mdmb_regression: Several Regression Models with Prior Distributions and...

In mdmb: Model Based Treatment of Missing Data

Several Regression Models with Prior Distributions and Sampling Weights

Description

Several regression functions which allow for sampling weights and prior distributions.

The function yjt_regression performs a linear regression in which the response variable is transformed according to the Yeo-Johnson transformation (Yeo & Johnson, 2000; see yjt_dist) and the residuals are distributed following the scaled t distribution. The degrees of freedom of the t distribution can be fixed or estimated (est_df=TRUE). The function bct_regression has same functionality like the Yeo-Johnson transformation but employs a Box-Cox transformation of the outcome variable.

The Yeo-Johnson transformation can be extended by a probit transformation (probit=TRUE) to cover the case of bounded variables on [0,1].

The function logistic_regression performs logistic regression for dichotomous data.

The function oprobit_regression performs ordinal probit regression for ordinal polytomous data.

Usage

#---- linear regression with Yeo-Johnson transformed scaled t distribution
yjt_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL,
        df=Inf, lambda_fixed=NULL, probit=FALSE, est_df=FALSE, df_min=0.5, df_max=100,
        use_grad=2, h=1e-5, optimizer="optim", maxiter=300, control=NULL)

## S3 method for class 'yjt_regression'
coef(object, ...)
## S3 method for class 'yjt_regression'
logLik(object, ...)
## S3 method for class 'yjt_regression'
predict(object, newdata=NULL, trafo=TRUE,  ...)
## S3 method for class 'yjt_regression'
summary(object, digits=4, file=NULL, ...)
## S3 method for class 'yjt_regression'
vcov(object, ...)

#---- linear regression with Box-Cox transformed scaled t distribution
bct_regression(formula, data, weights=NULL, beta_init=NULL, beta_prior=NULL,
        df=Inf, lambda_fixed=NULL, est_df=FALSE, use_grad=2, h=1e-5,
        optimizer="optim", maxiter=300, control=NULL)

## S3 method for class 'bct_regression'
coef(object, ...)
## S3 method for class 'bct_regression'
logLik(object, ...)
## S3 method for class 'bct_regression'
predict(object, newdata=NULL, trafo=TRUE, ...)
## S3 method for class 'bct_regression'
summary(object, digits=4, file=NULL, ...)
## S3 method for class 'bct_regression'
vcov(object, ...)

#---- logistic regression
logistic_regression(formula, data, weights=NULL, beta_init=NULL,
         beta_prior=NULL, use_grad=2, h=1e-5, optimizer="optim", maxiter=300,
         control=NULL)

## S3 method for class 'logistic_regression'
coef(object, ...)
## S3 method for class 'logistic_regression'
logLik(object, ...)
## S3 method for class 'logistic_regression'
predict(object, newdata=NULL, ...)
## S3 method for class 'logistic_regression'
summary(object, digits=4, file=NULL, ...)
## S3 method for class 'logistic_regression'
vcov(object, ...)

#---- ordinal probit regression
oprobit_regression(formula, data, weights=NULL, beta_init=NULL,
        use_grad=2, h=1e-5, optimizer="optim", maxiter=300,
        control=NULL, control_optim_fct=NULL)

## S3 method for class 'oprobit_regression'
coef(object, ...)
## S3 method for class 'oprobit_regression'
logLik(object, ...)
## S3 method for class 'oprobit_regression'
predict(object, newdata=NULL, ...)
## S3 method for class 'oprobit_regression'
summary(object, digits=4, file=NULL, ...)
## S3 method for class 'oprobit_regression'
vcov(object, ...)

Arguments

formula	Formula
data	Data frame. The dependent variable must be coded as 0 and 1.
weights	Optional vector of sampling weights
beta_init	Optional vector of initial regression coefficients
beta_prior	Optional list containing priors of all parameters (see Examples for definition of this list).
df	Fixed degrees of freedom for scaled t distribution
lambda_fixed	Optional fixed value for λ for scaled t distribution with Yeo-Johnson transformation
probit	Logical whether probit transformation should be employed for bounded outcome in yjt_regression
est_df	Logical indicating whether degrees of freedom in t distribution should be estimated.
df_min	Minimum value for estimated degrees of freedom
df_max	Maximum value for estimated degrees of freedom
use_grad	Computation method for gradients in stats::optim. The value 0 is the internal approximation of stats::optim and applies the settings in mdmb (≤0.3). The specification use_grad=1 uses the calculation of the gradient in CDM::numerical_Hessian. The value 2 is usually the most efficient calculation of the gradient.
h	Numerical differentiation parameter.
optimizer	Type of optimizer to be chosen. Options are "nlminb" (stats::nlminb) and the default "optim" (stats::optim)
maxiter	Maximum number of iterations
control	Optional arguments to be passed to optimization function (stats::nlminb) or stats::optim
control_optim_fct	Optional control argument for gradient in optimization
object	Object of class logistic_regression
newdata	Design matrix for predict function
trafo	Logical indicating whether fitted values should be on the transformed metric (trafo=TRUE) or the original metric (trafo=FALSE)
digits	Number of digits for rounding
file	File name if the summary output should be sunk into a file.
...	Further arguments to be passed.

Value

List containing values

coefficients	Estimated regression coefficients
vcov	Estimated covariance matrix
partable	Parameter table
y	Vector of values of dependent variable
Xdes	Design matrix
weights	Sampling weights
fitted.values	Fitted values in metric of probabilities
linear.predictor	Fitted values in metric of logits
loglike	Log likelihood value
logprior	Log prior value
logpost	Log posterior value
deviance	Deviance
loglike_case	Case-wise likelihood
ic	Information criteria
R2	Pseudo R-square value according to McKelvey and Zavoina

Author(s)

Alexander Robitzsch

References

McKelvey, R., & Zavoina, W. (1975). A statistical model for the analysis of ordinal level dependent variables. Journal of Mathematical Sociology, 4(1), 103-120. doi: 10.1080/0022250X.1975.9989847

Yeo, I.-K., & Johnson, R. (2000). A new family of power transformations to improve normality or symmetry. Biometrika, 87(4), 954-959. doi: 10.1093/biomet/87.4.954

See Also

See yjt_dist or car::yjPower for functions for the Yeo-Johnson transformation.

See stats::lm and stats::glm for linear and logistic regression models.

Examples

#############################################################################
# EXAMPLE 1: Simulated example logistic regression
#############################################################################

#--- simulate dataset
set.seed(986)
N <- 500
x <- stats::rnorm(N)
y <- 1*( stats::runif(N) < stats::plogis( -0.8 + 1.2 * x ) )
data <- data.frame( x=x, y=y )

#--- estimate logistic regression with mdmb::logistic_regression
mod1 <- mdmb::logistic_regression( y ~ x, data=data )
summary(mod1)

## Not run: 
#--- estimate logistic regression with stats::glm
mod1b <- stats::glm( y ~ x, data=data, family="binomial")
summary(mod1b)

#--- estimate logistic regression with prior distributions
b0 <- list( "dnorm", list(mean=0, sd=100) )  # first parameter
b1 <- list( "dcauchy", list(location=0, scale=2.5) )   # second parameter
beta_priors <- list( b0, b1 )  # order in list defines priors for parameters
#* estimation
mod2 <- mdmb::logistic_regression( y ~ x, data=data, beta_prior=beta_priors )
summary(mod2)

#############################################################################
# EXAMPLE 2: Yeo-Johnson transformed scaled t regression
#############################################################################

#*** create simulated data
set.seed(9865)
n <- 1000
x <- stats::rnorm(n)
y <- .5 + 1*x + .7*stats::rt(n, df=8 )
y <- mdmb::yj_antitrafo( y, lambda=.5 )
dat <- data.frame( y=y, x=x )
# display data
graphics::hist(y)

#--- Model 1: fit regression model with transformed normal distribution (df=Inf)
mod1 <- mdmb::yjt_regression( y ~ x, data=dat )
summary(mod1)

#--- Model 2: fit regression model with transformed scaled t distribution (df=10)
mod2 <- mdmb::yjt_regression( y ~ x, data=dat, df=10)
summary(mod2)

#--- Model 3: fit regression model with transformed normal distribution (df=Inf)
#             and fixed transformation parameter lambda of .5
mod3 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=.5)
summary(mod3)

#--- Model 4: fit regression model with transformed normal distribution (df=Inf)
#             and fixed transformation parameter lambda of 1
#             -> This model corresponds to least squares regression
mod4 <- mdmb::yjt_regression( y ~ x, data=dat, lambda_fixed=1)
summary(mod4)

# fit with lm function
mod4b <- stats::lm( y ~ x, data=dat )
summary(mod4b)

#--- Model 5: fit regression model with estimated degrees of freedom
mod5 <- mdmb::yjt_regression( y ~ x, data=dat, est_df=TRUE)
summary(mod5)

#** compare log-likelihood values
logLik(mod1)
logLik(mod2)
logLik(mod3)
logLik(mod4)
logLik(mod4b)
logLik(mod5)

#############################################################################
# EXAMPLE 3: Regression with Box-Cox and Yeo-Johnson transformations
#############################################################################

#*** simulate data
set.seed(985)
n <- 1000
x <- stats::rnorm(n)
y <- .5 + 1*x + stats::rnorm(n, sd=.7 )
y <- mdmb::bc_antitrafo( y, lambda=.5 )
dat <- data.frame( y=y, x=x )

#--- Model 1: fit regression model with Box-Cox transformation
mod1 <- mdmb::bct_regression( y ~ x, data=dat )
summary(mod1)
#--- Model 2: fit regression model with Yeo-Johnson transformation
mod2 <- mdmb::yjt_regression( y ~ x, data=dat )
summary(mod2)
#--- compare fit
logLik(mod1)
logLik(mod2)

#############################################################################
# EXAMPLE 4: Ordinal probit regression
#############################################################################

#--- simulate data
set.seed(987)
N <- 1500
x <- stats::rnorm(N)
z <- stats::rnorm(N)
# regression coefficients
b0 <- -.5 ; b1 <- .6 ; b2 <- .1
# vector of thresholds
thresh <- c(-1, -.3, 1)
yast <- b0 + b1 * x + b2*z + stats::rnorm(N)
y <- as.numeric( cut( yast, c(-Inf,thresh,Inf) ) ) - 1
dat <- data.frame( x=x, y=y, z=z )

#--- probit regression
mod <- mdmb::oprobit_regression( formula=y ~ x + z + I(x*z), data=dat)
summary(mod)

## End(Not run)

mdmb documentation built on March 7, 2023, 6:58 p.m.