# Calibrating to Estimated Control Totals

## Sample-based Calibration: An Introduction

Calibration weighting adjustments such as post-stratification or raking are often helpful for reducing sampling variance or non-sampling errors such as nonresponse bias. Typically, the benchmark data used for these calibration adjustments are estimates published by agencies such as the United States Census Bureau. For example, pollsters in the United States frequently rake polling data so that estimates for variables such as age or educational attainment match benchmark estimates from the American Community Survey (ACS).

While benchmark data (also known as control totals) for raking and calibration are often treated as the “true” population values, they are usually themselves estimates with their own sampling variance or margin of error. When we calibrate to estimated control totals rather than to “true” population values, we may need to account for the variance of the estimated control totals to ensure that calibrated estimates appropriately reflect sampling error of both the primary survey of interest and the survey from which the control totals were estimated. This is especially important if the control totals have large margins of error.

A handful of statistical methods have been developed for the problem of conducting replication variance estimation after sample-based calibration; see Opsomer and Erciulescu (2021) for a clear overview of the literature on this topic. All of these methods apply calibration weighting adjustment to full-sample weights and to each column of replicate weights. The key “trick” of these methods is to adjust each column of replicate weights to a slightly different set of control totals, varying the control totals used across all of the replicates in such a way that the variation across the columns is in a sense proportionate to the sampling variance of the control totals.

These statistical methods differ in the way that they generate different control totals for each column of replicate weights and in the type of data they require the analyst to use. The method of Fuller (1998) requires the analyst to have a variance-covariance matrix for the estimated control totals, while the method of Opsomer and Erciulescu (2021) requires the analyst to use the full dataset for the control survey along with associated replicate weights.

## Functions for Implementing Sample-Based Calibration

The ‘svrep’ package provides two functions to implement sample-based calibration.

With the function calibrate_to_estimate(), adjustments to replicate weights are conducted using the method of Fuller (1998), requiring a variance-covariance matrix for the estimated control totals.

calibrate_to_estimate(
rep_design = rep_design,
estimate = vector_of_control_totals,
vcov_estimate = variance_covariance_matrix_for_controls,
cal_formula = ~ CALIBRATION_VARIABLE_1 + CALIBRATION_VARIABLE_2 + ...,
)

With the function calibrate_to_sample(), adjustments to replicate weights are conducted using the method proposed by Opsomer and Erciulescu (2021), requiring a dataset with replicate weights to use for estimating control totals and their sampling variance.

calibrate_to_sample(
primary_rep_design = primary_rep_design,
control_rep_design = control_rep_design
cal_formula = ~ CALIBRATION_VARIABLE_1 + CALIBRATION_VARIABLE_2 + ...,
)

For both functions, it is possible to use a variety of calibration options from the survey package’s calibrate() function. For example, the user can specify a specific calibration function to use, such as calfun = survey::cal.linear to implement post-stratification or calfun = survey::cal.raking to implement raking. The bounds argument can be used to specify bounds for the calibration weights, and the arguments such as maxit or epsilon allow finer control over the Newton-Raphson algorithm used to implement calibration.

## An Example Using a Vaccination Survey

To illustrate the different methods for conducting sample-based calibration, we’ll use an example survey measuring Covid-19 vaccination status and a handful of demographic variables, based on a simple random sample of 1,000 residents of Louisville, Kentucky.

# Load the data
library(svrep)
data("lou_vax_survey")

# Inspect the first few rows
head(lou_vax_survey) |> knitr::kable()
RESPONSE_STATUS RACE_ETHNICITY SEX EDUC_ATTAINMENT VAX_STATUS SAMPLING_WEIGHT
Nonrespondent White alone, not Hispanic or Latino Female Less than high school NA 596.702
Nonrespondent Black or African American alone, not Hispanic or Latino Female High school or beyond NA 596.702
Respondent White alone, not Hispanic or Latino Female Less than high school Vaccinated 596.702
Nonrespondent White alone, not Hispanic or Latino Female Less than high school NA 596.702
Nonrespondent White alone, not Hispanic or Latino Female High school or beyond NA 596.702
Respondent White alone, not Hispanic or Latino Female High school or beyond Vaccinated 596.702

For the purpose of variance estimation, we’ll create jackknife replicate weights.

suppressPackageStartupMessages(
library(survey)
)

lou_vax_survey_rep <- svydesign(
data = lou_vax_survey,
ids = ~ 1, weights = ~ SAMPLING_WEIGHT
) |>
as.svrepdesign(type = "JK1", mse = TRUE)
#> Call: as.svrepdesign.default(svydesign(data = lou_vax_survey, ids = ~1,
#>     weights = ~SAMPLING_WEIGHT), type = "JK1", mse = TRUE)
#> Unstratified cluster jacknife (JK1) with 1000 replicates and MSE variances.

Because the survey’s key outcome, vaccination status, is only measured for respondents, we’ll do a quick nonresponse weighting adjustment to help make reasonable estimates for this outcome.

# Conduct nonresponse weighting adjustment

redistribute_weights(
reduce_if = RESPONSE_STATUS == "Nonrespondent",
increase_if = RESPONSE_STATUS == "Respondent"
) |>
subset(RESPONSE_STATUS == "Respondent")

# Inspect the result of the adjustment
rbind(
'Original' = summarize_rep_weights(lou_vax_survey_rep, type = 'overall'),
)[,c("nrows", "rank", "avg_wgt_sum", "sd_wgt_sums")]
#>             nrows rank avg_wgt_sum  sd_wgt_sums
#> Original     1000 1000      596702 0.000000e+00
#> NR-adjusted   502  502      596702 8.219437e-11

All of the work so far has given us the replicate design for the primary survey, prepared for calibration. Now we need to obtain benchmark data we can use for the calibration. We’ll use a Public-Use Microdata Sample (PUMS) dataset from the ACS as our source for benchmark data on race/ethnicity, sex, and educational attainment.

data("lou_pums_microdata")
# Inspect some of the rows/columns of data ----
tail(lou_pums_microdata, n = 5) |>
dplyr::select(AGE, SEX, RACE_ETHNICITY, EDUC_ATTAINMENT) |>
knitr::kable()
AGE SEX RACE_ETHNICITY EDUC_ATTAINMENT
20 Female Other Race, not Hispanic or Latino High school or beyond
50 Male Hispanic or Latino Less than high school
57 Female Other Race, not Hispanic or Latino High school or beyond
44 Male Hispanic or Latino High school or beyond
25 Female Hispanic or Latino High school or beyond

Next, we’ll prepare the PUMS data to use replication variance estimation using provided replicate weights.

# Convert to a survey design object ----
pums_rep_design <- svrepdesign(
data = lou_pums_microdata,
weights = ~ PWGTP,
repweights = "PWGTP\\d{1,2}",
type = "successive-difference",
variables = ~ AGE + SEX + RACE_ETHNICITY + EDUC_ATTAINMENT,
mse = TRUE
)

pums_rep_design
#> Call: svrepdesign.default(data = lou_pums_microdata, weights = ~PWGTP,
#>     repweights = "PWGTP\\d{1,2}", type = "successive-difference",
#>     variables = ~AGE + SEX + RACE_ETHNICITY + EDUC_ATTAINMENT,
#>     mse = TRUE)
#> with 80 replicates and MSE variances.

When conduction calibration, we have to make sure that the data from the control survey represent the same population as the primary survey. Since the Louisville vaccination survey only represents adults, we need to subset the control survey design to adults.

# Subset to only include adults
pums_rep_design <- pums_rep_design |> subset(AGE >= 18)

In addition, we need to ensure that the control survey design has calibration variables that align with the variables in the primary survey design of interest. This may require some data manipulation.

suppressPackageStartupMessages(
library(dplyr)
)

# Check that variables match across data sources ----
pums_rep_design$variables |> dplyr::distinct(RACE_ETHNICITY) #> RACE_ETHNICITY #> 1 Black or African American alone, not Hispanic or Latino #> 2 White alone, not Hispanic or Latino #> 3 Hispanic or Latino #> 4 Other Race, not Hispanic or Latino setdiff(lou_vax_survey_rep$variables$RACE_ETHNICITY, pums_rep_design$variables$RACE_ETHNICITY) #> character(0) setdiff(lou_vax_survey_rep$variables$SEX, pums_rep_design$variables$SEX) #> character(0) setdiff(lou_vax_survey_rep$variables$EDUC_ATTAINMENT, pums_rep_design$variables$EDUC_ATTAINMENT) #> character(0) # Estimates from the control survey (ACS) svymean( design = pums_rep_design, x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT ) #> mean #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 0.19950 #> RACE_ETHNICITYHispanic or Latino 0.04525 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 0.04631 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 0.70894 #> SEXMale 0.47543 #> SEXFemale 0.52457 #> EDUC_ATTAINMENTHigh school or beyond 0.38736 #> EDUC_ATTAINMENTLess than high school 0.61264 #> SE #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 0.0010 #> RACE_ETHNICITYHispanic or Latino 0.0002 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 0.0008 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 0.0007 #> SEXMale 0.0007 #> SEXFemale 0.0007 #> EDUC_ATTAINMENTHigh school or beyond 0.0033 #> EDUC_ATTAINMENTLess than high school 0.0033 # Estimates from the primary survey (Louisville vaccination survey) svymean( design = nr_adjusted_design, x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT ) #> mean #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 0.169323 #> RACE_ETHNICITYHispanic or Latino 0.033865 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 0.057769 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 0.739044 #> SEXFemale 0.535857 #> SEXMale 0.464143 #> EDUC_ATTAINMENTHigh school or beyond 0.458167 #> EDUC_ATTAINMENTLess than high school 0.541833 #> SE #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 0.0168 #> RACE_ETHNICITYHispanic or Latino 0.0081 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 0.0104 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 0.0196 #> SEXFemale 0.0223 #> SEXMale 0.0223 #> EDUC_ATTAINMENTHigh school or beyond 0.0223 #> EDUC_ATTAINMENTLess than high school 0.0223 ### Raking to estimated control totals We’ll start by raking to estimates from the ACS for race/ethnicity, sex, and educational attainment, first using the calibrate_to_sample() method and then using the calibrate_to_estimate() method. For the calibrate_to_sample() method, we need to obtain a vector of point estimates for the control totals, and an accompanying variance-covariance matrix for the estimates. acs_control_totals <- svytotal( x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT, design = pums_rep_design ) control_totals_for_raking <- list( 'estimates' = coef(acs_control_totals), 'variance-covariance' = vcov(acs_control_totals) ) # Inspect point estimates control_totals_for_raking$estimates
#> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino
#>                                                                119041
#>                                      RACE_ETHNICITYHispanic or Latino
#>                                                                 27001
#>                      RACE_ETHNICITYOther Race, not Hispanic or Latino
#>                                                                 27633
#>                     RACE_ETHNICITYWhite alone, not Hispanic or Latino
#>                                                                423027
#>                                                               SEXMale
#>                                                                283688
#>                                                             SEXFemale
#>                                                                313014
#>                                  EDUC_ATTAINMENTHigh school or beyond
#>                                                                231136
#>                                  EDUC_ATTAINMENTLess than high school
#>                                                                365566

# Inspect a few rows of the control totals' variance-covariance matrix
control_totals_for_raking$variance-covariance[5:8,5:8] |> colnames<-(NULL) #> [,1] [,2] [,3] [,4] #> SEXMale 355572.45 -29522.95 129208.95 196840.6 #> SEXFemale -29522.95 379494.65 81455.95 268515.8 #> EDUC_ATTAINMENTHigh school or beyond 129208.95 81455.95 4019242.10 -3808577.2 #> EDUC_ATTAINMENTLess than high school 196840.55 268515.75 -3808577.20 4273933.5 Crucially, we note that the vector of control totals has the same names as the estimates produced by using svytotal() with the primary survey design object whose weights we plan to adjust. svytotal(x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT, design = nr_adjusted_design) #> total #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 101035 #> RACE_ETHNICITYHispanic or Latino 20207 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 34471 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 440989 #> SEXFemale 319747 #> SEXMale 276955 #> EDUC_ATTAINMENTHigh school or beyond 273389 #> EDUC_ATTAINMENTLess than high school 323313 #> SE #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 10003.0 #> RACE_ETHNICITYHispanic or Latino 4824.4 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 6222.7 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 11713.1 #> SEXFemale 13301.6 #> SEXMale 13301.6 #> EDUC_ATTAINMENTHigh school or beyond 13289.2 #> EDUC_ATTAINMENTLess than high school 13289.2 To calibrate the design to the estimates, we supply the estimates and the variance-covariance matrix to calibrate_to_estimate(), and we supply the cal_formula argument with the same formula we would use for svytotal(). To use a raking adjustment, we specify calfun = survey::cal.raking. raked_design <- calibrate_to_estimate( rep_design = nr_adjusted_design, estimate = control_totals_for_raking$estimates,
vcov_estimate = control_totals_for_raking$variance-covariance, cal_formula = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT, calfun = survey::cal.raking, # Required for raking epsilon = 1e-9 ) #> Selection of replicate columns whose control totals will be perturbed will be done at random. #> For tips on reproducible selection, see help('calibrate_to_estimate') Now we can compare the estimated totals for the calibration variables to the actual control totals. As we might intuitively expect, the estimated totals from the survey now match the control totals, and the standard errors for the estimated totals match the standard errors of the control totals. # Estimated totals after calibration svytotal(x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT, design = raked_design) #> total #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 119041 #> RACE_ETHNICITYHispanic or Latino 27001 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 27633 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 423027 #> SEXFemale 313014 #> SEXMale 283688 #> EDUC_ATTAINMENTHigh school or beyond 231136 #> EDUC_ATTAINMENTLess than high school 365566 #> SE #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 633.63 #> RACE_ETHNICITYHispanic or Latino 107.98 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 472.41 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 594.14 #> SEXFemale 616.03 #> SEXMale 596.30 #> EDUC_ATTAINMENTHigh school or beyond 2004.80 #> EDUC_ATTAINMENTLess than high school 2067.35 # Matches the control totals! cbind( 'total' = control_totals_for_raking$estimates,
'SE' = control_totals_for_raking$variance-covariance |> diag() |> sqrt() ) #> total #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 119041 #> RACE_ETHNICITYHispanic or Latino 27001 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 27633 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 423027 #> SEXMale 283688 #> SEXFemale 313014 #> EDUC_ATTAINMENTHigh school or beyond 231136 #> EDUC_ATTAINMENTLess than high school 365566 #> SE #> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 633.6287 #> RACE_ETHNICITYHispanic or Latino 107.9829 #> RACE_ETHNICITYOther Race, not Hispanic or Latino 472.4107 #> RACE_ETHNICITYWhite alone, not Hispanic or Latino 594.1448 #> SEXMale 596.2990 #> SEXFemale 616.0314 #> EDUC_ATTAINMENTHigh school or beyond 2004.8048 #> EDUC_ATTAINMENTLess than high school 2067.3494 We can now see what effect the raking adjustment has had on our primary estimate of interest, which is the overall Covid-19 vaccination rate. The raking adjustment has reduced our estimate of the vaccination rate by about one percentage point and results in a similar standard error estimate. estimates_by_design <- svyby_repwts( rep_designs = list( "NR-adjusted" = nr_adjusted_design, "Raked" = raked_design ), FUN = svytotal, formula = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT ) t(estimates_by_design[,-1]) |> knitr::kable() NR-adjusted Raked RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 101035.199 119041.0000 RACE_ETHNICITYHispanic or Latino 20207.040 27001.0000 RACE_ETHNICITYOther Race, not Hispanic or Latino 34470.833 27633.0000 RACE_ETHNICITYWhite alone, not Hispanic or Latino 440988.928 423027.0000 SEXFemale 319746.689 313014.0000 SEXMale 276955.311 283688.0000 EDUC_ATTAINMENTHigh school or beyond 273389.363 231136.0000 EDUC_ATTAINMENTLess than high school 323312.637 365566.0000 se1 10002.951 633.6013 se2 4824.430 107.8910 se3 6222.719 471.9384 se4 11713.138 593.9867 se5 13301.627 615.9416 se6 13301.627 595.8601 se7 13289.206 2004.2516 se8 13289.206 2065.7822 Instead of doing the raking using a vector of control totals and their variance-covariance matrix, we could have instead done the raking by simply supplying the two replicate design objects to the function calibrate_to_sample(). This uses the Opsomer-Erciulescu method of adjusting replicate weights, in contrast to calibrate_to_estimate(), which uses Fuller’s method of adjusting replicate weights. raked_design_opsomer_erciulescu <- calibrate_to_sample( primary_rep_design = nr_adjusted_design, control_rep_design = pums_rep_design, cal_formula = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT, calfun = survey::cal.raking, epsilon = 1e-9 ) #> Matching between primary and control replicates will be done at random. #> For tips on reproducible matching, see help('calibrate_to_sample') We can see that the two methods yield identical point estimates from the full-sample weights, and the standard errors match nearly exactly for the calibration variables (race/ethnicity, sex, and educational attainment). However, there are small but slightly more noticeable differences in the standard errors for other variables, such as VAX_STATUS, resulting from the fact that the two methods have different methods of adjusting the replicate weights. Opsomer and Erciulescu (2021) explain the differences between the two methods and discuss why the the Opsomer-Erciulescu method used in calibrate_to_sample() may have better statistical properties than the Fuller method used in calibrate_to_estimate(). estimates_by_design <- svyby_repwts( rep_designs = list( "calibrate_to_estimate()" = raked_design, "calibrate_to_sample()" = raked_design_opsomer_erciulescu ), FUN = svytotal, formula = ~ VAX_STATUS + RACE_ETHNICITY + SEX + EDUC_ATTAINMENT ) t(estimates_by_design[,-1]) |> knitr::kable() calibrate_to_estimate() calibrate_to_sample() VAX_STATUSUnvaccinated 282904.4084 282904.4084 VAX_STATUSVaccinated 313797.5916 313797.5916 RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 119041.0000 119041.0000 RACE_ETHNICITYHispanic or Latino 27001.0000 27001.0000 RACE_ETHNICITYOther Race, not Hispanic or Latino 27633.0000 27633.0000 RACE_ETHNICITYWhite alone, not Hispanic or Latino 423027.0000 423027.0000 SEXFemale 313014.0000 313014.0000 SEXMale 283688.0000 283688.0000 EDUC_ATTAINMENTHigh school or beyond 231136.0000 231136.0000 EDUC_ATTAINMENTLess than high school 365566.0000 365566.0000 se1 13422.6415 13410.9076 se2 13376.7600 13410.1233 se3 633.6013 633.5867 se4 107.8910 107.9815 se5 471.9384 471.9864 se6 593.9867 594.0839 se7 615.9416 616.0278 se8 595.8601 596.2711 se9 2004.2516 2003.8989 se10 2065.7822 2066.5749 ### Post-stratification The primary difference between post-stratification and raking is that post-stratification essentially involves only a single calibration variable, with population benchmarks provided for each value of that variable. In the Louisville vaccination survey, that variable is called POSTSTRATUM and is based on combinations of race/ethnicity, sex, and educational attainment. # Create matching post-stratification variable in both datasets nr_adjusted_design <- nr_adjusted_design |> transform(POSTSTRATUM = interaction(RACE_ETHNICITY, SEX, EDUC_ATTAINMENT, sep = "|")) pums_rep_design <- pums_rep_design |> transform(POSTSTRATUM = interaction(RACE_ETHNICITY, SEX, EDUC_ATTAINMENT, sep = "|")) levels(pums_rep_design$variables$POSTSTRATUM) <- levels( nr_adjusted_design$variables$POSTSTRATUM ) # Estimate control totals acs_control_totals <- svytotal( x = ~ POSTSTRATUM, design = pums_rep_design ) poststratification_totals <- list( 'estimate' = coef(acs_control_totals), 'variance-covariance' = vcov(acs_control_totals) ) # Inspect the control totals poststratification_totals$estimate |>
as.data.frame() |>
colnames<-('estimate') |>
knitr::kable()
estimate
POSTSTRATUMBlack or African American alone, not Hispanic or Latino|Female|High school or beyond 12168
POSTSTRATUMHispanic or Latino|Female|High school or beyond 3998
POSTSTRATUMOther Race, not Hispanic or Latino|Female|High school or beyond 6190
POSTSTRATUMWhite alone, not Hispanic or Latino|Female|High school or beyond 84041
POSTSTRATUMBlack or African American alone, not Hispanic or Latino|Male|High school or beyond 17648
POSTSTRATUMHispanic or Latino|Male|High school or beyond 4132
POSTSTRATUMOther Race, not Hispanic or Latino|Male|High school or beyond 6687
POSTSTRATUMWhite alone, not Hispanic or Latino|Male|High school or beyond 96272
POSTSTRATUMBlack or African American alone, not Hispanic or Latino|Female|Less than high school 41944
POSTSTRATUMHispanic or Latino|Female|Less than high school 10321
POSTSTRATUMOther Race, not Hispanic or Latino|Female|Less than high school 6753
POSTSTRATUMWhite alone, not Hispanic or Latino|Female|Less than high school 118273
POSTSTRATUMBlack or African American alone, not Hispanic or Latino|Male|Less than high school 47281
POSTSTRATUMHispanic or Latino|Male|Less than high school 8550
POSTSTRATUMOther Race, not Hispanic or Latino|Male|Less than high school 8003
POSTSTRATUMWhite alone, not Hispanic or Latino|Male|Less than high school 124441

To post-stratify the design, we can either supply the estimates and their variance-covariance matrix to calibrate_to_estimate(), or we can supply the two replicate design objects to calibrate_to_sample(). With either method, we need to supply the cal_formula argument with the same formula we would use for svytotal(). To use a post-stratification adjustment (rather than raking), we specify calfun = survey::cal.linear.

# Post-stratify the design using the estimates
poststrat_design_fuller <- calibrate_to_estimate(
estimate = poststratification_totals$estimate, vcov_estimate = poststratification_totals$variance-covariance,
cal_formula = ~ POSTSTRATUM, # Specify the post-stratification variable
calfun = survey::cal.linear # This option is required for post-stratification
)
#> Selection of replicate columns whose control totals will be perturbed will be done at random.
#> For tips on reproducible selection, see help('calibrate_to_estimate')
# Post-stratify the design using the two samples
poststrat_design_opsomer_erciulescu <- calibrate_to_sample(
control_rep_design = pums_rep_design,
cal_formula = ~ POSTSTRATUM, # Specify the post-stratification variable
calfun = survey::cal.linear # This option is required for post-stratification
)
#> Matching between primary and control replicates will be done at random.
#> For tips on reproducible matching, see help('calibrate_to_sample')

As with the raking example, we can see that the full-sample post-stratified estimates are exactly the same for the two methods. The standard errors for post-stratification variables are essentially identical, while the standard errors for other variables differ slightly.

estimates_by_design <- svyby_repwts(
rep_designs = list(
"calibrate_to_estimate()" = poststrat_design_fuller,
"calibrate_to_sample()" = poststrat_design_opsomer_erciulescu
),
FUN = svymean,
formula = ~ VAX_STATUS + RACE_ETHNICITY + SEX + EDUC_ATTAINMENT
)

t(estimates_by_design[,-1]) |>
knitr::kable()
calibrate_to_estimate() calibrate_to_sample()
VAX_STATUSUnvaccinated 0.4779776 0.4779776
VAX_STATUSVaccinated 0.5220224 0.5220224
RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 0.1994982 0.1994982
RACE_ETHNICITYHispanic or Latino 0.0452504 0.0452504
RACE_ETHNICITYOther Race, not Hispanic or Latino 0.0463095 0.0463095
RACE_ETHNICITYWhite alone, not Hispanic or Latino 0.7089418 0.7089418
SEXFemale 0.4754266 0.4754266
SEXMale 0.5245734 0.5245734
EDUC_ATTAINMENTHigh school or beyond 0.3873558 0.3873558
EDUC_ATTAINMENTLess than high school 0.6126442 0.6126442
se1 0.0234553 0.0234357
se2 0.0234553 0.0234357
se3 0.0009762 0.0009765
se4 0.0001856 0.0001856
se5 0.0007811 0.0007802
se6 0.0007037 0.0007036
se7 0.0007464 0.0007464
se8 0.0007464 0.0007464
se9 0.0033292 0.0033324
se10 0.0033292 0.0033324

## Reproducibility

The calibration methods for calibrate_to_estimate() and calibrate_to_sample() involve one element of randomization: determining which columns of replicate weights are assigned to a given perturbation of the control totals. In the calibrate_to_sample() method of Fuller (1998), if the control totals are a vector of dimension $$p$$, then $$p$$ columns of replicate weights will be calibrated to $$p$$ different vectors of perturbed control totals, formed using the $$p$$ scaled eigenvectors from a spectral decomposition of the control totals’ variance-covariance matrix (sorted in order by the largest to smallest eigenvalues). To control which columns of replicate weights will be calibrated to each set of perturbed control totals, we can use the function argument col_selection.

# Randomly select which columns will be assigned to each set of perturbed control totals
dimension_of_control_totals <- length(poststratification_totals$estimate) columns_to_perturb <- sample(x = 1:ncol(nr_adjusted_design$repweights),
size = dimension_of_control_totals)

print(columns_to_perturb)
#>  [1] 586 316 548 240 182 617 619 557 993 125 377 450 578 828 900 925

# Perform the calibration
poststratified_design <- calibrate_to_estimate(
estimate = poststratification_totals$estimate, vcov_estimate = poststratification_totals$variance-covariance,
cal_formula = ~ POSTSTRATUM,
calfun = survey::cal.linear,
col_selection = columns_to_perturb # Specified for reproducibility
)

The calibrated survey design object contains an element perturbed_control_cols which indicates which columns were calibrated to the perturbed control totals; this can be useful to save and use as an input to col_selection to ensure reproducibility.

poststratified_design$perturbed_control_cols #> NULL For calibrate_to_sample(), matching is done between columns of replicate weights in the primary survey and columns of replicate weights in the control survey. The matching is done at random unless the user specifies otherwise using the argument control_col_matches. In the Louisville Vaccination Survey, the primary survey has 1,000 replicates while the control survey has 80 columns. So we can match these 80 columns to the 1,000 replicates by specifying 1,000 values consisting of NA or integers between 1 and 80. # Randomly match the primary replicates to control replicates set.seed(1999) column_matching <- rep(NA, times = ncol(nr_adjusted_design$repweights))
column_matching[sample(x = 1:1000, size = 80)] <- 1:80

str(column_matching)
#>  int [1:1000] NA NA NA 34 NA NA NA 68 NA NA ...

# Perform the calibration
poststratified_design <- calibrate_to_sample(
control_rep_design = pums_rep_design,
cal_formula = ~ POSTSTRATUM,
calfun = survey::cal.linear,
control_col_matches = column_matching
)

The calibrated survey design object contains an element control_column_matches which control survey replicate each primary survey replicate column was matched to.

str(poststratified_design\$control_column_matches)
#>  int [1:1000] NA NA NA 34 NA NA NA 68 NA NA ...

# References

Fuller, Wayne A. 1998. “Replication Variance Estimation for Two-Phase Samples.” Statistica Sinica 8 (4): 1153–64.
Opsomer, J. D., and A. L. Erciulescu. 2021. “Replication Variance Estimation After Sample-Based Calibration.” Survey Methodology, Statistics Canada Vol. 47 (No. 2).