treatment effect in instrumental variables regression I'm designing a randomized encouragement trial in which half of the sample will be randomly assigned to receive a special invitation to try a new intervention. This design will likely result in two-sided non-compliance with respect to random assignment:


*

*Everyone randomized to the encouragement arm will receive a special invitation to try the intervention, but only a subset of people in this group will take up this offer.

*People randomized to the control arm will NOT receive a special invitation to try the intervention, but some will learn about it through other channels and try it out on their own.
Encouragement designs account for this non-compliance by estimating the local average treatment effect (LATE). LATE is the effect of the intervention on 'compliers'—those who tried the intervention because they were randomly encouraged to do so but would not have tried if not encouraged. 
I simulated a basic dataset that mimics the planned study, and have a question about interpreting the treatment effect.
# setup
  library(dplyr)
  library(arm)
  library(AER)
  library(ivpack)
  library(stargazer)
  n <- 134

# https://rpubs.com/wsundstrom/t_ivreg
# function to calculate corrected SEs for OLS regression 
  cse = function(reg) {
      rob = sqrt(diag(vcovHC(reg, type = "HC1")))
      return(rob)
  }
# corrected SEs for IV regressions... slight difference from S&W method
  ivse = function(reg) {
      rob = robust.se(reg)[,2]
      return(rob)
  }

# create dataframe
  dat <- data.frame(partID=seq(1, n, 1),
                    trt=c(rep(0, n/2), 
                          rep(1, n/2)))

# set proportion use
  useT <- .8  # treatment group (encouraged)
  useC <- .2  # control group (not encouraged)

# create use variable
  set.seed(493)
  dat$use <- c(rbinom(n/2, 1, useC),
               rbinom(n/2, 1, useT))

# create covariate
# http://stackoverflow.com/questions/42147053/simulate-continuous-variable-that-is-correlated-to-existing-binary-variable

  x1    <- dat$use               # fixed given data
  rho   <- 0.1                   # desired correlation = cos(angle)
  theta <- acos(rho)             # corresponding angle
  x2    <- rnorm(n, 2, 0.5)      # new random data
  X     <- cbind(x1, x2)         # matrix
  Xctr  <- scale(X, center=TRUE, 
                 scale=FALSE)    # centered columns (mean 0)

  Id   <- diag(n)                           # identity matrix
  Q    <- qr.Q(qr(Xctr[ , 1, drop=FALSE]))  # QR-decomposition, just matrix Q
  P    <- tcrossprod(Q)          # = Q Q'   # projection onto space defined by x1
  x2o  <- (Id-P) %*% Xctr[ , 2]                 # x2ctr made orthogonal to x1ctr
  Xc2  <- cbind(Xctr[ , 1], x2o)                # bind to matrix
  Y    <- Xc2 %*% diag(1/sqrt(colSums(Xc2^2)))  # scale columns to length 1

  x <- Y[ , 2] + (1 / tan(theta)) * Y[ , 1]     # final new vector

  dat$age <- (1 + x) * 25 
  cor(dat$use, dat$age)
  dat$age <- round(dat$age, 0)

# outcome
  outT <- .35
  outC <- .05
  dat$y <- c(rbinom(n/2, 1, outC),
             rbinom(n/2, 1, outT))

# IV Regression
  ivR = ivreg(y ~ use + rescale(age) | rescale(age) + trt , data = dat)
  stargazer(ivR, 
            se=list(ivse(ivR)),
            title="IV Regression", 
            type="text", 
            df=FALSE, 
            digits=5,
            ci=TRUE) 

The coefficient on use is 0.43. This is the effect on the 'compliers'. The outcome y is binary. I want to be able to say that the intervention increased y by 43% points from a to b. 
How do I get a? The proportion among the control group is 0.03, but this includes compliers and never-takers.   
 A: I'm gonna need some help here, but here's one approach to an answer.
The parameter you want is not necessarily identifiable from your dataset.
Consider the following dataset, where out is the outcome, trt is the encouragement, use is the actual intervention, num says how many people fell into that pattern of encouragement/intervention, and comp is how we might categorize the patients in that group using the compliance classes outlined in this review.
id  out use trt comp
1   1   0   0   never_taker OR complier
2   1   1   1   complier
3   0   0   0   complier
4   1   1   0   always_taker

You'll observe the same data no matter whether participant 1 is a never-taker or a complier. But, participant 1's status affects the parameter you seek. If they are a complier, that pushes the untreated complier outcome probability from 0 to 1/2.
A: You've got 
E(y $\vert$ encouraged) = $\tfrac{18}{67}$
E(y $\vert$ !encouraged) = $\tfrac{2}{67}$
E(use $\vert$ encouraged) = $\tfrac{53}{67}$
E(use $\vert$ !encouraged) = $\tfrac{16}{67}$
Equation (2) from the source from other Eric in intuitive words:


*

*the total effect is the 53 that 'use' resulting in 18 'y'.

*substract from that the 16 that normally (without encouragement) 'use' and result in 2 'y'

*Then the causal effect of 'use', the 37 extra 'use' that stems from the encouragement, is 16 extra 'y', and "therefore" the causal effect is


$\frac{E(y\vert e)-E(y\vert !e)}{E(use\vert e)-E(use\vert !e)} =
   \frac{16}{37} = 0.43...$
This analysis is without the age parameter since it does not correlate anyway (then we do not need the regression to obtain the estimated values). For the model with the covariates (if you have a serious effect of age in the real data) you will need some regression model to obtain those estimated values. I suggest that you use a reasonable model, restricted to values between 0 and 1, such that the predicted values make at least some sense.

If you wish something like: "I want to be able to say that..." 
... - the intervention - increased - y - by 43% points - from a to b. 
What the double regression does is more something like (using the same structure):
... - the change in intervention use that is due to the encouragement (from 16 to 53) - increased - y - with a coefficient of 0.43 - from 2 to 18"
The encouragement increases the use and the use increases y. The 0.43 is not related to the relative increase of y (from a to b) but to to the coefficient that relates the change of 'use' with the change of 'y'.
