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Prior for R_0 -- suggestion #105

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maxbiostat opened this issue May 13, 2020 · 5 comments
Open

Prior for R_0 -- suggestion #105

maxbiostat opened this issue May 13, 2020 · 5 comments

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@maxbiostat
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Hi, first of all, many many thanks for the awesome work all of you have been doing!

Secondly, I have a minor question:

What's the justification for the prior currently employed for R_0? The hierarchy

kappa ~ half-normal(0, 0.5)
R_0 ~ half-normal(m, kappa)

leads to a "witch hat"-looking prior on R_0.

image

Would a simpler Gamma/log-normal prior not be better here? Easier to elicit and probably captures key features such as Pr(R_0 > 1) more easily. If you're interested I can help construct such a prior and then submit a PR, but I have no easy way of running the model to check results as of now.
@flaxter
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flaxter commented May 13, 2020 via email

@flaxter
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flaxter commented May 13, 2020 via email

@flaxter
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flaxter commented May 13, 2020 via email

@maxbiostat
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maxbiostat commented May 13, 2020 •
edited

Hi @flaxter ,

It's more likely I'm the buggy one:

Code

LogMean <- function(realMean, realSD){
  ## takes REAL mean and REAL standard deviation; returns LOG mean
  realVar <- realSD^2
  mu <- log(realMean/ sqrt(1 + (realVar/realMean^2)))
  return(mu)
}
LogVar <- function(realMean, realSD){
  ## takes REAL mean and REAL standard deviation; returns LOG variance
  realVar <- realSD^2
  sigmasq <- log(1 + (realVar/realMean^2))
  return(sigmasq)
}
###################
M <- 1e6
kappa_ic <- abs(rnorm(n = M, mean = 0, sd = 1/2))
R0_ic <- abs(rnorm(n = M, mean = 3.28, sd = kappa_ic))

kappa_bram <- abs(rnorm(n = M, mean = 0, sd = 1))
R0_bram <- abs(rnorm(n = M, mean = 2.4, sd = kappa_bram))

m <- 3 # E[R0]
v <- .9^2 # Var(R0)

# Gamma
alpha <- m^2/v
beta <- m/v

#Log-normal
mu <- LogMean(realMean = m, realSD = sqrt(v) )
sigma <- sqrt( LogVar(realMean = m, realSD = sqrt(v) )    ) 

R0_gamma <- rgamma(n = M, shape = alpha, rate = beta)
R0_ln <- rlnorm(n = M, meanlog = mu, sdlog = sigma)

forplot <- data.frame(R_0 = c(R0_ic,
                              R0_bram,
                              R0_gamma,
                              R0_ln),
                      study = rep(c("Imperial",
                                    "BRAM-COD",
                                    "Gamma",
                                    "Log-normal"), each = M))

library(ggplot2)

ggplot(forplot, aes(x = R_0, colour = study, fill = study)) +
  geom_density(alpha = .4) +
  scale_x_continuous(expression(R_0), expand = c(0, 0)) +
  scale_y_continuous("Density", expand = c(0, 0)) +
  geom_vline(xintercept = 1, linetype = "longdash") +
  theme_bw(base_size = 16)

summy <- function(x, na.rm = TRUE){
  ans <- data.frame(mean = mean(x, na.rm = na.rm),
                    sd = sd(x, na.rm = na.rm),
                    lwr = quantile(x, probs = .025, na.rm = na.rm),
                    uprr = quantile(x, probs = .975, na.rm = na.rm))
  return(ans)
} 


lapply(list(bramcod = R0_bram, imperial = R0_ic, Gamma = R0_gamma, LN = R0_ln), summy)

Last line gives

$bramcod
         mean       sd       lwr      upr
2.5% 2.435088 0.911255 0.5271803 4.580456

$imperial
         mean        sd      lwr      upr
2.5% 3.280113 0.4989465 2.190974 4.374173

$Gamma
         mean        sd      lwr      upr
2.5% 2.999272 0.8991958 1.503212 4.995466

$LN
        mean        sd      lwr      upr
2.5% 3.00032 0.8997761 1.617084 5.106452

BRAM-COD is a recent Brazilian study with a model based on your model from report 13.

@maxbiostat
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I realised the script above has a (as of yet inconsequential) bug in the generation of the random samples. Technically, the model specified in Stan is a truncated normal, not a folded normal, so the correct generation of the samples would be something like

M <- 1e6
library(truncnorm)
kappa_ic <- rtruncnorm(n = M, mean = 0, sd = 1/2, a = 0, b = Inf)
R0_ic <- rtruncnorm(n = M, mean = 3.28, sd = kappa_ic, a = 0, b = Inf)

kappa_bram <- rtruncnorm(n = M, mean = 0, sd = 1, a = 0, b = Inf)
R0_bram <- rtruncnorm(n = M, mean = 2.4, sd = kappa_bram, a = 0, b = Inf)

Giving

> lapply(list(bramcod = R0_bram, imperial = R0_ic, Gamma = R0_gamma, LN = R0_ln), summy)
$bramcod
         mean        sd       lwr    uprr
2.5% 2.471883 0.9082468 0.6756234 4.67399

$imperial
         mean       sd      lwr     uprr
2.5% 3.280978 0.497174 2.192721 4.371233

$Gamma
         mean        sd      lwr     uprr
2.5% 2.999731 0.9005028 1.502073 5.006691

$LN
        mean        sd      lwr     uprr
2.5% 3.00048 0.9005886 1.616915 5.110658

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@flaxter @maxbiostat