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Journal of Applied Mathematics and Decision Sciences
VolumeΒ 2009Β (2009), Article IDΒ 980706, 9 pages
http://dx.doi.org/10.1155/2009/980706
Research Article

Components of Pearson's Statistic for at Least Partially Ordered π‘š-Way Contingency Tables

School of Mathematical and Physical Sciences, University of Newcastle, NSW 2308, Australia

Received 1 May 2009; Accepted 22 October 2009

Academic Editor: DavidΒ Bulger

Copyright Β© 2009 J. C. W. Rayner and Eric J. Beh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For at least partially ordered three-way tables, it is well known how to arithmetically decompose Pearson's 𝑋2𝑃 statistic into informative components that enable a close scrutiny of the data. Similarly well-known are smooth models for two-way tables from which score tests for homogeneity and independence can be derived. From these models, both the components of Pearson's 𝑋2𝑃 and information about their distributions can be derived. Two advantages of specifying models are first that the score tests have weak optimality properties and second that identifying the appropriate model from within a class of possible models gives insights about the data. Here, smooth models for higher-order tables are given explicitly, as are the partitions of Pearson's 𝑋2𝑃 into components. The asymptotic distributions of statistics related to the components are also addressed.

1. Introduction

In [1, 2] it is shown how, for at least partially ordered three-way tables, to arithmetically decompose Pearson’s 𝑋2𝑃 statistic into informative components that enable a close scrutiny of the data. They focus on three-way tables as being indicative of higher-order tables. Here, we give models for arbitrary multiway tables that are at least partially ordered. We discuss the arithmetic decomposition of 𝑋2𝑃 into components, giving explicit formulae for these components. This enables 𝑋2𝑃 to be partitioned into meaningful 𝑋2-type statistics. Using extensions of models for two-way tables discussed in [3], the asymptotic distribution of statistics related to these components may be given.

At the onset, we should say what we mean by β€œordered.’’ A random variable is a mapping from the sample space to the real line. It is ordered if and only if the ordering of the range is meaningful. So, for example, a range containing only zero and one, denoting male and female, would not usually be considered meaningful. However, it would usually be considered meaningful if the zero and one denoted low and high, respectively. A variable is ordered if and only if it reflects a random variable that is ordered rather than not ordered, or nominal. A table is completely ordered if and only if all variables are ordered. It is partially ordered if and only if at least one but not all variables are ordered.

To give precedence, we observe here that the arithmetic decomposition of Pearson’s 𝑋2𝑃 statistic for two- and three-way tables can be shown quite compactly using results from [3, Chapter 4, Theorems  2.1 and  2.2, pages 90-91 and Theorem  5.2, page 101]. It is shown there that for contingency tables Lancaster’s πœ™2 is equal to the sum of the squares of the elements of a vector πœƒ in the subsequent models. In a parallel manner, working with observed proportions {𝑁𝑖𝑗/𝑛}, it can be shown that 𝑋2𝑃 is equal to the sum of the squares of components. This observation applies to verifying the results in [1, 2]. Moreover, precedence should be given to the work in [3, Chapter 12] for material throughout this paper.

We also note that the work in [4] considered models for ordered two-way contingency tables. In [4, Chapter 3], an extended hypergeometric model is used when both row and column marginal totals are known. This is not a smooth model and here we will not discuss either it or its extensions further. In [4, Chapter 8], doubly ordered models are considered, and these will be generalised in Section 2 in what follows.

In treating a singly ordered table, the work in [4, Chapter 4] assumed the total count for each treatment is known before sighting the data, and this leads to a smooth product multinomial model. If the treatment totals are not known before sighting the data, the resulting model is a single multinomial with cell probabilities modelled the same way as when the treatment totals are known before sighting the data. The models in [1, 2] are single multinomials, following the second approach. However, it is clear that, in general, for partially ordered tables, there are a multitude of possible models, depending on which marginal totals are assumed known before sighting the data. In all cases, the logarithms of the likelihoods are, apart from unimportant constants, the same. Henceforth, we will consistently work with product multinomials and note that the distributional results developed apply to the multitude of models indicated.

The outline of this paper is as follows. In Section 2, the more routine case of completely ordered multiway tables is discussed. The balance of the paper is about the more complicated partially ordered tables. In Section 3, the work of [4] on partially ordered tables is reviewed. In Section 4, the work in [1, 2] is reviewed and extended using smooth models. Section 5 gives the generalizations to arbitrary multiway partially ordered tables.

2. Completely Ordered Multiway Tables

For an m-way 𝐼1×𝐼2Γ—β‹―Γ—πΌπ‘š, completely ordered table of counts {𝑁𝑣1β‹―π‘£π‘š}, Pearson’s 𝑋2𝑃 is given by

𝑋2𝑃=𝐼1𝑣1=1β‹―ξ“πΌπ‘šπ‘£π‘š=1𝑁𝑣1β‹―π‘£π‘šβˆ’πΈ[𝑁𝑣1β‹―π‘£π‘š]ξ€Έ2𝐸𝑁𝑣1...π‘£π‘šξ€».(2.1) An extension of the approach in [1] demonstrates that 𝑋2𝑃 has an arithmetic decomposition:

𝑋2𝑃=𝐼1π‘’βˆ’11=0β‹―ξ“πΌπ‘šπ‘’βˆ’1π‘š=0𝑍2𝑒1β‹―π‘’π‘š,(2.2) in which the components 𝑍𝑒1β‹―π‘’π‘š, 𝑒1=0,…,𝐼1βˆ’1,…, π‘’π‘š=0,…,πΌπ‘šβˆ’1, are given by

𝑍𝑒1β‹―π‘’π‘š=βˆšπ‘›ξ“πΌ1𝑣1=1β‹―ξ“πΌπ‘šπ‘£π‘š=1π‘Žπ‘’1𝑣1ξ€Έβ‹―π‘Žπ‘’π‘šξ€·π‘£π‘šξ€Έπ‘π‘£1β‹―π‘£π‘š.(2.3) Here βˆ‘π‘›=𝐼1𝜈1=1β‹―βˆ‘πΌπ‘šπœˆπ‘š=1π‘πœˆ1β‹―πœˆπ‘š and for 𝑗=1,…,π‘š, {π‘Žπ‘’π‘—(β€’)} is orthonormal on {𝑝‒⋯‒𝑣𝑗‒⋯‒}, in which 𝑝𝑣1β‹―π‘£π‘š = 𝑁𝑣1β‹―π‘£π‘š/𝑛 and 𝑝‒⋯‒𝑣𝑗‒⋯‒ is obtained from 𝑝𝑣1β‹―π‘£π‘š by summing out all variables other than 𝑣𝑗. Furthermore, the orthonormal systems all have zeroth term identically one. This work builds on the iconic work of Oliver Lancaster, for which see [3], and [3, Chapter 12] in particular.

It is routine to show that the components 𝑍𝑒1β‹―π‘’π‘šare asymptotically multivariate normal, since an arbitrary linear combination of these variables is asymptotically normal by the central limit theorem. Utilizing the orthonormality of the {π‘Žπ‘’π‘—(β€’)}, it can be shown that all components have expectation zero, variance unity, and covariances zero. They are thus asymptotically mutually independent and asymptotically standard normal.

One possible smooth model for {𝑁𝑣1β‹―π‘£π‘š} is the multinomial with count total n and cell probabilities {π‘πœˆ1β‹―πœˆπ‘š} given by

π‘πœˆ1β‹―πœˆπ‘š=𝐼1π‘’βˆ’11=0β‹―ξ“πΌπ‘šπ‘’βˆ’1π‘š=0πœƒπ‘’1β‹―π‘’π‘šπ‘Žπ‘’1𝑣1ξ€Έβ‹―π‘Žπ‘’π‘šξ€·π‘£π‘šξ€Έξ‚‡π‘πœˆ1β€’β‹―β€’β‹―π‘β€’β‹―β€’πœˆπ‘š,(2.4) in which πœƒ0β‹―0=1 and πœƒ0β‹―0𝑒𝑗0β‹―0=0 for all 𝑒𝑗β‰₯1. This model includes all genuine two, three, and so forth m-way independence models. A routine extension of [4, Theorem  8.1] shows that the score test statistic for testing, that the πœƒπ‘’1β‹―π‘’π‘šare collectively zero against the negation of this is, as before, the sum of the squares of the 𝑍𝑒1β‹―π‘’π‘š. Moreover, these components have the distributional properties given in the previous paragraph. Generalising [4, Theorem  8.2], this score test statistic is 𝑋2𝑃. The score test has the advantage of weak optimality: see, for example, [5]. An additional advantage of this approach is that it can be shown that 𝑍2𝑒1β‹―π‘’π‘š is the score test statistic when testing πœƒπ‘’1β‹―π‘’π‘š=0 against πœƒπ‘’1β‹―π‘’π‘šβ‰ 0 in an appropriate model. Thus, in an informal sense, every 𝑍𝑒1β‹―π‘’π‘š is a detector of the corresponding πœƒπ‘’1β‹―π‘’π‘š.

The degrees of freedom associated with 𝑋2𝑃 are the number of πœƒπ‘’1β‹―π‘’π‘š (and hence 𝑍𝑒1β‹―π‘’π‘š) in the model, excluding those that are by convention always zero or one. The degrees of freedom are thus

𝑖<π‘—ξ€·πΌπ‘–πΌβˆ’1𝑗+ξ‘βˆ’1𝑖<𝑗<π‘˜ξ€·πΌπ‘–πΌβˆ’1ξ€Έξ€·π‘—πΌβˆ’1ξ€Έξ€·π‘˜ξ€Έξ€·πΌβˆ’1β‹―+1πΌβˆ’1ξ€Έξ€·2ξ€Έβ‹―ξ€·πΌβˆ’1π‘šξ€Έβˆ’1=𝐼1×𝐼2Γ—β‹―Γ—πΌπ‘šξ€·πΌβˆ’1βˆ’1ξ€Έβˆ’ξ€·πΌβˆ’12ξ€Έξ€·πΌβˆ’1βˆ’β‹―βˆ’π‘šξ€Έ.βˆ’1(2.5) The left-hand side consists of the degrees of freedom associated with all genuine two-way, three-way, and so forth π‘š-way models, while the right-hand side is the number of cells minus one for the constraint βˆ‘π‘›=𝐼1𝜈1=1β‹―βˆ‘πΌπ‘šπœˆπ‘š=1π‘πœˆ1β‹―πœˆπ‘š (reflecting that the sample size is known before sighting the data) minus the degrees of freedom associated with all one-way (essentially goodness of fit) models. For the happiness example in [1] 𝐼1=3, 𝐼2=4, 𝐼3=5 and substituting in the aforementioned formulae, there are 50 degrees of freedom.

3. Two-Way Singly Ordered Tables

  In [4, Section  4.4] two-way tables are discussed. We report on that discussion using our subsequent convention that ordered categories precede unordered categories. Tables {𝑁𝑀𝑧} are modelled by product multinomials, with the zth column being multinomial with total counts 𝑛‒𝑧 and cell probabilities:

𝑝𝑀𝑧=1+𝐼1𝑒=1πœƒπ‘’π‘§π‘Žπ‘’βˆš(𝑀)/𝑛‒𝑧𝑝𝑀‒,(3.1) for 𝑀=1,…,𝐼1βˆ’1. Note that the probabilities in the 𝐼1th row are found by difference: 𝑝𝐼1𝑧=1βˆ’π‘1π‘§βˆ’β‹―βˆ’π‘(𝐼1βˆ’1)𝑧 and 𝑧=1,…,𝐼2, where 𝑝𝑀‒=βˆ‘π‘§π‘π‘€π‘§/𝑛 in which βˆ‘π‘›=π‘€βˆ‘π‘§π‘π‘€π‘§. The efficient score contains random variables 𝑍𝑒𝑧=√(𝑛/𝑛‒𝑧)βˆ‘πΌ1𝑀=1π‘Žπ‘’(𝑀)𝑝𝑀𝑧 and the information matrix is found to be singular. In order to find a score test statistic in [4, Section  4.4], the model is modified by removing the πœƒs corresponding to the last column because the model is overparameterised: in any row, given the probabilities in the first 𝐼2βˆ’1 columns and the marginal probability for that row (the average of all probabilities in that row), the probability corresponding to the final column can be readily determined. A quicker approach is now outlined.

Write 𝑍𝑒=(𝑍𝑒1,…,𝑍𝑒𝐼2)T and 𝑍𝑇=(𝑍𝑇1,…,𝑍𝑇𝐼1βˆ’1). The 𝑛×𝑛 identity matrix is written as 𝐼𝑛; this will be clear from the context when this, and not the number of rows, and so forth, is intended. From the information matrix for 𝑍, the covariance matrix for 𝑍𝑒 is 𝐼𝐼2βˆšβˆ’(π‘›β€’π‘Žπ‘›β€’π‘)/𝑛. This is idempotent of rank𝐼2βˆ’1. There exists an 𝐼2Γ—(𝐼2βˆ’1) matrix A such that 𝐼𝐼2βˆšβˆ’(π‘›β€’π‘Žπ‘›β€’π‘)/𝑛=𝐴𝐴𝑇 and 𝐴𝑇𝐴=𝐼𝐼2βˆ’1. We now focus on a smooth model containing just one value of u (the full model is similar). Since the information matrix in terms of πœƒπ‘’=(πœƒπ‘’1,…,πœƒπ‘’πΌ2)𝑇=πœƒ say is singular, define πœ™ by π΄πœ™=πœƒ. Then using the results of the lemma in [6, Section  3], the efficient score and information in terms of πœƒ (π‘ˆπœƒ and πΌπœƒ) and πœ™ (π‘ˆπœ™ and πΌπœ™) are related by π‘ˆπœ™=π΄π‘‡π‘ˆπœƒ and πΌπœ™=π΄π‘‡πΌπœƒπ΄, respectively. It follows that since, in terms of πœ™, the efficient score is 𝐴𝑇𝑍𝑒=π‘Œπ‘’ say, and the information matrix is 𝐴𝑇{𝐼𝐼2βˆšβˆ’(π‘›β€’π‘Žπ‘›β€’π‘)/𝑛}𝐴=𝐼𝐼2βˆ’1, the score test statistic in terms of πœ™ is π‘Œπ‘‡π‘’π‘Œπ‘’=𝑍𝑇𝑒{𝐼𝐼2βˆšβˆ’(π‘›β€’π‘Žπ‘›β€’π‘)/𝑛}𝑍𝑒. Since π‘Œπ‘’ is asymptotically N𝐼2βˆ’1(0,𝐼𝐼2βˆ’1), the score test statistic has the πœ’2𝐼2βˆ’1 distribution, as is otherwise well known.

The columns of A are eigenvectors corresponding to the nonzero eigenvalues of {𝐼𝐼2βˆšβˆ’(π‘›β€’π‘Žπ‘›β€’π‘)/𝑛}. The eigenvector corresponding to the zero eigenvalue is (1,…,1)𝑇, so a typical eigenvector may be written 1βŸ‚. The elements of π‘Œπ‘’=𝐴𝑇𝑍𝑒 are of the form 1π‘‡βŸ‚π‘π‘’, that may fairly be called a contrast between the elements of 𝑍𝑒. They are mutually independent and standard normal. While the 𝑍𝑒𝑖 are immediately interpretable, they are slightly less convenient than π‘Œπ‘’π‘– that are orthogonal contrasts and are asymptotically mutually independent and asymptotically standard normal. These contrasts correspond to each order 𝑒, 𝑒=1,…,𝐼1βˆ’1, and reflect comparisons between the levels of the unordered factor. They may, for example, compare the means of the first two levels, the mean of the first two levels with that of the third level, the mean of the first three levels with that of the fourth level, and so on. Such contrasts may be described as Helmertian, from the Helmert matrix. In its simplest form, the Helmert matrix is an orthogonal (𝑛+1)Γ—(𝑛+1) matrix with all the elements of the first row 1/p(𝑛+1) and π‘Ÿth row 1/p[π‘Ÿ(π‘Ÿ+1)] (r times), βˆ’π‘Ÿ/p[π‘Ÿ(π‘Ÿ+1)], then all zeros.

4. Three-Way Partially Ordered Tables

4.1. Singly Ordered Three-Way Tables

For singly ordered 𝐼1×𝐼2×𝐼3 tables, a product multinomial model is assumed, with the counts corresponding to the 𝑧1th column and 𝑧2th layer, 𝑧1=1,…,𝐼2 and 𝑧2=1,…,𝐼3, being multinomial with total counts 𝑛‒𝑧1𝑧2 and cell probabilities:

𝑝𝑀𝑧1𝑧2=𝑝𝑀‒‒𝐼1βˆ’1𝑒=0πœƒπ‘’π‘§1𝑧2π‘Žπ‘’(𝑀),(4.1) for 𝑀=1,…,𝐼1, in which πœƒ0𝑧1𝑧2=1. Here and henceforth, the normalisation corresponding to the βˆšπ‘›β€’π‘§ factor in 𝑝𝑀𝑧 in Section 3 is absorbed into the πœƒπ‘’π‘§1𝑧2. The components are random variables:

𝑍𝑒𝑧1𝑧2=𝑛𝑝‒𝑧1‒𝑝‒‒𝑧2𝐼1𝑀=1π‘Žπ‘’(𝑀)𝑝𝑀𝑧1𝑧2,(4.2) where 𝑝‒𝑧1β€’=βˆ‘π‘€βˆ‘π‘§2𝑁𝑀𝑧1𝑧2/𝑛 and 𝑝‒‒𝑧2=βˆ‘π‘€βˆ‘π‘§1𝑁𝑀𝑧1𝑧2/𝑛. The 𝑍𝑒𝑧1𝑧2 are immediately interpretable [2]), and, by the multivariate central limit theorem, are asymptotically multivariate normal. This does not depend on the smooth model. As in Section 3, for each u, 𝑒=1,…,𝐼1βˆ’1, we may construct orthogonal contrasts that are asymptotically mutually independent and asymptotically standard normal. These contrasts reflect 𝑒th moment comparisons between the levels of the unordered factors.

In [2], without a model, it is shown that 𝑋2𝑃 is the sum of the squares of the 𝑍𝑒𝑧1𝑧2:

𝑋2𝑃=𝐼1βˆ’1𝑒=0𝐼2𝑧1=1𝐼3𝑧2=1𝑍2𝑒𝑧1𝑧2.(4.3) In 𝑋2𝑃, it is insightful to separate components corresponding to 𝑒=0 and 𝑒≠0. Thus

𝑋2𝑃=𝐼2𝑧1=1𝐼3𝑧2=1𝑍20𝑧1𝑧2+𝐼1βˆ’1𝑒=1𝐼2𝑧1=1𝐼3𝑧2=1𝑍2𝑒𝑧1𝑧2.(4.4) The first summand corresponds to a two-way completely unordered table obtained by summing over rows and may reasonably be denoted by 𝑋2𝑍1𝑍2. The second summation corresponds to a genuinely three-way singly ordered table and may reasonably be denoted by 𝑋2π‘ˆπ‘1𝑍2.

In [2] it is stated that the degrees of freedom associated with 𝑋2𝑃 are 𝐼1𝐼2𝐼3βˆ’πΌ1βˆ’πΌ2βˆ’πΌ3+2. This follows because there are (𝐼2βˆ’1)(𝐼3βˆ’1) degrees of freedom associated with 𝑋2𝑍1𝑍2, and (𝐼1βˆ’1)(𝐼2𝐼3βˆ’1) degrees of freedom associated with 𝑋2π‘ˆπ‘1𝑍2.

We can argue for these degrees of freedom by, when possible, counting the πœƒπ‘’π‘§1𝑧2 or the 𝑍𝑒𝑧1𝑧2. The table corresponding to 𝑋2𝑍1𝑍2 is completely unordered, so there are no πœƒπ‘’π‘§1𝑧2 to count. We propose no smooth model, and our components are not appropriate when there is no order. However, the degrees of freedom are known independently to be (𝐼2βˆ’1)(𝐼3βˆ’1). The table corresponding to 𝑋2π‘ˆπ‘1𝑍2 has degrees of freedom (𝐼1βˆ’1)(𝐼2𝐼3βˆ’1) since this is the number of parameters πœƒπ‘’π‘§1𝑧2 in the smooth model. There are 𝐼2𝐼3 multinomials, each of which has (𝐼1βˆ’1) parameters πœƒπ‘’π‘§1𝑧2 as the multinomials probabilities sum to one (so the final cell probability is given by difference). In addition, one of the 𝐼2𝐼3 multinomials is determined by {𝑝𝑀‒‒} and the remaining multinomials.

4.2. Doubly Ordered Three-Way Tables

For doubly ordered tables a product multinomial model is again assumed, with the counts corresponding to the 𝑧th layer being multinomial with total counts 𝑛‒‒𝑧 and cell probabilities:

𝑝𝑀1𝑀2𝑧=𝑝𝑀1𝑀2‒𝐼1π‘’βˆ’11=0𝐼2π‘’βˆ’12=0πœƒπ‘’1𝑒2π‘§π‘Žπ‘’1𝑀1ξ€Έπ‘Žπ‘’2𝑀2ξ€Έ,(4.5) for 𝑀1=1,…,𝐼1, 𝑀2=1,…,𝐼2, and 𝑧=1,…,𝐼3, in which πœƒ00𝑧=πœƒπ‘’10𝑧=πœƒ0𝑒2𝑧=1. The components are random variables:

𝑍𝑒1𝑒2𝑧=𝑛𝑝‒‒𝑧𝐼1𝑀1=1𝐼2𝑀2=1π‘Žπ‘’1𝑀1ξ€Έπ‘Žπ‘’2𝑀2𝑝𝑀1𝑀2𝑧(4.6) for 𝑒1=0,…,𝐼1βˆ’1, 𝑒2=0,…,𝐼2βˆ’1, and 𝑧=1,…,𝐼3, where 𝑝‒‒𝑧=βˆ‘π‘€1βˆ‘π‘€2𝑁𝑀1𝑀2z/𝑛. Again, by the multivariate central limit theorem, the 𝑍𝑒1𝑒2𝑧 are asymptotically multivariate normal. This does not depend on the smooth model. For each (𝑒1,𝑒2) pair, as in Section 3, we may construct orthogonal contrasts that are asymptotically mutually independent and asymptotically standard normal. These contrasts reflect bivariate moment comparisons between the levels of the unordered factor. A typical contrast may be (1st,2nd) moment differences between the first two levels reflected by layers.

In [2], without a model, it is shown that 𝑋2𝑃 is the sum of the squares of the 𝑍𝑒1𝑒2𝑧:

𝑋2𝑃=𝐼1π‘’βˆ’11=0𝐼2π‘’βˆ’12=0𝐼3𝑧=1𝑍2𝑒1𝑒2𝑧.(4.7) Again in 𝑋2𝑃, it is insightful to separate components corresponding to 𝑒𝑖=0 and 𝑒𝑖≠0. Thus,

𝑋2𝑃=𝐼3𝑧=1𝑍200𝑧+𝐼1π‘’βˆ’11=1𝐼3𝑧=1𝑍2𝑒10𝑧+𝐼2π‘’βˆ’12=1𝐼3𝑧=1𝑍20𝑒2𝑧+𝐼1π‘’βˆ’11=1𝐼2π‘’βˆ’12=1𝐼3𝑧=1𝑍2𝑒1𝑒2𝑧.(4.8) The first summand is identically zero. The second summand corresponds to a two-way singly ordered table obtained by summing over columns and may reasonably be denoted by 𝑋2π‘ˆ1𝑍. The third summation corresponds to another two-way singly ordered table obtained by summing over rows and may reasonably be denoted by 𝑋2π‘ˆ2𝑍. The final summation corresponds to a genuine three-way doubly ordered table and may reasonably be denoted by 𝑋2π‘ˆ1π‘ˆ2𝑍2.

In [2] it is incorrectly claimed that the associated degrees of freedom are, as in Section 4.1, 𝐼1𝐼2𝐼3βˆ’πΌ1βˆ’πΌ2βˆ’πΌ3+2. The one-way table corresponding to the components with 𝑒1=𝑒2=0 is uninformative, and should be ignored. The two-way tables corresponding to precisely one of the 𝑒1 or 𝑒2 zero are single-ordered, and, as in Section 3, have degrees of freedom (𝐼1βˆ’1)(𝐼3βˆ’1) and (𝐼2βˆ’1)(𝐼3βˆ’1), respectively. When neither 𝑒1 nor 𝑒2 is zero, the corresponding table is a genuine doubly ordered three-way table. There are 𝐼3 multinomials, each with (𝐼1βˆ’1)(𝐼2βˆ’1) parameters πœƒπ‘’1𝑒2𝑧 in their smooth model, but in fact the final of the 𝐼3 multinomials is determined by the {𝑝‒‒𝑧} and the remaining multinomials. So there are (𝐼1βˆ’1)(𝐼2βˆ’1)(𝐼3βˆ’1) degrees of freedom for this final table. In all, the degrees of freedom are

𝐼1πΌβˆ’1ξ€Έξ€·2πΌβˆ’1ξ€Έξ€·3ξ€Έ+ξ€·πΌβˆ’11πΌβˆ’1ξ€Έξ€·2ξ€Έ+ξ€·πΌβˆ’11πΌβˆ’1ξ€Έξ€·3ξ€Έ=ξ€·πΌβˆ’11𝐼2πΌβˆ’1ξ€Έξ€·3ξ€Έ.βˆ’1(4.9)

We note that although the degrees of freedom in the Happiness Example of [2] are in error, the P values and conclusions with the correct degrees of freedom are as given there. We recommend the reader refer to this example, examined from two different perspectives in [1, 2], to see the insight and interpretability the components give to data analysis.

5. m-Way Partially Ordered Tables

We consider now an m-way table that is at least partially ordered: without loss of generality the first r (β‰₯1) categorical variables are taken as ordered and the remaining 𝑠=π‘šβˆ’π‘Ÿ (β‰₯1) categorical variables are nominal. The notation reflects this convention; the subscripts w reflect ordered categories while the subscripts z reflect nominal categories. Accordingly, the table is denoted by {𝑁𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠}. As in [2] and [4, Chapter 4], we define components of the form

𝑍𝑒1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠=ξƒŽπ‘›ξ€½π‘β€’β‹―β€’π‘§1‒⋯‒×⋯×𝑝‒⋯‒𝑧𝑠𝐼1𝑀1=1β‹―ξ“πΌπ‘Ÿπ‘€π‘Ÿ=1π‘Žπ‘’1𝑀1ξ€Έβ‹―π‘Žπ‘’π‘Ÿξ€·π‘€π‘Ÿξ€Έπ‘π‘€1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠,(5.1) where 𝑝𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠= 𝑁𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠/𝑛 and where {π‘Žπ‘’π‘—(β€’)}, 𝑝‒⋯‒𝑧𝑗‒⋯‒ and 𝑝𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠 are defined similarly to the above. Again, by the multivariate central limit theorem, the 𝑍𝑒1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠 are asymptotically multivariate normal. This does not depend on the smooth model.

By manipulations similar to those for the three-way case, it is possible to argue that

𝑋2𝑃=𝐼1π‘’βˆ’11=0β‹―ξ“πΌπ‘Ÿπ‘’βˆ’1π‘Ÿ=0ξ“πΌπ‘Ÿ+1𝑧1=1β‹―ξ“πΌπ‘šπ‘§π‘ =1𝑍2𝑒1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠.(5.2) By separating components corresponding to 𝑒𝑖=0 and 𝑒𝑖≠0,𝑋2𝑃 can be partitioned as follows:

𝑋2𝑃=ξ“πΌπ‘Ÿ+1𝑧1=1β‹―ξ“πΌπ‘šπ‘§π‘ =1𝑍20β‹―0𝑧1⋯𝑧𝑠+ξ“π‘Ÿπ‘–=1ξ“πΌπ‘–π‘’βˆ’1𝑖=1ξ“πΌπ‘Ÿ+1𝑧1=1β‹―ξ“πΌπ‘šπ‘§π‘ =1𝑍20β‹―0𝑒𝑖0β‹―0𝑧1⋯𝑧𝑠+π‘Ÿξ“π‘–,𝑗=1π‘–β‰ π‘—ξ“πΌπ‘–π‘’βˆ’1𝑖=1ξ“πΌπ‘—π‘’βˆ’1𝑗=1ξ“πΌπ‘Ÿ+1𝑧1=1β‹―ξ“πΌπ‘šπ‘§π‘ =1𝑍20β‹―0𝑒𝑖0β‹―0𝑒𝑗0β‹―0𝑧1⋯𝑧𝑠++⋯𝐼1π‘’βˆ’11=1β‹―ξ“πΌπ‘Ÿπ‘’βˆ’1π‘Ÿ=1ξ“πΌπ‘Ÿ+1𝑧1=1β‹―ξ“πΌπ‘šπ‘§π‘ =1𝑍2𝑒1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠.(5.3) If 𝑠=1, the first term corresponds to a noninformative one-way table and contributes zero to the sum. The following term corresponds to all (𝑠+1)-way singly ordered tables obtained by summing over π‘Ÿβˆ’1 ordered marginals and may reasonably be denoted by βˆ‘π‘Ÿπ‘–=1𝑋2π‘ˆπ‘–π‘1⋯𝑍𝑠. The following term corresponds to all (𝑠+2)-way doubly ordered tables obtained by summing over π‘Ÿβˆ’2 ordered marginals and may reasonably be denoted by

π‘Ÿξ“π‘–,𝑗=1𝑖≠𝑗𝑋2π‘ˆπ‘–π‘ˆπ‘—π‘1⋯𝑍𝑠.(5.4) The subsequent terms involve components with successively more ordered marginals and correspond to tables that are of increasing size. The final term corresponds to a genuine m-way r-fold ordered table and may reasonably be denoted by 𝑋2π‘ˆ1β‹―π‘ˆπ‘Ÿπ‘1⋯𝑍𝑠. Thus,

𝑋2𝑃=ξ“π‘Ÿπ‘–=1𝑋2π‘ˆπ‘–π‘1⋯𝑍𝑠+π‘Ÿξ“π‘–,𝑗=1𝑖≠𝑗𝑋2π‘ˆπ‘–π‘ˆπ‘—π‘1⋯𝑍𝑠+β‹―+𝑋2π‘ˆ1β‹―π‘ˆπ‘Ÿπ‘1⋯𝑍𝑠.(5.5)

The smooth model envisaged here is product multinomial where for each (𝑧1,…,𝑧𝑠), the observations follow a multinomial distribution with total counts 𝑛‒⋯‒𝑧1⋯𝑧𝑠 and cell probabilities {𝑝𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠} given by

𝑝𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠=𝑝𝑀1β€’β‹―β€’Γ—β‹―Γ—π‘β€’β‹―β€’π‘€π‘Ÿβ€’β‹―β€’ξ‚†ξ“πΌ1π‘’βˆ’11=0β‹―ξ“πΌπ‘Ÿπ‘’βˆ’1π‘Ÿ=0πœƒπ‘’1β‹―π‘’π‘Ÿπ‘§1β‹―π‘§π‘ π‘Žπ‘’1𝑀1ξ€Έβ‹―π‘Žπ‘’π‘Ÿξ€·π‘€π‘Ÿξ€Έξ‚‡.(5.6)

An extension of the approach in [4, Section  4.4] investigates testing if the πœƒπ‘’1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠are collectively zero against the negation of this. Generalising the work in [4, Section  4.4], the efficient score statistic is 𝑍𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠. The information matrix is block diagonal but each block is singular. Nevertheless, the efficient score is asymptotically normal and appropriate orthogonal contrasts are asymptotically mutually independent and standard normal.

The degrees of freedom may be deduced either by counting πœƒπ‘’1β‹―π‘’π‘Ÿπ‘§1⋯𝑧𝑠 (or the corresponding components), or by the arguments in [2]. Consider a genuine m-way table with the first r categories ordered and the remaining 𝑠=π‘šβˆ’π‘Ÿ categories not ordered. This includes tables corresponding to 𝑋2π‘ˆπ‘–π‘ˆπ‘—π‘1⋯𝑍𝑠 say, resulting from summing out several of the ordered variables. This is now a doubly ordered (𝑠+2)-way table. The degrees of freedom for 𝑋2π‘ˆ1β‹―π‘ˆπ‘Ÿπ‘1⋯𝑍𝑠 are (𝐼1βˆ’1)(𝐼2βˆ’1)β‹―(πΌπ‘Ÿβˆ’1)(πΌπ‘Ÿ+1Γ—πΌπ‘Ÿ+2Γ—β‹―Γ—πΌπ‘šβˆ’1). There are πΌπ‘Ÿ+1Γ—πΌπ‘Ÿ+2Γ—β‹―Γ—πΌπ‘š multinomials (corresponding to 𝑍1=𝑧1,…,𝑍𝑠=𝑧𝑠) each with (𝐼1βˆ’1)(𝐼2βˆ’1)β‹―(πΌπ‘Ÿβˆ’1) degrees of freedom. However, one of these multinomials is determined by the marginals and the other multinomial models.

We decline to write out the contrasts corresponding to the asymptotically mutually independent standard normal variables that are linear combinations of the 𝑍𝑀1β‹―π‘€π‘Ÿπ‘§1⋯𝑧𝑠. The approach is similar to that employed in Section 3.

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