README.md

In zthuang0422/PURM-2021-Latent-Stratification: Maximum Likelihood Latent Stratification

README

library(ls)

1. Getting Started

• To install, devtools::install_github("eleafeit/latent_strat")
• This package is built from Berman & Feit (2019)
• Citation of the paper (APA): Berman, R. & Feit E.M. (2019, November 19). Principal Stratification for Advertising Experiments. The Wharton School Research Paper, Wharton Customer Analytics Initiative Research Paper, 1-37.

2. About the Package

This package contains the necessary functions of the latent stratification model from Berman & Feit (2019).In advertising experiments, noisy responses complicates the estimation of average treatment effect (ATE) and the assessment of the return of interest (ROI). Therefore, running normal t-tests will often yield estimates with high variance. The latent stratification model divides the customers into three strata: always buyer, who are positive under treatment and control (A); influenced buyer, who are positive only under treatment (B), and never buyer, who are zero under treatment and control (C). The model assumes that defiers, who are positive only under control, does not exist. It is able to improve the precision of the estimate by lowering the variance of the estimate. At the same time, it is also able to improve the accuracy by yielding an ATE closer to the true value.

3.1 Data Preparation

In order to demonstrate the package, we first simulate the data via the sim_latent_strat() function. The number following the strata (A for always buyer, B for influenced buyer, and C for never buyer) indicates whether the subject receives treatment. In this case, if the number is 1, then it means that the subject has received treatment, and 0 otherwise. The function takes in the sample size (n), proportion of always buyer (piA), proportion of influenced buyer (piB), mean of always buyer who received treatment (muA1), mean of always buyers who are in control, mean of the influenced buyer who received treatment, and the variance of group A1, A0, and B1. We assume that the treatment proportion is 50%. The sim_latent_strat() function generates a list containing a data frame data, the ATE ATE, and a vector par.

set.seed(20030601)
sim <- sim_latent_strat(n=10000, piA=0.2, piB=0.1, muA1=5, muA0=4.5, muB1=3, sigma=0.3)

The input parameters are stored as the par vector. The data will be simulated from these true parameters. For example, 20% of all observations belongs to strata A, who are positive under treatment and control. Observations in strata A who received treatment have a mean outcome of 5, and those who are in control have a mean outcome of 4.5, both with a variance of 0.3.

sim$par

##   piA   piB  muA1  muA0  muB1 sigma 
##   0.2   0.1   5.0   4.5   3.0   0.3

The true ATE can be calculated from these true parameters. In this example, the true ATE is 0.4.

sim$ATE

## [1] 0.4

Let’s now examine the simulated data.

head(sim$data)

##          y z s   sB   sC
## 1 3.356879 1 B  1.1 -0.5
## 2 0.000000 1 C  0.1  0.5
## 3 0.000000 1 C  0.1  0.5
## 4 0.000000 1 C  0.1  0.5
## 5 3.167482 1 B  1.1 -0.5
## 6 4.749002 1 A -0.9 -1.5

The data frame contains the 10000 randomly generated value (y) for each observation based on the strata (s; which can be A, always buyer; B, influenced buyer; or C, never buyer) and treatment (z; 1 for treatment, 0 for control). Since the treatment proportion is 50%, the first 5000 observations will have z = 1. For example, in the first entry, the subject belongs to strata B, who is an influenced buyer that only buys under treatment. Therefore, the subject in the first entry received treatment and has an outcome of 3.36. Following the same logic, the second entry is a subject in the never buyer strata. Therefore, even though he received treatment, his outcome is 0.

3.2 Estimating ATE Using Latent Stratification

To estimate ATE using latent stratification, we simply call the mle_ls() function and input the data frame as a parameter.

test = mle_ls(sim$data)

## Optimization runs took 0.1 secs
## Greatest ll was achieved in run 1 at -7008.303 (worst was -7008.303 )
## Computing variance-covariance matrix took 0.1  secs

test$pars

##      par        est          se
## 1    ATE  0.4032336 0.013019384
## 2 expATE 13.6626614 0.401495726
## 3    piA  0.2000032 0.004000203
## 4    piB  0.1000800 0.004225939
## 5   muA1  4.9938814 0.009284889
## 6   muA0  4.4878030 0.009460486
## 7   muB1  3.0177472 0.013298883
## 8  sigma  0.2961601 0.004195862

The function returned the analysis result as a data frame. It includes the estimates and standard errors of ATE, strata proportions, strata outcome means, and variance. The latent stratification model provides an accurate estimate of the ATE. This can be seen by ATE estimate of 0.403. In addition, the estimate is also precise. The standard error of the estimate under the latent stratification model is only 0.013.

zthuang0422/PURM-2021-Latent-Stratification documentation built on Dec. 23, 2021, 10:12 p.m.