model {dreamer}R Documentation

Model Creation

Description

Functions which set the hyperparameters, seeds, and prior weight for each model to be used in Bayesian model averaging via dreamer_mcmc().

See each function's section below for the model's details. In the following, y denotes the response variable and d represents the dose.

For the longitudinal specifications, see documentation on model_longitudinal.

Usage

model_linear(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_quad(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_loglinear(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_logquad(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_emax(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  mu_b4,
  sigma_b4,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_exp(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  shape,
  rate,
  w_prior = 1,
  longitudinal = NULL
)

model_beta(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  mu_b4,
  sigma_b4,
  shape,
  rate,
  scale = NULL,
  w_prior = 1,
  longitudinal = NULL
)

model_independent(
  mu_b1,
  sigma_b1,
  shape,
  rate,
  doses = NULL,
  w_prior = 1,
  longitudinal = NULL
)

model_linear_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_quad_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_loglinear_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_logquad_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_emax_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  mu_b4,
  sigma_b4,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_exp_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_beta_binary(
  mu_b1,
  sigma_b1,
  mu_b2,
  sigma_b2,
  mu_b3,
  sigma_b3,
  mu_b4,
  sigma_b4,
  scale = NULL,
  link,
  w_prior = 1,
  longitudinal = NULL
)

model_independent_binary(
  mu_b1,
  sigma_b1,
  doses = NULL,
  link,
  w_prior = 1,
  longitudinal = NULL
)

Arguments

mu_b1, sigma_b1, mu_b2, sigma_b2, mu_b3, sigma_b3, mu_b4, sigma_b4, shape, rate

models parameters. See sections below for interpretation in specific models.

w_prior

a scalar between 0 and 1 indicating the prior weight of the model.

longitudinal

output from a call to one of the model_longitudinal_*() functions. This is used to specify a longitudinal dose-response model.

scale

a scale parameter in the Beta model. Default is 1.2 * max(dose).

doses

the doses in the dataset to be modeled. The order of the doses corresponds to the order in which the priors are specified in mu_b1 and sigma_b1.

link

a character string of either "logit" or "probit" indicating the link function for binary model.

Value

A named list of the arguments in the function call. The list has S3 classes assigned which are used internally within dreamer_mcmc().

Linear

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * d

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

1 / \sigma^2 \sim Gamma(shape, rate)

Quadratic

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * d + b_3 * d^2

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

1 / \sigma^2 \sim Gamma(shape, rate)

Log-linear

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * log(d + 1)

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

1 / \sigma^2 \sim Gamma(shape, rate)

Log-quadratic

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * log(d + 1) + b_3 * log(d + 1)^2

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

1 / \sigma^2 \sim Gamma(shape, rate)

EMAX

y \sim N(f(d), \sigma^2)

f(d) = b_1 + (b_2 - b_1) * d ^ b_4 / (exp(b_3 * b_4) + d ^ b_4)

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

b_4 \sim N(mu_b4, sigma_b4 ^ 2), (Truncated above 0)

1 / \sigma^2 \sim Gamma(shape, rate)

Here, b_1 is the placebo effect (dose = 0), b_2 is the maximum treatment effect, b_3 is the log(ED50), and b_4 is the hill or rate parameter.

Exponential

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * (1 - exp(- b_3 * d))

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2), (truncated to be positive)

1 / \sigma^2 \sim Gamma(shape, rate)

Beta

y \sim N(f(d), \sigma^2)

f(d) = b_1 + b_2 * ((b3 + b4) ^ (b3 + b4)) / (b3 ^ b3 * b4 ^ b4) * (d / scale) ^ b3 * (1 - d / scale) ^ b4

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2), Truncated above 0

b_4 \sim N(mu_b4, sigma_b4 ^ 2), Truncated above 0

1 / \sigma^2 \sim Gamma(shape, rate)

Note that scale is a hyperparameter specified by the user.

Independent

y \sim N(f(d), \sigma^2)

f(d) = b_{1d}

b_{1d} \sim N(mu_b1[d], sigma_b1[d] ^ 2)

1 / \sigma^2 \sim Gamma(shape, rate)

Independent Details

The independent model models the effect of each dose independently. Vectors can be supplied to mu_b1 and sigma_b1 to set a different prior for each dose in the order the doses are supplied to doses. If scalars are supplied to mu_b1 and sigma_b1, then the same prior is used for each dose, and the doses argument is not needed.

Linear Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + b_2 * d

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

Quadratic Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + b_2 * d + b_3 * d^2

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

Log-linear Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + b_2 * log(d + 1)

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

Log-quadratic Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + b_2 * log(d + 1) + b_3 * log(d + 1)^2

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

EMAX Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + (b_2 - b_1) * d ^ b_4 / (exp(b_3 * b_4) + d ^ b_4)

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2)

b_4 \sim N(mu_b4, sigma_b4 ^ 2), (Truncated above 0)

Here, on the link(f(d)) scale, b_1 is the placebo effect (dose = 0), b_2 is the maximum treatment effect, b_3 is the log(ED50), and b_4 is the hill or rate parameter.

Exponential Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_1 + b_2 * (exp(b_3 * d) - 1)

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2), (Truncated below 0)

Beta Binary

y \sim Binomial(n, f(d)

link(f(d)) = b_1 + b_2 * ((b3 + b4) ^ (b3 + b4)) / (b3 ^ b3 * b4 ^ b4) * (d / scale) ^ b3 * (1 - d / scale) ^ b4

b_1 \sim N(mu_b1, sigma_b1 ^ 2)

b_2 \sim N(mu_b2, sigma_b2 ^ 2)

b_3 \sim N(mu_b3, sigma_b3 ^ 2), Truncated above 0

b_4 \sim N(mu_b4, sigma_b4 ^ 2), Truncated above 0

Note that scale is a hyperparameter specified by the user.

Independent Binary

y \sim Binomial(n, f(d))

link(f(d)) = b_{1d}

b_{1d} \sim N(mu_b1[d], sigma_b1[d]) ^ 2

Independent Binary Details

The independent model models the effect of each dose independently. Vectors can be supplied to mu_b1 and sigma_b1 to set a different prior for each dose in the order the doses are supplied to doses. If scalars are supplied to mu_b1 and sigma_b1, then the same prior is used for each dose, and the doses argument is not needed.

Longitudinal Linear

Let f(d) be a dose response model. The expected value of the response, y, is:

E(y) = g(d, t)

g(d, t) = a + (t / t_max) * f(d)

a \sim N(mu_a, sigma_a)

Longitudinal ITP

Let f(d) be a dose response model. The expected value of the response, y, is:

E(y) = g(d, t)

g(d, t) = a + f(d) * ((1 - exp(- c1 * t))/(1 - exp(- c1 * t_max)))

a \sim N(mu_a, sigma_a)

c1 \sim Uniform(a_c1, b_c1)

Longitudinal IDP

Increasing-Decreasing-Plateau (IDP).

Let f(d) be a dose response model. The expected value of the response, y, is:

E(y) = g(d, t)

g(d, t) = a + f(d) * (((1 - exp(- c1 * t))/(1 - exp(- c1 * d1))) * I(t < d1) + (1 - gam * ((1 - exp(- c2 * (t - d1))) / (1 - exp(- c2 * (d2 - d1))))) * I(d1 <= t <= d2) + (1 - gam) * I(t > d2))

a \sim N(mu_a, sigma_a)

c1 \sim Uniform(a_c1, b_c1)

c2 \sim Uniform(a_c2, b_c2)

d1 \sim Uniform(0, t_max)

d2 \sim Uniform(d1, t_max)

gam \sim Uniform(0, 1)

Examples

set.seed(888)
data <- dreamer_data_linear(
  n_cohorts = c(20, 20, 20),
  dose = c(0, 3, 10),
  b1 = 1,
  b2 = 3,
  sigma = 5
)

# Bayesian model averaging
output <- dreamer_mcmc(
  data = data,
  n_adapt = 1e3,
  n_burn = 1e2,
  n_iter = 1e3,
  n_chains = 2,
  silent = TRUE,
  mod_linear = model_linear(
    mu_b1 = 0,
    sigma_b1 = 1,
    mu_b2 = 0,
    sigma_b2 = 1,
    shape = 1,
    rate = .001,
    w_prior = 1 / 2
  ),
  mod_quad = model_quad(
    mu_b1 = 0,
    sigma_b1 = 1,
    mu_b2 = 0,
    sigma_b2 = 1,
    mu_b3 = 0,
    sigma_b3 = 1,
    shape = 1,
    rate = .001,
    w_prior = 1 / 2
  )
)
# posterior weights
output$w_post
# plot posterior dose response
plot(output)

# LONGITUDINAL
library(ggplot2)
set.seed(889)
data_long <- dreamer_data_linear(
  n_cohorts = c(10, 10, 10, 10), # number of subjects in each cohort
  doses = c(.25, .5, .75, 1.5), # dose administered to each cohort
  b1 = 0, # intercept
  b2 = 2, # slope
  sigma = .5, # standard deviation,
  longitudinal = "itp",
  times = c(0, 12, 24, 52),
  t_max = 52, # maximum time
  a = .5,
  c1 = .1
)

## Not run: 
ggplot(data_long, aes(time, response, group = dose, color = factor(dose))) +
   geom_point()

## End(Not run)

output_long <- dreamer_mcmc(
   data = data_long,
   n_adapt = 1e3,
   n_burn = 1e2,
   n_iter = 1e3,
   n_chains = 2,
   silent = TRUE, # make rjags be quiet,
   mod_linear = model_linear(
      mu_b1 = 0,
      sigma_b1 = 1,
      mu_b2 = 0,
      sigma_b2 = 1,
      shape = 1,
      rate = .001,
      w_prior = 1 / 2, # prior probability of the model
      longitudinal = model_longitudinal_itp(
         mu_a = 0,
         sigma_a = 1,
         a_c1 = 0,
         b_c1 = 1,
         t_max = 52
      )
   ),
   mod_quad = model_quad(
      mu_b1 = 0,
      sigma_b1 = 1,
      mu_b2 = 0,
      sigma_b2 = 1,
      mu_b3 = 0,
      sigma_b3 = 1,
      shape = 1,
      rate = .001,
      w_prior = 1 / 2,
      longitudinal = model_longitudinal_linear(
         mu_a = 0,
         sigma_a = 1,
         t_max = 52
      )
   )
)

## Not run: 
# plot longitudinal dose-response profile
plot(output_long, data = data_long)
plot(output_long$mod_quad, data = data_long) # single model

# plot dose response at final timepoint
plot(output_long, data = data_long, times = 52)
plot(output_long$mod_quad, data = data_long, times = 52) # single model

## End(Not run)
[Package dreamer version 3.1.0 Index]