Components of Pearson's Statistic for at Least Partially Ordered 𝑚-Way Contingency Tables

Abstract

For at least partially ordered three-way tables, it is well known how to arithmetically decompose Pearson's 𝑋^2𝑃 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 𝑋^2𝑃 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 𝑋^2𝑃 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 𝑋^2𝑃 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 𝑋^2𝑃 into components, giving explicit formulae for these components. This enables 𝑋^2𝑃 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 “ordered.’’ 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 𝑋^2𝑃 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 𝜙² is equal to the sum of the squares of the elements of a vector 𝜃 in the subsequent models. In a parallel manner, working with observed proportions {𝑁_𝑖𝑗/𝑛}, it can be shown that 𝑋^2𝑃 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×𝐼₂×⋯×𝐼_𝑚, completely ordered table of counts {𝑁_{𝑣1⋯𝑣𝑚}}, Pearson’s 𝑋^2𝑃 is given by

𝑋^2𝑃=^𝐼1_𝑣1=1⋯^𝐼𝑚_𝑣𝑚=1𝑁_{𝑣1⋯𝑣𝑚}−𝐸[𝑁_{𝑣1⋯𝑣𝑚}]²𝐸𝑁_{𝑣1...𝑣𝑚}.(2.1) An extension of the approach in [1] demonstrates that 𝑋^2𝑃 has an arithmetic decomposition:

𝑋^2𝑃=^{𝐼1𝑢−1}₁₌₀⋯^{𝐼𝑚𝑢−1}_𝑚=0𝑍^2𝑢_1⋯𝑢𝑚,(2.2) in which the components 𝑍_{𝑢1⋯𝑢𝑚}, 𝑢₁=0,…,𝐼₁−1,…, 𝑢_𝑚=0,…,𝐼_𝑚−1, are given by

𝑍_{𝑢1⋯𝑢𝑚}=√𝑛^𝐼1_𝑣1=1⋯^𝐼𝑚_𝑣𝑚=1𝑎_𝑢1𝑣₁⋯𝑎_𝑢𝑚𝑣_𝑚𝑝_{𝑣1⋯𝑣𝑚}.(2.3) Here 𝑛=∑^𝐼1_𝜈1=1⋯∑^𝐼𝑚_𝜈𝑚=1𝑁_{𝜈1⋯𝜈𝑚} and for 𝑗=1,…,𝑚, {𝑎_𝑢𝑗(•)} is orthonormal on {𝑝_{•⋯•𝑣𝑗•⋯•}}, in which 𝑝_{𝑣1⋯𝑣𝑚} = 𝑁_{𝑣1⋯𝑣𝑚}/𝑛 and 𝑝_{•⋯•𝑣𝑗•⋯•} is obtained from 𝑝_{𝑣1⋯𝑣𝑚} by summing out all variables other than 𝑣_𝑗. 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 𝑍_{𝑢1⋯𝑢𝑚}are asymptotically multivariate normal, since an arbitrary linear combination of these variables is asymptotically normal by the central limit theorem. Utilizing the orthonormality of the {𝑎_𝑢𝑗(•)}, 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 {𝑁_{𝑣1⋯𝑣𝑚}} is the multinomial with count total n and cell probabilities {𝑝_{𝜈1⋯𝜈𝑚}} given by

𝑝_{𝜈1⋯𝜈𝑚}=^{𝐼1𝑢−1}₁₌₀⋯^{𝐼𝑚𝑢−1}_𝑚=0𝜃_{𝑢1⋯𝑢𝑚}𝑎_𝑢1𝑣₁⋯𝑎_𝑢𝑚𝑣_𝑚𝑝_{𝜈1•⋯•}⋯𝑝_{•⋯•𝜈𝑚},(2.4) in which 𝜃_0⋯0=1 and 𝜃_{0⋯0𝑢𝑗0⋯0}=0 for all 𝑢_𝑗≥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 𝜃_{𝑢1⋯𝑢𝑚}are collectively zero against the negation of this is, as before, the sum of the squares of the 𝑍_{𝑢1⋯𝑢𝑚}. Moreover, these components have the distributional properties given in the previous paragraph. Generalising [4, Theorem  8.2], this score test statistic is 𝑋^2𝑃. 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 𝑍^2𝑢_1⋯𝑢𝑚 is the score test statistic when testing 𝜃_{𝑢1⋯𝑢𝑚}=0 against 𝜃_{𝑢1⋯𝑢𝑚}≠0 in an appropriate model. Thus, in an informal sense, every 𝑍_{𝑢1⋯𝑢𝑚} is a detector of the corresponding 𝜃_{𝑢1⋯𝑢𝑚}.

The degrees of freedom associated with 𝑋^2𝑃 are the number of 𝜃_{𝑢1⋯𝑢𝑚} (and hence 𝑍_{𝑢1⋯𝑢𝑚}) in the model, excluding those that are by convention always zero or one. The degrees of freedom are thus

_𝑖<𝑗𝐼_𝑖=−𝐼₁1𝐼_𝑗−1+_{𝑖<𝑗<𝑘}𝐼_𝑖−1𝐼_𝑗−1𝐼_𝑘−1⋯+𝐼₁−1𝐼₂−1⋯𝐼_𝑚−1×𝐼₂×⋯×𝐼_𝑚−1−𝐼₁−1−𝐼₂−1−⋯−𝐼_𝑚−1.(2.5) The left-hand side consists of the degrees of freedom associated with all genuine two-way, three-way, and so forth 𝑚-way models, while the right-hand side is the number of cells minus one for the constraint 𝑛=∑^𝐼1_𝜈1=1⋯∑^𝐼𝑚_𝜈𝑚=1𝑁_{𝜈1⋯𝜈𝑚} (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 {𝑁_𝑤𝑧} are modelled by product multinomials, with the zth column being multinomial with total counts 𝑛_•𝑧 and cell probabilities:

𝑝_𝑤𝑧=1+^𝐼1_𝑢=1𝜃_𝑢𝑧𝑎_𝑢(𝑤)/√𝑛_•𝑧𝑝^𝑤•,(3.1) for 𝑤=1,…,𝐼₁₋₁. Note that the probabilities in the 𝐼₁th row are found by difference: 𝑝_𝐼1𝑧=1−𝑝_1𝑧−⋯−𝑝_{(𝐼1−1)𝑧} and 𝑧=1,…,𝐼₂, where 𝑝_𝑤•=∑_𝑧𝑁_𝑤𝑧/𝑛 in which 𝑛=∑_𝑤∑_𝑧𝑁_𝑤𝑧. The efficient score contains random variables 𝑍_𝑢𝑧=√(𝑛/𝑛_•𝑧)∑^𝐼1_𝑤=1𝑎_𝑢(𝑤)𝑝_𝑤𝑧 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 𝜃s corresponding to the last column because the model is overparameterised: in any row, given the probabilities in the first 𝐼₂−1 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 𝑍_𝑢=(𝑍_𝑢1,…,𝑍_𝑢𝐼2)^T and 𝑍^𝑇=(𝑍^𝑇1,…,𝑍^𝑇𝐼₁₋₁). The 𝑛×𝑛 identity matrix is written as 𝐼_𝑛; 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 𝑍_𝑢 is 𝐼_𝐼2−(√𝑛_•𝑎𝑛_•𝑏)/𝑛. This is idempotent of rank𝐼₂−1. There exists an 𝐼₂×(𝐼₂−1) matrix A such that 𝐼_𝐼2−(√𝑛_•𝑎𝑛_•𝑏)/𝑛=𝐴𝐴^𝑇 and 𝐴^𝑇𝐴=𝐼_𝐼2−1. 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 𝜃_𝑢=(𝜃_𝑢1,…,𝜃_𝑢𝐼2)^𝑇=𝜃 say is singular, define 𝜙 by 𝐴𝜙=𝜃. Then using the results of the lemma in [6, Section  3], the efficient score and information in terms of 𝜃 (𝑈_𝜃 and 𝐼_𝜃) and 𝜙 (𝑈_𝜙 and 𝐼_𝜙) are related by 𝑈_𝜙=𝐴^𝑇𝑈_𝜃 and 𝐼_𝜙=𝐴^𝑇𝐼_𝜃𝐴, respectively. It follows that since, in terms of 𝜙, the efficient score is 𝐴^𝑇𝑍_𝑢=𝑌_𝑢 say, and the information matrix is 𝐴^𝑇{𝐼_𝐼2−(√𝑛_•𝑎𝑛_•𝑏)/𝑛}𝐴=𝐼_𝐼2−1, the score test statistic in terms of 𝜙 is 𝑌^𝑇𝑢𝑌_𝑢=𝑍^𝑇𝑢{𝐼_𝐼2−(√𝑛_•𝑎𝑛_•𝑏)/𝑛}𝑍_𝑢. Since 𝑌_𝑢 is asymptotically N_𝐼2−1(0,𝐼_𝐼2−1), the score test statistic has the 𝜒^2𝐼₂₋₁ distribution, as is otherwise well known.

The columns of A are eigenvectors corresponding to the nonzero eigenvalues of {𝐼_𝐼2−(√𝑛_•𝑎𝑛_•𝑏)/𝑛}. The eigenvector corresponding to the zero eigenvalue is (1,…,1)^𝑇, so a typical eigenvector may be written 1_⟂. The elements of 𝑌_𝑢=𝐴^𝑇𝑍_𝑢 are of the form 1^𝑇⟂𝑍_𝑢, that may fairly be called a contrast between the elements of 𝑍_𝑢. They are mutually independent and standard normal. While the 𝑍_𝑢𝑖 are immediately interpretable, they are slightly less convenient than 𝑌_𝑢𝑖 that are orthogonal contrasts and are asymptotically mutually independent and asymptotically standard normal. These contrasts correspond to each order 𝑢, 𝑢=1,…,𝐼₁−1, 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)×(𝑛+1) matrix with all the elements of the first row 1/p(𝑛+1) and 𝑟th row 1/p[𝑟(𝑟+1)] (r times), −𝑟/p[𝑟(𝑟+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:

𝑝_{𝑤𝑧1𝑧2}=𝑝_𝑤••^{𝐼1−1𝑢=0}𝜃_{𝑢𝑧1𝑧2}𝑎_𝑢(𝑤),(4.1) for 𝑤=1,…,𝐼₁, in which 𝜃_0𝑧1𝑧2=1. Here and henceforth, the normalisation corresponding to the √𝑛_•𝑧 factor in 𝑝_𝑤𝑧 in Section 3 is absorbed into the 𝜃_{𝑢𝑧1𝑧2}. The components are random variables:

𝑍_{𝑢𝑧1𝑧2}=𝑛𝑝_•𝑧1•𝑝_••𝑧2^𝐼1𝑤=1𝑎_𝑢(𝑤)𝑝_{𝑤𝑧1𝑧2},(4.2) where 𝑝_•𝑧1•=∑_𝑤∑_𝑧2𝑁_{𝑤𝑧1𝑧2}/𝑛 and 𝑝_••𝑧2=∑_𝑤∑_𝑧1𝑁_{𝑤𝑧1𝑧2}/𝑛. The 𝑍_{𝑢𝑧1𝑧2} 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, 𝑢=1,…,𝐼₁−1, we may construct orthogonal contrasts that are asymptotically mutually independent and asymptotically standard normal. These contrasts reflect 𝑢th moment comparisons between the levels of the unordered factors.

In [2], without a model, it is shown that 𝑋^2𝑃 is the sum of the squares of the 𝑍_{𝑢𝑧1𝑧2}:

𝑋^2𝑃=^{𝐼1−1𝑢=0}^𝐼2𝑧₁⁼¹^𝐼3𝑧₂⁼¹𝑍^2𝑢𝑧_1𝑧2.(4.3) In 𝑋^2𝑃, it is insightful to separate components corresponding to 𝑢=0 and 𝑢≠0. Thus

𝑋^2𝑃=^𝐼2_𝑧1=1^𝐼3𝑧₂⁼¹𝑍^20𝑧_1𝑧2+^{𝐼1−1𝑢=1}^𝐼2𝑧₁⁼¹^𝐼3𝑧₂⁼¹𝑍^2𝑢𝑧_1𝑧2.(4.4) The first summand corresponds to a two-way completely unordered table obtained by summing over rows and may reasonably be denoted by 𝑋^2𝑍_1𝑍2. The second summation corresponds to a genuinely three-way singly ordered table and may reasonably be denoted by 𝑋^2𝑈𝑍_1𝑍2.

In [2] it is stated that the degrees of freedom associated with 𝑋^2𝑃 are 𝐼₁𝐼₂𝐼₃−𝐼₁−𝐼₂−𝐼₃+2. This follows because there are (𝐼₂−1)(𝐼₃−1) degrees of freedom associated with 𝑋^2𝑍_1𝑍2, and (𝐼₁−1)(𝐼₂𝐼₃−1) degrees of freedom associated with 𝑋^2𝑈𝑍_1𝑍2.

We can argue for these degrees of freedom by, when possible, counting the 𝜃_{𝑢𝑧1𝑧2} or the 𝑍_{𝑢𝑧1𝑧2}. The table corresponding to 𝑋^2𝑍_1𝑍2 is completely unordered, so there are no 𝜃_{𝑢𝑧1𝑧2} 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 (𝐼₂−1)(𝐼₃−1). The table corresponding to 𝑋^2𝑈𝑍_1𝑍2 has degrees of freedom (𝐼₁−1)(𝐼₂𝐼₃−1) since this is the number of parameters 𝜃_{𝑢𝑧1𝑧2} in the smooth model. There are 𝐼₂𝐼₃ multinomials, each of which has (𝐼₁−1) parameters 𝜃_{𝑢𝑧1𝑧2} 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 {𝑝_𝑤••} 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:

𝑝_{𝑤1𝑤2𝑧}=𝑝_{𝑤1𝑤2•}^{𝐼1𝑢−1}₁⁼⁰^{𝐼2𝑢−1}₂⁼⁰𝜃_{𝑢1𝑢2𝑧}𝑎_𝑢1𝑤₁𝑎_𝑢2𝑤₂,(4.5) for 𝑤₁=1,…,𝐼₁, 𝑤₂=1,…,𝐼₂, and 𝑧=1,…,𝐼₃, in which 𝜃_00𝑧=𝜃_𝑢10𝑧=𝜃_0𝑢2𝑧=1. The components are random variables:

𝑍_{𝑢1𝑢2𝑧}=𝑛𝑝_••𝑧^𝐼1𝑤₁⁼¹^𝐼2𝑤₂⁼¹𝑎_𝑢1𝑤₁𝑎_𝑢2𝑤₂𝑝_{𝑤1𝑤2𝑧}(4.6) for 𝑢₁=0,…,𝐼₁−1, 𝑢₂=0,…,𝐼₂−1, and 𝑧=1,…,𝐼₃, where 𝑝_••𝑧=∑_𝑤1∑_𝑤2𝑁_𝑤1𝑤2z/𝑛. Again, by the multivariate central limit theorem, the 𝑍_{𝑢1𝑢2𝑧} are asymptotically multivariate normal. This does not depend on the smooth model. For each (𝑢₁,𝑢₂) 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 𝑋^2𝑃 is the sum of the squares of the 𝑍_{𝑢1𝑢2𝑧}:

𝑋^2𝑃=^{𝐼1𝑢−1}₁₌₀^{𝐼2𝑢−1}₂⁼⁰^𝐼3𝑧=1𝑍^2𝑢_1𝑢2𝑧.(4.7) Again in 𝑋^2𝑃, it is insightful to separate components corresponding to 𝑢_𝑖=0 and 𝑢_𝑖≠0. Thus,

𝑋^2𝑃=^𝐼3_𝑧=1𝑍^200𝑧+^{𝐼1𝑢−1}₁₌₁^𝐼3𝑧=1𝑍^2𝑢_10𝑧+^{𝐼2𝑢−1}₂₌₁^𝐼3𝑧=1𝑍^20𝑢_2𝑧+^{𝐼1𝑢−1}₁₌₁^{𝐼2𝑢−1}₂⁼¹^𝐼3𝑧=1𝑍^2𝑢_1𝑢2𝑧.(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 𝑋^2𝑈_1𝑍. The third summation corresponds to another two-way singly ordered table obtained by summing over rows and may reasonably be denoted by 𝑋^2𝑈_2𝑍. The final summation corresponds to a genuine three-way doubly ordered table and may reasonably be denoted by 𝑋^2𝑈_1𝑈2𝑍2.

In [2] it is incorrectly claimed that the associated degrees of freedom are, as in Section 4.1, 𝐼₁𝐼₂𝐼₃−𝐼₁−𝐼₂−𝐼₃+2. The one-way table corresponding to the components with 𝑢₁=𝑢₂=0 is uninformative, and should be ignored. The two-way tables corresponding to precisely one of the 𝑢₁ or 𝑢₂ zero are single-ordered, and, as in Section 3, have degrees of freedom (𝐼₁−1)(𝐼₃−1) and (𝐼₂−1)(𝐼₃−1), respectively. When neither 𝑢₁ nor 𝑢₂ is zero, the corresponding table is a genuine doubly ordered three-way table. There are 𝐼₃ multinomials, each with (𝐼₁−1)(𝐼₂−1) parameters 𝜃_{𝑢1𝑢2𝑧} 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 (𝐼₁−1)(𝐼₂−1)(𝐼₃−1) degrees of freedom for this final table. In all, the degrees of freedom are

𝐼₁−1𝐼₂−1𝐼₃−1+𝐼₁−1𝐼₂−1+𝐼₁−1𝐼₃−1=𝐼₁𝐼₂−1𝐼₃−1.(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 (≥1) categorical variables are taken as ordered and the remaining 𝑠=𝑚−𝑟 (≥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 {𝑁_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}}. As in [2] and [4, Chapter 4], we define components of the form

𝑍_{𝑢1⋯𝑢𝑟𝑧1⋯𝑧𝑠}=𝑛𝑝_{•⋯•𝑧1•⋯•}×⋯×𝑝_{•⋯•𝑧𝑠}^𝐼1𝑤1=1⋯^𝐼𝑟_𝑤𝑟=1𝑎_𝑢1𝑤₁⋯𝑎_𝑢𝑟𝑤_𝑟𝑝_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠},(5.1) where 𝑝_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}= 𝑁_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}/𝑛 and where {𝑎_𝑢𝑗(•)}, 𝑝_{•⋯•𝑧𝑗•⋯•} and 𝑝_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠} are defined similarly to the above. Again, by the multivariate central limit theorem, the 𝑍_{𝑢1⋯𝑢𝑟𝑧1⋯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

𝑋^2𝑃=^{𝐼1𝑢−1}₁₌₀⋯^{𝐼𝑟𝑢−1}_𝑟=0^{𝐼𝑟+1𝑧}₁⁼¹⋯^𝐼𝑚_𝑧𝑠=1𝑍^2𝑢_{1⋯𝑢𝑟𝑧1⋯𝑧𝑠}.(5.2) By separating components corresponding to 𝑢_𝑖=0 and 𝑢_𝑖≠0,𝑋^2𝑃 can be partitioned as follows:

𝑋^2𝑃=^𝐼𝑟+1_𝑧1=1⋯^𝐼𝑚_𝑧𝑠=1𝑍^20⋯0𝑧_1⋯𝑧𝑠+^𝑟𝑖=1^{𝐼𝑖𝑢−1}_𝑖⁼¹^{𝐼𝑟+1𝑧}₁⁼¹⋯^𝐼𝑚_𝑧𝑠=1𝑍^20⋯0𝑢_{𝑖0⋯0𝑧1⋯𝑧𝑠}+^𝑟_𝑖_{,𝑗=1𝑖≠𝑗}^{𝐼𝑖𝑢−1}_𝑖⁼¹^{𝐼𝑗𝑢−1}_𝑗⁼¹^{𝐼𝑟+1𝑧}₁⁼¹+⋯^𝐼𝑚_𝑧𝑠=1𝑍^20⋯0𝑢_{𝑖0⋯0𝑢𝑗0⋯0𝑧1⋯𝑧𝑠}+⋯_{𝐼1𝑢−11=1}⋯^{𝐼𝑟𝑢−1}_𝑟=1^{𝐼𝑟+1𝑧}₁⁼¹⋯^𝐼𝑚_𝑧𝑠=1𝑍^2𝑢_{1⋯𝑢𝑟𝑧1⋯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 𝑟−1 ordered marginals and may reasonably be denoted by ∑^𝑟𝑖=1𝑋^2𝑈_{𝑖𝑍1⋯𝑍𝑠}. The following term corresponds to all (𝑠+2)-way doubly ordered tables obtained by summing over 𝑟−2 ordered marginals and may reasonably be denoted by

^𝑟_{𝑖,𝑗=1𝑖≠𝑗}𝑋^2𝑈_{𝑖𝑈𝑗𝑍1⋯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 𝑋^2𝑈_{1⋯𝑈𝑟𝑍1⋯𝑍𝑠}. Thus,

𝑋^2𝑃=^𝑟𝑖=1𝑋^2𝑈_{𝑖𝑍1⋯𝑍𝑠}+^𝑟_𝑖_{,𝑗=1𝑖≠𝑗}𝑋^2𝑈_{𝑖𝑈𝑗𝑍1⋯𝑍𝑠}+⋯+𝑋^2𝑈_{1⋯𝑈𝑟𝑍1⋯𝑍𝑠}.(5.5)

The smooth model envisaged here is product multinomial where for each (𝑧₁,…,𝑧_𝑠), the observations follow a multinomial distribution with total counts 𝑛_{•⋯•𝑧1⋯𝑧𝑠} and cell probabilities {𝑝_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}} given by

𝑝_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}=𝑝_{𝑤1•⋯•}×⋯×𝑝_{•⋯•𝑤𝑟•⋯•}^{𝐼1𝑢−1}₁₌₀⋯^{𝐼𝑟𝑢−1}_𝑟=0𝜃_{𝑢1⋯𝑢𝑟𝑧1⋯𝑧𝑠}𝑎_𝑢1𝑤₁⋯𝑎_𝑢𝑟𝑤_𝑟.(5.6)

An extension of the approach in [4, Section  4.4] investigates testing if the 𝜃_{𝑢1⋯𝑢𝑟𝑧1⋯𝑧𝑠}are collectively zero against the negation of this. Generalising the work in [4, Section  4.4], the efficient score statistic is 𝑍_{𝑤1⋯𝑤𝑟𝑧1⋯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 𝜃_{𝑢1⋯𝑢𝑟𝑧1⋯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=𝑚−𝑟 categories not ordered. This includes tables corresponding to 𝑋^2𝑈_{𝑖𝑈𝑗𝑍1⋯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 𝑋^2𝑈_{1⋯𝑈𝑟𝑍1⋯𝑍𝑠} are (𝐼₁−1)(𝐼₂−1)⋯(𝐼_𝑟−1)(𝐼_𝑟+1×𝐼_𝑟+2×⋯×𝐼_𝑚−1). There are 𝐼_𝑟+1×𝐼_𝑟+2×⋯×𝐼_𝑚 multinomials (corresponding to 𝑍₁=𝑧₁,…,𝑍_𝑠=𝑧_𝑠) each with (𝐼₁−1)(𝐼₂−1)⋯(𝐼_𝑟−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 𝑍_{𝑤1⋯𝑤𝑟𝑧1⋯𝑧𝑠}. The approach is similar to that employed in Section 3.