# Two-phase designs In a two-phase design a sample is taken from a population and a subsample taken from the sample, typically stratified by variables not known for the whole population. The second phase can use any design supported for single-phase sampling. The first phase must currently be one-stage element or cluster sampling ```r twophase(id, strata = NULL, probs = NULL, weights = NULL, fpc = NULL, subset, data, method=c("full","approx","simple"), pps=NULL) twophasevar(x,design) twophase2var(x,design) ``` ## Arguments - `id`: list of two formulas for sampling unit identifiers - `strata`: list of two formulas (or `NULL`s) for stratum identifies - `probs`: list of two formulas (or `NULL`s) for sampling probabilities - `weights`: Only for `method="approx"`, list of two formulas (or `NULL`s) for sampling weights - `fpc`: list of two formulas (or `NULL`s) for finite population corrections - `subset`: formula specifying which observations are selected in phase 2 - `data`: Data frame will all data for phase 1 and 2 - `method`: `"full"` requires (much) more memory, but gives unbiased variance estimates for general multistage designs at both phases. `"simple"` or `"approx"` uses the standard error calculation from version 3.14 and earlier, which uses much less memory and is correct for designs with simple random sampling at phase one and stratified random sampling at phase two. - `pps`: With `method="full"`, an optional list of two PPS specifications for `svydesign`. At the moment, the phase-one element must be `NULL` - `x`: probability-weighted estimating functions - `design`: two-phase design ## Details The population for the second phase is the first-phase sample. If the second phase sample uses stratified (multistage cluster) sampling without replacement and all the stratum and sampling unit identifier variables are available for the whole first-phase sample it is possible to estimate the sampling probabilities/weights and the finite population correction. These would then be specified as `NULL`. Two-phase case-control and case-cohort studies in biostatistics will typically have simple random sampling with replacement as the first stage. Variances given here may differ slightly from those in the biostatistics literature where a model-based estimator of the first-stage variance would typically be used. Variance computations are based on the conditioning argument in Section 9.3 of Sarndal et al. Method `"full"` corresponds exactly to the formulas in that reference. Method `"simple"` or `"approx"` (the two are the same) uses less time and memory but is exact only for some special cases. The most important special case is the two-phase epidemiologic designs where phase 1 is simple random sampling from an infinite population and phase 2 is stratified random sampling. See the `tests` directory for a worked example. The only disadvantage of method="simple" in these cases is that standardization of margins (`marginpred`) is not available. For `method="full"`, sampling probabilities must be available for each stage of sampling, within each phase. For multistage sampling this requires specifying either `fpc` or `probs` as a formula with a term for each stage of sampling. If no `fpc` or `probs` are specified at phase 1 it is treated as simple random sampling from an infinite population, and population totals will not be correctly estimated, but means, quantiles, and regression models will be correct. The `pps` argument allows for PPS sampling at phase two (or eventually at phase one), and also for Poisson sampling at phase two as a model for non-response. ## Returns `twophase` returns an object of class `twophase2` (for `method="full"`) or `twophase`. The structure of `twophase2` objects may change as unnecessary components are removed. `twophase2var` and `twophasevar` return a variance matrix with an attribute containing the separate phase 1 and phase 2 contributions to the variance. ## References Sarndal CE, Swensson B, Wretman J (1992) "Model Assisted Survey Sampling" Springer. Breslow NE and Chatterjee N, Design and analysis of two-phase studies with binary outcome applied to Wilms tumour prognosis. "Applied Statistics" 48:457-68, 1999 Breslow N, Lumley T, Ballantyne CM, Chambless LE, Kulick M. (2009) Improved Horvitz-Thompson estimation of model parameters from two-phase stratified samples: applications in epidemiology. Statistics in Biosciences. doi 10.1007/s12561-009-9001-6 Lin, DY and Ying, Z (1993). Cox regression with incomplete covariate measurements. "Journal of the American Statistical Association" 88: 1341-1349. ## See Also `svydesign`, `svyrecvar` for multi*stage* sampling `calibrate` for calibration (GREG) estimators. `estWeights` for two-phase designs for missing data. The "epi" and "phase1" vignettes for examples and technical details. ## Examples ```r ## two-phase simple random sampling. data(pbc, package="survival") pbc$randomized<-with(pbc, !is.na(trt) & trt>0) pbc$id<-1:nrow(pbc) d2pbc<-twophase(id=list(~id,~id), data=pbc, subset=~randomized) svymean(~bili, d2pbc) ## two-stage sampling as two-phase data(mu284) ii<-with(mu284, c(1:15, rep(1:5,n2[1:5]-3))) mu284.1<-mu284[ii,] mu284.1$id<-1:nrow(mu284.1) mu284.1$sub<-rep(c(TRUE,FALSE),c(15,34-15)) dmu284<-svydesign(id=~id1+id2,fpc=~n1+n2, data=mu284) ## first phase cluster sample, second phase stratified within cluster d2mu284<-twophase(id=list(~id1,~id),strata=list(NULL,~id1), fpc=list(~n1,NULL),data=mu284.1,subset=~sub) svytotal(~y1, dmu284) svytotal(~y1, d2mu284) svymean(~y1, dmu284) svymean(~y1, d2mu284) ## case-cohort design: this example requires R 2.2.0 or later library("survival") data(nwtco) ## stratified on case status dcchs<-twophase(id=list(~seqno,~seqno), strata=list(NULL,~rel), subset=~I(in.subcohort | rel), data=nwtco) svycoxph(Surv(edrel,rel)~factor(stage)+factor(histol)+I(age/12), design=dcchs) ## Using survival::cch subcoh <- nwtco$in.subcohort selccoh <- with(nwtco, rel==1|subcoh==1) ccoh.data <- nwtco[selccoh,] ccoh.data$subcohort <- subcoh[selccoh] cch(Surv(edrel, rel) ~ factor(stage) + factor(histol) + I(age/12), data =ccoh.data, subcoh = ~subcohort, id=~seqno, cohort.size=4028, method="LinYing") ## two-phase case-control ## Similar to Breslow & Chatterjee, Applied Statistics (1999) but with ## a slightly different version of the data set nwtco$incc2<-as.logical(with(nwtco, ifelse(rel | instit==2,1,rbinom(nrow(nwtco),1,.1)))) dccs2<-twophase(id=list(~seqno,~seqno),strata=list(NULL,~interaction(rel,instit)), data=nwtco, subset=~incc2) dccs8<-twophase(id=list(~seqno,~seqno),strata=list(NULL,~interaction(rel,stage,instit)), data=nwtco, subset=~incc2) summary(glm(rel~factor(stage)*factor(histol),data=nwtco,family=binomial())) summary(svyglm(rel~factor(stage)*factor(histol),design=dccs2,family=quasibinomial())) summary(svyglm(rel~factor(stage)*factor(histol),design=dccs8,family=quasibinomial())) ## Stratification on stage is really post-stratification, so we should use calibrate() gccs8<-calibrate(dccs2, phase=2, formula=~interaction(rel,stage,instit)) summary(svyglm(rel~factor(stage)*factor(histol),design=gccs8,family=quasibinomial())) ## For this saturated model calibration is equivalent to estimating weights. pccs8<-calibrate(dccs2, phase=2,formula=~interaction(rel,stage,instit), calfun="rrz") summary(svyglm(rel~factor(stage)*factor(histol),design=pccs8,family=quasibinomial())) ## Since sampling is SRS at phase 1 and stratified RS at phase 2, we ## can use method="simple" to save memory. dccs8_simple<-twophase(id=list(~seqno,~seqno),strata=list(NULL,~interaction(rel,stage,instit)), data=nwtco, subset=~incc2,method="simple") summary(svyglm(rel~factor(stage)*factor(histol),design=dccs8_simple,family=quasibinomial())) ```

twophase function