Doubly robust estimates of for evaluating effects of treatments on ordinal outcomes.

The available parameters for evaluating treatment efficacy are:

Difference in (weighted) means: The outcome levels are treated numerically, with each level possibly assigned a weight. The difference in average outcomes is computed.
Log odds ratio: The comparison describes the average log-odds (treatment level 1 versus 0) of the cumulative probability for each level of the outcome.
Mann-Whitney: The probability that a randomly-selected individual receiving treatment 1 will have a larger outcome value than a randomly selected individual receiving treatment 0 (with ties assigned weight 1/2).

drord(
  out,
  treat,
  covar,
  out_levels = sort(unique(out)),
  out_form = paste0(colnames(covar), collapse = "+"),
  out_weights = rep(1, length(out_levels)),
  out_model = "pooled-logistic",
  treat_form = "1",
  param = c("weighted_mean", "log_odds", "mann_whitney"),
  ci = "wald",
  alpha = 0.05,
  nboot = 1000,
  return_models = TRUE,
  est_dist = TRUE,
  stratify = FALSE,
  ...
)

Arguments

out	A `numeric` vector containing the outcomes. Missing outcomes are allowed.
treat	A `numeric` vector containing treatment status. Missing values are not allowed unless the corresponding entry in `out` is also missing. Only values of 0 or 1 are treated as actual treatment levels. Any other value is assumed to encode a value for which the outcome is missing and the corresponding outcome value is ignored.
covar	A `data.frame` containing the covariates to include in the working proportional odds model.
out_levels	A `numeric` vector containing all ordered levels of the outcome.
out_form	The right-hand side of a regression formula for the working proportional odds model. NOTE: THIS FORMULA MUST NOT SUPPRESS THE INTERCEPT.
out_weights	A vector of `numeric` weights with length equal to the length of `out_levels`.
out_model	Which R function should be used to fit the proportional odds model. The recommended option is `"pooled-logistic"`. Other options available include `"polr"` (from the `MASS` package), "vglm" (from the `VGAM` package), or `"clm"` (from the `ordinal` package).
treat_form	The right-hand side of a regression formula for the working model of treatment probability as a function of covariates
param	A vector of `characters` indicating which of the three treatment effect parameters should be estimated (`"weighted_mean", "log_odds",` and/or `"mann_whitney"`).
ci	A vector of `characters` indicating which confidence intervals should be computed (`"bca"` and/or `"wald"`)
alpha	Confidence intervals have nominal level 1-`alpha`.
nboot	Number of bootstrap replicates used to compute bootstrap confidence intervals.
return_models	If `TRUE` the fitted working proportional odds models and treatment probability models are returned.
est_dist	A `boolean` indicating whether estimates of the CDF and PMF should be computed and returned. For real data analysis, we generally recommend leaving as `TRUE`; however, when studying performance in simulations, it can save time to set to `FALSE`.
stratify	If `TRUE`, then a fully stratified estimator is computed, i.e., the empirical CDF of each treatment arm is estimated stratifying by levels of `covar`. For now, this option is limited to univariate covariates.
...	Other options (not currently used).

Value

An object of class drord. In addition to information related to how drord was called, the output contains the following:

log_odds: inference pertaining to the log-odds parameter. NULL if this parameter not requested in call to drord.
mann_whitney: inference pertaining to the Mann-Whitney parameter. NULL if this parameter not requested in call to drord.
weighted_mean: inference pertaining to weighted mean parameter. NULL if this parameter not requested in call to drord.
cdf: inference pertaining to the treatment-specific CDFs. See the plot method for a convenient way of visualizing this information. NULL if est_dist = FALSE in call to drord.
pmf: inference pertaining to the treatment-specific PMFs. See the plot method for a convenient way of visualizing this information. NULL if est_dist = FALSE in call to drord.
treat_mod: the fitted model for the probability of treatment as a function of covariates. NULL if return_models = FALSE
out_mod: the proportional odds model fit in each treatment arm. named entries in list indicate the corresponding treatment arm. NULL if return_models = FALSE or stratify = TRUE.

Details

In each case, estimates are constructed by obtaining a doubly robust estimate of the cumulative distribution function (CDF) for each treatment group. This is achieved by fitting a (working) proportional odds model that includes inverse probability of treatment weights. The inclusion of these weights ensures that, so long as the working model includes intercept terms, the resultant estimate of the CDF is an augmented inverse probability of treatment weighted estimate. This implies that the estimate is nonparametric efficient if the working model contains the truth; however, even if the working model does not contain the truth, the CDF estimates are consistent and asymptotically normal with variance expected to dominate that of an unadjusted estimate of the same treatment effect.

The CDF estimates are subsequently mapped into estimates of each requested parameter for evaluating treatment effects. The double robustness and efficiency properties of the CDF estimates extend to these quantities as well. Confidence intervals and hypothesis tests can be carried out in closed form using Wald-style intervals and tests or using a nonparametric corrected and accelerated bootstrap (BCa). Inference for the CDF and probability mass function is also returned and can be used for subsequent visualizations (see plot.drord).

Examples

data(covid19)

# get estimates of all parameters based on main-effects 
# proportional odds model and intercept-only propensity model
fit <- drord(out = covid19$out, treat = covid19$treat, 
             covar = covid19[, "age_grp", drop = FALSE])

# get estimates of all parameters based on proportional odds and
# propensity model that treats age_grp as categorical
fit2 <- drord(out = covid19$out, treat = covid19$treat, 
              covar = covid19[, "age_grp", drop = FALSE],
         out_form = "factor(age_grp)",
         treat_form = "factor(age_grp)")

# obtain estimator stratified by age group 
fit3 <- drord(out = covid19$out, treat = covid19$treat, 
              covar = covid19[, "age_grp", drop = FALSE],
         stratify = TRUE)

# demonstration with missing outcome data
covid19$out[1:5] <- NA

# propensity model should now adjust for covariates to address
# the potential for informative missingness
fit4 <- drord(out = covid19$out, treat = covid19$treat, 
              covar = covid19[, "age_grp", drop = FALSE],
         treat_form = "age_grp")