# Navigated weighting (NAWT) estimation `nawt` estimates a pre-specified parameter of interest (e.g., the average treatment effects (ATE) or the average treatment effects on the treated (ATT)) with the inverse probability weighting where propensity scores are estimated using estimating equations suitable for the parameter of interest. It includes the covariate balancing propensity score proposed by Imai and Ratkovic (2014), which uses covariate balancing conditions in propensity score estimation. `nawt` can also be used to estimate average outcomes in missing outcome cases. ```r nawt( formula, outcome, estimand = "ATT", method = "score", data, weights = NULL, alpha = 2, twostep = TRUE, boot = FALSE, B = 2000, clevel = 0.95, message = TRUE ) ``` ## Arguments - `formula`: an object of class `formula` (or one that can be coerced to that class): a symbolic description of the model to be fitted. - `outcome`: a character string specifying the name of outcome values in `data`. - `estimand`: a character string specifying a parameter of interest. Choose "ATT" for the average treatment effects on the treated estimation, "ATE" for the average treatment effects estimation, "ATC" for the average outcomes estimation in missing outcome cases. You can choose "ATEcombined" for the combined estimation for the average treatment effects estimation. - `method`: a character string specifying a type of weighting functions in propensity score estimation ($\omega(\pi)$). Choose "score" for a power function of propensity scores (need to specify the value for alpha), "cb" for a covariate balancing weighting function, or "both" to use both the above weighting functions (need to specify the value for alpha). - `data`: a data frame (or one that can be coerced to that class) containing the outcomes and the variables in the model. - `weights`: an optional vector of ‘prior weights’ (e.g. sampling weights) to be used in the fitting process. Should be NULL or a numeric vector. - `alpha`: a positive value for an exponent in a power weighting function ($\omega(\pi) = \pi^\alpha$, in the ATT estimation, for example). Default is 2. Set to 0 to use the standard logistic regression for propensity score estimation. Note that `nawt` with alpha being one of the pre-specified values (0, 0.5, 1, , 5) runs substantially faster than with any other values, and the latter case requires `hypergeo` package. - `twostep`: a logical value indicating whether to use a two-step estimator when `method = "both"`. Default is `TRUE`. Set to `FALSE` to use a continuously-updating GMM estimator, which is substantially computationally intensive. - `boot`: a logical value indicating whether to use a non-parametric bootstrapping method to estimate the variance-covariance matrix and confidence intervals for parameters. Default is `FALSE`. Set to `FALSE` to use a sandwich-type asymptotic covariance estimator. - `B`: the number of bootstrap replicates. Default is 2,000. - `clevel`: confidence level. Default is 0.95. - `message`: a logical value indicating whether messages are shown or not. ## Returns `nawt` returns an object of class inheriting from "nawt". The function summary (i.e., `summary.nawt`) can be used to obtain or print a summary of the results. An object of class "nawt" is a list containing the following components: - **est**: the point estimate of the parameter of interest. - **weights**: the estimated inverse probability weights. - **ps**: the estimated propensity scores. A matrix of two sets of the estimated propensity scores is returned when `estimand = "ATE"`. - **coefficients**: a named vector of coefficients. A matrix of two sets of coefficients for two sets of propensity scores is returned when `estimand = "ATE"`. - **varcov**: the variance-covariance matrix of the coefficients and parameter of interest. - **converged**: logical. Was the algorithm judged to have converged? - **naive_weights**: the estimated inverse probability weights with the standard logistic regression for the propensity score estimation. - **naive_coef**: a named vector of coefficients with the standard logistic regression for the propensity score estimation. - **scratio**: an optimal ratio of the covariate balancing weighting function to the power weighting function in taking the weighted average weights for the weighted score conditions when `method = "both"` and `twostep = TRUE`. A vector of length two for two propensity score estimation is returned when `estimand = "ATE"`. - **estimand**: the parameter of interest specified. - **method**: the method specified. - **outcome**: the outcome vector. - **alpha**: alpha specified. - **names.x**: names of the explanatory variables in propensity score estimation. - **prior.weights**: the weights initially supplied, a vector of 1s if none were. - **treat**: the treatment vector. The missingness vector when the missing outcome cases. - **ci**: a matrix of the confidence intervals for the parameter of interest. - **omega**: a vector of weights for the weighted score conditions ($\omega$). A matrix of two sets of omega is returned when `estimand = "ATE"`. - **effN_ps**: the effective sample size for the propensity score estimation. A vector of length two for two propensity score estimation is returned when `estimand = "ATE"`. - **effN_est**: the effective sample size for the parameter of interest estimation. - **effN_original**: the effective sample size with the initial weights. - **formula**: formula specified. - **call**: the matched call. - **data**: the data argument. ## Details The treatment variable (or, missingness variable in missing outcome cases) must be binary and coded as 0 (for controlled or non-missing observations) or 1 (for treated or missing observations). When the data frame has incomplete cases, which have NAs for either of the treatment variable, explanatory variables for propensity score estimation, or the outcome variable, `nawt` conducts listwise deletion. Returned values (e.g., `weights`, `ps`, `data`) do not contain values for these deleted cases. The parameter of interest is estimated by the Hajek estimator, where inverse probability weights are standardized to sum to 1 within each treatment group after being calculated as $t_i / \pi_i - (1 - t_i) / (1 - \pi_i)$ for the ATE estimation, $(t_i - \pi_i) / (1 - \pi_i)$ for the ATT estimation, $(t_i - \pi_i) / \pi_i$ for the ATC estimation, and $(1 - t_i) / (1 - \pi_i)$ for the missing outcome cases. For the ATE estimation, it is recommended to specify the `estimand` as `"ATE"` but you may specify it as `"ATEcombined"`. The former utilizes the separated estimation whereas the latter utilizes the combined estimation, and the former should produce smaller biases and variances. Note that the former estimates two propensity scores for each observation by estimating two propensity score functions with different estimating equations. When a two-step estimator is used in `nawt` with `method = "both"`, `scratio` ($r$) is calculated in the first step. `scratio` is a ratio of accuracy in propensity score estimation in the NAWT with a power weighting function with a specified `alpha` to that with a covariate balancing weighting function. It determines the mixture weight in the second step, like the weighting matrix in the two-step over-identified GMM estimation, where weighted estimating equations of those with the power weighting function and the covariate balancing function is used. This mixture weight is proportional to the `scratio` (e.g., $\omega(\pi) = r \pi^\alpha + (1 - r) / (1 - \pi)$, in the ATT estimation). Since the NAWT utilizes weighted estimating equations in propensity score estimation, it may sometimes become unstable especially when only a few observations have extremely large weights in propensity score estimation. `nawt` generates a warning when the effective sample size for propensity score estimation is smaller than a quarter of the effective sample size with the initial weights. In that case, carefully look at the estimated coefficients to check whether the estimation fails or not and `cbcheck` will be helpful. ## Examples ```r # Simulation from Kang and Shafer (2007) and Imai and Ratkovic (2014) # ATT estimation # True ATT is 10 tau <- 10 set.seed(12345) n <- 1000 X <- matrix(rnorm(n * 4, mean = 0, sd = 1), nrow = n, ncol = 4) prop <- 1 / (1 + exp(X[, 1] - 0.5 * X[, 2] + 0.25 * X[, 3] + 0.1 * X[, 4])) treat <- rbinom(n, 1, prop) y <- 210 + 27.4 * X[, 1] + 13.7 * X[, 2] + 13.7 * X[, 3] + 13.7 * X[, 4] + tau * treat + rnorm(n) df <- data.frame(X, treat, y) colnames(df) <- c("x1", "x2", "x3", "x4", "treat", "y") # A misspecified model Xmis <- data.frame(x1mis = exp(X[, 1] / 2), x2mis = X[, 2] * (1 + exp(X[, 1]))^(-1) + 10, x3mis = (X[, 1] * X[, 3] / 25 + 0.6)^3, x4mis = (X[, 2] + X[, 4] + 20)^2) # Data frame and formulas for propensity score estimation df <- data.frame(df, Xmis) formula_c <- as.formula(treat ~ x1 + x2 + x3 + x4) formula_m <- as.formula(treat ~ x1mis + x2mis + x3mis + x4mis) # Correct propensity score model # Power weighting function with alpha = 2 fits2c <- nawt(formula = formula_c, outcome = "y", estimand = "ATT", method = "score", data = df, alpha = 2) summary(fits2c) # Covariate balancing weighting function fitcbc <- nawt(formula = formula_c, outcome = "y", estimand = "ATT", method = "cb", data = df) summary(fitcbc) # Standard logistic regression fits0c <- nawt(formula = formula_c, outcome = "y", estimand = "ATT", method = "score", data = df, alpha = 0) summary(fits0c) # Misspecified propensity score model # Power weighting function with alpha = 2 fits2m <- nawt(formula = formula_m, outcome = "y", estimand = "ATT", method = "score", data = df, alpha = 2) summary(fits2m) # Covariate balancing weighting function fitcbm <- nawt(formula = formula_m, outcome = "y", estimand = "ATT", method = "cb", data = df) summary(fitcbm) # Standard logistic regression fits0m <- nawt(formula = formula_m, outcome = "y", estimand = "ATT", method = "score", data = df, alpha = 0) summary(fits0m) # Empirical example # Load the LaLonde data data(LaLonde) formula_l <- as.formula("exper ~ age + I(age^2) + educ + I(educ^2) + black + hisp + married + nodegr + I(re75 / 1000) + I(re75 == 0) + I(re74 / 1000)") # Experimental benchmark mean(subset(LaLonde, exper == 1 & treat == 1)$re78) - mean(subset(LaLonde, exper == 1 & treat == 0)$re78) # Power weighting function with alpha = 2 fits2l <- nawt(formula = formula_l, estimand = "ATT", method = "score", outcome = "re78", data = LaLonde, alpha = 2) mean(subset(LaLonde, exper == 1 & treat == 1)$re78) - with(LaLonde, sum((1 - exper) * re78 * fits2l$weights) / sum((1 - exper) * fits2l$weights)) # Covariate balancing weighting function fitcbl <- nawt(formula = formula_l, estimand = "ATT", method = "cb", outcome = "re78", data = LaLonde) mean(subset(LaLonde, exper == 1 & treat == 1)$re78) - with(LaLonde, sum((1 - exper) * re78 * fitcbl$weights) / sum((1 - exper) * fitcbl$weights)) # Standard logistic regression fits0l <- nawt(formula = formula_l, estimand = "ATT", method = "score", outcome = "re78", data = LaLonde, alpha = 0) mean(subset(LaLonde, exper == 1 & treat == 1)$re78) - with(LaLonde, sum((1 - exper) * re78 * fits0l$weights) / sum((1 - exper) * fits0l$weights)) ``` ## References Imai, Kosuke and Marc Ratkovic. 2014. "Covariate Balancing Propensity Score." Journal of the Royal Statistical Society, Series B (Statistical Methodology) 76 (1): 243--63. Christian Fong, Marc Ratkovic and Kosuke Imai (2019). CBPS: Covariate Balancing Propensity Score. R package version 0.21. https://CRAN.R-project.org/package=CBPS Katsumata, Hiroto. 2020. "Navigated Weighting to Improve Inverse Probability Weighting for Missing Data Problems and Causal Inference." arXiv preprint arXiv:2005.10998. ## See Also `summary.nawt` ## Author(s) Hiroto Katsumata

nawt function