The file "bound_ps_function_submission.R" includes all the R codes for the functions needed for
the sensitivity analysis. There are two main functions most uers will use: 

1. 	"bound.ps" is the main function for the sensitivity analysis. By default it outputs the results based
    	on both the general framework where no assumption is made for the true propensity score model and 
    	probit model for the true propensity score.

 	Input:
 	y: outcome vector, 0<=y<=1
 	z: treatment assignment vector, 1: treated; 0:control
 	s: "proposed" or pseudo propensity score vector, 0<s<1
 	y.bin: whether or not the outcome is binary
	r1: the correlation coefficient between the multiplicative error of s and the potential outcome under treatment
 	r0: the correlation coefficient between the multiplicative error of 1-s and the potential outcome under control
 	lambda1: the variance of the multiplicative error of s as a proportion of its maximum
 	lambda0: the variance of the multiplicative error of 1-s as a proportion of its maximum 
 	probit: whether or not to carry out sensitivity analysis based on the probit model
 	tau: the regression coefficient for the standardized uncontrolled confounders in the probit model
 	prob.se: whether or not to calculate the standard error of the bound estimates using large sample theory for the
                 probit model
 	beta.cov: if prob.se=T, provide the variance-covariance matrix of the regression coefficient estimator
 	x: the design matrix of the probit model

 	Output: a list of three components
 	result: a matrix with columns about the mean of the potential outcome under treatment ("y1.mean") and control ("y0.mean"),
                average treatment effect for the treated ("att"), control ("atc") and the whole population ("ate"). The rows are
                estimate assuming no uncontrolled confounding ("est"), range under the general framework without assumptions about
                the functional form of the true propensity score ("min" and "max"), range under the assumption that the true propensity
                score is a probit model ("min.pb" and "max.pb"), and the standard errors of the bounds under the probit model 
                ("se.min.pb" and "se.max.pb")            
 	distri: estimable parameters used in constructing the "result" matrix
 	parameter: non-estimable parameters provided by the user


2.	 The function "svar.lim" calculated the allowable values of lambda0 for a given lambda1.
 	 s: the pseudo propensity score vector
 	 lambda: value of lambda1


The file "lindner_application_submission.R" include the R codes for application of the sensitivity analysis to the data in the second
         example in the paper. The data is included in an R package "USPS", which needs to be installed first.