# 1. Analysis of Response Rates and Response Propensity

The starting point of any nonresponse bias analysis is to calculate response rates, since nonresponse bias can only occur when a survey or census’s response rate is below 100%. When response rates are below 100%, nonresponse bias will arise if both of the following patterns occur:

- Certain subpopulations are less likely to respond to the survey than others, and

- Those subpopulations differ in outcomes we are trying to measure.

In this vignette, we show how to compare response rates across subpopulations and check whether observed differences reflect statistically significant differences in likelihoods of responding to a survey (referred to as “response propensity”).

# Calculating response rates

## Data Needs

To calculate response rates, it’s not enough to look at just the data from individuals who ultimately responded to the survey; we need a dataset that includes every individual from whom a response was sought. For example, if a school sent out an email survey to parents, then to calculate response rates we would need a list of all the parents to whom an email was sent, regardless of whether that parent ultimately responded. For example, we would need a data file with a response status variable for each parent, as in the example table below:

UNIQUE_ID | RESPONSE_STATUS
--- | ---
ID_00781 | 2 (Nonrespondent)
ID_13053 | 1 (Respondent)
ID_06582 | 3 (Ineligible)
ID_08389 | 2 (Nonrespondent)
ID_11177 | 4 (Unknown Eligibility)
ID_08446 | 3 (Ineligible)
ID_19211 | 1 (Respondent)
ID_12273 | 4 (Unknown Eligibility)

With such a dataset, we need to classify each individual’s response status into one of four categories:

- Ineligible: This individual was asked to complete the survey, but it was discovered that they were not eligible. For example, if a person was invited to complete a school parent survey and that person replied that they are not in fact a parent, that person would be classified as ineligible.

- Eligible Respondent: This individual completed the survey and was in fact eligible to do so.

- Eligible Nonrespondent: This individual did not complete the survey, but it is known that they were eligible to do so. For example, a parent may respond to a school’s emails for purposes such as communicating with teachers, but they may not respond to a survey invitation sent to their email. In this example, it is thus known that the person is a nonrespondent and that they were in fact eligible for the survey.

- Unknown Eligibility: It is unknown whether this individual was eligible to complete the survey. For example, suppose a mail survey was sent to a random sample of addresses in a school district, where the survey was meant to include only parents of school-aged children. If a given address never replies to the survey and it is thus unknown whether any parents live at that address, then the case would be classified as having unknown eligibility.

## Specifying a Response Rate Analysis

The function `calculate_response_rates()` can be used to calculate the response rate for a survey once all of the records in the data have been grouped into the four categories. The argument `status` identifies the variable in the data used to indicate response status, and the argument `status_codes` allows the user to specify how the categories of that variable should be interpreted. The result is a data frame, with the response rate given in the column `RR3_Unweighted` and the underlying counts given in the columns `n`, `n_ER`, `n_EN`, etc.

    # Load example data
    data('involvement_survey_srs', package = "nrba")

    # Calculate overall response rates for the survey
    calculate_response_rates(
      data = involvement_survey_srs,
      status = "RESPONSE_STATUS",
      status_codes = c(
        'ER' = 'Respondent',
        'EN' = 'Nonrespondent',
        'IE' = 'Ineligible',
        'UE' = 'Unknown'
      ),
      rr_formula = 'RR1'
    )
    #>   RR1_Unweighted    n n_ER n_EN n_IE n_UE
    #> 1      0.6316342 5000 3011 1521  233  235

If the survey uses weights to account for unequal probabilities of selection, then the name of a weight variable can be supplied to the `weights` argument. The output variables `Nhat`, `Nhat_ER`, etc. provide weighted versions of the variables `n`, `n_ER`, etc.

    # Load example data
    data('involvement_survey_str2s', package = "nrba")

    # Calculate overall response rates for the survey
    calculate_response_rates(
      data = involvement_survey_str2s,
      weights = "BASE_WEIGHT",
      status = "RESPONSE_STATUS",
      status_codes = c(
        'ER' = 'Respondent',
        'EN' = 'Nonrespondent',
        'IE' = 'Ineligible',
        'UE' = 'Unknown'
      ),
      rr_formula = 'RR1'
    )
    #>   RR1_Unweighted RR1_Weighted    n    Nhat n_ER  Nhat_ER n_EN Nhat_EN n_IE
    #> 1      0.6516736    0.6286321 1000 20012.2  623 11967.32  284 6066.26   44
    #>   Nhat_IE n_UE Nhat_UE
    #> 1  975.12   49  1003.5

## Calculate response rates by group

To calculate response rates separately by groups, we can first group the input data using the `group_by()` function from the popular ‘dplyr’ package.

    library(dplyr)

    involvement_survey_srs |>
      group_by(STUDENT_RACE) |>
      calculate_response_rates(
        status = "RESPONSE_STATUS",
        status_codes = c(
          'ER' = 'Respondent',
          'EN' = 'Nonrespondent',
          'IE' = 'Ineligible',
          'UE' = 'Unknown'
        ),
        rr_formula = 'RR1'
      )
    #> # A tibble: 7 × 7
    #>   STUDENT_RACE                      RR1_Unweighted     n  n_ER  n_EN  n_IE  n_UE
    #>   <chr>                                      <dbl> <int> <int> <int> <int> <int>
    #> 1 AM7 (American Indian or Alaska N…          0.730    37    27     9     0     1
    #> 2 AS7 (Asian)                                0.708    52    34    11     4     3
    #> 3 BL7 (Black or African American)            0.682   723   465   178    41    39
    #> 4 HI7 (Hispanic or Latino Ethnicit…          0.341   896   291   523    43    39
    #> 5 MU7 (Two or More Races)                    0.709   112    73    28     9     2
    #> 6 PI7 (Native Hawaiian or Other Pa…          0.771    37    27     7     2     1
    #> 7 WH7 (White)                                0.696  3143  2094   765   134   150

## Formulas for calculating response rates

When every person invited to participate in a survey is known to be eligible for the survey, it is quite easy to calculate a response rate: simply count the number of respondents and divide this count by the total number of respondents and nonrespondents. Response rate calculations become more complicated when there are cases with unknown eligibility.

When there are cases with unknown eligibility, the common convention is to use one of the response rate formulas promulgated by the American Association for Public Opinion Research (AAPOR); see @theamericanassociationforpublicopinionresearchStandardDefinitionsFinal2016. The three most commonly-used formulas are referred to as “RR1”, “RR3”, and “RR5”. Response rates can be calculated using one or more formulas by supplying the formula names to the `rr_formula` argument of `calculate_response_rates()`.

    #>   RR1_Unweighted RR3_Unweighted RR5_Unweighted    n n_ER n_EN n_IE n_UE
    #> 1      0.6316342      0.6331604      0.6643866 5000 3011 1521  233  235
    #>     e_unwtd
    #> 1 0.9511018

These formulas differ only in how many cases with unknown eligibility are estimated to in fact be eligible for the survey.

\[ \begin{aligned} RR1 &= ER / (ER + EN + UE) \\ RR3 &= ER / (ER + EN + (e \times UE)) \\ RR5 &= ER / (ER + EN) \\ &\text{where:} \\ ER &\text{ is the total number of eligible respondents} \\ EN &\text{ is the total number of eligible nonrespondents} \\ UE &\text{ is the total number of cases whose eligibility is unknown} \\ &\text{and} \\ e &\text{ is an *estimate* of the percent of unknown eligibility cases} \\ &\text{which are in fact eligible} \end{aligned} \] For the RR3 formula, it is necessary to produce an estimate of the share of unknown eligibility cases who are in fact eligible, denoted \(e\). One common estimation method is the “CASRO” method: among cases with known eligibility status, calculate the percent who are known to be eligible.

\[ \begin{aligned} \text{CASRO}&\text{ method:} \\ e &= 1 - \frac{IE}{IE + ER + EN} \\ &\text{where:} \\ IE &\text{ is the total number of sampled cases known to be ineligible} \\ \end{aligned} \]

When calculating response rates for population subgroups, one can either assume the eligibility rate \(e\) is constant across all subgroups or one can estimate the eligibility rate separately for each subgroup. When using the CASRO method to estimate the eligibility rate, the former approach is referred to as the “CASRO overall” method, while the latter approach is referred to as the “CASRO subgroup” method. Either option can be used by specifying either `elig_method='CASRO-overall'` or `elig_method='CASRO-subgroup'`.

    involvement_survey_srs |>
      group_by(PARENT_HAS_EMAIL) |>
      calculate_response_rates(
        status = "RESPONSE_STATUS",
        status_codes = c(
          'ER' = 'Respondent',
          'EN' = 'Nonrespondent',
          'IE' = 'Ineligible',
          'UE' = 'Unknown'
        ),
        rr_formula = 'RR3',
        elig_method = "CASRO-subgroup"
      )
    #> # A tibble: 2 × 8
    #>   PARENT_HAS_EMAIL RR3_Unweighted     n  n_ER  n_EN  n_IE  n_UE e_unwtd
    #>   <chr>                     <dbl> <int> <int> <int> <int> <int>   <dbl>
    #> 1 Has Email                 0.637  4234  2564  1271   201   198   0.950
    #> 2 No Email                  0.610   766   447   250    32    37   0.956

Alternatively, the user may specify a specific value of \(e\) to use for response rates or a variable in the data to use which specifies a value of \(e\) for different groups.

    involvement_survey_srs %>%
      mutate(e_by_email = ifelse(PARENT_HAS_EMAIL == 'Has Email', 0.75, 0.25)) %>%
      group_by(PARENT_HAS_EMAIL) %>%
      calculate_response_rates(status = "RESPONSE_STATUS",
                               status_codes = c(
                                 'ER' = 'Respondent',
                                 'EN' = 'Nonrespondent',
                                 'IE' = 'Ineligible',
                                 'UE' = 'Unknown'
                               ),
                               rr_formula = "RR3",
                               elig_method = "specified",
                               e = "e_by_email")
    #> # A tibble: 2 × 8
    #>   PARENT_HAS_EMAIL RR3_Unweighted     n  n_ER  n_EN  n_IE  n_UE e_unwtd
    #>   <chr>                     <dbl> <int> <int> <int> <int> <int>   <dbl>
    #> 1 Has Email                 0.644  4234  2564  1271   201   198    0.75
    #> 2 No Email                  0.633   766   447   250    32    37    0.25

# Testing for differences in response propensity by subpopulation

To check whether observed differences in response rates are attributable to random sampling, we can use a Chi-Squared test. This test evaluates whether the observed differences in response rates between categories of an auxiliary variable are attributable simply to random sampling rather than subpopulations having different likelihoods of responding to the survey. If the p-value for this test is quite small, then there is evidence that an observed difference in response rates between subpopulations in the sample is unlikely to have arisen simply because of random sampling.

To ensure that the Chi-Squared test correctly takes into account the sample design, it is necessary to create a survey design object using the ‘survey’ package. The following example demonstrates the creation of a survey design object for a stratified multistage sample.

    library(survey)

    # Create a survey design object with the 'survey' package
    involvement_svy <- svydesign(
      data = involvement_survey_str2s,
      weights = ~ BASE_WEIGHT,
      strata =  ~ SCHOOL_DISTRICT,
      ids =     ~ SCHOOL_ID             + UNIQUE_ID, # School ID and Student ID
      fpc =     ~ N_SCHOOLS_IN_DISTRICT + N_STUDENTS_IN_SCHOOL # Population sizes at each sampling stage
    )

With the survey design object thus created, we can use the function `chisq_test_ind_response()` to test whether response status is independent of auxiliary variables using a Chi-Square test (with Rao-Scott’s second-order adjustment for complex survey designs).

    chisq_test_ind_response(
      survey_design = involvement_svy,
      # Specify the response status variable
      status = "RESPONSE_STATUS",
      # Specify how to interpret categories of response status variable
      status_codes = c(
        'ER' = 'Respondent',
        'EN' = 'Nonrespondent',
        'IE' = 'Ineligible',
        'UE' = 'Unknown'
      ),
      # Specify variable(s) to use for the Chi-Square test(s)
      aux_vars = c("STUDENT_RACE", "PARENT_HAS_EMAIL")
    )
    #> Subsetting to only compare eligible respondents to eligible nonrespondents: `RESPONSE_STATUS` in ('Respondent','Nonrespondent')
    #>   auxiliary_variable  statistic      ndf      ddf      p_value
    #> 1       STUDENT_RACE 13.6560181 4.109016 328.7213 1.634543e-10
    #> 2   PARENT_HAS_EMAIL  0.2941381 1.000000  80.0000 5.890885e-01
    #>                                           test_method variance_method
    #> 1 Rao-Scott Chi-Square test (second-order adjustment)   linearization
    #> 2 Rao-Scott Chi-Square test (second-order adjustment)   linearization

# Modeling response propensity using logistic regression

To better understand the relationship between response propensity and auxiliary variables, it can be helpful to model response status directly. The function `predict_response_status_via_glm()` facilitates the modeling process.

    predict_response_status_via_glm(
      survey_design = involvement_svy,
      status = "RESPONSE_STATUS",
      status_codes = c("ER" = "Respondent",
                       "EN" = "Nonrespondent",
                       "IE" = "Ineligible",
                       "UE" = "Unknown"),
      # Specify models
      model_selection = 'main-effects',
      # Specify predictor variables for the model
      numeric_predictors = c("STUDENT_AGE"),
      categorical_predictors = c("PARENT_HAS_EMAIL", "STUDENT_RACE")
    )
    #> # A tibble: 9 × 12
    #>   variable        variable_level_p_value variable_category estimated_coefficient
    #>   <chr>                            <dbl> <chr>                             <dbl>
    #> 1 (Intercept)                   NA       <NA>                            1.76
    #> 2 PARENT_HAS_EMA…                9.82e-1 No Email                       -0.00709
    #> 3 STUDENT_AGE                    7.13e-1 <NA>                           -0.0113
    #> 4 STUDENT_RACE                   7.45e-9 AS7 (Asian)                     0.679
    #> 5 STUDENT_RACE                   7.45e-9 BL7 (Black or Af…              -0.298
    #> 6 STUDENT_RACE                   7.45e-9 HI7 (Hispanic or…              -2.14
    #> 7 STUDENT_RACE                   7.45e-9 MU7 (Two or More…               1.47
    #> 8 STUDENT_RACE                   7.45e-9 PI7 (Native Hawa…              13.3
    #> 9 STUDENT_RACE                   7.45e-9 WH7 (White)                    -0.690
    #> # ℹ 8 more variables: se_coefficient <dbl>, conf_intrvl_lower <dbl>,
    #> #   conf_intrvl_upper <dbl>, p_value_coefficient <dbl>,
    #> #   LRT_chisq_statistic <dbl>, LRT_DEff <dbl>, LRT_df_numerator <int>,
    #> #   LRT_df_denominator <dbl>