Binomial Distribution

TODO: Add simulation study code

Review and run the code below to better understand the data distribution.

Generate cross sectional data

library(SPR)
library(MASS)

N = 5000 #this should be divisible by however many groups you use!
number.groups <- 2
number.timepoints <- 1
set.seed(2092021)

dat <- data.frame(
                  'USUBJID' = rep(paste0('Subject_', formatC(1:N, width = 4, flag = '0')), length.out= N*number.timepoints),
                  'Group' = rep(paste0('Group_', 1:number.groups), length.out = N*number.timepoints),
                  stringsAsFactors=F)


# Create Beta parameters for these design matrix:
X <- model.matrix( ~ Group  , data = dat) 

# Create Beta
Beta <- matrix(0, nrow = ncol(X), dimnames=list(colnames(X), 'param'))
Beta[] <- c(-0.5, 2)

# Matrix multiply:
XB <- X %*% Beta
p <- exp(XB)/(1 + exp(XB))

# Dichotomize
dat$Y_binom <- 1*(p > runif(n = N))

# Plot
aggregate(Y_binom  ~ Group, FUN = function(x) round(mean(x, na.rm = T),2), dat = dat, na.action = na.pass)
barplot(100*table(dat$Y_binom )/sum(table(dat$Y_binom )), ylim = c(0, 100), ylab = 'Percentage', col = 'grey', main = 'Binomial')


library(ggplot2)
ggplot(data = dat, aes(x= as.factor(Y_binom), fill = Group)) +
      geom_bar(aes(y = (..count..)/sum(..count..)), color="#e9ecef", alpha=0.6, position="dodge", stat="count") +
      scale_y_continuous(labels=scales::percent_format(accuracy = 1)) +
      scale_fill_manual(name = 'Groups', values=c("blue2", "red2")) +
      theme_minimal() +
      theme(legend.position = 'bottom') +
      labs(x = 'Score', y = 'Percentage',
           title = 'Scores with a Binomial Distribution',
           caption = 'Note: Simulated data')

mod <- glm(Y_binom ~ Group, data = dat, family = 'binomial')
summary(mod)
exp(1.99821)
xtabs(~ Y_binom + Group, data = dat)
G2 <- 2062/438
G1 <- 974/1526
G2/G1

# Probability that Group 1 is sick:
exp(-0.448)/(1 + exp(-0.448))
# Probability that Group 2 is sick:
exp(-0.448 + 1.99821)/(1 + exp(-0.448 + 1.99821))
# Compare to observed proportions:
prop.table(xtabs(~ Y_binom + Group, data = dat), 2)

mod <- glm(Y_binom ~ Group, data = dat, family = 'binomial')
summary(mod)
exp(1.99821)
xtabs(~ Y_binom + Group, data = dat)
G2 <- 2062/438
G1 <- 974/1526
G2/G1

# Probability that Group 1 is sick:
exp(-0.448)/(1 + exp(-0.448))
# Probability that Group 2 is sick:
exp(-0.448 + 1.99821)/(1 + exp(-0.448 + 1.99821))
# Compare to observed proportions:
prop.table(xtabs(~ Y_binom + Group, data = dat), 2)

Run custom glm() estimation code

This is helpful because it shows you the likelihood

loglike <- function(vP, X, Y){
  
  Y <- cbind(1-Y, Y)
  XB <- X %*% vP
  p <- 1/(1 + exp(-1*(XB)))
  P <- cbind(1-p, p)
  # Loglikelihood:
  loglike <- Y*log(P)
  loglike <- -1*loglike # optimization will minimize function
  loglike <- sum(loglike)
  return(loglike)
  
}

Optimize

vP <- rep(0, length(Beta))
out <- optim(par = vP, fn = loglike, hessian = T,  method = 'BFGS', X = X, Y = dat$Y_binom)
# Compare Generating Parameter to my estimate and the glm() estimate
cbind(Beta, round(out$par,4), round(coef(mod),4))
# Compare the standard errors:
sqrt(diag(solve(out$hessian)))

Compute with Covariate included

N = 5000 #this should be divisible by however many groups you use!
number.groups <- 2
number.timepoints <- 1
set.seed(2092021)

dat <- data.frame(
                  'USUBJID' = rep(paste0('Subject_', formatC(1:N, width = 4, flag = '0')), length.out= N*number.timepoints),
                  'Group' = rep(paste0('Group_', 1:number.groups), length.out = N*number.timepoints),
                  'Covariate' = rnorm(n = N, mean = 0, sd = 1),
                  stringsAsFactors=F)


# Create Beta parameters for these design matrix:
X <- model.matrix( ~ Group + Covariate , data = dat)

# Create Beta
Beta <- matrix(0, nrow = ncol(X), dimnames=list(colnames(X), 'param'))
Beta[] <- c(-0.5, 2, 1)

# Matrix multiply:
XB <- X %*% Beta
p <- exp(XB)/(1 + exp(XB))

# Dichotomize
dat$Y_binom <- 1*(p > runif(n = N))

# Plot
library(ggplot2)
ggplot(data = dat, aes(x= as.factor(Y_binom), fill = Group)) +
      geom_bar(aes(y = (..count..)/sum(..count..)), color="#e9ecef", alpha=0.6, position="dodge", stat="count") +
      scale_y_continuous(labels=scales::percent_format(accuracy = 1)) +
      scale_fill_manual(name = 'Latent Classes', values=c("blue2", "red2")) +
      theme_minimal() +
      theme(legend.position = 'bottom') +
      labs(x = 'Score', y = 'Percentage',
           title = 'Scores with a Binomial Distribution',
           caption = 'Note: Simulated data')

# Estimate
mod <- glm(Y_binom ~ Group, data = dat, family = 'binomial')
summary(mod)
exp(1.70170)
xtabs(~ Y_binom + Group, data = dat)
G2 <- 1963/537
G1 <- 1000/1500
G2/G1

# Estimate the Group parameter adjusting for covariate
mod2 <- glm(Y_binom ~ Group + Covariate, data = dat, family = 'binomial')
summary(mod2)
exp(2.01047)
# Cannot recreate this conditional/adjusted odds ratio for Groups using
# hand computations.