HypoTest {CompRandFld} R Documentation

## 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).

### 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 Y with probability density function f(y;theta), where theta in Theta is a q-dimensional vector of parameters. Suppose that theta=(psi, 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 <> 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' x 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_{psi}{psi_0, hat{tau}(psi_0)}^T H^{psi psi}(hat{theta}_psi) {G^{psi psi}(hat{theta}_psi)}^{-1} H^{psi psi}(hat{theta}_psi) s_{psi}{psi_0, hat{tau}(psi_0)},

where H^{psi psi} is the q' x 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[Cl(hat{theta};y) - Cl{psi_0, hat{tau}(psi_0);y}].

The asymptotic distribution of the composite likelihood ratio statistic is given by

W ~ sum_i lambda_i Chi^2_i,

for i=1,...,q', where Chi^2_i are q' iid copies of a chi-square one random variable and lambda_1,...,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 / sum_i lambda^2_i for i=1...,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`.

### 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.

### 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.

`FitComposite`.

### Examples

```# 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')

```

[Package CompRandFld version 1.0.3-6 Index]