Nonparametric exponential tilting (EM) engine for NMAR — exptilt_nonparam

Constructs a configuration for the nonparametric exponential tilting estimator under nonignorable nonresponse (NMAR). This engine implements the "Fully Nonparametric Approach" from **Appendix 2** of Riddles et al. (2016). The estimator uses an Expectation-Maximization (EM) algorithm to directly estimate the nonresponse odds $O(x_1, y)$ for aggregated, categorical data.

Usage

exptilt_nonparam_engine(refusal_col = "", max_iter = 100, tol_value = 1e-06)

Arguments

refusal_col: character; the column name in data that contains the aggregated counts of non-respondents (refusals).
max_iter: integer; the maximum number of iterations for the EM algorithm.
tol_value: numeric; the convergence tolerance for the EM algorithm. The loop stops when the sum of absolute changes in the odds matrix is less than this value.

Value

An engine object of class c("nmar_engine_exptilt_nonparam","nmar_engine"). This is a configuration list; it is not a fit. Pass it to nmar.

Details

This engine is designed for cases where all variables (outcomes $Y$, response predictors $X_1$, and instrumental variables $X_2$) are categorical, and the input data is pre-aggregated into strata.

The method assumes an instrumental variable $X_2$ is available. The response probability is assumed to depend on $X_1$ and $Y$, but *not* on $X_2$.

The EM algorithm iteratively solves for the nonresponse odds: $$ O^{(t+1)}(x_1^*, y^*) = \frac{M_{y^*x_1^*}^{(t)}}{N_{y^*x_1^*}} $$ where $M_{y^*x_1^*}^{(t)}$ is the expected count of non-respondents (calculated in the E-step) and $N_{y^*x_1^*}$ is the observed count of respondents for a given stratum $(x_1, y)$.

The final output from the nmar call is an object containing data_to_return, an aggregated data frame where the original 'refusal' counts have been redistributed into the outcome columns (e.g., 'Voted_A', 'Voted_B') as expected non-respondent counts.

References

Minsun Kim Riddles, Jae Kwang Kim, Jongho Im A Propensity-score-adjustment Method for Nonignorable Nonresponse Journal of Survey Statistics and Methodology, Volume 4, Issue 2, June 2016, Pages 215–245. (See **Appendix 2** for this specific method).

Examples

# Test data (Riddles 2016, Table 9)
voting_data_example <- data.frame(
  Gender = rep(c("Male", "Male", "Male", "Male", "Female", "Female", "Female", "Female"), 1),
  Age_group = c("20-29", "30-39", "40-49", ">=50", "20-29", "30-39", "40-49", "50+"),
  Voted_A = c(93, 104, 146, 560, 106, 129, 170, 501),
  Voted_B = c(115, 233, 295, 350, 159, 242, 262, 218),
  Other = c(4, 8, 5, 3, 8, 5, 5, 7),
  Refusal = c(28, 82, 49, 174, 62, 70, 69, 211),
  Total = c(240, 427, 495, 1087, 335, 446, 506, 937)
)

np_em_config <- exptilt_nonparam_engine(
  refusal_col = "Refusal",
  max_iter = 100,
  tol_value = 0.001
)

# Formula: Y1 + Y2 + ... ~ X1_vars | X2_vars
# Here, Y = Voted_A, Voted_B, Other
#      x1 = Gender (response model)
#      x2 = Age_group (instrumental variable)
em_formula <- Voted_A + Voted_B + Other ~ Gender | Age_group

# \donttest{
results_em_np <- nmar(
  formula = em_formula,
  data = voting_data_example,
  engine = np_em_config,
  trace_level = 0
)

# View the final adjusted counts
# (Original counts + expected non-respondent counts)
print(results_em_np$data_final)
#>    Voted_A  Voted_B    Other Gender Age_group
#> 1 108.4560 118.8595 12.68447   Male     20-29
#> 2 137.3696 248.0971 41.53330   Male     30-39
#> 3 172.4092 305.7756 16.81518   Male     40-49
#> 4 705.4611 368.3589 13.18002   Male      >=50
#> 5 149.2433 166.3526 19.40414 Female     20-29
#> 6 180.9255 253.0418 12.03269 Female     30-39
#> 7 224.0130 271.4359 10.55110 Female     40-49
#> 8 693.1421 227.4771 16.38086 Female       50+
# }