CPP Data Analysis

We illustrate our method by using data from the Collaborative Perinatal Project. It is an epidemiological study where investigators were interested in studying in utero exposure to polychlorinated biphenyls (PCB) in relation to various health outcomes including IQ, among children born in the CPP study. Subjects were enrolled through 12 university-

affiliated medical clinics, with the centers contributing unequal numbers of subjects from 23 to 204. One of the hypotheses is that the total PCB level is related to the performance on the Weschler intelligence scale for children at 7 years of age. The total PCB was measured by analyzing the third-trimester blood serum specimens preserved from mothers in the CPP study. In our CPP data, total PCB was ascertained on all 850 subjects. We assume that they consist of the full study cohort. Some baseline characteristics of the CPP table are presented in Table 5.2.

To implement the probability dependent sampling, we seta = 1so that the domain of total PCB is divided into 3 non-overlapping strata: A1 = (−∞,1.24], A2 = (1.24,5.05], A3 = (5.05,∞). A simple random sample of size 200 is selected from the full study cohort. We fit two separate logistic regression models with random intercept to predictprˆ(PCB∈ A1|IQ, Z) andprˆ(PCB ∈ A3|IQ, Z), where Z is the vector of covariates available on all subjects in the full cohort. Specifically, Z includes standardized maternal education at birth of child, socioeconomic index, standardized maternal age at registration, race (black = 1), gender (female = 1), DDT, total cholesterol and triglycerides. Based on the predicted probabilities, we conduct supplementary sampling withc = 85% and select 100 subjects from both probabilistic tails. Usingi = 1, . . . ,12to index medical clinics andj to index subject within the clinic, we postulate the following linear mixed effect model on IQ

IQ_ij =β0+β1PCBij +β2Educij +ηi+ij, (5.12)

where Educ is the standardized maternal education at birth of child. We also implemented the same model (5.12) on the data from (1) the full cohort and (2) a simple random sample of size 400. Analysis results are reported in Table 5.3. All three estimators identify a highly significant positive relationship between maternal education and IQ-score, that is, higher maternal education is significantly associated with higher IQ-score of the born child. On the other hand, none of them demonstrate a significant total PCB effect on IQ-score. The

corresponding P values were 0.8455, 0.6508 and 0.7848, respectively for full cohort, SRS and PDS estimators. Nonetheless, βˆP DS has a smaller standard error compared to βˆSRS

and will result in a narrower confidence interval. For example, the 95% confidence interval for PDS estimator is(2.00,4.47), while the counterpart for SRS estimator is(2.40,5.18).

5.5. Discussion

Compared to the classical ODS, PDS does not require assuming a true underlying linear relationship between response and exposure, hence offers more flexibility. In this paper, we proposed a new random effect model for a cluster-based probability dependent sam- pling scheme. The implementation of PDS scheme involved drawing a first-stage simple random sample. Using this validation sample, we obtained the predicted probabilities of X falling in pre-specified lower and upper tails, then performed supplementary probabil- ity sampling. The estimation and inference procedures were based on a profile likelihood function. The procedure also properly accounted for the cluster-level heterogeneity. We observed improved efficiency over the estimator from a complete simple random sample in both simulation studies and real data analysis.

One potential drawback of our method lies in the added computational burden to ad- dress the cluster level random effect. In order to compute double integrals in (5.8), we applied the trapezoidal rule on a fairly fine grid of responseY and random effectη. The computing speed for this naive method was acceptable, because we do not need the numer- ical approximation to be extremely precise. We also considered other numerical integral methods like adaptive cubature. However, we failed to observe improved performance over the trapezoidal rule. Another approach to handle the numerical integral is Laplace approximation similar to the technique used in (Xu and Zhou 2012). By using Laplace approximation, each cluster-specific random effectηi becomes an unknown parameter to

clusters (e.g. in CPP), the Newton-Raphson algorithm will be intractable when we have many small clusters.

While reviewing the literature, we noticed there was a lack of methodology for cluster- based ODS scheme. It is important to fill this gap. We can then further assess the perfor- mance of our cluster-based PDS estimator by comparing it with its ODS counterpart. PDS scheme also has the potential to be extended beyond the scope of continuous responses. One possible extension is to time-to-event data. PDS may be implemented in combination with the generalized case-cohort sampling design (Cai and Zeng 2007). Specifically, PDS can be embedded into the case-sampling stage of generalized case-cohort design. We can postulate a model for the failure times on the first stage simple random sample. We then obtain the predicted probabilities of a failure time is smaller or larger than a pre-specified threshold. We can sample the remaining cases based on the predicted probabilities and get a more representative second-stage supplementary sample, which may lead to improved statistical efficiency.