Abstract
Epidemiologic cross-sectional, case-cohort, or case–control studies often select augmentation samples to supplement an existing (baseline) sample, primarily for the two reasons: (1) to increase the sample sizes from certain subdomains of interest that were not originally considered in the design of the baseline study and (2) to obtain samples from an extension of the target population. To address these two objectives, two-stage stratified sample designs are considered, where the stratification based on the expanded population at the second stage is not nested in the first stage strata. The sample weighting and Taylor linearization variance estimation for the two-stage stratified sample designs, involving re-stratification and population expansion, are provided for estimating population totals and logistic regression coefficients. Results from limited simulation studies and a logistic regression analysis of a study of human papillomavirus serology are provided.
Keywords: Pseudo-likelihood function, Sample weighting, Taylor linearization variance estimation, Two-stage stratified sampling
1. Introduction
Epidemiologic cross-sectional, case-cohort, or case–control studies are sometimes found to be inadequate for future investigations that utilize the original (baseline) study sample. The original study may fail to have adequate sample sizes to obtain precise estimates within population subdomains of interest to future studies. Or, the target population of the original study may no longer be considered adequate for future studies. To address these limitations, augmentation samples are often collected primarily: (1) to increase the sample sizes from certain subdomains of interest and (2) to obtain samples from an extension of the target population. In this paper, we consider a two-stage stratified sample design. In the first stage, a baseline sample is selected using stratified simple random sampling (SSRS). Upon evaluation of the baseline sample with regard to the sample sizes of the subdomains of interest and the extension of the target population, a second stage stratified sample using a different set of stratification variables (that identify the subdomains) is selected from the remaining original target population (i.e. the target population without the baseline sample) plus an extension of the target population. Similar types of two-stage stratified sample designs, also called re-stratification sampling, but without additional sampling from an extension of the target population, have been described in the ecology literature (Stehman and others, 2012). Re-stratification sample designs differ from two-phase designs used in epidemiology studies, in which a subsample from the first phase sample is selected at the second phase (Breslow and Cain, 1988); whereas in augmentation sampling, a sample of additional individuals is selected.
This paper is motivated by a cross-sectional study on human papillomavirus (HPV) serology. There are over 100 HPV types; approximately 40 infect the genital tract, of which 12 are considered carcinogenic because infections with these are a necessary cause of cervical cancer (Vaccarella and others, 2010; Schiffman and others, 2009). The genital HPV types are sexually transmitted, and therefore infections with more than one genotype are common, especially among young women (Vaccarella and others, 2010; Schiffman and others, 2009). Two of the 12 carcinogenic types cause 70% of cervical cancer worldwide, with HPV16 causing 50% and HPV18 causing 20% of cancers. For a natural history study, a baseline sample based on HPV16 infection status and enrollment serology was firstly selected, and then augmented by the sample based on HPV18 infection status and enrollment serology.
Considering the two-stage augmentation sample design described above, this paper develops (1) sample weighting method to account for differential sampling rates across the strata in both stages of sampling and (2) variance estimation method for estimating finite population totals under the two-stage sample design considered in this paper. Because nearly all estimators can be expressed as explicit or implicit functions (through estimating equations) of estimated finite population totals, including adjusted odds ratios using logistic regression, our variance estimators will have wide application to many estimators of interest in epidemiology.
2. Methods
2.1. Motivating study
The HPV serological study that motivated our methods is nested within the control (unvaccinated) arm of an HPV vaccine trial in Guanacaste, Costa Rica (Herrero and others, 2008). Using a sample of size 
collected by a two-stage re-stratification sample design, we estimated the association between age and lifetime number of sex partners, and seropositivity by the glutathione S-transferase (GST) multiplex Luminex assay (which can be used as a measure of current or past HPV exposure) based on laboratory cutoffs (
1 if seropositive by GST; 0 otherwise) (Coseo and others, 2010; 2011). In addition, the associations of GST with incident cervical HPV infection, with and without adjusting for the number of sex partners, are also evaluated. All analyses were performed separately for both HPV16 and HPV18 infections.
At the first stage of sampling, 388 unvaccinated women were selected from the initial study population of 2814 women who were HPV16 DNA negative at enrollment, using SSRS with stratum defined by “baseline HPV16 ELISA levels (
8, 8–86, or 
86 EU/ml)” and “HPV16 incident infection status over follow-up (
1 if HPV16 infected; 0 otherwise)”. This baseline sample, however, resulted in insufficient sample sizes for estimation of the association of anti-HPV18 antibodies (seropositivity) with HPV18 infection status. For example, there were only 9 women (out of an available 42) who were HPV18 seropositive with HPV18 infection in the baseline sample. Therefore, to improve the precision for estimating the association between seropositivity and incident infection for HPV18, an augmentation sample of 112 women was selected in the second stage using a re-stratified sample design from the 2582 women who were HPV18 DNA negative at enrollment and not selected in the baseline sample (
). The strata in the augmented sample are formed by a new set of stratification variables of “HPV18 ELISA levels” (
7, 7–40, or 
40 EU/ml) and “HPV18 incident infection status” (
1 if HPV18 infected; 0 otherwise). After assays were performed for the entire sample (
), 12 women for whom an assay failed were dropped from the analysis to give the final sample of size 
. This study differs from typical studies with augmentation sampling, in which the augmentation sample is selected from the same target population as is the baseline sample selected from, our study population was expanded from control women who were HPV16 DNA negative only at enrollment in the HPV trial to include women who were HPV16 and/or 18 DNA negative at enrollment. Figure 1 shows the two-stage sample design involving re-stratification and the population expansion described above.
Fig. 1.
In Stage 1, 388 women were selected using SSRS from 2814 women who are HPV16 DNA negative (HPV16
) at baseline. In Stage 2, 122 women were augmented using SSRS design with a different set of stratification variables from 2582 women who are HPV18 DNA negative (HPV18
) at baseline, excluding the first-stage sample. The overlapped population was the women who are both HPV16
and HPV18
at baseline, denoted by 
; 
denotes the first-stage population but excluding 
; and 
denotes the second-stage population but excluding 
. After assays were performed for the entire sample (
), 12 women for whom an assay failed were dropped from the analysis to give the final sample of size 
.
2.2. Sample weighting
This section describes the computation of the sample weights for the re-stratification sample design with population expansion. The population represented by the baseline sample is different from but overlapped with the population represented by the augmented sample. Therefore, we partition the combined population into three domains—
and 
—corresponding to the overlapped population, the population represented by the baseline sample (
) excluding 
, and the population represented by the augmented sample (
) excluding 
. During the augmentation sampling, the baseline sample is excluded from the population to avoid duplicate selections.
Suppose that the baseline population (i.e. 
) is stratified into 
strata, the 
th stratum consisting of 
subjects (
), from which 
subjects are selected by simple random sampling (SRS). The sample weights under stratified sampling are calculated as the population size divided by the sample size in each stratum, i.e. 
for subject 
sampled in stratum 
in the baseline sample, and the sampling fractions are the inverse of the sample weights, i.e. 
for stratum 
. During augmentation sampling, the population (i.e. 
) is re-stratified into 
strata with stratum 
consisting of 
subjects (
), from which, excluding the baseline sample selected in 
, 
subjects are selected by SRS. Define
 |
The sample weights for the augmentation sample are 
for the subject 
selected in stratum 
and the sampling fractions are 
, where 
denotes the collection of subjects sampled in stratum 
in the baseline (augmented) sample. Note that the sample weights in the augmented sample depend on the baseline sample via 
, i.e. the number of subjects in the baseline sample falling in stratum 
, which is random.
To make the augmented sample fully represent the population in 
, we adjust the sample weights by adding the value of one to the sample weights for subjects selected in 
in the baseline sample. The populations in 
and 
are well represented by the baseline and augmentation sample weights, respectively. The population in 
, however, is overrepresented (represented twice) because the weighted baseline sample and the weighted augmentation sample in 
are representing the same population in 
. To account for this over-representation, we multiply both the adjusted baseline sample weights and the augmentation sample weights by 0.5 for subjects selected in 
.
In summary, the final weights for a subject 
selected in 
, 
or 
are, respectively,
 |
and
 |
2.3. Estimation of population totals
The parameters of interest, such as proportions, odds ratios, or logistic regression coefficients, can be expressed explicitly or implicitly as functions of totals. Therefore, in this section we describe how to estimate population totals using sample weights described in the preceding section. A population total 
for the three domains (
, 
, and 
) is 
is, where 
is the variable of interest for the subject 
. We define the indicator functions 
if subject 
is in domain 
corresponding to 
, 
, 
, respectively, and 
otherwise, and then 
can be estimated by 
, where 
and 
can be re-expressed as 
where 
and 
. Note that 
is calculated using data selected from only the baseline sample, whereas 
depends on the augmented sample and the baseline sample via 
, which has the random term of 
.
The variance of 
is estimated using standard stratified variance estimation for totals (Cochran, 1977). In order to account for the random variability in 
in the estimation of 
we decompose the variance into the sum of the variance of the conditional expectation of 
and the expectation of the conditional variance of 
, where we condition on the baseline sample. In addition, the variance of 
includes a covariance between 
and 
because the re-stratification depends on the (random) number of sampled subjects in the baseline sample that are in the stratum 
. See supplementary material available at Biostatistics online for further details.
2.4. Estimation of logistic regression coefficients
Let 
be a binary variable 
and be a 
-vector of covariates or risk factors. Consider a logistic regression model
 |
where 
is the intercept and 
a 
-vector of unknown model parameters. The corresponding score equations can be derived by taking the derivatives of the (weighted) pseudo-loglikelihood
 |
with respect to the model parameters, given by
 |
(1) |
with 
being the final weight described in Section 2.2 “Sample Weighting” and 
. The parameters 
and 
can be estimated by solving 
for unknown parameter 
using iterative methods, such as Newton–Raphson iterative method. We denote the resulting estimates by 
and 
.
A convenient way of approximating the variance for non-linear estimators, such as 
, involves calculating the Taylor deviate for each observation. Taylor deviates for an estimator are observation-level terms whose sample weighted sum is a way to write the first-order Taylor approximation of the estimator. Expressing estimators as a sample weighted sum of the Taylor deviates is convenient for complex samples because variance estimators of weighted sums are readily available from standard sample survey method theory. These variance estimators are called Taylor linearization variance estimators. Shah (2004) describes how the Taylor deviate can be obtained by differentiating the weighted-estimator with respect to its weights. For further applications of Taylor deviates to obtain Taylor linearization estimators for logistic and linear regression coefficients, see Graubard and others (2005).
Evaluating the score equations (2.1) at 
and 
, we obtain
 |
where 
is 
evaluated at 
and 
. To obtain the Taylor deviate associated with the 
th subject, we first take the derivative of 
with respect to its final sample weight 
, leading to
 |
Solving 
for 
, we have
 |
since 
.
The Taylor deviate for the 
th sampled subject is 
, and the Var
can be approximated by the variance of the weighted total 
. The variance estimator of 
depends on the sample design of the stratified baseline sample and the augmentation sample, as discussed in Section 2.3.
3. Results
3.1. Simulation study
We conducted a limited simulation study to evaluate the finite-sample performance of the proposed estimators of the population total and the logistic regression coefficients under a stratified baseline and augmentation sample design with re-stratification and population expansion. To reflect the design of the HPV study in Costa Rica, we partition our population into three domains 
, 
, and 
, with population size of 
, respectively. Five equal-sized strata are randomly formed in 
and 
, and four equal-sized strata are randomly formed in 
and 
. We generated a binary outcome variable 
based on the logistic regression model
 |
where the covariate 
is generated as a surrogate of 
as 
if 
; 
otherwise, and 
is assumed to follow an independent normal distribution with differing means 
, corresponding to each of the five equal-sized strata that form our baseline population of size 
. The 
in the expanded population of 
size 
follows an independent normal distribution with mean zero. A common standard deviation 
is specified for all of the independent normal distributions. We set the value of 
and 
to be 
1 and log(1.5), respectively, so that the marginal probability of the outcome, 
, is approximately 28.5%.
In each simulation, we randomly sample 50, 100, 30, 80, and 40 subjects from the five strata in the baseline population 
and 
. Excluding the sampled subjects in 
, an augmented re-stratified sample of size 30, 130, 50, and 30 is randomly selected from the four reformed strata in 
and 
. Note that although the sample sizes across strata in both the baseline sample of size 
and the augmented sample of size 
are fixed, the distribution of the 300 selected baseline subjects across the four strata, which are formed for the selection of the augmentation sample in the populations 
and 
, are random, whose variability is accounted for in the variance of estimates of the population total of 
and the regression coefficients 
and 
.
To evaluate the performance of the proposed methods, we calculate (1) the relative bias, i.e. RelBias 
average of the estimates minus the true value, and then divided by the true value and (2) the ratio of the estimated variances, i.e. VarRatio 
ratios of the mean of the estimated Taylor linearization variances to the empirical variances of estimates.
Based on 2500 replications, we observe that the proposed estimators are approximately unbiased and the ratios of the two estimated variances are close to one for both the total of 
(with 
and 
) and regression coefficient 
(with 
and 
), implying that the Taylor linearization variance estimators perform well in approximating the true variances.
3.2. Analyses of HPV16/18 serological study in Costa Rica
We apply our proposed methods using data from the HPV16/18 study described in Section 2.1. Sampling weights for the baseline and augmentation samples were calculated as described in Section 2.2 “Sample Weighting”.
Using the weighted sample of size 
collected from the baseline sample and the augmented sample, we estimated the association between age and lifetime number of sex partners and GST (Coseo and others, 2010; 2011). In addition, the associations of GST with incident cervical HPV infection, with and without adjusting for the number of sex partners, are also evaluated. All analyses were performed separately for both HPV16 and HPV18.
Our analysis results showed, HPV 16 seropositivity by GST increased with age (
for a 1-year increase, 95% CI 1.03–1.40) and the number of sexual partners (
for a 1-partner increase, 95% CI 1.49–2.83). Incident HPV16 infection was more likely among women HPV 16 seropositive by GST (
, 95% CI 1.36–3.37). We adjusted for number of sexual partners in the logistic regression model. This resulted in a statistically non-significant association of HPV16 incidence with HPV 16 seropositivity by GST (
, 95% CI 0.95–3.13). Logistic regression analyses showed HPV 18 seropositivity by GST increased with age (
for a 1-year increase, 95% CI 1.04–1.38) and number of sexual partners (
for a 1-partner increase, 95% CI 1.18–2.15). In contrast to HPV16, seropositivity by GST for HPV18 showed a statistically non-significant inverse association with incident infection (
, 95% CI 0.12–1.46). This result was not altered by adjustment for number of sexual partners (
, 95% CI 0.12–1.23). For more details about the study on which this example is based, see (Robbins and others, 2014) who applied the proposed methods to the same data and evaluated the GST as a measure of cumulative HPV infection and future immune protection among HPV-unvaccinated women.
4. Discussion
In this paper, we have developed innovative statistical methods for estimating population totals and logistic regression coefficients under a two-stage stratified sample design. Specifically, a stratified baseline sample is selected at the first stage and a re-stratified augmentation sample from an expanded population is selected at the second stage. Under the two-stage stratified sample design, the sample sizes from certain subdomains of interest that were not originally considered in the design of the baseline study are increased; and the original target population is expanded so that an additional random sample of individuals is included in the augmentation sample. Our simulation results showed that for finite samples the proposed estimators are approximately unbiased and the Taylor linearization variance estimators perform well with finite samples in approximating the true variances. The proposed methods are applied to an HPV 16/18 serological study.
The two-stage augmentation sample design should not be confused with two-phase designs used in epidemiologic studies to reduce the costs by limiting ascertainment of expensive covariates to a small subsample selected from the baseline sample (Breslow and Cain, 1988; White, 1982). Outcomes and some covariate information, which may be cheaply gathered for all subjects in the baseline sample, can be used for informative sampling of subjects at phase II. In such a setting, a subsample from the baseline sample is selected based on sample information gathered in the first phase. In contrast, the augmentation sample designs considered in this paper are used to increase the baseline sample size to improve estimation of subdomains that may have insufficient sample sizes in the baseline sample and/or to make the sample representative of an expanded target population from the target population represented by the baseline sample.
Somewhat related to our work on augmentation sampling is dual frame or multiple frame sampling (Hartley, 1962) that is used in survey methodology. In dual frame sampling two sample frames, A and B, each consisting of sampling units, e.g. a list of telephone numbers and a list of household addresses, are used to sample the target population of individuals. The combination of the two frames covers the entire (target) population where often the frames A and B overlap, which is called the overlap domain. Information from samples selected from frames A and B, is combined to estimate the quantities of interest. In our example, the individuals who are DNA HPV16
that are sampled from in the first stage and the individuals who are HPV18
that are sampled from in the second stage might be viewed as two frames. However, different from our augmentation sampling, dual frame sampling independently select samples from each of frame A and frame B, and therefore the part of the population covered by units in the overlap of frame A and frame B may be selected twice; whereas our augmentation sampling excludes the baseline sample that is selected in the first stage, and therefore depends on the first stage sampling, which introduces a random component and a covariance between estimates from each stage that needs to be accommodated in the variance estimation.
In dual frame sampling, various approaches have been discussed for estimating the population total in the overlap domain (Lohr and Rao, 2000). The population total in the overlap domain is usually estimated by a linear combination of the estimated total using the sample from frame A only and the estimated total using the sample from frame B only, where a factor 
, 
is used to linearly combine these two estimated totals. Different methods have been proposed to choose or estimate the 
(Hartley, 1962; Fuller and Burmeister, 1972; Skinner and Rao, 1992). The reader is referred to (Lohr and Rao, 2000) for a thorough comparison of different methods. In Section 2.3, we chose the value of 
. Obtaining an optimal value of 
to achieve the minimal asymptotic variance is a topic for future research interest.
During the augmentation sampling for the HPV16/18 serological study in Costa Rica, no observations were selected from the stratum of women who were HPV18 seronegative at enrollment and had no incident HPV18 infection, because it was expected that this stratum would be like the stratum of women who were HVP16 seronegative at enrollment and had no incident HPV16 infection (were less risky, with low likelihood of exposure) based on HPV natural history studies (Coseo and others, 2010). To account for the non-representation in this stratum, we randomly sampled 10 individuals from the baseline sample of 25 women who were HPV16 seronegative and had no incident HPV16 infection, and then considered this sample as a pseudo-sample that had been selected during augmentation sampling from the stratum of HPV18 seronegative women without an incident HPV18 infection. Stratified sampling weights for these women were calculated as if they had been sampled from that stratum. We verified that the 10 selected women were HPV18 DNA negative at enrollment, HPV18 seronegative at enrollment, and did not develop an incident HPV18 infection. This is consistent with their assignment to the stratum of HPV18 seronegative women without an incident HPV18 infection.
Re-stratification sample designs addressed in this paper can be useful to augment subcohort samples of case-cohort studies. For instance, the subcohort sample that was obtained for an original case series may not be appropriate for a newly considered case series that has quite different risk factor and/or confounder distributions than the original cases. As an example, consider a case-cohort study of primarily Caucasian cancers such as breast cancer and skin cancer where the subcohort was sampled based on age and gender. Suppose that the investigators want to study kidney cancer, which has a higher incidence rate among African Americans. The subcohort sample could be augmented using re-stratification sampling based on race. The methods of Li and others (2012) for estimating variances for regression coefficients using Poisson regression analyses of case-cohort studies with sample weighting of the subcohort can be extended to re-stratification sampling of the subcohort. Similar augmentation sampling could also be considered to enhance a control sample from an existing case–control study to be used for a new study of a different case series type.
Supplementary material
Supplementary Material is available at http//biostatistics.oxfordjournals.org.
Funding
This work was supported by the Intramural Research Program of the National Cancer Institute at the National Institutes of Health.
Supplementary Material
Acknowledgements
Authors are grateful to anonymous AE and referees for their constructive comments. The authors are grateful to the investigators and staff of the Costa Rica Vaccine Trial at the National Cancer Institute, USA, and Proyecto Epidemiologico Guanacaste, Costa Rica. We thank Brian Befano and Greg Rydzak of Information Management Services, Inc. for their assistance with data management. Conflict of Interest: None declared.
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