Multinomial Sampling Research Articles

Power and sample size calculation for 2×2 tables under multinomial sampling with random loss

Multinomial sampling, in which the total number of sampled subjects is fixed, is probably one of the most commonly used sampling schemes in categorical data analysis. When we apply multinomial sampling to collect subjects who are subject to a random exclusion from our data analysis, the number of subjects falling into each comparison group is random and can be small with a positive probability. Thus, the application of the traditional statistics derived from large sample theory for testing equality between two independent proportions can sometimes be theoretically invalid. On the other hand, using Fisher's exact test can always assure that the true type I error is less than or equal to a nominal \alpha-level. Thus, we discuss here power and sample size calculation based on this exact test. For a desired power at a given \alpha-level, we develop an exact sample size calculation procedure, that accounts for a random loss of sampled subjects, for testing equality between two independent proportions under multinomial sampling. Because the exact sample size calculation procedure requires intensive computations when the underlying required sample size is large, we also present an approximate sample size formula using large sample theory. On the basis of Monte Carlo simulation, we note that the power of using this approximate sample size formula generally agrees well with the desired power on the basis of the exact test. Finally, we propose a trial-and-error procedure using the approximate sample size as an initial estimate and Monte Carlo simulation to expedite the procedure for searching the minimum required sample size.

Adaptive Tests for Ordered Categorical Data

Consider testing for independence against stochastic order in an ordered 2xJ contingency table, under product multinomial sampling. In applications one may wish to exploit prior information concerning the direction of the treatment effect, yet ultimately end up with a testing procedure with good frequentist properties. As such, a reasonable objective may be to simultaneously maximize power at a specified alternative and ensure reasonable power for all other alternatives of interest. For this objective, none of the available testing approaches are completely satisfactory. A new class of admissible adaptive tests is derived. Each test in this class strictly preserves the Type I error rate and strikes a balance between good global power and nearly optimal (envelope) power to detect a specific alternative of most interest. Prior knowledge of the direction of the treatment effect, the level of confidence in this prior information, and possibly the marginal totals might be used to select a specific test from this class.

Unconditional small-sample confidence intervals for the odds ratio.

The traditional approach to 'exact' small-sample interval estimation of the odds ratio for binomial, Poisson, or multinomial samples uses the conditional distribution to eliminate nuisance parameters. This approach can be very conservative. For two independent binomial samples, we study an unconditional approach with overall confidence level guaranteed to equal at least the nominal level. With small samples this interval tends to be shorter and have coverage probabilities nearer the nominal level.

A Gamma/Dirichlet model for estimating uncertainty in age-specific abundance of Norwegian spring-spawning herring

An approach to estimating the statistical uncertainty in age-specific abundance of Norwegian spring-spawning herring is presented. The method is applied to data from a survey on the overwintering stock in the Vestfjord system of northern Norway in December 1996. It is based on building separate statistical models for acoustic and biological data. The acoustic data are modelled as a gamma-transformed spatial random field and fitted to echo readings from the survey. The biological data are fitted to a two-stage model using multinomial sampling from Dirichlet-distributed age proportions. Uncertainty in the abundance by age distribution is obtained by bootstrapping. Previous assessments have a priori taken the abundance by age from a skewed distribution, such as gamma or log-normal families. In contrast, this analysis results in an estimated abundance by age that seems symmetric and does not display heavy tails.

Exact credal treatment of missing data

This paper proposes an exact, no-assumptions approach to dealing with incomplete sets of multivariate categorical data. An incomplete data set is regarded as a finite collection of complete data sets, and a joint distribution is obtained from each of them, at a descriptive level. The tools to simultaneously treat all the possible joint distributions compatible with an incomplete set of data are given. In particular, a linear description of the set of distributions is formulated, and it is shown that the computation of bounds on the expectation of real-valued functions under such distributions is both possible and efficient, by means of linear programming. Specific algorithms are also developed whose complexity grows linearly in the number of observations. An analysis is then carried out to estimate population probabilities from incomplete multinomial samples. The descriptive tool extends in a straightforward way to the inferential problem by exploiting Walley's imprecise Dirichlet model.

Model checking in loglinear models using φ-divergences and MLEs

Consider the loglinear model for categorical data under the assumption of either Poisson, multinomial, or product-multinomial sampling. We are interested in testing between various hypotheses on the parameter space. In this paper, the usual likelihood ratio test, with maximum likelihood estimators for the unspecified parameters, is generalized to tests based on φ-divergences, still using maximum likelihood estimators. These tests yield the likelihood ratio test as a special case. Asymptotic distributions for the new test statistics are derived under both the null and the alternative hypotheses. Then it is shown how the φ-divergences can be used to test nested hypotheses, yielding a type of “analysis of divergence”.

Sampling Schemes and the Selection of Log-Linear Models for Longitudinal Data

Because the selection of a sampling scheme can have strong effects on the admissibility of log-linear models, the authors discuss these effects for the multinomial and the product multinomial sampling schemes for prospective and retrospective sampling.In multi-nomial sampling, marginal frequencies are not fixed a priori.In product multinomial sampling, uni-or multidimensional marginal frequencies are considered fixed.Therefore, models must fit these marginal frequencies exactly.A data example illustrates model selection using data from a project of family development.

An Analysis of Demand for Living Space by Housing Tenure Type in Korea

This study, using a multinomial logit sample selection model, estimates the determinants of tenure choice and demand for living space with a sample of 2,615 urban households in Korea. Housing tenure in Korea seems to be hierarchically structured by household income. Two-stage OLS estimation results of the living space demand equations suggest that the elasticities of both income and price are inelastic. However, the absolute magnitude of price elasticity is greater than that of income elasticity. This implies that the problem of insufficient living space of the lower income group will not be improved if the increasing rate of housing price is greater than that of income. The accessibility to formal mortgage markets is another important determinant of demand for living space. Thus enhancing accessibility to mortgage markets could be one of the policy devices to improve the problem of limited living space of renters effectively.

Bayesian analysis of mutational spectra.

Studies that examine both the frequency of gene mutation and the pattern or spectrum of mutational changes can be used to identify chemical mutagens and to explore the molecular mechanisms of mutagenesis. In this article, we propose a Bayesian hierarchical modeling approach for the analysis of mutational spectra. We assume that the total number of independent mutations and the numbers of mutations falling into different response categories, defined by location within a gene and/or type of alteration, follow binomial and multinomial sampling distributions, respectively. We use prior distributions to summarize past information about the overall mutation frequency and the probabilities corresponding to the different mutational categories. These priors can be chosen on the basis of data from previous studies using an approach that accounts for heterogeneity among studies. Inferences about the overall mutation frequency, the proportions of mutations in each response category, and the category-specific mutation frequencies can be based on posterior distributions, which incorporate past and current data on the mutant frequency and on DNA sequence alterations. Methods are described for comparing groups and for assessing dose-related trends. We illustrate our approach using data from the literature.

Asymptotic Properties of Weighted M-estimators for variable probability samples

I provide a systematic treatment of the asymptotic properties of weighted M-estimators under variable probability stratified sampling. The characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis. Simple, consistent asymptotic variance matrix estimators are proposed for a large class of problems. When stratification is based on exogenous variables, I show that the unweighted M-estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality. When population frequencies are known, a more efficient weighting is possible. I also show how the results carry over to multinomial sampling.

Miscellanea. An extension of Morel-Nagaraj's finite mixture distribution for modelling multinomial clustered data

Morel & Nagaraj (1993) proposed a finite mixture distribution for modelling multinomial extra variation when the extra variation is believed to be caused by clumped multinomial sampling with one clumping. We propose an extension of the model to incorporate possible multiple clumping. We show that the extended finite mixture distribution is a multinomial mixing distribution with different mixing coefficients. We derive maximum likelihood estimating equations and an asymptotic approximation to the Fisher information matrix for the parameters of the model. A real-life dataset is analysed.

Updating exact P- values for the conditional likelihood ratio test of independence

We consider the exact likelihood ratio test of independence conditioned on row and column margins in an r × c contingency table with multinomial sampling. We develop an update algorithm to compute the exact Pvalue of the test and show it is better than the network algorithm in terms of computing speed. In the algorithm the P-value is reduced to a sum of probabilities for 2 × 2 contingency tables, which we compute using the hyper-geometric distribution. The same algorithm can also be used for testing homogeneity of independent multinomial populations.

Fitting Markov Chain Models to Discrete State Series such as DNA Sequences

SUMMARY Discrete state series such as DNA sequences can often be modelled by Markov chains. The analysis of such series is discussed in the context of log-linear models. The data produce contingency tables with similar margins due to the dependence of the observations. However, despite the unusual structure of the tables, the analysis is equivalent to that for data from multinomial sampling. The reason why the standard number of degrees of freedom is correct is explained by using theoretical arguments and the asymptotic distribution of the deviance is verified empirically. Problems involved with fitting high order Markov chain models, such as reduced power and computational expense, are also discussed.

Test for Equality of Intraclass Correlation Coefficients under Unequal Family Sizes

The likelihood ratio test, the large-sample z-test, and the large-sample z-test have been proposed to test for the equality of two intraclass correlation coefficients under unequal family sizes based on two independent multinomial samples. It has been found on the basis of simulation studies that the likelihood ratio test consistently and reliably produces results superior to those of the large-sample z-test and large-sample z-test in terms of power for various combinations of intraclass correlation coefficient values.

A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context

Abstract For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an unrestricted alternative. We show that in all cases the test statistics are asymptotically distributed as a mixture of chi-squared distributions, with weights determined by the information matrix. By exploiting the special structure of this matrix in these three cases, we find tight upper and lower bounds to the distribution of all test statistics. These bounding distributions are free of nuisance parameters and relatively easy to compute. Two examples are presented to illustrate the methodology and the required computations needed to apply these techniques.

A Unified Approach to Likelihood Inference on Stochastic Orderings in a Nonparametric Context

For data in a two-way contingency table with ordered margins, we consider various hypotheses of stochastic orders among the conditional distributions considered by rows and show that each is equivalent to requiring that an invertible transformation of the vectors of conditional row probabilities satisfies an appropriate set of linear inequalities. This leads to the construction of a general algorithm for maximum likelihood estimation under multinomial sampling and provides a simple framework for deriving the asymptotic distribution of log-likelihood ratio tests. The usual stochastic ordering and the so called uniform and likelihood ratio orderings are considered as special cases. In particular, for each of these three orderings we determine the transformation required to apply the estimation algorithm; we then consider testing the hypothesis that the rows are identically distributed against the alternative that they are stochastically ordered, as well as testing each stochastic order against an unrestricted alternative. We show that in all cases the test statistics are asymptotically distributed as a mixture of chi-squared distributions, with weights determined by the information matrix. By exploiting the special structure of this matrix in these three cases, we find tight upper and lower bounds to the distribution of all test statistics. These bounding distributions are free of nuisance parameters and relatively easy to compute. Two examples are presented to illustrate the methodology and the required computations needed to apply these techniques.

Prior information in optimal allocation for multinomial population

This paper investigates a criterion to determine the less informative prior between two Dirichlet priors for multinomial samples based on optimal allocation. The uniform and Jeffreys' priors are shown to be more informative than other Dirichlet priors under certain conditions. The least informative Dirichlet priors among two subclasses and the whole class of Dirichlet priors are found.

Modelling population heterogeneity in sensitivity and specificity of a multi-stage screen for obstructive sleep apnoea.

We propose a model-based approach for modelling population heterogeneity in terms of sensitivity and specificity of multi-stage screening procedures that consists of multiple tests ordered according to criteria such as cost and invasiveness. It is assumed that a patient proceeds to the next test only if they test positive for the current test. An overall positive result occurs if a patient tests positive for all tests. A dropout occurs when a subject tests positive for a test but does not proceed to subsequent tests. Chinchilli proposed estimates of sensitivity and specificity based upon ratios of multinomial sample probabilities for such a multi-stage procedure in a homogeneous population. The method proposed here accommodates population heterogeneity with generalized linear models and transformation-based confidence intervals. In contrast to the approach of Chinchilli, such an approach provides model-based tests of population differences, narrower confidence intervals that satisfy boundary constraints, and a method for accommodating dropouts without the need for prespecified weights. The proposed method is motivated by the need to assess age differences in the accuracy of a multi-stage test for obstructive sleep apnoea.

Log-Linear Modeling with the Negative Multinomial Distribution

We develop a negative multinomial sampling plan in which observed cell counts are positively correlated. We show that maximum likelihood estimates of cell means are the same as those found under independent Poisson sampling. There is no maximum likelihood estimate for the shape parameter in general. We propose an estimate of the shape parameter based on the mean and quantiles of Pearson's chi-squared statistic. These techniques are applied to models of cancer incidence for three cities in Ohio and longitudinal health care utilization by a group of senior citizens.

Multinomial-sampling models for random genetic drift.

Three different derivations of models with multinomial sampling of genotypes in a finite population are presented. The three derivations correspond to the operation of random drift through population regulation, conditioning on the total number of progeny, and culling, respectively. Generations are discrete and nonoverlapping; the diploid population mates at random. Each derivation applies to a single multiallelic locus in a monoecious or dioecious population; in the latter case, the locus may be autosomal or X-linked. Mutation and viability selection are arbitrary; there are no fertility differences. In a monoecious population, the model yields the Wright-Fisher model (i.e., multinomial sampling of genes) if and only if the viabilities are multiplicative. In a dioecious population, the analogous reduction does not occur even for pure random drift. Thus, multinomial sampling of genotypes generally does not lead to multinomial sampling of genes. Although the Wright-Fisher model probably lacks a sound biological basis and may be inaccurate for small populations, it is usually (perhaps always) a good approximation for genotypic multinomial sampling in large populations.