2. Aspectos referenciales
2.1 Antecedentes Investigativos
As discussed in section 3.1, X2 and Dmay be used to assess the fit of a model to observed data when the number of covariate patterns is fixed. In this case the two statistics are asymptotically equivalent. However, under some circumstances the asymptotic distributions of these statistics can differ. For example, if the number of covariate patterns grows at a similar rate to the overall number of observations, but their ratio remains fixed, then the asymptotic means and variances
of X2 and D may differ (Read and Cressie 1988). This condition is referred to as “sparseness”. Which of these two statistics is optimal depends on the particular situation.
Cressie and Read (1984) introduced a family of power-divergence statistics that gives a unifying approach to testing the fit of models with discrete multivariate data. These statistics take
the form
1 2 observed observed 1 1 expected J i j j j
(3.26)where J is the fixed number of possible outcomes and is a real-valued parameter chosen by the user. They define the two cases that can result in division by zero, that is 0 and 1, as the limits where 0 and 1, respectively. Which value of gives the optimal test statistic depends on the particular circumstances, such as if there is a condition of sparseness or whether the null hypothesis is true. They suggest (Read, et al. 1988) that a reasonable choice for
the value of is a value that lies in the range
1,2
. This range includes both X2, for which 1, and D, for which 0. When certain details of the circumstances are unknown, for example the alternative model, then they suggest a test statistic which lies between X2 andD, with 2 3, as a compromise with excellent properties when the sample size is small (see Read and Cressie, 1988, chapter 5). Read, et al. (1988) report the asymptotic distribution of the
power-divergence tests are central chi-squared with degrees of freedom equal to G J
1
K, when all three parameters are fixed.In a series of papers, McCullagh(1985, 1986) considered the effect of sparse data on both X2
and D. Specifically he considers the case when the number of cells is increasing to infinity, rather than the mean count within the cells increasing to infinity. McCullagh argues that the
conditional distribution of X2 and D, rather than their marginal distributions, are relevant for assessing goodness-of-fit of GLMs when the parameters have been estimated with reference to the data, rather than fixed in advance. To remove the distributional dependence of the statistics on the parameter estimates, he conditions on the sufficient statistic for the parameter estimates.
He gives an approximate analytical solution for the conditional distributions of both X2 and D
for GLM with canonical link functions. He found that for binary data, D was uninformative as a goodness-of-fit test because it is a function of the sufficient statistic, and when every
observation has its own covariate pattern, D is completely independent of the observations. He instead recommends the use of X2as a goodness-of-fit test for binary data when data are sparse, and presents a standardized Pearson statistic for goodness-of-fit that is conditional on the sufficient statistic of the unknown parameters. He derives the first three unconditional and
conditional moments of X2, which are necessary for the calculation of his generalized statistic. McCullagh shows, using the first order correction term to X2 , that X2 and the sufficient statistic are independent (i.e. they are orthogonal), thus accommodating the estimation of parameters referencing the observed data rather than determining them in advance. A second- order correction is applied using Edgeworth expansion to obtain improved approximations for
the distributions of X2. Alternately, Farrington (1996) suggests a comparison of McCullagh’s statistic to a N
0,1 .Osius and Rojek (1992) applied the work of Cressie, et al. (1984), Read, et al. (1988),
increasing. They derived a statistic similar to McCullagh’s statistic, which when applied to binary data is a score test for the fit of the hypothesized model against a particular enlarged model alternative. Their derivation is based on the calculation of the first two moments of the Cressie-Read power-divergence statistic, and which, under certain conditions, has an asymptotic standard normal distribution.
Pulkstenis and Robinson (2002) suggested two alternative goodness-of-fit statistics that are also intended to overcome the problems created when groups contain subjects with a wide range of
values of the covariate. Their two statistics are similar to X2 and D, but can be applied to logistic regression models containing both categorical and continuous covariates. They use the Hosmer-Lemeshow strategy of grouping data, but also cross-classify categorical variables to allow the structure of the individual covariate patterns to remain intact. First observations are sorted by unique covariate patterns based only on the categorical covariates. Next, within each of the first level groups the observations are sorted by fitted probabilities. Finally, the groups are split again into two, with division at on the median categorical response. If the median response is an actual value, then it is placed in the lower group. Under the null hypothesis, the statistic is reported to have an approximate asymptotic chi-squared distribution of two times the number of unique covariate patterns (based only on the categorical covariates), minus the number of categorical variables in the model, minus two. Pulkstenis, et al. (2002) found that the power of their statistic was greater than that of HL, with regard to its ability to detect the omission of an intercept term from the true model. In addition, their grouping method allows for an analysis of observed and expected cell counts based on the covariate classification, which can aid in identifying poor fit within the covariate space. However, Pulkstenis and Robinson point out that their model is applicable only in certain situations - when a model contains both continuous and categorical covariates, and there are not too many cross-classifications among the categorical covariates in the model. They suggest applying their test in conjunction with HL.