Chapter 3 Types of Causal Effects

Sometimes our experiments because quasi-experiments because of crossover or non-compliance.

Simulating data on assignment, compliance, and potential outcomes:

set.seed(1)
assign = rbinom(100, 1, 0.5) # half half

Simulating non-compliance as a function of gender:

set.seed(101)
gender = rbinom(100, 1, 0.4) # 40% are men
comply = rep(); 
comply[gender==1] = rbinom(sum(gender==1), 1, 0.5) # prob of men complying is 0.5
comply[gender==0] = rbinom(sum(gender==0), 1, 0.7) # prob of women complying is 0.7

treatment effect: 100 (from 400 –> 500) fixed for everyone:

notutor = rnorm(100, 400, 12) # Y0
tutor = notutor + 100 + rnorm(100, 0, 20) # Y1
simdata = data.frame(assign, notutor, tutor)
head(simdata)

##   assign  notutor    tutor
## 1      0 403.2168 499.9361
## 2      0 392.8935 465.2280
## 3      1 425.6018 534.0721
## 4      1 414.0730 498.2632
## 5      0 408.9611 533.1596
## 6      1 397.2339 515.1242

actual treatment:

treat = ifelse(comply==1, assign, 1-assign)
table(assign, treat)

##       treat
## assign  0  1
##      0 31 21
##      1 22 26

observed value:

obs = treat*tutor + (1-treat)*notutor # Y1 if T=1 and Y0 if T=0

when treat=1: Y = 1tutor + 0notutor = Y1
when treat=0: Y = 0tutor + 1notutor = Y0

Simulated data:

simdata = data.frame(assign, comply, treat, tutor, notutor, obs, gender)
head(simdata)

##   assign comply treat    tutor  notutor      obs gender
## 1      0      1     0 499.9361 403.2168 403.2168      0
## 2      0      1     0 465.2280 392.8935 392.8935      0
## 3      1      0     0 534.0721 425.6018 425.6018      1
## 4      1      0     0 498.2632 414.0730 414.0730      1
## 5      0      1     0 533.1596 408.9611 408.9611      0
## 6      1      1     1 515.1242 397.2339 515.1242      0

3.1 ATT, ATE, ATU, … what the what?

ATE/ACE: Average Treatment Effect in the population
ATT: Average Treatment Effect on the Treated
ATU: Average Treatment Effect on the Untreated
LATE: Local Average Treatment Effect for Compilers
CATE: Conditional Average Treatment Effect
ITT: Intention To Treat. Effect of simply the intention (instead of receipt) of treatment

ATE = mean(simdata$tutor - simdata$notutor)  # whole population
ATT = mean((simdata$tutor - simdata$notutor)[treat==1])
ATU = mean((simdata$tutor - simdata$notutor)[treat==0])
LATE = mean((simdata$tutor - simdata$notutor)[comply==1])
CATE0 = mean((simdata$tutor - simdata$notutor)[gender==0])
CATE1 = mean((simdata$tutor - simdata$notutor)[gender==1])

cbind(ATE, ATT, ATU, LATE, CATE0, CATE1)

##           ATE      ATT      ATU     LATE    CATE0    CATE1
## [1,] 100.0414 101.1523 99.05629 99.88937 102.4829 97.28824

# all close to ATE

3.2 Calculating different causal effects in practice

ATE_obs = lm(obs ~ treat, data = simdata)
coef(ATE_obs) # 99.82048

## (Intercept)       treat 
##   400.12192    99.82048

ATT & ATU

no way to estimate ATT & ATU from data

LATE if compliance status is known

LATE_obs = lm(obs ~ treat, data = simdata[simdata$comply==1,])
coef(LATE_obs) # 101.1594

## (Intercept)       treat 
##    398.5844    101.1594

CATE if gender is included as a confounder:

CATE0_obs = lm(obs ~ treat, data = simdata[simdata$gender==0,])
coef(CATE0_obs) # 106.5414

## (Intercept)       treat 
##    400.1959    106.5414

CATE1_obs = lm(obs ~ treat, data = simdata[simdata$gender==1,])
coef(CATE1_obs) # 90.71861

## (Intercept)       treat 
##   400.05067    90.71861

Intention To Treat (ITT)

ITT = lm(obs ~ assign)
coef(ITT) # 15.460 whereas the true effect is 100!!

## (Intercept)      assign 
##   439.61675    15.45999

Intention To Treat on Compliers

ITT_c = lm(obs ~ assign, data = simdata[simdata$comply==1,])
coef(ITT_c) # 101.159 which is pretty close to the true effect 100

## (Intercept)      assign 
##    398.5844    101.1594

Generally recommended: LATE or ITT on Compliers!