wals: Weighted-Average Least Squares for linear regression models

Description

Performs model averaging for linear regression models using the Weighted-Average Least Squares method by magnus2010growth;textualWALS. See also deluca2011stata;textualWALS, kumar2013normallocation;textualWALS and magnus2016wals;textualWALS.

Usage

wals(x, ...)
# S3 method for formula
wals(
  formula,
  data,
  subset = NULL,
  na.action = NULL,
  weights = NULL,
  offset = NULL,
  prior = weibull(),
  model = TRUE,
  keepY = TRUE,
  keepX = FALSE,
  sigma = NULL,
  ...
)
# S3 method for matrix
wals(
  x,
  x2,
  y,
  subset = NULL,
  na.action = NULL,
  weights = NULL,
  offset = NULL,
  prior = weibull(),
  keepY = TRUE,
  keepX = FALSE,
  sigma = NULL,
  ...
)
# S3 method for default
wals(x, ...)

Value

wals.formula() returns an object of class "wals". This is a list that contains all elements returned from walsFit and additionally

y: If keepY = TRUE, contains the response vector.
x: list. If keepX = TRUE, then it is a list with elements x1 and x2 containing the design matrices of the focus and auxiliary regressors, respectively.
weights: returns the argument weights.
offset: returns the argument offset.
cl: Call of the function.
formula: formula used.
terms: List containing the model terms of the focus and auxiliary regressors separately, as well as for the full model.
levels: List containing the levels of the focus and auxiliary regressors separately, as well as for the full model.
contrasts: List containing the contrasts of the design matrices of focus and auxiliary regressors.
model: If model = TRUE, contains the model frame.

See returns of walsFit for more details.

wals.matrix() returns an object of class "walsMatrix", which inherits from "wals". This is a list that contains all elements returned from walsFit and additionally the response y, the list x with model matrices x1 and x2, the call cl, offset and weights.

wals.default() raises an error if x is not an object of class "matrix" or a class that extends "matrix". Otherwise returns an object of class "walsMatrix". See above for more details.

Arguments

x: Design matrix of focus regressors. Usually includes a constant (column full of 1s) and can be generated using model.matrix.
...: Arguments for workhorse walsFit.
formula: an object of class "Formula" (or one that can be coerced to that class, e.g. "formula"): a symbolic description of the model to be fitted. The details of model specification are given under ‘Details’.
data: an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment which the function is called from.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
na.action: not implemented yet.
weights: not implemented yet.
offset: not implemented yet.
prior: Object of class "familyPrior". For example weibull or laplace.
model: if TRUE (default), then the model.frame is stored in the return.
keepY: if TRUE (default), then the response is stored in the return.
keepX: if TRUE, then the model matrices are stored in the return. the return.
sigma: if NULL (default), then the variance of the error term is estimated. See walsFit for more details.
x2: Design matrix of auxiliary regressors. Usually does not include a constant column and can also be generated using model.matrix.
y: Response as vector.

Details

R port of MATLAB code wals.m (version 2.0, revision 18 December 2013) by J.R. Magnus and G. De Luca, available from https://www.janmagnus.nl/items/WALS.pdf. Calculates WALS estimates when focus regressors (X1) are present in all submodels and model averaging takes place over the auxiliary regressors (X2).

Formulas typically contain two parts, i.e. they are of the form "y ~ X11 + X12 | X21 + X22", where the variables before "|" are the focus regressors (includes a constant by default) and the ones after "|" are the auxiliary regressors. If only a one-part formula is specified, then all regressors are considered as auxiliary regressors and only a constant is employed as focus regressor, i.e. "y ~ X1 + X2" is equivalent to "y ~ 1 | X1 + X2".

WARNING: Interactions in formula do not work properly yet. It is recommended to manually create the interactions beforehand and then to insert them as 'linear terms' in the formula.

wals.default() raises an error if x is not an object of class "matrix" or a class that extends "matrix". Otherwise it calls wals.matrix(). It is a modified version of glmboost.default from the mboost package version 2.9-8 (2023-09-06) mboostWALS.

References

Examples

Run this code

## Replicate table on p. 534 of De Luca & Magnus (2011)
fitDM <- wals(gdpgrowth ~ lgdp60 + equipinv + school60 + life60 + popgrowth |
                  law + tropics + avelf + confucian, data = GrowthMPP,
                prior = laplace())
tableDM <- cbind("coef" = coef(fitDM), "se" = sqrt(diag(vcov(fitDM))))
print(round(tableDM, 7))


## Replicate first panel of Table I in Amini & Parmeter (2012)
data("datafls", package = "BMS")

# NOTE: Authors manually scale data, then rescale the resulting coefs and se.
X <- model.matrix(y ~ ., data = datafls)
Xscaled <- apply(X, MARGIN = 2, function(x) x/max(x))
Xscaled <- Xscaled[,-1]
scaleVector <- apply(X, MARGIN = 2, function(x) max(x))
flsScaled <- as.data.frame(cbind(y = datafls$y, Xscaled))

# NOTE: prescale = FALSE, still used old version of WALS in Magnus et al. (2010).
# Not recommended anymore!
fitFLS <- wals(y ~ 1 | ., data = flsScaled, prescale = FALSE, eigenSVD = FALSE,
               prior = laplace())
tableFLS <- cbind('coef' = coef(fitFLS)/scaleVector,
                  'se' = sqrt(diag(vcov(fitFLS)))/scaleVector)
printVars <- c("(Intercept)", "GDP60", "Confucian", "LifeExp", "EquipInv",
               "SubSahara", "Muslim", "RuleofLaw")
print(round(tableFLS[printVars,], 4))


## Replicate third panel of Table I in Amini & Parmeter (2012)
data("SDM", package = "BayesVarSel")

# rescale response
SDM$y <- SDM$y / 100

# NOTE: Authors manually scale data, then rescale the resulting coefs and se.
X <- model.matrix(y ~ ., data = SDM)
Xscaled <- apply(X, MARGIN = 2, function(x) x/max(x))
Xscaled <- Xscaled[,-1]
scaleVector <- apply(X, MARGIN = 2, function(x) max(x))
SDMscaled <- as.data.frame(cbind(y = SDM$y, Xscaled))

# NOTE: prescale = FALSE, still used old version of WALS in Magnus et al. (2010).
# Not recommended anymore!
fitDW <- wals(y ~ 1 | ., data = SDMscaled, prescale = FALSE, eigenSVD = FALSE,
              prior = laplace())
tableDW <- cbind(coef(fitDW)/scaleVector, sqrt(diag(vcov(fitDW)))/scaleVector)
printVars <- c("(Intercept)", "EAST", "P60", "IPRICE1", "GDPCH60L", "TROPICAR")
print(round(tableDW[printVars,], 5))

## Example for wals.matrix()
X <- model.matrix(mpg ~ disp + hp + wt + vs + am + carb, data = mtcars)
X1 <- X[,c("(Intercept)", "disp", "hp", "wt")] # focus
X2 <- X[,c("vs", "am", "carb")] # auxiliary
y <- mtcars$mpg

wals(X1, X2, y, prior = weibull())

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