeria.

5. Department of Haematology, University of Port Harcourt Teaching Hospital, Port Harcourt, Rivers State Nigeria. Corresponding author: Okeh Uchechukwu Marius

Email Id:

Web Address: http://jresearchbiology.com/ documents/RA0391.pdf

Keywords: Traditional odds ratio, prevalence, sensitivity, specificity, false rates.

Article Citation: Oyeka Ikewelugo Cyprian Anaene, Okeh Uchechukwu Marius, Igwebuike Victor Onyiaorah, Adaora Amaoge Onyiaorah and Chilota Chibuife Efobi Estimates of Sensitivity, Specificity, False Rates and Expected Proportion of Population Testing Positive in Screening Tests Journal of Research in Biology (2014) 4(8): 1498-1504 Date: Received: 06 Nov 2013

Accepted: 15 Jan 2014

Published: 15 Nov 2014

This article is governed by the Creative Commons Attribution License (http://creativecommons.org/ licenses/by/4.0), which gives permission for unrestricted use, non-commercial, distribution and reproduction in all medium, provided the original work is properly cited. Journal of Research in Biology An International Scientific Research Journal

1498-1504 | JRB | 2014 | Vol 4 | No 8

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Anaene et al., 2014 previous results from a gold standard test to actually

INTRODUCTION In diagnostic screening tests indices used as

have a certain condition in nature from a population and

measures to evaluate or assess results obtained include,

also takes a second random sample of n.2 subjects from

sensitivity and specificity of the test and if the prevalence

the same population Keeping in mind that known or

rate of a condition of interest in a population is known or

believed not to actually have the same condition in

can be estimated from a previous study, also the false

nature, thus giving a total random sample of size

positive and false negative rates of the test as well as the

n=n..=n.1+n.2 subjects to be studied. It is always treated

proportion of the population expected to test positive to

to confirm through a diagnostic screening test for

the condition (Fleiss, 1973; Pepe, 2003). Hence research

whether or not each sampled subjects have or does not

interest is often in statistical methods for estimating

have the condition of interest. Further interest is also to

sensitivity, specificity, false rates and the proportion of a

estimate the sensitivity and specificity of the test and

population expected to test positive to a condition in

then the false rates as functions of sensitivity and

these screening tests. The sensitivity of a test is the

specificity given knowledge or availability of an estimate

proportion of subjects testing positive among the subjects

of the prevalence rate of a condition in a population.

known or believed to actually have a condition in nature,

Now suppose B and B are respectively the

while the specificity of a test is the proportion of subjects

events that a randomly selected subject from a

who actually test negative to a condition among the

population has and does not have a condition in nature.

subjects known or believed not to actually have the

Also let A and A be respectively the events that the

condition in nature. False positive rate of a test is the

randomly selected subject tests positive, and negative to

proportion of subjects who are known or believed not to

the condition in the test. We here assume that the

actually have a condition in nature among the subjects

prevalence rate P(B) of the condition in the population is

testing positive, while false negative rate is the

either known or can be reliably estimated from previous

proportion of subjects who are known or believed to

studies. The results of such a screening test may be

actually have a condition in nature among the subjects

presented in the form of a four fold Table (Table 1).

who never-the-less test negative (Fleiss,1973;Greenberg

In Table 1 above, of the n=n.. sample subjects

et al., 2001;Linn, 2004).Sensitivity and Specificity of a

studied, n.1 subjects are known or believed to have the

test are independent of the population being studied and

condition in nature, that is in B and n.2 are known or

hence independent of the prevalence rate of a condition

B and believed not to have the condition in nature, that is B .

in the population. False rates of a test on the other hand

Also n1. subjects respond positive that is in A and n2.

are functions of the prevalence rate of a condition in a

subjects respond negative, thatAisand in A . Of the n.1

population and hence are dependent on the population of

subjects in B, n11 subjects actually have the condition

interest (Fleiss, 1973;Linn, 2004).

and test positive that is in AB and n21 subjects actually

here

present

statistical

methods

for

have the condition but test negative, that is in AB .

estimating these rates and for testing hypotheses

Of the n.2 subjects who are known or believed not to

concerning them. An estimate of the proportion of a

have the condition in nature,n12 subjects who do not have

population expected to test positive in a diagnostic

the condition test positive, that is in A B w h i l e n22 subjects who do not have the condition in nature also

screening test is also provided. Given that a researcher collects a random sample

test negative that is in A B. In an actual screening

of n.1 subjects known or believed, perhaps on the basis of

test usually only the total sample size n=n..,n.1 subjects

1499

Journal of Research in Biology (2014) 4(8): 1498-1504

Anaene et al., 2014 Table 1.Format for Presentation of Results of a Diagnostic Screening Test Screening Test Results

Condition Present

Condition Absent

(A)

(B) n11

B n12

Negative (Ᾱ)

n21

n22

n2.

Total (n.j)

n.1

n.2

n..(=n)

Positive

in B, n11 subjects in AB,n.2 subjects in B ¯ a n d n

2 2

subjects in A B are observed and actually known. The values n12 in AB and n21 in AB are not

Total (ni.) n1.

known or believed not to have the condition in nature. Notationally, we have that P(A) = P (AB) + P (AB )

known and hence also are n1. and n2., the overall number

3 Now to develop sample estimates of these indices,

of subjects who would test positive and negative

sensitivity for instance, we may let,

respectively in the screening test. Hence only the known values namely total sample size n, the number of

1, if the ith randomly selected and screened subject known or believed to actually have a condition in nature tests positive

ui1

subjects, n.1 known to have the condition in nature, n11,

0, otherwise

the number of subjects who test positive among these

for i

known to have the condition in nature, the number of

Let Let π1=1 P (ui1 =P 1) ui1 and and

subjects n.2 known not to have the condition in nature and n22 subjects who test negative among the subjects known not to have the condition in nature are used here to estimate the required indices and test statistics. Now

1, 2,...n.1 subjects

n1.

Wi = W 1

ui1

the sensitivity (Se) and specificity (Sp) of a screening test expressed in terms of conditional probabilities or

Now the expected value and variances of

specific rates of events A and B are respectively

E ui1

P ( A / B ); Sp

P( A / B )

The higher Se and Sp are more sensitive and specific is the screening test, the lower these rates, the weaker are the sensitivity and specificity of the test. The false positive rate

: Var ui1

ui1

are

Similarly the expected value and variance of are respectively n1.

E W1

E (ui1 )

n.1

i 1

and the false negative rate of a

;Var W1

n1. i 1

Var (ui1 )

n.1

screening test also expressed in terms of conditional

Now π1 is the probability that a randomly

probabilities or specific rates of events A and B are

selected and screened subject known or believed to have

respectively

a condition in nature in a population tests positive; that is

F ve P( B / A)

1 P( A / B ) P( B ) P( A)

; F ve P( B / A)

1 P( A / B ) P ( B ) P( A)

the proportion of subjects testing positive among the 2

subjects in the population known or believed to actually

Where P(A) consists of the probability of

have a condition in nature. This is in fact a measure of

composition of the events AB and which is the

the sensitivity Se of the screening test. The sample

probability of the union of events that a randomly

estimate

selected subject tests positive and is known or believed to have a condition in nature or tests positive and is Journal of Research in Biology (2014) 4(8): 1498-1504

ˆ Se

of π1 is W1 n.1

f n.1

1500

Anaene et al., 2014 Where f+ is the number of subjects who test

condition given that the subject is known or believed not

positive among subjects in the population known or

to actually have the condition in nature. In other words,

believed to have the condition of interest in nature. In

π2 is the proportion of subjects testing negative among

other words, f is the number of 1s in the frequency

the population of subject known or believed not to have a

distribution of the n.1values of 1s and 0s in ui1,for i=1,2,

condition in nature. Thus π2 is actually a measure of the

…,n.1.Hence f =n11 of Table 1.

specificity Sp of the screening test. Its sample estimate is

The corresponding variance of is from equation (8)

from equation (18)

Var W1

ˆ Var Se

Var ˆ1

n.11

ˆ (1 Se ˆ ) Se 10 n.1

ˆ1 (1 ˆ1 ) n.1

ˆ2

W2 n.2

10 Se

A researcher may sometimes wish to test a null

Where f

f n.2 -

is the number of subjects whose test

hypothesis that sensitivity of a screening test is at most

negative among the n.2 subjects in the sampled

some

population known or believed not to have a condition in

value π10 = Se0 . That is the null hypothesis, 11

11other words f - is the total number of 1s in the nature. In

This null hypothesis may be tested using the test statistic

frequency distribution of the n.2 values of 0s and 1s in

H 0 : Se

Seo

n.1 Seo

versus H1 : Se

Seo(0

ˆ n.1 Se

Seo

ˆ (1 Se

ˆ ) Se

Var W1

Seo

ˆ1

ˆ10

ˆ1 (1

ˆ1 )

n.1

ui2,for i=1,2,…n.2.Thus f - = n22 in Table 1. The variance

Which under Ho has approximately the chi-square distribution with one (1) degree of freedom for

of equation ˆ 2 Var

ˆ Sp

Var (W2 ) n.22

ˆ Var Sp

ˆ2

(18) ˆ (1 Sp ˆ ) Sp n.2

ˆ 2 (1 ˆ 2 ) n.2

sufficiently large n.1.the null hypothesis Ho is rejected at

A researcher may also wish to test a null hypothesis that

the level of significance if

specificity Sp of a diagnostic screening test is at least

2 1

some value That is the null hypothesis Ho : Sp

Similarly to develop a sample estimate of the specificity

ui 2

0, otherwise for i

1, 2,...n.2 subjects

And W2

n.2

ui 2

;Var ui 2

And n.2

;Var W2

n.2

)

Note that π2 is the probability that a randomly selected and screened subject tests negative to the 1501

n.2 Spo

Var W2

ˆ n.2 Sp

Spo

ˆ (1 Sp

ˆ ) Sp

n.2

ˆ2

ˆ 20

ˆ 2 (1

ˆ2 )

Which under Ho has approximately the chisquare distribution with one (1) degree of freedom for at the level of significance if equation (13) is satisfied, 15 otherwise Ho is accepted. a population expected to 16test positive to a condition in a diagnostic screening test, we note that when expressed in

Now

E W2

To develop sample estimate of the proportion of

i 1

E ui 2

sufficiently large n.2. The null hypothesis Ho is rejected

Define P ui 2

Spo, (0

This null hypothesis is tested using test statistic

Sp of a screening test, we may let 1, if the ith randomly selected and screened subject in the population is known or believed not to actually have a condition in nature tests negative

Spo versus H1 : Sp

terms of conditional probability using Bayes rule 17 equation (3) becomes P( A) P( A / B).P( B) P( A / B ).P( B ) P( A / B).P( B)

1 P( A / B ) 1 P( B)

18 Or when expressed in terms of sensitivity Se and specificity Sp of the screening test and prevalence rate P(B) of a condition in a population becomes Journal of Research in Biology (2014) 4(8): 1498-1504

Anaene et al., 2014 24

The null hypothesis Ho is expected at the α level of

The sampled estimate of P(A) is using equation ( 9) and

significance if equation (13) is satisfied; otherwise Ho is

equation (19) in equation (24)

accepted.

P( A) SeP( B) (1 Sp).P( B) 1 P( A / B) 1 P( A / B) P( A / B) .P( B) 1 Sp 1 Se Sp .P( B)

Pˆ ( A)

P( A)

ˆ ( B) SeP

ˆ ).P( B ) (1 Sp

ˆ 1 Sp

ˆ 1 Se

ˆ .P( B) Sp

The corresponding sample variance is ˆ )( P( B)) Var ( P( A)) Var (Se

ˆ ) P(( B)) Var (Sp

of false rates in a diagnostic screening test if the

ˆ ˆ ) 2P( B).P( B).Cov SeSp

The researcher may also wish to obtain sample estimates

It is easily shown that

prevalence rate P(B) of a condition in a population is known or can be determined.

ˆ ; Sp ˆ ) Cov ( Se

Cov

W1 W2 ; n.1 n.2

Now from equations (2) and (25), the sample estimate of

false positive rate in terms of sample estimates of

To prove this it is sufficient to show that

sensitivity and specificity and the known or estimated

Cov ui1 ; ui 2

prevalence rate is

Now Cov ui1 ; ui 2

E ui1 .ui 2

E (ui1 ).E (ui 2 )

E ui1 .ui1

1. 2 .

Now ui1 .ui1can assume only the values 1 and 0 . It assumes the value 1 if ui1 and ui2both assume the value 1 with probability it assumes the value 0 if assumes the values 1 and ui2 assumes the value 0 or ui1 assumes

Fˆ

(1 ve

ˆ ).P( B ) Sp P( A)

ˆ Sp

ˆ Se

ˆ Sp

P( B) ˆ .P( B) Sp

Similarly the sample estimate of false negative rate is from equations (2) and (25) ˆ F

ˆ ).P( B) Se P ( A)

(1 ve

ˆ Sp

ˆ P( B) Se

ˆ Se

ˆ .P( B) Sp

the value 0 and ui2 assumes the value 1 with probability

Where Ŝe and Ŝp are given in equation (30).

π1(1-π2) - π2(1-π1) Hence

Finally with further interest the researcher may use some

Cov ui1 ; ui 2 ˆ ) Var ( Sp ˆ ) Var ( P( A)) Var ( Se

ˆ (1 Se ˆ ) Sp ˆ (1 Sp ˆ ) Se n.1 n.2

0 so that ˆ1 (1 ˆ1 ) n.1

elementary calculus or apply Fiellers convenience

ˆ 2 (1 ˆ 2 ) n.2

Theory to obtain approximate estimates of the variances of Fˆ ve and Fˆ ve and also test any desired hypotheses.

The researcher may also wish to test the null hypothesis that the proportion P(A) of

subjects in a

ILLUSTRATIVE EXAMPLE

population expected to test positive to a condition in a

It a clinician is collecting a random sample of 98

diagnostic screening test is at most some value Po(A).

subjects from a certain population; twelve of whom are

That is the null hypothesis

doubted for having prostrate cancer and 86 of whom are

Ho : P( A)

Po( A) versus H1 : P( A)

Po( A), (0

Po( A) 1)

This null hypothesis is tested using the test statistic P ( A)

Var

Po( A)

assumed not to28have the disease. The clinician’s interest is to confirm through a diagnosis screening test whether or not each of the sampled subjects are actually prostrate

P ( A)

cancer positive or negative. The results of the screening test are presented in Table 2.

Which under Ho has approximately the chi-square

Now from Table 2 we have that the sample

distribution with one (1) degree of freedom where P(A)

estimate of the sensitivity and specificity of the test are

and Var (P(A)) are given by equations (25) and (26)

respectively

respectively and from Table 1 ˆ Se

ˆ1

f1 n.1

n11 ˆ ; Sp n.1

ˆ Se

ˆ2

f2 n.2

n22 n.2

Journal of Research in Biology (2014) 4(8): 1498-1504

f n.1

and ˆ Sp

f n.2

98 4

98 12

n.1 n

f n.2

8.167

0.041

0.335

30 n

98 84 86

1.140

0.857

0.977.

1502

Anaene et al., 2014 These results show that the screening test is low

Table 2: Result of Prostrate Cancer Screening Test

in sensitivity but has high specificity.

Clinical diagnosis

Present (B)

Now from equations (13) and (14) the sample estimates

Prostrate Cancer

n11=f + +=4

Absent B n12=f+ - =4

n1.= 6

of are respectively

Positive (A)

n21=f - +=4

n22=f - - =4

n2.=92

Negative (Ā)

n.1=12

n.2=86

n..=n=98

12 0.335

86 0.977

0.898

and 12 1 0.335

86 1 0.977

0.102

Hence from equations (15) and (17) we have that

0.898 0.102

0.796

expected to be correctly informed. This probably makes more difficulty in understanding than the simple information conveyed by the simple difference in rates, ˆ

0.796,

namely, the proportion of subjects

With estimated variance obtained from equations (16)

testing positive among subjects who have prostrate

and (18) as

cancer or testing negative among subjects who do not

Var ( ˆ )

Var ( ˆ

ˆ )

1 (0.796) 98

0.004

have prostrate cancer is 79.6 percent higher than the proportion of subjects testing positive among subjects

Hence the test statistic of no association between

who do not have prostrate cancer or testing negative

screening test results and state of nature or condition

among subjects who have the disease.

(Prostrate Cancer) of equation (19) are obtained from equation (20) as

0.796

Se(O)

1 n11

1 n12

1 n21

1 n22

21.00

1 4

1 2

1 8

1 84

(21.00)(0.942) 19.782

0.634 158.500( P value 0.0000) 0.004 0.004 Which with one (1) degree of freedom is highly

This measure of the error of ‘O’ namely 19.782

statistically significant indicating a strong degree of

association between screening test results and state of

sample data. The chi-square test statistic for the

nature or condition (presence of Prostrate cancer in

significance of ‘O’ is

the population). Also since ˆ

0.796

is positive, the association is positive and direct.

is clearly much larger than the error of only ˆ

X2=

0.004

0.063

of the estimated value of for our

n(n11n22–n12 n21)2 98(94)(84)-(2)(8)2 =

n1.n2.n1.n2

=17.616 (p-value=0.0000)

(6)(92)(12)(86)

It is commendable to compare the present results with

Which is also statistically significant again leading to a

what would have been obtained if we have used the

rejection of the null hypothesis of no association.

traditional odds ratio to analyze the data of Table 2. In

However, the proposed method and the traditional odds

spite of odds ratio’s short comings as already pointed out

ratio approach explained here (both) lead to a rejection of

above, when used in the analysis of screening test results.

the null hypothesis, the relative sizes of the calculated

The sample estimate of the traditional odds ratio for the

chi-square values suggest that the traditional odds ratio

data of Table 2 is

method is less efficient and likely to lead to an

n11 n22 n12 n21

(4)(84) (2)(8)

21.00

This means, for every subject who has prostrate cancer

acceptance of a false null hypothesis (Type II Error) more frequently and hence is likely to be less powerful than the proposed method.

among tested subjects and erroneously informed that they are free of the disease (21 subjects) among those tested and found to have prostrate cancer would be 1503

Journal of Research in Biology (2014) 4(8): 1498-1504

Anaene et al., 2014 CONCLUSION In this paper, we proposed, developed and presented a statistical method for measuring the strength of association between test results and state of nature or condition in a population expressed to a diagnostic screening test. The proposed measure is based on only the sensitivity and specificity of the screening test which are independent of the population of interest and estimated using only observed sample values. The proposed measure which ranges from -1 to 1 can be used to establish whether an association is strong and direct, strong and indirect or zero estimates of the standard error. Test statistics for the significance of the proposed measure are provided. The proposed measure of association is shown to be easier to interpret and explain than the traditional odds ratio, and the sample data used suggest that the measure is at least as efficient and powerful as the traditional odds ratio. REFERENCE Baron

and

Sorensen

HT.

2010.

Clinical

epidemiology, in Teaching Epidemiology: A guide for teachers in epidemiology, public health and clinical medicine. eds. Olsen J; Saracci R and Trichopoulos D; Oxford: Oxford University Press, 237-249. Fleiss JL. 1973. Statistical Method for Rates and Proportions. John Wiley, New York. Greenberg RS, Daniels SR, Flanders WD, Eley JW and Boring JR. 2001. Medical Epidemiology, London: Lange-McGraw- Hill.

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