Continuation of Missing Data Handling: What is the relationship between MCAR/MAR/Conditional MCAR?

TODO: check stat sig of the test below

Conditional MCAR with correlation = 0.8

Will use this to demonstrate the test of MCAR.

Notice that you don’t even need the MMRM - you just need to control for the covariate governing missingness.

library(SPR)
library(MASS)
library(emmeans)
library(nlme)

 set.seed(03232021)

  out <- sim_dat(N = 2000  , 
                 number.groups = 2  , 
                 number.timepoints = 4  , 
                 cond.mcar = T  ,
                 #reg.formula = formula( ~ Time + Group + Time*Group),
                 Beta = 2  ,
                 corr = 'ar1'  , 
                 cor.value = 0.8  , 
                 var.values = 2  )
                
  
  dat <- out$dat
  dat <- dropout(dat = dat, 
                 type_dropout  = c('cmcar'), 
                 prop.miss = 0.3)

Test for MCAR

Does your score at a previous timepoint predict your score at the next timepoint?

dat1 <- dat[dat$Time == 'Time_3', 'Y_cmcar']
dat2 <- dat[dat$Time == 'Time_4', 'Y_cmcar']
dropped.out <- !is.na(dat1) & is.na(dat2) # these subjects dropped out at timepoint 4
table(dropped.out)

# Does the score at timepoint 3 predict drop-out at timepoint 4?
mod <- glm(dropped.out ~ dat1, family = 'binomial')
summary(mod)  
# # TODO 3.23.21 - why is this not stat sig? resolve

Return to the Covariate adjustment later

Correctly specified model

What is the correctly specified model? Check data generation:

  out$reg.formula
  
#  OLS Regression Model 
  mod.ols1 <- lm(Y_comp ~ Group + Time + Covariate + Time * Group + Covariate * Time,  data = dat)

# Correctly specified OLS model with missing data:
  mod.ols2 <- lm(Y_cmcar ~ Group + Time + Covariate + Time * Group + Covariate * Time, data = dat)
  
  # Misspecified OLS Model:
  mod.ols3 <- lm(Y_cmcar ~ Group + Time + Time * Group, data = dat)

  # Compute marginal mean contrasts
  # Don't worry about the degrees of freedom, we aren't focused on the Type I error or Power
  # Just look at the parameter estimates to check for bias
  library(emmeans)
  mod.emms.ols1 <- emmeans(mod.ols1, pairwise ~ Group | Time, adjust = 'none',  mode = 'df.error')
  mod.emms.ols2 <- emmeans(mod.ols2, pairwise ~ Group | Time, adjust = 'none',  mode = 'df.error')
  mod.emms.ols3 <- emmeans(mod.ols3, pairwise ~ Group | Time, adjust = 'none',  mode = 'df.error')

 # Examine output:
  out$Beta 
  # Data generation done specifically to demonstrate this point
  
  # Look at drop-out:
  aggregate(Y_cmcar ~ Time, FUN = function(x) mean(is.na(x)), dat = dat, na.action = na.pass)
  aggregate(Y_cmcar ~ Group + Time, FUN = function(x) mean(is.na(x)), dat = dat, na.action = na.pass)
  # Substntially different drop out rates across the treatment arms 
  
  # Descriptive Statistics 
  aggregate(Y_comp ~ Group + Time, FUN = function(x) round(mean(x, na.rm = T),2), dat = dat, na.action = na.pass)
  aggregate(Y_cmcar ~ Group + Time, FUN = function(x) round(mean(x, na.rm = T),2), dat = dat, na.action = na.pass)
 # You can't adjust for the Covariate with descriptive statistics!
  
  # OLS
  mod.emms.ols1$contrasts
  mod.emms.ols2$contrasts
  mod.emms.ols3$contrasts
 # Misspecified model leads to biased estimates
 # With conditional MCAR data, the key is to control for the covariate causing dropout
 # Once you do that, you're back to MCAR data, and a simple OLS works fine
 # Of course, we can confirm that MMRM also yields unbiased estimates (not shown here)
  

Test for MCAR

Return to this - adjust for Covariate, recheck Test for MCAR:

dat1 <- dat[dat$Time == 'Time_3', 'Y_cmcar']
dat2 <- dat[dat$Time == 'Time_4', 'Y_cmcar']
dropped.out <- !is.na(dat1) & is.na(dat2) # these subjects dropped out at timepoint 4
table(dropped.out)

# Does the score at the previous timepoint predict drop-out AFTER adjusting for Covariate?
cov <- dat$Covariate[dat$Time == 'Time_4']
mod.mcar <- glm(dropped.out ~ dat1 + cov, family = 'binomial')
summary(mod.mcar)