Question

A thousand (1000) passengers in a cruise started having symptoms of a food poisoning after a dinner on the ship. They were rushed to a hospital that was using a relatively new test for Hepatitis A: The following data was available for the researchers: SICK PASSENGERS HEALTHY PASSENGERS TEST POSITIVE 480 TEST POSITIVE 50 TEST NEGATIVE 30 TEST NEGATIVE 440 510 490 Based on the data: Calculate: A) The sensitivity of the test. B) The specificity of the test. C) The false positives and false negatives.D) The predictive value+ E) The predicted value -F) The prevalence. G) The accuracy. Based on the results, do you think the test is useful in the diagnosis of Hepatitis A?

Answer 1

GIVEN:

The confusion matrix/ classification table:

	ACTUAL
PREDICTED		Sick Passengers	Healthy Passengers	TOTAL
	Test Positive	480 (TP)	50 (FP)	530
	Test Negative	30 (FN)	440 (TN)	470
	TOTAL	510	490	1000

True Positives (TP) = 480

True Negatives (TN) = 440

False Positives (FP) = 50

False Negatives (FN) = 30

A) SENSITIVITY:

The formula for sensitivity is,

Sensitivity

TP/(TP FN)

480/(48030)

0.94

Thus
94
% of passengers with Hepatitis A disease are correctly identified as positive by the diagnostic test.

B) SPECIFICITY:

The formula for specificity is,

Specificity $=TN/(TN+FP)$

440/(440+50

0.90

Thus
90
% of passengers without Hepatitis A disease are correctly identified as negative by the diagnostic test.

C) FALSE POSITIVES AND FALSE NEGATIVES:

A false-positive test result indicates that a person has a specific disease or condition when the person actually does not have it.

A false negative is a test result that indicates a person does not have a disease or condition when the person actually does have the disease.

From the classification table,

Number of passengers whose test result indicated that they have Hepatitis A when they actually does not have the disease (False Positives)

50

Number of passengers whose test result indicated that they does not have Hepatitis A when they actually have the disease (False Negatives)

30

D) POSITIVE PREDICTIVE VALUE:

The formula for positive predictive value is,

$PP+=TP/(TP+FP)$

$=480/(480+50)$

$=0.91$

Thus % of passengers with positive hepatitis A test result actually have the condition.

E) NEGATIVE PREDICTIVE VALUE:

The formula for negative predictive value is,

$PP-=TN/(TN+FN)$

$=440/(440+30)$

0.94

Thus
94
% of passengers with negative hepatitis A test result actually do not have the condition.

F) PREVALENCE:

The formula for prevalence is,

Prevalence $=(TP+FN)/(TP+TN+FP+FN)$

$=(480+30)/1000$

$=0.51$

Thus % of passengers are actually sick in the given sample.

G) ACCURACY:

The formula for accuracy is,

Accuracy $=(TP+TN)/(TP+TN+FP+FN)$

  $=(480+440)/1000$

  $=0.92$

The overall accuracy of classifer is %.

Thus the test is useful in the diagnosis of Hepatitis A, since the accuracy of classifier is high.