AT: Average treatment effect of a binary or continuous endogenous variable

Description

AT can be used to calculate the sample average treatment effect of a binary or continuous endogenous predictor/treatment, with corresponding interval obtained using mainly posterior simulation.

Usage

AT(x, nm.end, E = TRUE, treat = TRUE, type = "bivariate", ind = NULL, 
   sub.l = 50, delta = FALSE, n.sim = 100, prob.lev = 0.05, hd.plot = FALSE,
   prob.plot = FALSE, main = "Histogram and Kernel Density of Simulated Average Effects", 
   xlab = "Simulated Average Effects", ...)

Arguments

A fitted SemiParBIVProbit object as produced by SemiParBIVProbit().

nm.end

Name of the endogenous variable.

If TRUE then AT calculates the sample ATE. If FALSE then it calculates the sample AT for the treated individuals only.

treat

If TRUE then AT calculates the AT using the treated only. If FALSE then it calculates the effect on the control group. This only makes sense if E = FALSE.

type

This argument can take three values: "naive" (the effect is calculated ignoring the presence of observed and unobserved confounders), "univariate" (the effect is obtained from the univariate model whic

ind

Binary logical variable. It can be used to calculate the AT for a subset of the data. Note that it does not make sense to use ind when some observations are excluded from the AT calculation (e.g., when using E = FALSE

sub.l

Length of subset of observations to use in the calculation of the treatment effect when the outcome of interest is continuous.

delta

If TRUE then the delta method is used for confidence interval calculations, otherwise Bayesian posterior simulation (the most reliable option, despite a bit slower) is employed. This option is only available for binary treatmen

n.sim

Number of simulated coefficient vectors from the posterior distribution of the estimated model parameters. This is used when delta = FALSE. It may be increased if more precision is required.

prob.lev

Overall probability of the left and right tails of the AT distribution used for interval calculations.

hd.plot

If TRUE then a plot of the histogram and kernel density estimate of the simulated average effects is produced. This can only be produced when delta = FALSE.

prob.plot

For the case of continuous endogenous variable and binary outcome, if TRUE then a plot showing probability that the binary outcome is equal to 1 for each value of the endogenous variable and respective i

main

Title for the plot.

xlab

Title for the x axis.

...

Other graphics parameters to pass on to plotting commands. These are used only when hd.plot = TRUE.

Value

resIt returns three values: lower confidence interval limit, estimated AT and upper interval limit.
prob.levProbability level used.
sim.ATIf delta = FALSE then it returns a vector containing simulated values of the average treatment effect. This is used to calculate intervals.
AT.soIt returns a vector containing the estimated effect for each single observation. This may not be available in some cases.
EffectsFor the case of continuous endogenous variable and binary outcome, it returns a matrix made up of three columns containing the effects for each unit increase in the endogenous variable and respective intervals.
PrFor the case of continuous endogenous variable and binary outcome, it returns a matrix made up of three columns containing the probability that the binary outcome is equal to 1 for each value of the endogenous variable and respective intervals.

Details

AT measures the sample average difference in outcomes under treatment (the binary predictor or treatment assumes value 1) and under control (the binary treatment assumes value 0). Posterior simulation and delta method can be used to obtain a confidence/credible interval. The former is more reliable, especially at small sample sizes. See the references below for details. For the case of continuous response and binary treatment, the AT is calculated via numerical integration, which can result in costly computation depending on the size of the dataset and number of simulated coefficient vectors used for interval calculations. In the model building process, we recommend using a subset of observations to calculate the AT (see sub.l). In our experience this still provides representative average effects. Once a preferred model has been found, the AT can be calculated using the entire dataset by setting sub.l to the number of observations (note that this can be time consuming). AT can also calculate the effect that a continuous endogenous variable has on a binary outcome. In this case the effect will depend on the unit increment chosen (as shown by the plot produced). An overall AT is provided as usual.

References

Radice R., Marra G. and M. Wojtys (in press), Copula Regression Spline Models for Binary Outcomes. Statistics and Computing.

Examples

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## see examples for SemiParBIVProbit

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