# 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-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)
    #> Successive difference 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 | 633.6287
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 | 13425.0629 | 13409.3779
se2 | 13396.6464 | 13431.8766
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 | 0.0234766 | 0.0234450
se2 | 0.0234766 | 0.0234450
se3 | 0.0009764 | 0.0009766
se4 | 0.0001856 | 0.0001856
se5 | 0.0007810 | 0.0007809
se6 | 0.0007040 | 0.0007039
se7 | 0.0007461 | 0.0007464
se8 | 0.0007461 | 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]  85 856  82 131 451 952  46 113 524 377 916 685 901 717 960 610

    # 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.

    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(
      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.

    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).