Help for package unmconf

Type:

Package

Title:

Modeling with Unmeasured Confounding

Version:

1.0.0

Maintainer:

David Kahle <david@kahle.io>

Description:

Tools for fitting and assessing Bayesian multilevel regression models that account for unmeasured confounders.

SystemRequirements:

JAGS >= 4.3.0 (http://mcmc-jags.sourceforge.net)

License:

MIT + file LICENSE

Encoding:

UTF-8

RoxygenNote:

7.3.2

Depends:

R (≥ 2.10), rjags

Imports:

stats, glue, janitor, coda

Suggests:

bayesplot, posterior, ggplot2, dplyr, tidyr, tibble, knitr, rmarkdown, testthat (≥ 3.0.0)

VignetteBuilder:

knitr

Config/testthat/edition:

NeedsCompilation:

Packaged:

2024-09-09 21:40:15 UTC; david_kahle

Author:

Ryan Hebdon [aut], James Stamey

[aut], David Kahle

[aut, cre], Xiang Zhang [aut]

Repository:

CRAN

Date/Publication:

2024-09-09 22:00:02 UTC

unmconf: Modeling with Unmeasured Confounding

Description

Tools for fitting and assessing Bayesian multilevel regression models that account for unmeasured confounders.

Convert to Greek expressions

Description

Convert to Greek expressions for plotting

Usage

expand_labels(labs)

greek_expander(s)

make_greek_coefs(mod)

drop_nulls(x)

Arguments

labs

A character vector of greek symbols of the form ga_x and be_1.

s

A character vector of Greek short hand codes, e.g. "si".

mod

Output from unm_glm().

x

Character vector.

Value

A character vector.

Examples


labs <- c("ga_1", "ga_treatment", "ga_x", "be_1",
  "be_treatment", "be_x", "la_u", "al_y", "si")
expand_labels(labs)

Generate synthetic data

Description

runm() generates synthetic data for use of modeling with unmeasured confounders. Defaults to the case of one unmeasured confounder present and fixed parameter values. Can be customized. Currently set up to have at most two unmeasured confounders to pair with unm_glm().

Usage

runm(
  n,
  type = "int",
  missing_prop = 0.8,
  response = "bin",
  response_param,
  response_model_coefs = c(int = -1, z1 = 0.5, z2 = 0.5, z3 = 0.5, u1 = 0.5, x = 0.5),
  treatment_model_coefs = c(int = -1, z1 = 0.5, z2 = 0.5, z3 = 0.5, u1 = 0.5),
  covariate_fam_list = list("norm", "bin", "norm"),
  covariate_param_list = list(c(mean = 0, sd = 1), prob = 0.3, c(0, 2)),
  unmeasured_fam_list = list("norm"),
  unmeasured_param_list = list(c(mean = 0, sd = 1))
)

Arguments

n

Number of observations. When type = "int", n is a vector of length 1. When type = "ext", n can either be a vector of length 1 or 2. For the case when n is of length 2, ⁠n = (n_main, n_external)⁠, where n_main corresponds to the main study sample size and n_external corresponds to the external validation sample size. For the case when n is of length 1, n will be split evenly between main study and external validation observations, with the main study getting the additional observation when n is odd.

type

Type of validation source. Can be "int" for internal validation or "ext" for external validation. Defaults to "int".

missing_prop

Proportion of missing values for internal validation scenario (i.e., when type = "int").

response

"norm", "bin", "pois", or "gam". Defaults to "bin".

response_param

Nuisance parameters for response type. For "norm", the default standard deviation is 1. For "gam", the default shape parameter is 2. For "pois", an offset variable is added to the dataset that is uniformly distributed from 1 to 10.

response_model_coefs

A named vector of coefficients to generate data from the response model. This must include an intercept (⁠"int" = ⁠), a coefficient for each covariate specified, a coefficient for each unmeasured confounder, and a treatment coefficient (⁠"x" = ⁠). The coefficients for the covariates and treatment will be denoted with "beta[.]" and the unmeasured confounders with "lambda[.]".

treatment_model_coefs

A named vector of coefficients to generate data from the treatment model. This must include an intercept (⁠"int" = ⁠), a coefficient for each covariate specified, and a coefficient for each unmeasured confounder. The coefficients for the covariates and unmeasured confounders will be denoted with "eta[.]".

covariate_fam_list

A list of either "norm" or "bin", where the length of the list matches the number of covariates in the model.

covariate_param_list

A list of parameters for the respective distributions in covariate_fam_list, where the length of the list matches the length of covariate_fam_list.

unmeasured_fam_list

A list of either "norm" or "bin", where the length of the list matches the number of unmeasured confounders in the model. This can be at most a length of 2 to pair with unm_glm().

unmeasured_param_list

A list of parameters for the respective distributions in unmeasured_fam_list, where the length of the list matches the length of unmeasured_fam_list.

Value

A tibble

Examples


runm(100)
runm(n = 100, type = "int", missing_prop = .75)
runm(n = 100, type = "int", missing_prop = .75) |> attr("params")
runm(100, type = "int", response = "norm")
runm(100, type = "int", response = "norm") |> attr("params")
runm(100, type = "int", response = "norm", response_param = 3) |> attr("params")
runm(100, type = "int", response = "gam")
runm(100, type = "int", response = "gam", response_param = 5) |> attr("params")
runm(100, type = "int", missing_prop = .5, response = "pois")

runm(n = 100, type = "ext")
runm(n = 100, type = "ext") |> attr("params")
runm(n = c(10, 10), type = "ext")
runm(100, type = "ext", response = "norm")
runm(100, type = "int", response = "norm", response_param = 3) |> attr("params")
runm(100, type = "ext", response = "gam")
runm(100, type = "ext", response = "pois")

runm(
  n = 100,
  type = "int",
  missing_prop = .80,
  response = "norm",
  response_param = c("si_y" = 2),
  response_model_coefs = c("int" = -1, "z" = .4,
                           "u1" = .75, "u2" = .75, "x" = .75),
  treatment_model_coefs = c("int" = -1, "z" = .4,
                            "u1" = .75, "u2" = .75),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm", "bin"),
  unmeasured_param_list = list(c(mean = 0, sd = 1), c(.3))
)

runm(
  n = c(20, 30),
  type = "ext",
  response = "norm",
  response_param = c("si_y" = 2),
  response_model_coefs = c("int" = -1, "z1" = .4, "z2" = .5, "z3" = .4,
                           "u1" = .75, "u2" = .75, "x" = .75),
  treatment_model_coefs = c("int" = -1, "z1" = .4, "z2" = .5, "z3" = .4,
                            "u1" = .75, "u2" = .75),
  covariate_fam_list = list("norm", "bin", "norm"),
  covariate_param_list = list(c(mean = 0, sd = 1), c(.3), c(0, 2)),
  unmeasured_fam_list = list("norm", "bin"),
  unmeasured_param_list = list(c(mean = 0, sd = 1), c(.3))
)

Fitting Multilevel Bayesian Regression Model with Unmeasured Confounders

Description

unm_glm() fits a multilevel Bayesian regression model that accounts for unmeasured confounders. Users can input model information into unm_glm() in a similar manner as they would for the standard stats::glm() function, providing arguments like formula, family, and data. Results are stored as MCMC iterations.

Usage

unm_glm(
  form1,
  form2 = NULL,
  form3 = NULL,
  family1 = binomial(),
  family2 = NULL,
  family3 = NULL,
  data,
  n.iter = 2000,
  n.adapt = 1000,
  thin = 1,
  n.chains = 4,
  filename = tempfile(fileext = ".jags"),
  quiet = getOption("unm_quiet"),
  progress.bar = getOption("unm_progress.bar"),
  code_only = FALSE,
  priors,
  response_nuisance_priors,
  response_params_to_track,
  confounder1_nuisance_priors,
  confounder1_params_to_track,
  confounder2_nuisance_priors,
  confounder2_params_to_track,
  ...
)

jags_code(mod)

## S3 method for class 'unm_int'
print(x, digits = 3, ..., print_call = getOption("unm_print_call"))

## S3 method for class 'unm_int'
coef(object, ...)

Arguments

form1

The formula specification for the response model (Level I)

form2

The formula specification for the first unmeasured confounder model (Level II)

form3

The formula specification for the second unmeasured confounder model (Level III)

family1, family2, family3

The family object, communicating the types of models to be used for response (form1) and unmeasured confounder (⁠form2, form3⁠) models. See stats::family() for details

data

The dataset containing all variables (this function currently only supports a single dataset containing internally validated data)

n.iter

n.iter argument of rjags::coda.samples()

n.adapt

n.adapt argument of rjags::jags.model()

thin

thin argument of rjags::coda.samples()

n.chains

n.chains argument of rjags::jags.model()

filename

File name where to store jags code

quiet

The quiet parameter of rjags::jags.model(). Defaults to TRUE, but you can change it on a per-session basis with options(unm_quiet = FALSE).

progress.bar

The progress.bar parameter of rjags::update.jags(). Defaults to "none", but you can change it on a per-session basis with options(unm_progress.bar = "text").

code_only

Should only the code be created?

priors

Custom priors to use on regression coefficients, see examples.

response_nuisance_priors, confounder1_nuisance_priors, confounder2_nuisance_priors

JAGS code for the nuisance priors on parameters in a JAGS model (see examples)

response_params_to_track, confounder1_params_to_track, confounder2_params_to_track

Additional parameters to track when nuisance parameter priors are used (see examples)

...

Additional arguments to pass into rjags::jags.model(), such as inits

mod

The output of unm_glm()

x

Object to be printed

digits

Number of digits to round to; defaults to 3

print_call

Should the call be printed? Defaults to TRUE, but can be turned off with options("unm_print_call" = FALSE)

object

Model object for which the coefficients are desired

Value

(Invisibly) The output of rjags::coda.samples(), an object of class mcmc.list, along with attributes code containing the jags code used and file containing the filename of the jags code.

Examples


# ~~ One Unmeasured Confounder Examples (II-Level Model) ~~


# normal response, normal confounder model with internally validated data
(df <- runm(20, response = "norm"))

(unm_mod <- unm_glm(
  form1 = y ~ x + z1 + z2 + z3 + u1,  family1 = gaussian(),
  form2 = u1 ~ x + z1 + z2 + z3,      family2 = gaussian(),
  data = df
))

## Not run:  # reduce cran check time

(unm_mod <- unm_glm(
  y ~ .,      family1 = gaussian(),
  u1 ~ . - y, family2 = gaussian(),
  data = df
))
glm(y ~ x + z1 + z2 + z3, data = df)
coef(unm_mod)

jags_code(unm_mod)
unm_glm(
  y ~ .,
  u1 ~ . - y,
  family1 = gaussian(),
  family2 = gaussian(),
  data = df, code_only = TRUE
)




# a normal-normal model - external validation
(df <- runm(c(10, 10), type = "ext", response = "norm"))

unm_glm(
  y ~ x + z1 + z2 + z3 + u1,  family1 = gaussian(),
  u1 ~ x + z1 + z2 + z3,      family2 = gaussian(),
  data = df
)



# setting custom priors
unm_glm(
  y ~ .,      family1 = gaussian(),
  u1 ~ . - y, family2 = gaussian(),
  data = df,
  code_only = TRUE
)

unm_glm(
  y ~ .,      family1 = gaussian(),
  u1 ~ . - y, family2 = gaussian(),
  data = df,
  code_only = FALSE,
  priors = c("lambda[u1]" = "dnorm(1, 10)"),
  response_nuisance_priors = "tau_{y} <- sigma_{y}^-2; sigma_{y} ~ dunif(0, 100)",
  response_params_to_track = "sigma_{y}",
  confounder1_nuisance_priors = "tau_{u1} <- sigma_{u1}^-2; sigma_{u1} ~ dunif(0, 100)",
  confounder1_params_to_track = "sigma_{u1}"
)


# turn progress tracking on
options("unm_progress.bar" = "text")



# more complex functional forms _for non-confounder predictors only_
# zero-intercept model
unm_glm(
  y ~ . - 1,
  u1 ~ . - y,
  family1 = gaussian(),
  family2 = gaussian(),
  data = df
)
glm(y ~ . - 1, data = df)

# polynomial model
unm_glm(
  y ~ x + poly(z1, 2) + u1,
  u1 ~ x + z1,
  family1 = gaussian(),
  family2 = gaussian(),
  data = df
)
glm(y ~ x + poly(z1, 2), data = df)

# interaction model
unm_glm(
  y ~ x*z1 + u1, family1 = gaussian(),
  u1 ~ x*z1,     family2 = gaussian(),
  data = df
)
glm(y ~ x*z1, data = df)



# a binomial-binomial model
(df <- runm(
  50,
  missing_prop = .75,
  response = "bin",
  unmeasured_fam_list = list("bin"),
  unmeasured_param_list = list(.5)
))
(unm_mod <- unm_glm(
  y ~ .,
  u1 ~ . - y,
  family1 = binomial(),
  family2 = binomial(),
  data = df
))
glm(y ~ . - u1, family = binomial(), data = df)



# a poisson-normal model
(df <- runm(
  25,
  response = "pois",
  response_model_coefs = c("int" = -1, "z" = .5, "u1" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5, "u1" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm"),
  unmeasured_param_list = list(c(0, 1))
))

(unm_mod <- unm_glm(
  y ~ x + z + u1 + offset(log(t)),
  u1 ~ x + z,
  family1 = poisson(),
  family2 = gaussian(),
  data = df
))
glm(y ~ x + z + offset(log(t)), family = poisson(), data = df)



# a poisson-binomial model
(df <- runm(
  25,
  response = "pois",
  response_model_coefs = c("int" = -1, "z" = .5, "u1" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5, "u1" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("bin"),
  unmeasured_param_list = list(.5)
))

(unm_mod <- unm_glm(
  y ~ x + z + u1 + offset(log(t)), family1 = poisson(),
  u1 ~ x + z,                      family2 = binomial(),
  data = df
))

glm(y ~ x + z + offset(log(t)), family = poisson(), data = df)



# a gamma-normal model
(df <- runm(
  25,
  response = "gam",
  response_model_coefs = c("int" = -1, "z" = .5, "u1" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5, "u1" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm"),
  unmeasured_param_list = list(c(0, 1))
))

(unm_mod <- unm_glm(
  y ~ x + z + u1, family1 = Gamma(),
  u1 ~ x + z,     family2 = gaussian(),
  data = df
))

glm(y ~ x + z, family = Gamma(link = "log"), data = df)



# a gamma-binomial model
(df <- runm(
  25,
  response = "gam",
  response_model_coefs = c("int" = -1, "z" = .5, "u1" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5, "u1" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("bin"),
  unmeasured_param_list = list(.5)
))

(unm_mod <- unm_glm(
  y ~ x + z + u1, family1 = Gamma(),
  u1 ~ x + z,     family2 = binomial(),
  data = df
))
print(df, n = 25)

glm(y ~ x + z, family = Gamma(link = "log"), data = df)



# the output of unm_glm() is classed jags output

(df <- runm(20, response = "norm"))
(unm_mod <- unm_glm(
  y ~ .,
  u1 ~ . - y,
  family1 = gaussian(),
  family2 = gaussian(),
  data = df)
)
class(unm_mod)
jags_code(unm_mod)
unm_glm(y ~ ., u1 ~ . - y, data = df, code_only = TRUE)





# visualizing output
library("ggplot2")
library("bayesplot"); bayesplot_theme_set(ggplot2::theme_minimal())
mcmc_hist(unm_mod, facet_args = list(labeller = label_parsed))
mcmc_hist(unm_mod)
mcmc_trace(unm_mod, facet_args = list(labeller = label_parsed))

# more extensive visualization with the tidyverse
mcmc_intervals(unm_mod, prob = .90) +
  geom_point(
    aes(value, name), data = tibble::enframe(attr(df, "params")),
    color = "red", fill = "pink", size = 4, shape = 21
  )


library("dplyr")
library("tidyr")
unm_mod %>%
  as.matrix() %>%
  as_tibble() %>%
  pivot_longer(everything(), names_to = "var", values_to = "val") %>%
  ggplot(aes("0", val)) +
  geom_jitter() +
  geom_point(
    aes("0", value), data = tibble::enframe(attr(df, "params"), name = "var"),
    color = "red", fill = "pink", size = 4, shape = 21
  ) +
  coord_flip() +
  facet_grid(var ~ ., scales = "free_y", labeller = label_parsed) +
  theme_bw() +
  theme(
    axis.title = element_blank(),
    axis.text.y = element_blank(), axis.ticks.y = element_blank(),
    strip.text.y = element_text(angle = 0)
  )


# getting draws out
(samps <- posterior::as_draws_df(unm_mod))
samps$`.chain`
samps$`.iteration`
samps$`.draw`



# implementation is variable-name independent
(df <- runm(100, response = "norm"))
df$ht <- df$y
df$age <- df$u1
df$biom <- df$x
(unm_mod <- unm_glm(
  ht ~ x + biom + age,
  age ~ x + biom,
  data = df,
  family1 = gaussian(),
  family2 = gaussian()
))
jags_code(unm_mod)

# ~~ Two Unmeasured Confounders Examples (III-Level Model) ~~
# a normal-normal-normal model - internal validation
(df <- runm(
  50,
  missing_prop = .75,
  response = "norm",
  response_model_coefs = c("int" = -1, "z" = .5,
                           "u1" = .5, "u2" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5,
                            "u1" = .5, "u2" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm", "norm"),
  unmeasured_param_list = list(c(0, 1), c(0, 1))
))
(unm_mod <- unm_glm(
  y ~ x + z + u1 + u2,
  u1 ~ x + z + u2,
  u2 ~ x + z,
  family1 = gaussian(),
  family2 = gaussian(),
  family3 = gaussian(),
  data = df
))

glm(y ~ x + z, data = df)
coef(unm_mod)

unm_glm(
  y ~ x + z + u1 + u2, family1 = gaussian(),
  u1 ~ x + z + u2,   family2 = gaussian(),
  u2 ~ x + z,        family3 = gaussian(),
  data = df,
  code_only = TRUE
)



# a normal-normal-normal model - external validation
(df <- runm(
  c(20, 20),
  type = "ext",
  response = "norm",
  response_model_coefs = c("int" = -1, "z" = .5,
                           "u1" = .5, "u2" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5,
                            "u1" = .5, "u2" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm", "norm"),
  unmeasured_param_list = list(c(0, 1), c(0, 1))
))
(unm_mod <- unm_glm(
  y ~ x + z + u1 + u2,  family1 = gaussian(),
  u1 ~ x + z + u2,      family2 = gaussian(),
  u2 ~ x + z,           family3 = gaussian(),
  data = df
))



# a binomial-binomial-binomial model - internal validation
(df <- runm(
  25,
  response = "bin",
  response_model_coefs = c("int" = -1, "z" = .5,
                           "u1" = .5, "u2" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5,
                            "u1" = .5, "u2" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("bin", "bin"),
  unmeasured_param_list = list(.5, .75)
))
unm_glm(y ~ x + z + u1 + u2, family1 = binomial(),
        u1 ~ x + z + u2,     family2 = binomial(),
        u2 ~ x + z,          family3 = binomial(),
        data = df,
        code_only = TRUE
)



## End(Not run)

Generate synthetic data

Description

unm_summary() produces result summaries of the results from the model fitting function, unm_glm(). The table of results are summarized from the MCMC draws of the posterior distribution.

Usage

unm_summary(mod, data, quantiles = c(0.025, 0.5, 0.975))

unm_backfill(data, mod)

unm_dic(mod)

Arguments

mod

Output from unm_glm().

data

The data mod was generated with.

quantiles

A numeric vector of quantiles.

Value

A tibble

Examples


# ~~ One Unmeasured Confounder Examples (II-Level Model) ~~

# normal response, normal confounder model with internally validated data
(df <- runm(20, response = "norm"))

(unm_mod <- unm_glm(
  y ~ x + z1 + z2 + z3 + u1, family1 = gaussian(),
  u1 ~ x + z1 + z2 + z3,     family2 = gaussian(),
  data = df
))

glm(y ~ x + z1 + z2 + z3, data = df)

coef(unm_mod)
jags_code(unm_mod)
unm_summary(unm_mod)
unm_summary(unm_mod, df) # true values known df

# impute missing values with model
unm_backfill(df, unm_mod)

## Not run:  # reduce cran check time

# a normal-normal model - external validation
(df <- runm(c(10, 10), type = "ext", response = "norm"))

(unm_mod <- unm_glm(
  y ~ x + z1 + z2 + z3 + u1,
  u1 ~ x + z1 + z2 + z3,
  family1 = gaussian(),
  family2 = gaussian(),
  data = df
))

unm_backfill(df, unm_mod)


# a binomial-binomial model
(df <- runm(
  50,
  missing_prop = .75,
  response = "bin",
  unmeasured_fam_list = list("bin"),
  unmeasured_param_list = list(.5)
))

(unm_mod <- unm_glm(
  y ~ .,
  u1 ~ . - y,
  family1 = binomial(),
  family2 = binomial(),
  data = df
))

glm(y ~ . - u1, family = binomial(), data = df)
unm_backfill(df, unm_mod)
unm_summary(unm_mod, df)

# computing the dic. penalty = effective number of parameters
unm_dic(unm_mod)
coef(unm_mod)
unm_backfill(df, unm_mod)


# ~~ Two Unmeasured Confounders Examples (III-Level Model) ~~
# a normal-normal-normal model - internal validation
(df <- runm(
  50,
  missing_prop = .75,
  response = "norm",
  response_model_coefs = c("int" = -1, "z" = .5,
                           "u1" = .5, "u2" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5,
                            "u1" = .5, "u2" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm", "norm"),
  unmeasured_param_list = list(c(0, 1), c(0, 1))
))

(unm_mod <- unm_glm(
  y ~ x + z + u1 + u2, family1 = gaussian(),
  u1 ~ x + z + u2, family2 = gaussian(),
  u2 ~ x + z, family3 = gaussian(),
  data = df
))
glm(y ~ x + z, data = df)
coef(unm_mod)
unm_summary(unm_mod)
unm_summary(unm_mod, df) # true values known df
unm_backfill(df, unm_mod)

# a normal-normal-normal model - external validation
(df <- runm(
  c(20, 20),
  type = "ext",
  response = "norm",
  response_model_coefs = c("int" = -1, "z" = .5,
                           "u1" = .5, "u2" = .5, "x" = .5),
  treatment_model_coefs = c("int" = -1, "z" = .5,
                            "u1" = .5, "u2" = .5),
  covariate_fam_list = list("norm"),
  covariate_param_list = list(c(mean = 0, sd = 1)),
  unmeasured_fam_list = list("norm", "norm"),
  unmeasured_param_list = list(c(0, 1), c(0, 1))
))

(unm_mod <- unm_glm(
  y ~ x + z + u1 + u2, family1 = gaussian(),
  u1 ~ x + z + u2, family2 = gaussian(),
  u2 ~ x + z, family3 = gaussian(),
  data = df
))
unm_backfill(df, unm_mod)

## End(Not run)

unmconf: Modeling with Unmeasured Confounding

Description

Convert to Greek expressions

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Value

Examples

Generate synthetic data

Description

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Value

Examples

Fitting Multilevel Bayesian Regression Model with Unmeasured Confounders

Description

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See Also

Examples

Generate synthetic data

Description

Usage

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Examples