For Example 11.4, we are interested in the sensitivity and specificity of the test.

(a) Compute the MLEs of sensitivity and specificity and their asymptotic variances based on the likelihood (11.9).

(b) Another way to parametrize the distribution is to use \(\operatorname{Pr}(t=1), \operatorname{Pr}(d=1 \mid t=1)\) \((\mathrm{PPV})\), and \(\operatorname{Pr}(d=0 \mid t=0)(\mathrm{NPV})\). Write down the likelihood and compute the MLEs of sensitivity and specificity and their asymptotic variances.

(c) Compare the estimates in

(a) and (b). They should be the same, since MLE does not depend on how the model is parameterized.

 Example 11.4

 (11.9) is:

Transcribed Image Text:

Example 11.4 In Example 11.1, since both the disease status and test result are binary, their joint dis- tribution follows a multinomial with four categories. Let Pkj = Pr(di = k and t = j) for k, j = 0, 1, where d = 1 (0) for the diseased (nondiseased) status and t = 1 (0) for the positive (negative) test result. Since the missingness of d; depends on the observed value of the test ti, it is MAR and the observed-data likelihood is (P11+P01) (P10+ Poo)P11 P10 Pol Poo (11.9) where nkj is the number of subjects with observed d = k and t = j (k, j = 0,1), and mj is the number of subjects with di missing and observed t = j (j = 0, 1). The MLE for the prevalence is 0.5, with an estimated standard error of 0.017. Note that in this simple example, the MLE and associated asymptotic variance estimate are readily evaluated. But, as discussed in Chapter 9, computation of estimates based on likelihood- based methods is generally quite complex, even for complete data. [] Because missing values can be ignored under MAR, this mechanism is commonly as- sumed in real data applications. However, as argued in Section 11.2.2.1, it is important to assess the plausibility of this missing-value model within the particular context of the study, since appearances can be deceiving, and mechanisms that appear MAR may actually follow MNAR.