Confidence interval for a binomial proportion

Data

We’re looking at deaths as a proportion of all hospital discharges.

Binomial experiment assumptions:

• Each discharge is a “trial” with 2 possible outcomes - death/no death
• Trials are independent (this may be questionable)
• Constant probability of “success”¹ on each trial (this may be questionable)

df1.data <- 
  tibble::tribble(
    ~fiscal_year, ~site, ~discharges, ~in_hospital_deaths,
         "16/17", "VGH",        4288,                  32,
         "17/18", "VGH",        4476,                  31,
         "18/19", "VGH",        4450,                  39
    )

df1.data <- 
  df1.data %>% 
  mutate(p_hat = in_hospital_deaths/discharges)  

df1.data %>% 
  kable() %>% 
  kable_styling(bootstrap_options = c("striped",
              "condensed", 
              "responsive"))

Confidence interval for binomial proportion:

See Brown, Cai, DasGupta (2001), Statistical Science

The standard “textbook” CI for binomial proportion has poor coverage when the actual proportion is close to 0 or 1

Better option: use the Wilson interval.

Reference: http://math.furman.edu/~dcs/courses/math47/R/library/Hmisc/html/binconf.html

“Following Agresti and Coull, the Wilson interval is to be preferred and so is the default.”

binconf(32, 4288, method = "wilson")

##     PointEst       Lower      Upper
##  0.007462687 0.005291247 0.01051583

# binconf(32, 4288, method = "wilson")[2]  # lower est 
# binconf(32, 4288, method = "wilson")[3]  # upper est

# is this a symmetric CI? No. 
# binconf(32, 4288, method = "wilson")[1] - binconf(32, 4288, method = "wilson")[2]
# binconf(32, 4288, method = "wilson")[3] - binconf(32, 4288, method = "wilson")[1]

# for comparison, using the "textbook interval"
binconf(32, 4288, method = "asymptotic")

##     PointEst       Lower      Upper
##  0.007462687 0.004886711 0.01003866

# is this a symmetric CI? Yes. 
# binconf(32, 4288, method = "asymptotic")[1] - binconf(32, 4288, method = "asymptotic")[2]
# binconf(32, 4288, method = "asymptotic")[3] - binconf(32, 4288, method = "asymptotic")[1]

Updating the data

df2.data_with_ci <-
  df1.data %>% 
  mutate(ci_lower = map2_dbl(in_hospital_deaths, 
                             discharges, 
                             function(x, y){
                               binconf(x, y, 
                                       method = "wilson")[2]
                               
                             }), 
         ci_upper = map2_dbl(in_hospital_deaths, 
                             discharges, 
                             function(x, y){
                               binconf(x, y, 
                                       method = "wilson")[3]
                               
                             }))

df2.data_with_ci %>% 
  kable() %>% 
  kable_styling(bootstrap_options = c("striped",
              "condensed", 
              "responsive"))

Confidence interval for Poisson mean (todo):

• Using normal approx: https://stats.stackexchange.com/questions/15371/how-to-calculate-a-confidence-level-for-a-poisson-distribution
• Without normal approx: http://ms.mcmaster.ca/peter/s743/poissonalpha.html