In an individual clinical trial, the possible results of a test are either positive or negative. However, if the trial is for the efficacy of a new treatment or drug, for example, the relevant results would be whether the drug is truly effective or not, such as for removing specific symptoms. The possibility of error should also be taken into consideration, thus the tests could produce true positives, true negatives, false positives, and false negatives. Therefore, the confusion matrix of such a trial would be in the following format:Positive TestNegative TestEffectiveTrue PositiveFalse NegativeNot EffectiveFalse PositiveTrue NegativeThe same type of confusion matrix can be used for other types of clinical tests such as safety.(ii) State definitions for sensitivity and specificityThe terms sensitivity and specificity in this context refer to the efficiency of the test itself, and not its specific results. The sensitivity of the test refers to its ability to correctly recognize the factor being tested for when it is actually present. On the other hand, the specificity of the test refers to its ability to recognize the absence of the factor being tested when it is truly absent.(iii) State definitions for sensitivity and specificity in terms of conditional probability using the following terms:Pos : positive test outcomeNeg : negative test outcomeD : Disease actually present~D : Disease actually absent (i.e. healthy individual)In terms of conditional probability when detecting a disease, for instance, sensitivity is the probability that the test would correctly recognize the presence of the disease. In other words, sensitivity is the ratio of the true positives (TP), when a positive test outcome (Pos) coincides with the disease actually present (D), over the sum of the TP and the false negatives (FN), when a negative test outcome (Neg) coincides with the disease actually present; or, sensitivity = TP / (TP + FN). As one will notice, sensitivity only applies to the values that involve the disease actually being present, with the denominator the total number of instances of D. On the other hand, specificity is the probability that the test would correctly recognize the absence of the disease. Thus, it is the ratio of the true negatives (TN), when a Neg coincides with the disease actually absent (~D), over the sum of the TN and the false positives (FP), when a Pos coincides with the disease actually absent; or, specificity = TN / (TN + FP). Similar to sensitivity, specificity only deals with values that involve the absence of the disease, with the numerator the total number of instances that the disease was absent. (b) A screening test based on ultrasound imaging is proposed for testicular cancer. It is low cost (i.e. affordable), readily available and no other technology or technique can offer this level of performance: advocates of the test claim it has a sensitivity of 99% and a specificity of 95%. Given that the underlying incidence of testicular cancer is thought to be 2000 cases in 100,000, calculate the positive predictive value, i.e. the probability of correctly identifying an individual with testicular cancer, given a positive test result.Given:Sensitivity = TP / (TP + FN) = 99% = 0.99Specificity = TN / (TN + FP) = 95% = 0.95Incidence of cancer = 2000 in 100,000 The hypothesis (H) is that the patient has cancer, while the evidence (E) is a positive result. We are looking for P(H|E), or the probability of the patient having cancer given a positive result.Using Bayes’s Rule, we have:P(H|E) =P(E|H) P(H)P(E)P(E|H) = sensitivity = 99% = 0.99P(H) = 2000/100,000 = 0.02However, we do not know P(E), thus we employ normalization, which transforms the above equation to:P(H|E) =P(E|H) P(H)P(E|H) P(H) + P(E|~H) P(~H)P(E|~H) = 0.05 This is taken from the specificity, (P~E|~H), because P(E|~H) + P(~E|~H) = 1.P(~H) = (100,000 – 2000)/100,000 = 0.98Replacing the variables of the second equation with the known values, then, we have:P(H|E) =(0.99)(0.02)(0.99)(0.02) + (0.05) (0.98)P(H|E) =0.0198=0.01980.0198 + 0.0490.0688P(H|E) = 0.2878 = 28.78% ~ 29%(c)(i) Assuming the estimated underlying incidence is true, and given the stated performance test above (in terms of sensitivity and specificity), estimate approximately how many cases might be expected to be correctly classified as true positives, given a target screening population of several million individuals. State whether this is an acceptable level of performance. The given sensitivity of 99% means that for every 100 patients with cancer, 99 of those would achieve a positive test result. Given that in 100,000 cases, 2000 have testicular cancer, first:We obtain 99% of 2000.True positive cases per 100,000 = (0.99)(2000) = 1980Thus, the percentage of true positive cases from the total number of cases is:% True positive cases = 1980/100,000% True positive cases = 1.98%This means that for every 1 million individuals tested, 19,800 would be correctly classified as true positives and 200 individuals with cancer would have false negative results. This is quite a decent percentage of correctly classified true positives. However, the number of individuals with cancer that have false negative results will of course increase the larger the screening population.(ii) If this level of screening performance (in terms of sensitivity and specificity) needed to be improved, how might this process be improved?Of course, the most common suggestion with regard to procedures such as this is the improvement of the technology for better accuracy. However, assuming that the technology used in the process is already at optimal possible performance, another suggestion is multiple screenings. Individuals with negative test results can be screened again. The smaller population would yield another percentage of true positives, and thus minimize the chances of misdiagnosis. Especially since the screenings are low cost, the small added cost of multiple screenings of those with negative test results would save the individuals money in the long run if the cancer can be halted through early detection.