Fitting Weighted Generalized Linear Models

lm_weightit() and glm_weightit() are used to fit (generalized) linear models with a variance matrix that accounts for estimation of weights, if supplied. By default, these functions use M-estimation to construct a robust covariance matrix using the estimation equations for the weighting model and the outcome model. lm_weightit() is a wrapper for glm_weightit() with the Gaussian family and identity link (i.e., a linear model).

Usage

glm_weightit(
  formula,
  data,
  family = gaussian,
  weightit,
  vcov = NULL,
  cluster,
  R = 500,
  offset,
  start = NULL,
  etastart,
  mustart,
  control = list(...),
  x = FALSE,
  y = TRUE,
  contrasts = NULL,
  fwb.args = list(),
  ...
)

lm_weightit(
  formula,
  data,
  weightit,
  vcov = NULL,
  cluster,
  R = 500,
  offset,
  start = NULL,
  etastart,
  mustart,
  control = list(...),
  x = FALSE,
  y = TRUE,
  contrasts = NULL,
  ...
)

# S3 method for glm_weightit
summary(object, ci = FALSE, level = 0.95, ...)

Arguments

formula: an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.
data: a data frame containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which the function is called.
family: a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. See family for details of family functions. Can also be the string "multinomial" for multinomial logistic regression.
weightit: a weightit or weightitMSM object; the output of a call to weightit() or weightitMSM(). If not supplied, an unweighted model will be fit.
vcov: string; the method used to compute the variance of the estimated parameters. Allowable options include "asympt", which uses the asymptotically correct M-estimation-based method that accounts for estimation of the weights when available; "const", which uses the usual maximum likelihood estimates (only available when weightit is not supplied); "HC0", which computes the robust sandwich variance treating weights (if supplied) as fixed; "BS", which uses the traditional bootstrap (including re-estimation of the weights, if supplied); "FWB", which uses the fractional weighted bootstrap as implemented in fwb::fwb() (including re-estimation of the weights, if supplied); and "none" to omit calculation of a variance matrix. If NULL (the default), will use "asympt" if weightit is supplied and M-estimation is available and "HC0" otherwise. See the vcov_type component of the outcome object to see which was used.
cluster: optional; for computing a cluster-robust variance matrix, a variable indicating the clustering of observations, a list (or data frame) thereof, or a one-sided formula specifying which variable(s) from the fitted model should be used. Note the cluster-robust variance matrix uses a correction for small samples, as is done in sadwich::vcovCL() by default. Cluster-robust variance calculations are available only when vcov is "asympt", "HC0", "BS", or "FWB".
R: the number of bootstrap replications when vcov is "BS" or "FWB". Default is 500. Ignored otherwise.
offset: optional; a numeric vector contain the model offset. See offset(). An offset can also be preset in the model formula.
start: optional starting values for the coefficients.
etastart, mustart: optional starting values for the linear predictor and vector of means when family is not "multinomial". Passed to glm().
control: a list of parameters for controlling the fitting process.
x, y: logical values indicating whether the response vector and model matrix used in the fitting process should be returned as components of the returned value.
contrasts: an optional list define contrasts for factor variables. See model.matrix().
fwb.args: an optional list of further arguments to supply to fwb::fwb() when vcov = "FWB".
...: for glm_weightit() and lm_weightit(), arguments to be used to form the default control argument if it is not supplied directly. Ignored otherwise.
object: a glm_weightit object.
ci: logical whether to display Wald confidence intervals for estimated coefficients. Default is FALSE.
level: when ci = TRUE, the desired confidence level.

Value

A glm_weightit object, which inherits from glm. Unless vcov = "none", the vcov component contains the covariance matrix adjusted for the estimation of the weights if requested and a compatible weightit object was supplied. The vcov_type component contains the type of variance matrix requested. If cluster is supplied, it will be stored in the "cluster" attribute of the output object, even if not used.

print(), vcov(), predict(), and confint() methods are also available; these generally follow the same pattern as the respect method for glm objects. confint() uses Wald confidence intervals (internally calling confint.lm()). When family = "multinomial", predict() produces a matrix of predicted probabilities, one for each level of the outcome, and the type argument is ignored. model.frame() output (also the model component of the output object) will include two extra column when weightit is supplied: (weights) containing the weights used in the model (the product of the estimated weights and the sampling weights, if any) and (s.weights) containing the sampling weights, which will be 1 if s.weights is not supplied in the original weightit() call.

Details

glm_weightit() is essentially a wrapper for glm() that optionally computes a coefficient variance matrix that can be adjusted to account for estimation of the weights if a weightit or weightitMSM object is supplied to the weightit argument. When no argument is supplied to weightit or there is no "Mparts" attribute in the supplied object, the default variance matrix returned will be the "HC0" sandwich variance matrix, which is robust to misspecification of the outcome family (including heteroscedasticity). Otherwise, the default variance matrix uses M-estimation to additionally adjust for estimation of the weights. When possible, this often yields smaller (and more accurate) standard errors. See the individual methods pages to see whether and when an "Mparts" attribute is included in the supplied object. To request that a variance matrix be computed that doesn't account for estimation of the weights even when a compatible weightit object is supplied, set vcov = "HC0", which treats the weights as fixed.

Bootstrapping can also be used to compute the coefficient variance matrix; when vcov = "BS" or vcov = "FWB", which implement the traditional resampling-based and fractional weighted bootstrap, respectively, the entire process of estimating the weights and fitting the outcome model is repeated in bootstrap samples (if a weightit object is supplied). This accounts for estimation of the weights and can be used with any weighting method. It is important to set a seed using set.seed() to ensure replicability of the results. The fractional weighted bootstrap is more reliable but requires the weighting method to accept sampling weights (which most do, and you'll get an error if it doesn't). Setting vcov = "FWB" and supplying fwb.args = list(wtype = "multinom") also performs the resampling-based bootstrap but with the additional features fwb provides (e.g., a progress and parallelization) at the expense of needing to have fwb installed.

When family = "multinomial", multinomial logistic regression is fit using a custom function in WeightIt that uses M-estimation to estimate the model coefficients. This implementation is less robust to failures than other multinomial logistic regression solvers and should be used with caution. Estimation of coefficients should align with that from mlogit::mlogit() and mclogit::mblogit().

Functions in the sandwich package can be to compute standard errors after fitting, regardless of how vcov was specified, though these will ignore estimation of the weights, if any. When no adjustment is done for estimation of the weights (i.e., because no weightit argument was supplied or there was no "Mparts" component in the supplied object), the default variance matrix produced by glm_weightit() should align with that from sandwich::vcovHC(. type = "HC0") or sandwich::vcovCL(., type = "HC0", cluster = cluster) when cluster is supplied.

Examples


data("lalonde", package = "cobalt")

# Logistic regression ATT weights
w.out <- weightit(treat ~ age + educ + married + re74,
                  data = lalonde, method = "glm",
                  estimand = "ATT")

# Linear regression outcome model that adjusts
# for estimation of weights
fit1 <- lm_weightit(re78 ~ treat, data = lalonde,
                    weightit = w.out)

summary(fit1)
#> 
#> Call:
#> lm_weightit(formula = re78 ~ treat, data = lalonde, weightit = w.out)
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   5515.9      376.5  14.649   <2e-16 ***
#> treat          833.2      669.1   1.245    0.213    
#> Standard error: HC0 robust (adjusted for estimation of weights)

# Linear regression outcome model that treats weights
# as fixed
fit2 <- lm_weightit(re78 ~ treat, data = lalonde,
                    weightit = w.out, vcov = "HC0")

summary(fit2)
#> 
#> Call:
#> lm_weightit(formula = re78 ~ treat, data = lalonde, weightit = w.out, 
#>     vcov = "HC0")
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   5515.9      353.5  15.604   <2e-16 ***
#> treat          833.2      676.6   1.232    0.218    
#> Standard error: HC0 robust

# Linear regression outcome model that bootstraps
# estimation of weights and outcome model fitting
# using fractional weighted bootstrap with "Mammen"
# weights
set.seed(123)
fit3 <- lm_weightit(re78 ~ treat, data = lalonde,
                    weightit = w.out,
                    vcov = "FWB",
                    R = 50,
                    fwb.args = list(wtype = "mammen"))

summary(fit3)
#> 
#> Call:
#> lm_weightit(formula = re78 ~ treat, data = lalonde, weightit = w.out, 
#>     vcov = "FWB", R = 50, fwb.args = list(wtype = "mammen"))
#> 
#> Coefficients:
#>             Estimate Std. Error z value Pr(>|z|)    
#> (Intercept)   5515.9      361.5  15.257   <2e-16 ***
#> treat          833.2      644.4   1.293    0.196    
#> Standard error: fractional weighted bootstrap

# Multinomial logistic regression outcome model
# that adjusts for estimation of weights
lalonde$re78_3 <- factor(findInterval(lalonde$re78,
                                      c(0, 5e3, 1e4)))

fit4 <- glm_weightit(re78_3 ~ treat, data = lalonde,
                     weightit = w.out,
                     family = "multinomial")

summary(fit4)
#> 
#> Call:
#> glm_weightit(formula = re78_3 ~ treat, data = lalonde, family = "multinomial", 
#>     weightit = w.out)
#> 
#> Coefficients:
#>               Estimate Std. Error z value Pr(>|z|)    
#> 2~(Intercept) -0.90398     0.1491  -6.064 1.32e-09 ***
#> 2~treat        0.05006     0.2363   0.212    0.832    
#> 3~(Intercept) -1.02170     0.1570  -6.508 7.62e-11 ***
#> 3~treat        0.12015     0.2384   0.504    0.614    
#> Standard error: HC0 robust (adjusted for estimation of weights)

Usage

Arguments

Value

Details

See also

Examples