# 4. Analysis

#### Yuki Atsusaka and Seo-young Silvia Kim

    library(rankingQ)
    library(estimatr)
    data(identity)

The IPW workflow is easiest to think of in two steps. First, `imprr_weights()` estimates respondent-level correction weights. Those weights can then be used for point estimation in downstream analyses.

    out_weights <- imprr_weights(
      identity,
      main_q = c("party", "religion", "gender", "race"),
      anc_correct = "anc_correct_identity"
    )
    #> No weight column supplied; using equal weights for all observations.

    out_weights$est_p_random
    #> [1] 0.3163224

For example, to estimate the average rank of party as a point estimate, one can leverage weighted linear regression as follows:

    lm_robust(
      party ~ 1,
      data = out_weights$results,
      weights = out_weights$results$weights
    ) |>
      tidy()
    #>          term estimate  std.error statistic p.value conf.low conf.high   df
    #> 1 (Intercept) 3.220388 0.02790142  115.4202       0 3.165641  3.275135 1081
    #>   outcome
    #> 1   party

That gives a valid point estimate, but its standard error treats the estimated IPW weights as fixed. For built-in ranking summaries, use `imprr_weights_boot()`, which resamples respondents, reruns `imprr_weights()` inside each resample, and summarizes the resulting quantities of interest.

## Computing Average Ranks

The `avg_rank` function remains a convenient way to compute point estimates from the IPW-adjusted respondent-level data:

    items_df <- data.frame(
      variable = c("party", "religion", "gender", "race"),
      item = c("Party", "Religion", "Gender", "Race")
    )

    avg_rank(out_weights$results, items = items_df, weight = "weights", raw = FALSE)
    #>       item          qoi     mean         se    lower    upper method
    #> 1    Party Average Rank 3.220388 0.02790142 3.165641 3.275135    IPW
    #> 2 Religion Average Rank 2.609023 0.04051924 2.529518 2.688529    IPW
    #> 3   Gender Average Rank 1.706054 0.02448379 1.658013 1.754095    IPW
    #> 4     Race Average Rank 2.464535 0.02616016 2.413204 2.515865    IPW

For bootstrap uncertainty on the same quantities, `imprr_weights_boot()` returns summaries in the same general format as `imprr_direct()`:

    out_boot <- imprr_weights_boot(
      identity,
      main_q = c("party", "religion", "gender", "race"),
      anc_correct = "anc_correct_identity",
      n_bootstrap = 10,
      seed = 123
    )
    #> No weight column supplied; using equal weights for all observations.

    out_boot$est_p_random
    #>        mean     lower     upper
    #> 1 0.3157438 0.2982159 0.3334646
    subset(out_boot$results, qoi == "average rank")
    #> # A tibble: 4 × 6
    #>   item     qoi          outcome        mean lower upper
    #>   <chr>    <chr>        <chr>         <dbl> <dbl> <dbl>
    #> 1 gender   average rank Avg: gender    1.70  1.66  1.73
    #> 2 party    average rank Avg: party     3.21  3.15  3.30
    #> 3 race     average rank Avg: race      2.47  2.41  2.55
    #> 4 religion average rank Avg: religion  2.61  2.52  2.71

The same object also contains bootstrap summaries for pairwise, top-k, and marginal ranking quantities:

    subset(out_boot$results, qoi == "pairwise ranking")
    #> # A tibble: 12 × 6
    #>    item     qoi              outcome      mean lower upper
    #>    <chr>    <chr>            <chr>       <dbl> <dbl> <dbl>
    #>  1 gender   pairwise ranking v. party    0.867 0.842 0.893
    #>  2 gender   pairwise ranking v. race     0.770 0.740 0.792
    #>  3 gender   pairwise ranking v. religion 0.658 0.631 0.680
    #>  4 party    pairwise ranking v. gender   0.133 0.107 0.158
    #>  5 party    pairwise ranking v. race     0.275 0.242 0.307
    #>  6 party    pairwise ranking v. religion 0.381 0.348 0.426
    #>  7 race     pairwise ranking v. gender   0.230 0.208 0.260
    #>  8 race     pairwise ranking v. party    0.725 0.693 0.758
    #>  9 race     pairwise ranking v. religion 0.571 0.532 0.612
    #> 10 religion pairwise ranking v. gender   0.342 0.320 0.369
    #> 11 religion pairwise ranking v. party    0.619 0.574 0.652
    #> 12 religion pairwise ranking v. race     0.429 0.388 0.468

If you want uncertainty for an arbitrary downstream weighted analysis, the same principle applies: resample respondents and rerun `imprr_weights()` within each resample before recomputing the target estimand.