Testing order restrictions in contingency tables

Abstract

Though several interesting models for contingency tables are defined by a system of inequality constraints on a suitable set of marginal log-linear parameters, the specific features of the corresponding testing problems and the related procedures are not widely well known. After reviewing the most common difficulties which are intrinsic to inequality restricted testing problems, the paper concentrates on the problem of testing a set of equalities against the hypothesis that these are violated in the positive direction and also on testing the corresponding inequalities against the saturated model; we argue that valid procedures should consider these two testing problems simultaneously. By reformulating and adapting procedures appeared in the econometric literature, we propose a likelihood ratio and a multiple comparison procedure which are both based on the joint distribution of two relevant statistics; these statistics are used to divide the sample space into three regions: acceptance of the assumed equality constraints, rejection towards inequalities in the positive direction and rejection towards the unrestricted model. A simulation study indicates that the likelihood ratio based procedure perform substantially better. Our procedures are applied to the analysis of two real data sets to clarify how they work in practice.

Appendices

Appendix 1: Computation of probability weights

In order to compute the weights $w_i(\varvec{V},\mathcal {C})$, it may be useful to summarize the geometry of the projection of a random vector $\varvec{y} \sim \mathcal {N}(\varvec{0},\varvec{V})$ onto a convex cone $\mathcal {C}=\{\varvec{\eta }:\varvec{D}\varvec{\eta }\ge \varvec{0}\}$, where $\varvec{D}$ is a $k\times (t-1)$ matrix of rank $k$. Let $\varvec{H}$ be the left component of the Cholesky decomposition of the positive definite matrix $\varvec{\varPsi } = \varvec{D} \varvec{V} \varvec{D}^{\prime }$ then $\varvec{z} = \varvec{H}^{-1} \varvec{D}\varvec{y}\,\sim \,N(\varvec{0},\varvec{I}_k)$, the transformation $\varvec{\lambda }= \varvec{H}^{-1} \varvec{D}\varvec{\eta }$ defines the cone $\mathcal {C}^* =\{\varvec{\lambda }:\varvec{H} \varvec{\lambda }\ge \varvec{0}, \, \mathcal {C}^* \in \mathfrak {R}^k\}$, it follows that: $\min _{{\varvec{D}}{\varvec{\eta }}\ge {\varvec{0}}}(\varvec{y}-\varvec{\eta })^{\prime }\varvec{V}^{-1}(\varvec{y}-\varvec{\eta })=\min _{{\varvec{H}}{\varvec{\lambda }}\ge {\varvec{0}}}(\varvec{z}-\varvec{\lambda })^{\prime }(\varvec{z}-\varvec{\lambda })$. The cone $\mathcal {C}^*$ may also be defined by its generating vectors which are the columns of $\varvec{U} = \varvec{H}^{-1}$: a vector $\varvec{z}$ belongs to $\mathcal {C}^*$ if $\varvec{z}=\varvec{U}\varvec{u}$ where $\varvec{u}\ge \varvec{0}$. In a similar way the dual cone $\mathcal {C}^{*0}$ is generated by the columns of $\varvec{W}=-\varvec{H}^{\prime }$ and note that $\varvec{U}^{\prime }\varvec{W} = -\varvec{I}$.

Within the euclidian metric, $\mathfrak {R}^k$ can be partitioned into $2^k$ convex cones as follows: let $\mathcal {J}$ be the collection of all possible subsets of $(1, \dots , k)$, including the empty set and the whole set. For any pair $\varvec{i},\,\varvec{j}\in \mathcal {J},\, \varvec{i}\cup \varvec{j} = (1, \dots , k)$, let $\left( \varvec{U}_{{\varvec{i}}},\, \varvec{W}_{{\varvec{j}}}\right)$, be the matrix whose columns are, respectively, the columns of $\varvec{U}$ with index in $\varvec{i}$ and the columns of $\varvec{W}$ with index in $\varvec{j}$; the columns of this matrix generate the convex cone $\mathcal {C}^*(\varvec{i})$ whose elements, when projected onto $\mathcal {C}^*$, belong to the face generated by the columns of $\varvec{U}_{{\varvec{i}}}$, this face is itself a convex cone of dimension equal to the cardinality $|\varvec{i}|$ of $\varvec{i}$. Thus

$$\begin{aligned} w_{i+q}(\varvec{V},\mathcal {C}) =w_i(\varvec{I},\mathcal {C}^*) =\sum _{\,|\,{\varvec{i}}\,|\,=i} P[\varvec{z}\in \mathcal {C}^*(\varvec{i})], \quad i=0,1,\ldots ,k \end{aligned}$$

where $q$ is the dimension of $\mathcal {L}_0$.

To compute $P[\varvec{z}\in \mathcal {C}^*(\varvec{i})]$ note that $\varvec{z}\in \mathcal {C}^*(\varvec{i})$ if and only if $\varvec{t} = \left( \varvec{U}_{{\varvec{i}}},\, \varvec{W}_{{\varvec{j}}}\right) ^{-1}\varvec{z}\ge \varvec{0}$, in other words, the linear transformation above reduces $\mathcal {C}^*(\varvec{i})$ into the positive orthant for the multivariate normal random variable $\varvec{t}$; thus, to compute $P[\varvec{t}\in \mathcal {R}^{k+}]$, the only quantity we need is $Var(\varvec{t}) = \varvec{\varOmega }$. Let $\varvec{\varPsi } = \varvec{W}^{\prime }\varvec{W}$ and $\varvec{\varPhi } = (\varvec{U}^{\prime }\varvec{U})$ and note that $\varvec{\varPsi } = \varvec{D}\varvec{V}\varvec{D}^{\prime }= \varvec{\varPhi }^{-1}$. It can be shown that $\varvec{\varOmega }$ is block diagonal with elements given by $(\varvec{\varPhi }_{{\varvec{i}}{\varvec{i}}})^{-1}$ and $(\varvec{\varPsi }_{{\varvec{j}}{\varvec{j}}})^{-1}$, which are related by the well known formulas for the inverse of a partitioned matrix:

$$\begin{aligned} (\varvec{\varPhi }_{{\varvec{i}}{\varvec{i}}})^{-1}= & {} \varvec{\varPsi }_{{\varvec{i}}{\varvec{i}}}-\varvec{\varPsi }_{{\varvec{i}}{\varvec{j}}}(\varvec{\varPsi }_{{\varvec{j}}{\varvec{j}}})^{-1}\varvec{\varPsi }_{{\varvec{j}}{\varvec{i}}}\\ (\varvec{\varPsi }_{{\varvec{j}}{\varvec{j}}})^{-1}= & {} \varvec{\varPhi }_{{\varvec{j}}{\varvec{j}}}-\varvec{\varPhi }_{{\varvec{j}}{\varvec{i}}}(\varvec{\varPhi }_{{\varvec{i}}{\varvec{i}}})^{-1}\varvec{\varPhi }_{{\varvec{i}}{\varvec{j}}}. \end{aligned}$$

So, if $\,|\,\varvec{i}\,|\,\le \,|\,j\,|\,$, it is convenient to compute $(\varvec{\varPhi }_{{\varvec{i}}{\varvec{i}}})^{-1}$ directly and $(\varvec{\varPsi }_{{\varvec{j}}{\varvec{j}}})^{-1}$ from the second expression above, instead, when $\,|\,\varvec{i}\,|\,> \,|\,j\,|\,$, compute $(\varvec{\varPsi }_{{\varvec{j}}{\varvec{j}}})^{-1}$ directly and $(\varvec{\varPhi }_{{\varvec{i}}{\varvec{i}}})^{-1}$ from the first expression above. In any case, because $\varvec{\varOmega }$ is block diagonal, $P[\varvec{t}\in \mathcal {R}^{k+}]$ factorizes into the product of two lower dimensional integrals.

Because Proposition 3.6.1(3) in Silvapulle and Sen (2005) says that the weights with index $j$ even or odd sum to 0.5, we may avoid computing the two weight which correspond to the largest number of side cones; these correspond to $(k/2-1,\,k/2)$ when $k$ is even and to $((k-1)/2,\,(k+1)/2)$ when $k$ is odd.

Appendix 2: Simulations results

See Tables 3, 4 and 5.

Table 3 Likelihood ratio procedures in $3\times 3$ tables

Table 4 Likelihood ratio procedures in $3\times 4$ tables

Table 5 Multiple comparisons procedures in $3\times 3$ tables