CompRandFld (version 1.0.3-6)

HypoTest: Statistical Hypothesis Tests for Nested Models

Description

The function performs statistical hypothesis tests for nested models based on composite likelihood versions of: Wald-type, score-type and Wilks-type (likelihood ratio) statistics.

Usage

HypoTest(object1, object2, …, statistic)

Arguments

object1

An object of class FitComposite.

object2

An object of class FitComposite that is a nested model within object1.

…

Further successively nested objects.

statistic

String; the name of the statistic used within the hypothesis test (see Details).

Value

An object of class c("data.frame"). The object contain a table with the results of the tested models. The rows represent the responses for each model and the columns the following results:

Num.Par

The number of the model's parameters.

Diff.Par

The difference between the number of parameters of the model in the previous row and those in the actual row.

Df

The effective number of degree of freedom of the chi-square distribution.

Chisq

The observed value of the statistic.

Pr(>chisq)

The p-value of the quantile Chisq computed using a chi-squared distribution with Df degrees of freedom.

Details

The implemented hypothesis tests for nested models are based on the following statistics:

1. Wald-type (Wald);

2. Score-type, also known as Rao-type (Rao);

3. Wilks-type; also known as the composite likelihood ratio statistic. Available are variants of the basic version, in particular:

   • Rotnitzky and Jewell adjustment (WilksRJ);

   • Satterhwaite adjustment (WilksS);

   • Chandler and Bate adjustment (WilksCB);

   • Pace, Salvan and Sartori adjustment (WilksPSS);

More specifically, consider an \(p\)-dimensional random vector \(\mathbf{Y}\) with probability density function \(f(\mathbf{y};\mathbf{\theta})\), where \(\mathbf{\theta} \in \Theta\) is a \(q\)-dimensional vector of parameters. Suppose that \(\mathbf{\theta}=(\mathbf{\psi},\mathbf{\tau})\) can be partitioned in a \(q'\)-dimensional subvector \(\psi\) and \(q''\)-dimensional subvector \(\tau\). Assume also to be interested in testing the specific values of the vector \(\psi\). Then, one can use some statistical hypothesis tests for testing the null hypothesis \(H_0: \psi=\psi_0\) against the alternative \(H_1: \psi \neq \psi_0\). Composite likelihood versions of 'Wald' and 'score' statistics have the usual asymptotic chi-square distribution with \(q'\) degree of freedom. The Wald-type statistic is $$W=(\hat{\psi}-\psi_0)^T (G^{\psi \psi})^{-1}(\hat{\theta})(\hat{\psi}-\psi_0),$$ where \(G_{\psi \psi}\) is the \(q' \times q'\) submatrix of the Godambe information pertaining to \(\psi\) and \(\hat{\theta}\) is the maximum likelihood estimator from the full model. The score-type statistic (Rao-type) is $$W=s_{\mathbf{\psi}}\{\mathbf{\psi}_0, \hat{\mathbf{\tau}}(\mathbf{\psi}_0)\}^T H^{\psi \psi}(\hat{\theta}_\psi) \{G^{\psi \psi}(\hat{\theta}_\psi)\}^{-1} H^{\psi \psi}(\hat{\theta}_\psi) s_{\mathbf{\psi}}\{\mathbf{\psi}_0, \hat{\mathbf{\tau}}(\mathbf{\psi}_0)\},$$ where \(H^{\psi \psi}\) is the \(q' \times q'\) submatrix of the inverse of \(H(\theta)\) pertaining to \(\psi\) (the same for \(G\)) and \(\hat{\theta}_\psi\) is the constrained maximum likelihood estimate of \(\theta\) for fixed \(\psi\). These two statistics can be called from the routine HypoTest assigning at the argument statistic respectively the values: Wald and Rao.

Alternatively to the Wald-type and score-type statistics one can use the composite version of the Wilks-type or likelihood ratio statistic, given by $$W=2[C \ell(\hat{\mathbf{\theta}};\mathbf{y}) - C \ell\{\mathbf{\psi}_0, \hat{\mathbf{\tau}}(\mathbf{\psi}_0);\mathbf{y}\}].$$ The asymptotic distribution of the composite likelihood ratio statistic is given by $$W \dot{\sim} \sum_{i} \lambda_i \chi^2,$$ for \(i=1,\ldots,q'\), where \(\chi^2_i\) are \(q'\) iid copies of a chi-square one random variable and \(\lambda_1,\ldots,\lambda_{q'}\) are the eigenvalues of the matrix \((H^{\psi \psi})^{-1} G^{\psi \psi}\). There exist several adjustments to the composite likelihood ratio statistic in order to get an approximated \(\chi^2_{q'}\). For example, Rotnitzky and Jewell (1990) proposed the adjustment \(W'= W / \bar{\lambda}\) where \(\bar{\lambda}\) is the average of the eigenvalues \(\lambda_i\). This statistic can be called within the routine by the value: WilksRJ. A better solution is proposed by Satterhwaite (1946) defining \(W''=\nu W / (q' \bar{\lambda})\), where \(\nu=(\sum_i \lambda)^2 / \sum_i \lambda^2_i\) for \(i=1\ldots,q'\), is the effective number of the degree of freedom. Note that in this case the distribution of the likelihood ratio statistic is a chi-square random variable with \(\nu\) degree of freedom. This statistic can be called from the routine assigning the value: WilksS. For the adjustments suggested by Chandler and Bate (2007) and Pace, Salvan and Sartori (2011) we refere to the articles (see References), these versions can be called from the routine assigning respectively the values: WilksCB and WilksPSS.

References

Chandler, R. E., and Bate, S. (2007). Inference for Clustered Data Using the Independence log-likelihood. Biometrika, 94, 167--183.

Pace, L., Salvan, A. and Sartori, N. (2011). Adjusting Composite Likelihood Ratio Statistics. Statistica Sinica, 21, 129--148.

Rotnitzky, A. and Jewell, N. P. (1990). Hypothesis Testing of Regression Parameters in Semiparametric Generalized Linear Models for Cluster Correlated Data. Biometrika, 77, 485--497.

Satterthwaite, F. E. (1946). An Approximate Distribution of Estimates of Variance Components. Biometrics Bulletin, 2, 110--114.

Varin, C., Reid, N. and Firth, D. (2011). An Overview of Composite Likelihood Methods. Statistica Sinica, 21, 5--42.

See Also

FitComposite.

Examples

# NOT RUN {
# Please remove the symbol hashtag to run the code

library(CompRandFld)
library(RandomFields)
set.seed(3451)

# Define the spatial-coordinates of the points:
x <- runif(300, 0, 10)
y <- runif(300, 0, 10)

################################################################
###
### Example 1. Composite likelihood-based hypothesis testing.
### Simulation of a Gaussian spatial random field with
### stable correlation.
### Estimation by composite likelihood using the setting:
### marginal pairwise likelihood objects.
###
###############################################################

# Set the model's parameters:
corrmodel <- "stable"
mean <- 0
sill <- 1
nugget <- 1
scale <- 1
power <- 1.3

# Simulation of the spatial Gaussian random field:
data <- RFsim(x, y, corrmodel=corrmodel, param=list(mean=mean,
              sill=sill,nugget=nugget,scale=scale,power=power))$data

# Maximum composite-likelihood fitting of the random field, full model:
fit1 <- FitComposite(data, x, y, corrmodel=corrmodel, maxdist=5,
                    varest=TRUE,start=list(mean=mean,power=power,scale=scale,sill=sill),
                    fixed=list(nugget=1))

# Maximum composite-likelihood fitting of the random field, first nasted model:
fit2 <- FitComposite(data, x, y, corrmodel=corrmodel, maxdist=5,
                    varest=TRUE,start=list(mean=mean,power=power,scale=scale),
                    fixed=list(nugget=1,sill=1))

# Maximum composite-likelihood fitting of the random field, second nasted model:
fit3 <- FitComposite(data, x, y, corrmodel=corrmodel, maxdist=5,
                    varest=TRUE,start=list(scale=scale),
                    fixed=list(nugget=1,sill=1,mean=0,power=1.3))

# Hypothesis testing results:
# composite Wald-type statistic:
HypoTest(fit1, fit2, fit3, statistic='Wald')

# composite score-type statistic:
HypoTest(fit1, fit2, fit3, statistic='Rao')

# composite likelihood ratio statistic with RJ adjustment:
HypoTest(fit1, fit2, fit3, statistic='WilksRJ')

# composite likelihood ratio statistic with S adjustment:
HypoTest(fit1, fit2, fit3, statistic='WilksS')

# composite likelihood ratio statistic with CB adjustment:
HypoTest(fit1, fit2, fit3, statistic='WilksCB')

# composite likelihood ratio statistic with PSS adjustment:
HypoTest(fit1, fit2, fit3, statistic='WilksPSS')

# }