mipfp (version 3.2.1)

gof.estimates: Wald, Log-likelihood ratio and Person Chi-square statistics for mipfp object

Description

This method computes three statistics to perform a test wheter the seed agrees with the target data. The statistics are the Wilk's log-likelihood ratio statistic, the Wald statistic and the Person Chi-square statistic.

The method also returns the associated degrees of freedom.

Usage

# S3 method for mipfp
gof.estimates(object, seed = NULL, target.data = NULL, 
              target.list = NULL, replace.zeros = 1e-10, ...)

Arguments

object

The object of class mipfp containing.

seed

The seed used to compute the estimates (optional). If not provided, the method tries to determine the seed automatically.

target.data

A list containing the data of the target margins. Each component of the list is an array storing a margin. The list order must follow the one defined in target.list. Note that the cells of the arrays must be non-negative (and can even be NA if method = ipfp) (optional). If not provided, the method tries to dermine target.data automatically.

target.list

A list of the target margins provided in target.data. Each component of the list is an array whose cells indicates which dimension the corresponding margin relates to (optional). If not provided, the method tries to determine target.list automatically.

replace.zeros

If 0-cells are to be found, then they are replaced with this value.

…

Not used.

Value

A list whose elements are detailed below.

G2

The Log-likelihood statistic.

W2

The Wald statistic.

X2

The Pearson chi-squared statistic.

stats.df

The degrees of freedom for the G2, W2 and X2 statistics.

Details

The test is formally expressed as: $$H_0 ~ : ~ h(\pi) = 0 \quad vs \quad H_1~:~h(\pi) \neq 0$$ where \(\pi\) is the vector of the seed probabilities and \(h(x) = A^T x - m\) with \(A\) and \(m\) being respectively the marginal matrix and the margins vector of the estimation problem.

The three statistics are then defined as:

• Wilk's log-likelihoold ratio $$G^2 = 2 \sum x_i \ln \frac{\pi_i}{\hat{\pi}_i}$$

• Wald's statistic $$W^2 = h(x)^T ( H^T_x D_x H_x)^{-1} h(x)$$

• Pearson Chi-square $$\chi^2 = (x - n \hat{\pi})^T D^{-1}_{n\hat{\pi}} (x - n \hat{\pi})$$

where \(x\) is the vectorization of the seed, \(n = \sum x_i\), \(D_v\) is a diagonal matrix derived from the vector \(v\) and \(H\) denotes the Jacobian evaluated in \(\hat{\pi}\) (the vector of the estimated probabilities) of the function \(h(x)\).

The degrees of freedom for these statistics corresponds to the number of components in \(m\).

References

Lang, J.B. (2004) Multinomial-Poisson homogeneous models for contingency tables. Annals of Statistics 32(1): 340-383.

See Also

Estimate function to create an object of class mipfp and to update an initial multidimensional array with respect to given constraints. summary.mipfp can also retrieve the statistics and their associated p-values.

Examples

# NOT RUN {
# loading the data
data(spnamur, package = "mipfp")
# subsetting the data frame, keeping only the first 3 variables
spnamur.sub <- subset(spnamur, select = Household.type:Prof.status)
# true table
true.table <- table(spnamur.sub)
# extracting the margins
tgt.v1        <- apply(true.table, 1, sum)
tgt.v1.v2     <- apply(true.table, c(1,2), sum)
tgt.v2.v3     <- apply(true.table, c(2,3), sum)
tgt.list.dims <- list(1, c(1,2), c(2,3))
tgt.data      <- list(tgt.v1, tgt.v1.v2, tgt.v2.v3)
# creating the seed, a 10 pct sample of spnamur
seed.df <- spnamur.sub[sample(nrow(spnamur), round(0.10*nrow(spnamur))), ]
seed.table <- table(seed.df)
# applying one fitting method (ipfp)
r.ipfp <- Estimate(seed=seed.table, target.list=tgt.list.dims, 
                   target.data = tgt.data)
# printing the G2, X2 and W2 statistics
print(gof.estimates(r.ipfp))
# alternative way (pretty printing, with p-values)
print(summary(r.ipfp)$stats.gof)
# }