Question

Prostate-specific antigen. Prostate-specific antigen (PSA) is a protein produced by the cells
of the prostate gland. Blood PSA level is often elevated in men with prostate cancer, but a number of
benign (not cancerous) conditions can also cause a man’s PSA level to rise. The PSA test for prostate
cancer is a laboratory test that measures PSA levels from a blood sample. The test measures the
amount of PSA in ng/ml (nanograms per milliliter of blood).
The sensitivity and specificity of the PSA test depend on the cutoff value used to label a PSA
level as abnormally high. In the last decade, 4.0 ng/ml has been considered the upper limit of
normal, and values 4.1 and higher were used to classify a PSA test as positive. Using this value, the
sensitivity of the PSA test is 20% and the specificity is 94%.
The likelihood that a man has undetected prostate cancer depends on his age. This likelihood
is also called the prevalence of undetected cancer in the male population. The following table shows
the prevalence of undetected prostate cancer by age group.

Age Group	Prevalence	PPV	NPV
< 50 years 50 - 60 years 61 – 70 years 71 -80 years	0.001 0.020 0.060 0.100

(a) Calculate the missing PPV and NPV values.

(b) Describe any trends you see in the PPV and NPV values.
(c) Explain the reason for the trends in part b), in language that someone who has not taken a
statistics course would understand.
(d) The cutoff for a positive test is somewhat controversial. Explain, in your own words, how lowering
the cutoff for a positive test from 4.1 ng/ml to 2.5 ng/ml would affect sensitivity and
specificity.

Answer 1

we can calculate PPV and NPV from prevalence from the following formulas

and

then the missing PPV and NPV values can be tabulated as follows

Age Group	Prevalence	PPV	NPV
<50 years	0.001	0.0033	0.9991
50-60 years	0.020	0.063	0.9829
61-70years	0.060	0.175	0.9484
71-80 years	0.100	0.270	0.9136

b)We can see that as the prevalence is increasing the PPV(positive predictive value) is also increasing but the NPV(negetive predictive value is decreasing)

c)before going to the ecplanation part we will go through the meaning of PPV and NPV

PPV-positive predictive value is the probability that a person following a positive test result will actually have the disease

Mathematically

  

NPV-negetive predictive value is the probability that a person following a negetive test result will actually not have the disease

Mathematically

True positive:- no. of people who are tested positive and are actually diseased

True negetive:-no. of people who are tested negetive and are actually disease free

False positive:-no. of people who are tested positive and they do not have the disease

False negetive:-no. of people who are tested negetive but are actually diseased

this can be tabulated as follows

	Diseased(D+)	Disease Free(D-)
Test positive(T+)	TP(true positive)	FP(False positive)
Test negative(T-)	FN(False negative)	TN(True Negetive)

prevalence means total no. of people who are diseased in a population at a specific period or a point of time

then

prevalence=TP+FN --------------------(i)

now if prevalence decreases then according to the defination the no. of people who are actually  diseased and are tested positive(TP) will also decrease this means in turn the false positives will increase for every decrease in the true positives(see the table).This increases the denominator of the PPV which makes a PPV decrease.

Again if Prevalence decreases means that total no. of people who are diseased in a population at a specific period or a point of time this includes the False negetives too (see the (i) also the definition of false negetive) meaning that if prevalence decrease the False negetive also decreases.This decrease the denominator of the NPV(see the formula for NPV) which makes NPV to rise as prevalence decreses

d) sensitivity means ability of a test to corretly classify the individual diseased when the person actually has the disease

Specificity means ability of a test to correctly classify an individual as not diseased when the person dosen't have the disease

Here in this question person whose PSA levels are more than 4.1 are tested positive now if we lower the value from 4.1 to 2.5 then the no. of people tested with PSA levels more than 2.5 will be tested positive hence this will increase the sensitivity

again no. of people who had PSA levels less than 4.1 were tested negetive now if we lower the value from 4.1 to 2.5 meaning the difference of 4.1 to 2.5 won't be tested as negetive hence lowering the specificity value