15-prop-scores-r.Rmd

# 15. Outcome regression and propensity scores{-}

```{r setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = '#>')
```

## Program 15.1

- Estimating the average causal effect within levels of confounders under the assumption of effect-measure modification by smoking intensity ONLY
- Data from NHEFS

```{r, results='hide', message=FALSE, warning=FALSE}
library(here)
```

```{r}
#install.packages("readxl") # install package if required
library("readxl")

nhefs <- read_excel(here("data", "NHEFS.xls"))
nhefs$cens <- ifelse(is.na(nhefs$wt82), 1, 0)

# regression on covariates, allowing for some effect modification
fit <- glm(wt82_71 ~ qsmk + sex + race + age + I(age*age) + as.factor(education)
           + smokeintensity + I(smokeintensity*smokeintensity) + smokeyrs
           + I(smokeyrs*smokeyrs) + as.factor(exercise) + as.factor(active)
           + wt71 + I(wt71*wt71) + I(qsmk*smokeintensity), data=nhefs)
summary(fit)

# (step 1) build the contrast matrix with all zeros
# this function builds the blank matrix
# install.packages("multcomp") # install packages if necessary
library("multcomp")
makeContrastMatrix <- function(model, nrow, names) {
  m <- matrix(0, nrow = nrow, ncol = length(coef(model)))
  colnames(m) <- names(coef(model))
  rownames(m) <- names
  return(m)
}
K1 <-
  makeContrastMatrix(
    fit,
    2,
    c(
      'Effect of Quitting Smoking at Smokeintensity of 5',
      'Effect of Quitting Smoking at Smokeintensity of 40'
    )
  )
# (step 2) fill in the relevant non-zero elements
K1[1:2, 'qsmk'] <- 1
K1[1:2, 'I(qsmk * smokeintensity)'] <- c(5, 40)

# (step 3) check the contrast matrix
K1

# (step 4) estimate the contrasts, get tests and confidence intervals for them
estimates1 <- glht(fit, K1)
  summary(estimates1)
  confint(estimates1)

# regression on covariates, not allowing for effect modification
fit2 <- glm(wt82_71 ~ qsmk + sex + race + age + I(age*age) + as.factor(education)
           + smokeintensity + I(smokeintensity*smokeintensity) + smokeyrs
           + I(smokeyrs*smokeyrs) + as.factor(exercise) + as.factor(active)
           + wt71 + I(wt71*wt71), data=nhefs)

summary(fit2)
```


## Program 15.2

- Estimating and plotting the propensity score
- Data from NHEFS
```{r, out.width="85%", fig.align='center'}
fit3 <- glm(qsmk ~ sex + race + age + I(age*age) + as.factor(education)
            + smokeintensity + I(smokeintensity*smokeintensity) + smokeyrs
            + I(smokeyrs*smokeyrs) + as.factor(exercise) + as.factor(active)
            + wt71 + I(wt71*wt71), data=nhefs, family=binomial())
summary(fit3)
nhefs$ps <- predict(fit3, nhefs, type="response")

summary(nhefs$ps[nhefs$qsmk==0])
summary(nhefs$ps[nhefs$qsmk==1])

# # plotting the estimated propensity score
# install.packages("ggplot2") # install packages if necessary
# install.packages("dplyr")
library("ggplot2")
library("dplyr")

nhefs <- nhefs %>% mutate(qsmklabel = ifelse(qsmk == 1,
                       yes = 'Quit Smoking 1971-1982',
                       no = 'Did Not Quit Smoking 1971-1982'))

ggplot(nhefs, aes(x = ps, fill = qsmklabel, color = qsmklabel)) + 
  geom_density(alpha = 0.2) +
  xlab('Probability of Quitting Smoking During Follow-up') +
  ggtitle('Propensity Score Distribution by Treatment Group') +
  theme(legend.position = 'bottom', legend.direction = 'vertical',
        legend.title = element_blank())

# alternative plot with histograms
ggplot(nhefs, aes(x = ps, fill = as.factor(qsmk), color = as.factor(qsmk))) +
  geom_histogram(alpha = 0.3, position = 'identity', bins=15) +
  facet_grid(as.factor(qsmk) ~ .) +
  xlab('Probability of Quitting Smoking During Follow-up') +
  ggtitle('Propensity Score Distribution by Treatment Group') +
  scale_fill_discrete('') +
  scale_color_discrete('') +
  theme(legend.position = 'bottom', legend.direction = 'vertical')
```

```{r, eval=FALSE, out.width="85%", fig.align='center'}
# attempt to reproduce plot from the book
nhefs %>%
  mutate(ps.grp = round(ps/0.05) * 0.05) %>%
  group_by(qsmk, ps.grp) %>%
  summarize(n = n()) %>%
  ungroup() %>%
  mutate(n2 = ifelse(qsmk == 0, yes = n, no =  -1*n)) %>%
  ggplot(aes(x = ps.grp, y = n2, fill = as.factor(qsmk))) +
  geom_bar(stat = 'identity', position = 'identity') +
  geom_text(aes(label = n, x = ps.grp, y = n2 + ifelse(qsmk == 0, 8, -8))) +
  xlab('Probability of Quitting Smoking During Follow-up') +
  ylab('N') +
  ggtitle('Propensity Score Distribution by Treatment Group') +
  scale_fill_discrete('') +
  scale_x_continuous(breaks = seq(0, 1, 0.05)) +
  theme(legend.position = 'bottom', legend.direction = 'vertical',
        axis.ticks.y = element_blank(),
        axis.text.y = element_blank())
```

## Program 15.3

- Stratification on the propensity score
- Data from NHEFS
```{r}
# calculation of deciles
nhefs$ps.dec <- cut(nhefs$ps,
                    breaks=c(quantile(nhefs$ps, probs=seq(0,1,0.1))),
                    labels=seq(1:10),
                    include.lowest=TRUE)

#install.packages("psych") # install package if required
library("psych")
describeBy(nhefs$ps, list(nhefs$ps.dec, nhefs$qsmk))

# function to create deciles easily
decile <- function(x) {
  return(factor(quantcut(x, seq(0, 1, 0.1), labels = FALSE)))
}

# regression on PS deciles, allowing for effect modification
for (deciles in c(1:10)) {
  print(t.test(wt82_71~qsmk, data=nhefs[which(nhefs$ps.dec==deciles),]))
}

# regression on PS deciles, not allowing for effect modification
fit.psdec <- glm(wt82_71 ~ qsmk + as.factor(ps.dec), data = nhefs)
summary(fit.psdec)
confint.lm(fit.psdec)
```

## Program 15.4

- Standardization using the propensity score
- Data from NHEFS
```{r}
#install.packages("boot") # install package if required
library("boot")

# standardization by propensity score, agnostic regarding effect modification
std.ps <- function(data, indices) {
  d <- data[indices,] # 1st copy: equal to original one`
  # calculating propensity scores
  ps.fit <- glm(qsmk ~ sex + race + age + I(age*age)
                + as.factor(education) + smokeintensity
                + I(smokeintensity*smokeintensity) + smokeyrs
                + I(smokeyrs*smokeyrs) + as.factor(exercise)
                + as.factor(active) + wt71 + I(wt71*wt71),
                data=d, family=binomial())
  d$pscore <- predict(ps.fit, d, type="response")

  # create a dataset with 3 copies of each subject
  d$interv <- -1 # 1st copy: equal to original one`
  d0 <- d # 2nd copy: treatment set to 0, outcome to missing
  d0$interv <- 0
  d0$qsmk <- 0
  d0$wt82_71 <- NA
  d1 <- d # 3rd copy: treatment set to 1, outcome to missing
  d1$interv <- 1
  d1$qsmk <- 1
  d1$wt82_71 <- NA
  d.onesample <- rbind(d, d0, d1) # combining datasets

  std.fit <- glm(wt82_71 ~ qsmk + pscore + I(qsmk*pscore), data=d.onesample)
  d.onesample$predicted_meanY <- predict(std.fit, d.onesample)

  # estimate mean outcome in each of the groups interv=-1, interv=0, and interv=1
  return(c(mean(d.onesample$predicted_meanY[d.onesample$interv==-1]),
           mean(d.onesample$predicted_meanY[d.onesample$interv==0]),
           mean(d.onesample$predicted_meanY[d.onesample$interv==1]),
           mean(d.onesample$predicted_meanY[d.onesample$interv==1])-
             mean(d.onesample$predicted_meanY[d.onesample$interv==0])))
}

# bootstrap
results <- boot(data=nhefs, statistic=std.ps, R=5)

# generating confidence intervals
se <- c(sd(results$t[,1]), sd(results$t[,2]),
        sd(results$t[,3]), sd(results$t[,4]))
mean <- results$t0
ll <- mean - qnorm(0.975)*se
ul <- mean + qnorm(0.975)*se

bootstrap <- data.frame(cbind(c("Observed", "No Treatment", "Treatment",
                                "Treatment - No Treatment"), mean, se, ll, ul))
bootstrap
```

```{r}
# regression on the propensity score (linear term)
model6 <- glm(wt82_71 ~ qsmk + ps, data = nhefs) # p.qsmk
summary(model6)

# standarization on the propensity score
# (step 1) create two new datasets, one with all treated and one with all untreated
treated <- nhefs
  treated$qsmk <- 1

untreated <- nhefs
  untreated$qsmk <- 0

# (step 2) predict values for everyone in each new dataset based on above model
treated$pred.y <- predict(model6, treated)
untreated$pred.y <- predict(model6, untreated)

# (step 3) compare mean weight loss had all been treated vs. that had all been untreated
mean1 <- mean(treated$pred.y, na.rm = TRUE)
mean0 <- mean(untreated$pred.y, na.rm = TRUE)
mean1
mean0
mean1 - mean0

# (step 4) bootstrap a confidence interval
# number of bootstraps
nboot <- 100
# set up a matrix to store results
boots <- data.frame(i = 1:nboot,
                    mean1 = NA,
                    mean0 = NA,
                    difference = NA)
# loop to perform the bootstrapping
nhefs <- subset(nhefs, !is.na(ps) & !is.na(wt82_71)) # p.qsmk
for(i in 1:nboot) {
  # sample with replacement
  sampl <- nhefs[sample(1:nrow(nhefs), nrow(nhefs), replace = TRUE), ]

  # fit the model in the bootstrap sample
  bootmod <- glm(wt82_71 ~ qsmk + ps, data = sampl) # ps

  # create new datasets
  sampl.treated <- sampl %>%
    mutate(qsmk = 1)

  sampl.untreated <- sampl %>%
    mutate(qsmk = 0)

  # predict values
  sampl.treated$pred.y <- predict(bootmod, sampl.treated)
  sampl.untreated$pred.y <- predict(bootmod, sampl.untreated)

  # output results
  boots[i, 'mean1'] <- mean(sampl.treated$pred.y, na.rm = TRUE)
  boots[i, 'mean0'] <- mean(sampl.untreated$pred.y, na.rm = TRUE)
  boots[i, 'difference'] <- boots[i, 'mean1'] - boots[i, 'mean0']

  # once loop is done, print the results
  if(i == nboot) {
    cat('95% CI for the causal mean difference\n')
    cat(mean(boots$difference) - 1.96*sd(boots$difference),
        ',',
        mean(boots$difference) + 1.96*sd(boots$difference))
  }
}
```

A more flexible and elegant way to do this is to write a function to perform the model fitting, prediction, bootstrapping, and reporting all at once.