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
nr_adjusted_design <- lou_vax_survey_rep |>
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'),
'NR-adjusted' = summarize_rep_weights(nr_adjusted_design, 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-11All 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.
# 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.
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 0.0223Raking 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$`variance-covariance`[5:8,5:8] |>
`colnames<-`(NULL)
#> [,1] [,2] [,3] [,4]
#> SEXMale 355572.45 -29522.95 129208.95 196840.6
#> SEXFemale -29522.95 379494.65 81455.95 268515.8
#> EDUC_ATTAINMENTHigh school or beyond 129208.95 81455.95 4019242.10 -3808577.2
#> EDUC_ATTAINMENTLess than high school 196840.55 268515.75 -3808577.20 4273933.5Crucially, we note that the vector of control totals has the same
names as the estimates produced by using svytotal() with
the primary survey design object whose weights we plan to adjust.
svytotal(x = ~ RACE_ETHNICITY + SEX + EDUC_ATTAINMENT,
design = nr_adjusted_design)
#> total
#> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 101035
#> RACE_ETHNICITYHispanic or Latino 20207
#> RACE_ETHNICITYOther Race, not Hispanic or Latino 34471
#> RACE_ETHNICITYWhite alone, not Hispanic or Latino 440989
#> SEXFemale 319747
#> SEXMale 276955
#> EDUC_ATTAINMENTHigh school or beyond 273389
#> EDUC_ATTAINMENTLess than high school 323313
#> SE
#> RACE_ETHNICITYBlack or African American alone, not Hispanic or Latino 10003.0
#> RACE_ETHNICITYHispanic or Latino 4824.4
#> RACE_ETHNICITYOther Race, not Hispanic or Latino 6222.7
#> RACE_ETHNICITYWhite alone, not Hispanic or Latino 11713.1
#> SEXFemale 13301.6
#> SEXMale 13301.6
#> EDUC_ATTAINMENTHigh school or beyond 13289.2
#> EDUC_ATTAINMENTLess than high school 13289.2To calibrate the design to the estimates, we supply the estimates and
the variance-covariance matrix to calibrate_to_estimate(),
and we supply the cal_formula argument with the same
formula we would use for svytotal(). To use a raking
adjustment, we specify calfun = survey::cal.raking.
raked_design <- calibrate_to_estimate(
rep_design = nr_adjusted_design,
estimate = control_totals_for_raking$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 2067.3494We 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 | 633.6287 |
| se2 | 4824.430 | 107.9829 |
| se3 | 6222.719 | 472.4107 |
| se4 | 11713.138 | 594.1448 |
| se5 | 13301.627 | 616.0314 |
| se6 | 13301.627 | 596.2990 |
| se7 | 13289.206 | 2004.8048 |
| se8 | 13289.206 | 2067.3494 |
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 | 13416.0747 | 13398.6829 |
| se2 | 13411.0008 | 13427.8218 |
| se3 | 633.6287 | 633.6287 |
| se4 | 107.9829 | 107.9829 |
| se5 | 472.4107 | 472.4107 |
| se6 | 594.1448 | 594.1448 |
| se7 | 616.0314 | 616.0314 |
| se8 | 596.2990 | 596.2990 |
| se9 | 2004.8048 | 2004.8048 |
| se10 | 2067.3494 | 2067.3494 |
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(
rep_design = nr_adjusted_design,
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(
primary_rep_design = nr_adjusted_design,
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 | 0.0234586 | 0.0234216 |
| se2 | 0.0234586 | 0.0234216 |
| se3 | 0.0009765 | 0.0009766 |
| se4 | 0.0001857 | 0.0001856 |
| se5 | 0.0007811 | 0.0007809 |
| se6 | 0.0007040 | 0.0007039 |
| se7 | 0.0007464 | 0.0007464 |
| se8 | 0.0007464 | 0.0007464 |
| se9 | 0.0033341 | 0.0033339 |
| se10 | 0.0033341 | 0.0033339 |
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] 514 344 51 970 396 367 699 552 323 168 272 679 107 500 986 933
# Perform the calibration
poststratified_design <- calibrate_to_estimate(
rep_design = nr_adjusted_design,
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.
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(
primary_rep_design = nr_adjusted_design,
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.