plaqr {plaqr}

Partially Linear Additive Quantile Regression

Description

Returns an object of class "plaqr" and "rq" that represents a quantile regression fit. A nonlinear term z is transformed using bs(z) before fitting the model. The formula of the model (as it appears in R) becomes y~ x1 + x2 + bs(z1) + bs(z2) where bs(z1) is a B-spline.

Usage

plaqr(formula, nonlinVars=NULL, tau=.5, data=NULL, subset,   
            weights, na.action, method = "br", model = TRUE, 
            contrasts = NULL, splinesettings=NULL, ...)

Arguments

formula

a formula object, with the response on the left of a ~ operator, and the linear terms, separated by + operators, on the right. Any terms on the right of the ~ operator that also appear in nonlinVars will be included in the model as spline terms, not linear terms.

nonlinVars

a one-sided formula object, with a ~ operator to the left of the nonlinear terms seperated by + operators. A term appearing in both formula and nonlinVars will be treated as a nonlinear term. If nonlinVars is not NULL, then an intercept will automatically be included in the model (despite a -1 or 0 term included in formula).

tau

the quantile to be estimated, this is a number strictly between 0 and 1 (for now).

data

a data.frame in which to interpret the variables named in the formula, or in the subset and the weights argument. If this is missing, then the variables in the formula should be on the search list. This may also be a single number to handle some special cases – see below for details.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

weights

vector of observation weights; if supplied, the algorithm fits to minimize the sum of the weights multiplied into the absolute residuals. The length of weights must be the same as the number of observations. The weights must be nonnegative and it is strongly recommended that they be strictly positive, since zero weights are ambiguous.

na.action

a function to filter missing data. This is applied to the model.frame after any subset argument has been used. The default (with na.fail) is to create an error if any missing values are found. A possible alternative is na.omit, which deletes observations that contain one or more missing values.

model

if TRUE then the model frame is returned. This is essential if one wants to call summary subsequently.

method

the algorithmic method used to compute the fit. There are several options: The default method is the modified version of the Barrodale and Roberts algorithm for l_1-regression, used by l1fit in S, and is described in detail in Koenker and d'Orey(1987, 1994), default = "br". This is quite efficient for problems up to several thousand observations, and may be used to compute the full quantile regression process. It also implements a scheme for computing confidence intervals for the estimated parameters, based on inversion of a rank test described in Koenker(1994). For larger problems it is advantagous to use the Frisch–Newton interior point method "fn". And very large problems one can use the Frisch–Newton approach after preprocessing "pfn". Both of the latter methods are described in detail in Portnoy and Koenker(1997). There is a fifth option "fnc" that enables the user to specify linear inequality constraints on the fitted coefficients; in this case one needs to specify the matrix R and the vector r representing the constraints in the form Rb \geq r. See the examples. Finally, there are two penalized methods: "lasso" and "scad" that implement the lasso penalty and Fan and Li's smoothly clipped absolute deviation penalty, respectively. These methods should probably be regarded as experimental.

contrasts

a list giving contrasts for some or all of the factors default = NULL appearing in the model formula. The elements of the list should have the same name as the variable and should be either a contrast matrix (specifically, any full-rank matrix with as many rows as there are levels in the factor), or else a function to compute such a matrix given the number of levels.

splinesettings

a list of length equal to the number of nonlinear effects containing arguments to pass to the bs function for each term. Each element of the list is either NULL or a list with named elements correpsonding to the arguments in bs. If not NULL, the first element of splinesettings corresponds to the first nonlinear effect and so on.

...

additional arguments for the fitting routines (see the rq function in the ‘quantreg’ package ).

Value

Returns the following:

coefficients

Coefficients from the fitted model

x

optionally the model matrix, if x=TRUE.

y

optionally the response, if y=TRUE.

residuals

the residuals from the fit.

dual

the vector dual variables from the fit.

fitted.values

fitted values from the fit.

formula

the formula that was used in the rq function.

rho

the value of the objective function at the solution.

model

optionally the model frame, if model=TRUE

linear

the linear terms used in the model fit.

nonlinear

the nonlinear terms used in the model fit.

z

the values of the nonlinear terms.

Author(s)

Adam Maidman

References

Hastie, T. J. (1992) Generalized additive models. Chapter 7 of Statistical Models in S eds J. M. Chambers and T. J. Hastie, Wadsworth & Brooks/Cole.

Koenker, R. W. (2005). Quantile Regression, Cambridge U. Press.

Sherwood, B. and Wang, L. (2016). Partially linear additive quantile regression in ultra-high dimension. The Annals of Statistics 44, 288-317.

Maidman, A., Wang, L. (2017). New Semiparametric Method for Predicting High-Cost Patients. Preprint.

Examples

data(simData)

ss <- vector("list", 2)
ss[[2]]$degree <- 5
ss[[2]]$Boundary.knots <- c(-1, 1)

plaqr(y~., nonlinVars=~z1+z2, data=simData) 
#same as plaqr(formula= y~x1+x2+x3, nonlinVars=~z1+z2, data=simData)

plaqr(y~0, nonlinVars=~z1+z2, data=simData, splinesettings=ss) #no linear terms in the model

plaqr(y~., data=simData) #all linear terms