Prediction with a residual bias correction estimator
Description
This method combines the regression estimator with a residual bias correction for estimating a parametric ADRF.
Usage
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 
Arguments
Y 
is the the name of the outcome variable contained in 
treat 
is the name of the treatment variable contained in

covar_formula 
is the formula to describe the covariates needed
to estimate the constant term:

covar_lin_formula 
is the formula to describe the covariates needed
to estimate the linear term, t:

covar_sq_formula 
is the formula to describe the covariates needed
to estimate the quadratic term, t^2:

data 
is a dataframe containing 
e_treat_1 
a vector, representing the conditional expectation of

e_treat_2 
a vector, representing the conditional expectation of

e_treat_3 
a vector, representing the conditional expectation of

e_treat_4 
a vector, representing the conditional expectation of

degree 
is 1 for linear and 2 for quadratic outcome model. 
wt 
is weight used in lsfit for outcome regression. Default is wt = NULL. 
method 
is "same" if the same set of covariates are used to estimate the constant, linear, and/or quadratic term. If method = "different", then different sets of covariates can be used to estimate the constant, linear, and/or quadratic term. covar_lin_formula and covar_sq_formula must be specified if method = "different". 
spline_df 
degrees of freedom. The default, spline_df = NULL, corresponds to no knots. 
spline_const 
is the number of spline terms needed to estimate the constant term. 
spline_linear 
is the number of spline terms needed to estimate the linear term. 
spline_quad 
is the number of spline terms needed to estimate the quadratic term. 
Details
This estimator bears a strong resemblance to general regression estimators in the survey literature, part of a more general class of calibration estimators (Deville and Sarndal, 1992). It is doubly robust, which means that it is consistent if either of the models is true (Scharfstein, Rotnitzky and Robins 1999). If the Ymodel is correct, then the first term in the previous equation is unbiased for ΞΎ and the second term has mean zero even if the Tmodel is wrong. If the Ymodel is incorrect, the first term is biased, but the second term gives a consistent estimate of (minus one times) the bias from the Ymodel if the Tmodel is correct.
This function is a doublyrobust estimator that fits an outcome regression model with a bias correction term. For details see Schafer and Galagate (2015).
Value
aipwee_est
returns an object of class "causaldrf_lsfit",
a list that contains the following components:
param 
parameter estimates for a add_spl fit. 
t_mod 
the result of the treatment model fit. 
out_mod 
the result of the outcome model fit. 
call 
the matched call. 
References
Schafer, J.L., Galagate, D.L. (2015). Causal inference with a continuous treatment and outcome: alternative estimators for parametric doseresponse models. Manuscript in preparation.
Schafer, Joseph L, Kang, Joseph (2008). Average causal effects from nonrandomized studies: a practical guide and simulated example. Psychological methods, 13.4, 279.
Robins, James M and Rotnitzky, Andrea (1995). Semiparametric efficiency in multivariate regression models with missing data Journal of the American Statistical Association, 90.429, 122β129.
Scharfstein, Daniel O and Rotnitzky, Andrea and Robins, James M (1999). Adjusting for nonignorable dropout using semiparametric nonresponse models Journal of the American Statistical Association, 94.448, 1096β1120.
Deville, JeanClaude and Sarndal, CarlErik (1992). Calibration estimators in survey sampling Journal of the American Statistical Association, 87.418, 376β380.
See Also
iptw_est
, ismw_est
,
reg_est
, wtrg_est
,
##' etc. for other estimates.
t_mod
, overlap_fun
to prepare the data
for use in the different estimates.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54  ## Example from Schafer (2015).
example_data < sim_data
t_mod_list < t_mod(treat = T,
treat_formula = T ~ B.1 + B.2 + B.3 + B.4 + B.5 + B.6 + B.7 + B.8,
data = example_data,
treat_mod = "Normal")
cond_exp_data < t_mod_list$T_data
full_data < cbind(example_data, cond_exp_data)
aipwee_list < aipwee_est(Y = Y,
treat = T,
covar_formula = ~ B.1 + B.2 + B.3 + B.4 + B.5 + B.6 + B.7 + B.8,
covar_lin_formula = ~ 1,
covar_sq_formula = ~ 1,
data = example_data,
e_treat_1 = full_data$est_treat,
e_treat_2 = full_data$est_treat_sq,
e_treat_3 = full_data$est_treat_cube,
e_treat_4 = full_data$est_treat_quartic,
degree = 1,
wt = NULL,
method = "same",
spline_df = NULL,
spline_const = 1,
spline_linear = 1,
spline_quad = 1)
sample_index < sample(1:1000, 100)
plot(example_data$T[sample_index],
example_data$Y[sample_index],
xlab = "T",
ylab = "Y",
main = "aipwee estimate")
abline(aipwee_list$param[1],
aipwee_list$param[2],
lty = 2,
lwd = 2,
col = "blue")
legend('bottomright',
"aipwee estimate",
lty = 2,
lwd = 2,
col = "blue",
bty='Y',
cex=1)
rm(example_data, t_mod_list, cond_exp_data, full_data, aipwee_list, sample_index)
