Testing the equality of critical points

Description

This function can be used to test the equality of the M critical points estimated from the respective level-specific curves.

Usage

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localtest(formula, data = data, na.action = "na.omit", der,
  smooth = "kernel", weights = NULL, nboot = 500, h0 = -1, h = -1,
  nh = 30, kernel = "epanech", p = 3, kbin = 100, rankl = NULL,
  ranku = NULL, seed = NULL, cluster = TRUE, ncores = NULL, ...)

Arguments

formula

An object of class formula: a sympbolic description of the model to be fitted. The details of model specification are given under 'Details'.

data

A data frame or matrix containing the model response variable and covariates required by the formula.

na.action

A function which indicates what should happen when the data contain 'NA's. The default is 'na.omit'.

der

Number which determines any inference process. By default der is NULL. If this term is 0, the testing procedures is applied for the estimate. If it is 1 or 2, it is designed for the first or second derivative, respectively.

smooth

Type smoother used: smooth = "kernel" for local polynomial kernel smoothers and smooth = "splines" for splines using the mgcv package.

weights

Prior weights on the data.

nboot

Number of bootstrap repeats.

h0

The kernel bandwidth smoothing parameter for the global effect (see references for more details at the estimation). Large values of the bandwidth lead to smoothed estimates; smaller values of the bandwidth lead lo undersmoothed estimates. By default, cross validation is used to obtain the bandwidth.

h

The kernel bandwidth smoothing parameter for the partial effects.

nh

Integer number of equally-spaced bandwidth on which the h is discretised, to speed up computation.

kernel

A character string specifying the desired kernel. Defaults to kernel = "epanech", where the Epanechnikov density function kernel will be used. Also, several types of kernel funcitons can be used: triangular and Gaussian density function, with "triang" and "gaussian" term, respectively.

p

Degree of polynomial to be used. Its value must be the value of derivative + 1. The default value is 3 due to the function returns the estimation, first and second derivative.

kbin

Number of binning nodes over which the function is to be estimated.

rankl

Number or vector specifying the minimum value for the interval at which to search the x value which maximizes the estimate, first or second derivative (for each level). The default is the minimum data value.

ranku

Number or vector specifying the maximum value for the interval at which to search the x value which maximizes the estimate, first or second derivative (for each level). The default is the maximum data value.

seed

Seed to be used in the bootstrap procedure.

cluster

A logical value. If TRUE (default), the bootstrap procedure is parallelized (only for smooth = "splines". Note that there are cases (e.g., a low number of bootstrap repetitions) that R will gain in performance through serial computation. R takes time to distribute tasks across the processors also it will need time for binding them all together later on. Therefore, if the time for distributing and gathering pieces together is greater than the time need for single-thread computing, it does not worth parallelize.

ncores

An integer value specifying the number of cores to be used in the parallelized procedure. If NULL (default), the number of cores to be used is equal to the number of cores of the machine - 1.

...

Other options.

Details

localtest can be used to test the equality of the M critical points estimated from the respective level-specific curves. Note that, even if the curves and/or their derivatives are different, it is possible for these points to be equal.

For instance, taking the maxima of the first derivatives into account, interest lies in testing the following null hypothesis

H_0: x_{01} = … = x_{0M}

versus the general alternative

H_1: x_{0i} \ne x_{0j} \quad {\rm{for}} \quad {\rm{some}} \quad \emph{i}, \emph{j} \in \{ 1, …, M\}.

The above hypothesis is true if d=x_{0j}-x_{0k}=0 where

(j,k)= argmax \quad (l,m) \quad \{1 ≤q l<m ≤q M\} \quad |x_{0l}-x_{0m}|,

otherwise H_0 is false. It is important to highlight that, in practice, the true x_{0j} are not known, and consequently neither is d, so an estimate \hat d = \hat x_{0j}-\hat x_{0k} is used, where, in general, \hat x_{0l} are the estimates of x_{0l} based on the estimated curves \hat m_l with l = 1, … , M.

Needless to say, since \hat d is only an estimate of the true d, the sampling uncertainty of these estimates needs to be acknowledged. Hence, a confidence interval (a,b) is created for d for a specific level of confidence (95%). Based on this, the null hypothesis is rejected if zero is not contained in the interval.

Note that if this hypothesis is rejected (and the factor has more than two levels), one option could be to use the maxp.diff function in order to obtain the differences between each pair of factor's levels.

Note that the models fitted by localtest function are specified in a compact symbolic form. The \~ operator is basic in the formation of such models. An expression of the form y ~ model is interpreted as a specification that the response y is modelled by a predictor specified symbolically by model. The possible terms consist of a variable name or a variable name and a factor name separated by : operator. Such a term is interpreted as the interaction of the continuous variable and the factor. However, if smooth = "splines", the formula is based on the function formula.gam of the mgcv package.

Value

The estimate of d value is returned and its confidence interval for a specific-level of confidence, i.e. 95%. Additionally, it is shown the decision, accepted or rejected, of the local test. Based on the null hypothesis is rejected if a zero value is not within the interval.

Author(s)

Marta Sestelo, Nora M. Villanueva and Javier Roca-Pardinas.

References

Sestelo, M. (2013). Development and computational implementation of estimation and inference methods in flexible regression models. Applications in Biology, Engineering and Environment. PhD Thesis, Department of Statistics and O.R. University of Vigo.

Examples

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library(npregfast)
data(barnacle)
localtest(DW ~ RC : F, data = barnacle, der = 1, seed = 130853, nboot = 100)

# localtest(height ~ s(age, by = sex), data = children, seed = 130853, 
# der = 1, smooth = "splines")