Two-way model with random cell sizes

Abstract

We consider inference for row effects in the presence of possible interactions in a two-way fixed effects model when the numbers of observations are themselves random variables. Let N_ij be the number of observations in the (i, j) cell, π_ij be the probability that a particular observation is in that cell and μ_ij be the expected value of an observation in that cell. We assume that the {N_ij} have a joint multinomial distribution with parameters n and {π_ij}. Then μ̄_i. = Σ_jπ_ijμ_ij/Σ_jπ_ij is the expected value of a randomly chosen observation in the ith row. Hence, we consider testing that the μ̄_i. are equal. With the {π_ij} unknown, there is no obvious sum of squares and F-ratio computed by the widely available statistical packages for testing this hypothesis. Let Ȳ_i.. be the sample mean of the observations in the ith row. We show that Ȳ_i.. is an MLE of μ̄_i., is consistent and is conditionally unbiased. We then find the asymptotic joint distribution of the Ȳ_i.. and use it to construct a sensible asymptotic size α test of the equality of the μ̄_i. and asymptotic simultaneous (1 − α) confidence intervals for contrasts in the μ̄_i..

1. Introduction

The two-way model with fixed effects for unbalanced data has received considerable attention in the past 30 years. There has been much discussion of the definition of the effects, calculation of sums of squares and definitions of hypotheses. The model is the subject of many texts, prominent among those Searle (1971, 1982), Graybill (1976), Scheffè (1959), and Arnold (1981). Several decompositions of sums of squares are presented in ANOVA tables, and testing the parameters of the model is routinely accomplished by subtracting sums of squares to isolate the contribution of a certain parameter. Searle’s (1971) R-notation, for example R(α/μ, β) is usually involved in testing the contribution of including α after μ and β in the model. The R-notation implies orthogonalizations of the design matrix that lead to sums of squares that test a host of hypotheses that depend on the cell sizes. Many hypotheses have interpretation problems and several articles have been written on the hypotheses implied by using Searle’s R(·/·) notation, e.g. see, Speed et al. (1978), Kutner (1974); for an extensive list see Macnaugton (1998).

This paper considers the same two-way model under the assumption that the overall sample size n is fixed but the cell sizes are themselves random variables following the multinomial distribution with unknown parameters π_ij. The setting has been introduced in Moschopoulos and Davidson (1985). Applications of the new setting arise mainly in survey studies. In these studies it is often not possible to sample from the individual cells. Instead, a sample of size n is taken from the overall population and is divided after the fact into the two-way cells according to sample characteristics. The following two examples will help in understanding this setting:

Example 1

A random sample size n of women is drawn from a metropolitan area for the purpose of studying fertility, measured by the number of children born to women (the response y-variable). It is assumed that fertility is dependent on religion, the row factor (Catholic (1) or Protestant (2)) and educational level, the column factor (Elementary (1), High School (2), College (3)). Sampling 300 women from the overall population results in 2 × 3 = 6 cell sizes n_ij, i = 1, 2; j = 1, 2, 3 that are random variables following the multinomial distribution (for large n). Note that sampling from each of the six classifications is not possible since religion and education are only known after the sample is drawn. In this example, let i represent the woman’s religion and j represent the woman’s educational level. In this situation, π₂₁ represents the proportion of women who are Protestant with Elementary education; μ₁₂ represents the expected number of children in Catholic women with High School education; ${\overline{\mu }}_{2.}^{\pi }$ represents the expected number of children by Protestant women averaged across educational levels. In testing that the ${\overline{\mu }}_{i.}^{\pi }$, i = 1, 2 are all equal in this setting, we are testing the equality of the mean number of children for the two religions averaged over educational levels. Testing for no column effects with these weights is testing the equality between educational levels averaged over religion. For a real situation related to this, see Groat and Neal (1967).

Example 2

Consider an academic class size n viewed as a random sample from a population of similar students with the response y being the student performance during an examination. The purpose of the investigation here is to relate class performance to student anxiety, the row factor (High (1), Moderate (2), Low (3)) and student attitude towards the subject the column factor (Positive (1), Negative (2)). Note that sampling from each of the six classifications here is not possible as the student classifications are known only after the sample is drawn. Thus, the cell sizes are random following the multinomial distribution (for large n). As in the first example, here testing the equality of ${\overline{\mu }}_{i.}^{\pi }$, i = 1, 2, 3 is testing the equality of performances of High, Moderate and Low anxiety students averaged over student attitudes. For a real situation on this, see Galassi et al. (1981).

Random cell sizes but from a different perspective are considered in Psychology, consideration there given to random cell sizes arising as a result of the underlying treatment, see Weiss (1999, 2006).

As stated above, the two-way model with fixed effects and unbalanced data is not free of controversies. The main problems are the following: (1) the definition of main effects, i.e row, column and interaction effects, and (2) hypotheses tested by typical ANOVA decompositions and interpretations of such hypotheses. Concerning (1) above, as it turns out, in the case of the two-way model with multinomially distributed cell sizes we can provide a natural definition of main effects. This definition is not arbitrary but is a natural consequence of the sampling scheme and it is expressed in terms of parameters (i.e cell population proportions and the cell population means), see the next section. As for (2) above, we are concerned here with one particular hypothesis, namely the equality of row (column) means. Let μ_ij, i = 1, …, r; j = 1, …, c be the mean of cell (i,j), and let π_ij be the probability of obtaining an observation in cell (i,j). We show that the i-th row mean is

$${\mu }_i=\sum _{j=1}^c\frac{{\pi }_{ij}}{{\pi }_{i.}}{\mu }_{ij},$$ (1.1)

where ${\pi }_{i.}={\sum }_{j=1}^c{\pi }_{ij}$. The hypothesis of equality of row means is

$$H_o:{\mu }_i\text{equal},i=1,\dots ,r.$$ (1.2)

The hypothesis (1.2) above is equivalent to the hypothesis that the α-main effects (defined in Section 2) are zero. This hypothesis is well known in the literature but the means are defined with cell size weights, i.e. (see Searle, 1971)

$${\stackrel{\sim }{\mu }}_i=\sum _{j=1}^c\frac{N_{ij}}{N_{i.}}{\mu }_{ij}.$$ (1.3)

Obviously, when the cell sizes are random variables, testing the equality of means in (1.3) is not proper. Testing of the right hypothesis (1.2) is the second goal of this paper. However, there is no test derived from standard ANOVA sums of squares decompositions that is proper for testing (1.2). We develop an asymptotic (n → ∞) test for (1.2) and asymptotic 100 × (1 − α)% confidence intervals for contrasts of row means. In addition, we provide some numerical evaluations of the performance of the proposed test under the null hypothesis. Under the multinomial assumption for the cell sizes, it is possible that a cell size may be zero. The paper is rigorously considering the case of zero cells.

2. The two-way model with random cell sizes

The two-way model is often modeled in the following over-parameterized form that includes terms for main effects and interactions. We observe Y_ijk independent,

$$Y_{\mathit{ijk}}~N(\theta +{\alpha }_i+{\beta }_j+{\gamma }_{ij},{\sigma }^2),i=1,\dots ,r;j=1,\dots ,c;k=1,\dots ,N_{ij},$$ (2.1)

where θ, {α_i}, {β_j}, {γ_ij} and σ² are unknown parameters. In order to make these parameters identifiable, weights w_ij ≥ 0 are often chosen and it is assumed in addition that

$$\sum _i\sum _j{w}_{ij}{\alpha }_i=0,\sum _i\sum _j{w}_{ij}{\beta }_j=0,\sum _i{w}_{ij}{\gamma }_{ij}=0,\sum _j{w}_{ij}{\gamma }_{ij}=0.$$ (2.2)

It is well known that the problem of testing that the interactions γ_ij = 0 does not depend on the weights, nor does the problem of testing that the main effects α_i = 0 when it is assumed that the γ_ij = 0 (e.g. see Arnold, 1981, pp. 93–96). Unfortunately, however, the problem of testing that the α_i = 0 when the γ_ij may be non-zero depends on the weights chosen. Different weights lead to different hypotheses. Let

$${\mu }_{ij}=\theta +{\alpha }_i+{\beta }_j+{\gamma }_{ij},{\overline{\mu }}_i^W=\sum _j{w}_{ij}{\mu }_{ij}/\sum _j{w}_{ij}.$$ (2.3)

Then testing that α_i = 0 with the weights w_ij is the same as testing the equality of the ${\overline{\mu }}_i^W$. Therefore, the weights w_ij represent the relative importance of the observations in the ith row, jth cell out of all the observations in the ith row.

Now, suppose that the individuals observed are themselves a sample of size n( = Σ_i Σ_jN_ij) from a very large population. In that case, we may assume that the cell sizes {N_ij} have a joint multinomial distribution with parameters n, and {π_ij} written as

$$\{N_{ij}\}~M_{rc}(n,\{{\pi }_{ij}\}),$$

where π_ij is the probability that a randomly chosen individual is in the ith row and the jth column. We assume that the π_ij are unknown parameters for this model. In this case, naturally, we use the weights

$$w_{ij}={\pi }_{ij}.$$ (2.4)

Using these weights, we see that the conditional expectation of an observation in row i is

$$\begin{array}{l}E(Y_{\mathit{ijk}}\mid \text{row}i)=\sum _{j=1}^{c}Pr(\text{column}j\mid \text{row}i)E(Y_{\mathit{ijk}}\mid \text{row}i,\text{column}j)\\ =\sum _{j=1}^c\frac{{\pi }_{ij}}{{\pi }_{i.}}{\mu }_{ij}={\overline{\mu }}_i\end{array}$$ (2.5)

since

$$P(\text{column}j\mid \text{row}i)={\eta }_{ij}={\pi }_{ij}/\sum _j{\pi }_{ij}=\frac{{\pi }_{ij}}{{\pi }_i}.$$ (2.6)

Therefore, in testing the equality of the μ̄_i. we are testing that the expected value of the response variable is the same in all the rows.

Note

The expectation in (2.5) is approximate if some cells are empty in row i; the case of empty cells is treated in this paper, see Basic results. It is now interesting to note that if the artificial parameters θ, α_i, β_j and γ_ij above are defined using the weights w_ij = π_ij, then

$$\theta =\sum _i\sum _j{\pi }_{ij}{\mu }_{ij},{\alpha }_i={\overline{\mu }}_{i.}-\theta ,{\beta }_j={\overline{\mu }}_{.j}-\theta ,{\gamma }_{ij}={\mu }_{ij}-{\overline{\mu }}_{i.}-{\overline{\mu }}_{.j}+\theta .$$ (2.7)

3. Basic results

For simplicity we consider here the cell means μ_ij instead of the over-parameterized model. Let Y_ijk be independent,

$$Y_{\mathit{ijk}}~N({\mu }_{ij},{\sigma }^2),i=1,\dots ,r;j=1,\dots ,c;k=1,\dots ,N_{ij},$$

where

$$\{N_{ij}\}~M_{rc}(n,\{{\pi }_{ij}\},)$$

i.e., the {N_ij} are (jointly) multinomially distributed with parameters n and {π_ij}. We assume that

$${\pi }_{ij}>0\text{for}\text{all}i\text{and}j$$

and that the π_ij are unknown and must be estimated from the data, but that n is known and fixed in advance. It is important to remember that the N_ij are random variables, and may in fact be 0.

Let q be the number of cells (i,j) in which N_ij > 0. (Note that q is a random variable dependent on the N_ij). Let

$${\overline{Y}}_{ij.}=\sum _k{Y}_{\mathit{ijk}}/N_{ij}\text{if}{N}_{ij}>0,{\overline{Y}}_{ij.}=0\text{if}{N}_{ij}=0,$$ (3.1)

$$S^2=\sum _i\sum _j\sum _k{(Y_{\mathit{ijk}}-{\overline{Y}}_{ij.})}^2/(n-q).$$ (3.2)

Then it is easily shown that the sets {Ȳ_ij.}, {N_ij} together with S² form a sufficient statistic for this model and that Ȳ_ij., N_ij/n and (n − q)S²/n are the MLE’s of μ_ij, π_ij and σ² respectively. (Note that the MLE for μ_ij is only unique if N_ij > 0.) Furthermore, N_ij/n is an unbiased estimator of π_ij. Unbiased estimators for the μ_ij do not appear possible, due to the possibility that N_ij = 0. Finally, note that conditionally on q,

$$(n-q)S^2/{\sigma }^2\mid q~{\chi }_{n-q}^2$$ (3.3)

so that S² is consistent and unbiased.

Now, with μ̄_i. as in (2.5) and π_i. and η_ij as in (2.6), let N_i = Σ_jN_ij and

$${\widehat{\pi }}_{i.}=N_i/n,{\widehat{\eta }}_{ij}=N_{ij}/N_i\text{if}{N}_i>0\text{and}{\widehat{\eta }}_{ij}=0\text{if}{N}_i=0$$

be MLE’s of π_i. and η_ij respectively. Further, let

$${\overline{Y}}_{i\mathrm{..}}=\sum _j{\widehat{\eta }}_{ij}{\overline{Y}}_{ij.}.$$

Note that Ȳ_i.. is the MLE for μ̄_i. by the invariance principle, and is unique as long as N_i > 0. Note also that as long as N_i > 0,

$${\overline{Y}}_{i\mathrm{..}}=\sum _j\sum _k{Y}_{\mathit{ijk}}/N_i.$$ (3.4)

Finally, note that Ȳ_i.. has a point mass at 0, so is not a continuous random variable.

$$P({\overline{Y}}_{i\mathrm{..}}=0)=P(N_i=0)={(1-{\pi }_{i.})}^n.$$ (3.5)

The Distribution of Ȳ_i

We are primarily interested in inference about the μ̄_i. In particular, we are interested in testing that the μ̄_i. are all equal. In order to draw inference about the μ̄_i., we need to learn about the distribution of Ȳ_i... For notational convenience we let

$${\overrightarrow{\eta }}_i^{\prime }={({\eta }_{i1},\dots ,{\eta }_{ic})}^{\prime },{\widehat{\overrightarrow{\eta }}}_i^{\prime }={({\widehat{\eta }}_{i1},\dots ,{\widehat{\eta }}_{ic})}^{\prime },{\overrightarrow{m}}_i^{\prime }=({\mu }_{i1},\dots ,{\mu }_{ic}).$$

The following random variables are useful in our development here and represent row-means weighted by cell sizes. For i = 1, …, r, let

$${\stackrel{\sim }{\mu }}_i=\sum _j\frac{N_{ij}}{N_i}{\mu }_{ij}=\sum _j{\widehat{\eta }}_{ij}{\mu }_{ij}={\overrightarrow{m}}_i^{\prime }{\widehat{\overrightarrow{\eta }}}_i.$$ (3.6)

We note here that testing for main effects in the presence of interaction and using the usual conditional F-test entails testing the hypothesis that the μ̃_i’s are equal, see Searle, 1971, pp. 292–231, Arnold, 1981, pp. 93–96. Obviously this hypothesis makes no sense especially if the cell sizes are random variables. Define

$$e_{\mathit{ijk}}=Y_{\mathit{ijk}}-{\mu }_{ij},{\overline{e}}_{i\mathrm{..}}=\sum _{jk}{e}_{\mathit{ijk}}/N_i\text{if}{N}_i>0,{\overline{e}}_{i\mathrm{..}}=0\text{if}{N}_i=0.$$ (3.7)

If N_i > 0, then

$${\overline{Y}}_{i\mathrm{..}}-{\stackrel{\sim }{\mu }}_i=\sum _{jk}(Y_{\mathit{ijk}}-{\mu }_{ij})/N_i={\overline{e}}_{i\mathrm{..}}.$$ (3.8)

When N_i = 0,

$${\overline{Y}}_{i\mathrm{..}}={\stackrel{\sim }{\mu }}_i={\overline{e}}_{i\mathrm{..}}=0.$$

Therefore, for all N_i,

$${\overline{Y}}_{i\mathrm{..}}={\stackrel{\sim }{\mu }}_i+{\overline{e}}_{i\mathrm{..}}={\overline{e}}_{i\mathrm{..}}+{\overrightarrow{m}}_i^{\prime }{\widehat{\overrightarrow{\eta }}}_i.$$ (3.9)

Lemma 1

Conditionally on N_i, ē_i.. and η̂_i are independent.

Proof

If N_i = 0, then both ē_i.. and η̂_i are degenerate and hence independent. If N_i > 0, then ē_i.. is the average of all N_i of the errors in the ith row, irrespective of which cell they are in. Since the errors are independently identically distributed, ē_i.. does not depend on the relative proportions in the columns, and hence on η̂_i.

This result implies that conditionally on N⃗(N₁, N₂, …, N_r), the Ȳ_i.., i = 1, …, r are not normally distributed; by (3.9), Ȳ_i.. is the sum of two independent random variables and if it were normal then both components should be normal, which obviously is not the case. This result is of importance in our development for the following reasons: The definition of μ̄_i. depends only on μ_ij and η_ij and the distribution of N⃗ depends only on the π_i.. Therefore, N⃗ is an ancillary statistic for this problem and it makes sense to work conditionally on N⃗. Note also that the N_ij are not ancillary for the μ̄_i., so that we should not work conditionally on the N_ij.

Comment

Working conditionally on N⃗ = (N₁, N₂, …, N_r), is of course equivalent to making inferences about r means of non-normal sub-populations with the overall population being a mixture of these r sub-populations, each with mixing probability π_i. This will lead to an approximate test statistic for the problem for large n and essentially reduces the two-way problem to a ‘one-way problem’. However, preliminary numerical evaluations showed that the ‘heuristic’ one-way ANOVA test of equality of row means in this case is totally inappropriate, because the distribution of the row-sample means is NOT normal. Obviously, any test statistic for testing the hypothesis that the μ_i.’s in (2.5) are equal would have to rely on the distribution of the row-sample mean. We first study the conditional distribution of the Ȳ_i.., given N⃗.

Theorem 1

1. In the conditional distribution given N⃗, the Ȳ_i.., i = 1, …, r are independent.

2. If N_i = 0, then E(Ȳ_i.. | N⃗) = Var(Ȳ_i.. | N⃗) = 0.

3. If N_i > 0, then

   $$E({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})={\overline{\mu }}_{i.},\text{Var}({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})={\delta }_i{\sigma }^2/N_i,$$
   where

   $${\delta }_i=1+\sum _j{\eta }_{ij}{({\mu }_{ij}-{\overline{\mu }}_{i.})}^2/{\sigma }^2.$$ (3.10)

Proof

1. Conditionally on the N⃗, the ē_i.. are independent as are the η̂_i, i = 1, …, r. Hence, using the lemma, the ${\overline{Y}}_{i\mathrm{..}}={\overline{e}}_{i\mathrm{..}}+{\overrightarrow{m}}_i^{\prime }{\widehat{\eta }}_i$ are also independent.

2. When N_i = 0, then Ȳ_i.. is degenerate at 0, so these results follow.

3. If N_i > 0, then $N_i{\widehat{\overrightarrow{\eta }}}_i$ conditional on N⃗ has the following multinomial distribution:

   $$N_i\widehat{\overrightarrow{\eta }}\mid \overrightarrow{N}~M_c(N_i,{\overrightarrow{\eta }}_i).$$ (3.11)

Now,

$$\begin{array}{l}E({\overline{e}}_{i\mathrm{..}}\mid \overrightarrow{N})=0,\\ E({\stackrel{\sim }{\mu }}_i\mid \overrightarrow{N})={\overrightarrow{m}}_i^{\prime }E({\widehat{\overrightarrow{\eta }}}_i\mid \overrightarrow{N})={\overrightarrow{m}}_i^{\prime }{\overrightarrow{\eta }}_i={\overline{\mu }}_{i.}\end{array}$$ (3.12)

and therefore,

$$E({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})=E({\overline{e}}_{i\mathrm{..}}\mid \overrightarrow{N})+E({\stackrel{\sim }{\mu }}_i\mid \overrightarrow{N})=0+{\overline{\mu }}_i.$$ (3.13)

To see the formula for the variance, note that

$$\text{Var}({\overline{e}}_{i\mathrm{..}}\mid \overrightarrow{N})={\sigma }^2/N_i$$

and using the variance–covariance matrix of the multinomial proportions

$${\widehat{\overrightarrow{\eta }}}_i^{\prime }={({\widehat{\eta }}_{i1},\dots ,{\widehat{\eta }}_{ic})}^{\prime }$$

we obtain

$$\text{Var}({\stackrel{\sim }{\mu }}_i\mid \overrightarrow{N})={\overrightarrow{m}}_i^{\prime }(\text{Cov}({\widehat{\overrightarrow{\eta }}}_i\mid \overrightarrow{N})){\overrightarrow{m}}_i={\overrightarrow{m}}_i^{\prime }{V}_i{\overrightarrow{m}}_i/N_i,$$

where V_i has (j, g) element V_ijk given by

$$V_{\mathit{ijj}}={\eta }_{ij}(1-{\eta }_{ij}),V_{\mathit{ijg}}=-{\eta }_{ij}{\eta }_{ig},j\ne g.$$ (3.14)

Since ē_i.. and μ̃_i are independent (conditionally on N⃗)

$$\text{Var}({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})=\text{Var}({\overline{e}}_{i\mathrm{..}}\mid \overrightarrow{N})+\text{Var}({\stackrel{\sim }{\mu }}_i\mid N)=({\sigma }^2/N_i)(1+{\overrightarrow{m}}_i^{\prime }{V}_i{\overrightarrow{m}}_i/{\sigma }^2)={\sigma }^2{\delta }_i/N_i$$

and the theorem is proved.

Note from part (c) that Ȳ_i.. is a conditionally unbiased estimator of μ̄_i > 0 but is not unbiased. In fact,

$$E{\overline{Y}}_{i\mathrm{..}}={\overline{\mu }}_{i.}P(N_i>0)={\overline{\mu }}_{i.}(1-{(1-{\pi }_{i.})}^n).$$

The asymptotic distribution of Ȳ_i... is given in the following.

Theorem 2

1. Ȳ_i.. is a consistent estimator of μ̄_i

2. $N_i^{1/2}({\overline{Y}}_{i\mathrm{..}}-{\overline{\mu }}_{i.})\mid \overrightarrow{N}\stackrel{d}{\to }{U}_i~N(0,{\delta }_i{\sigma }^2)$ as N_i → ∞.

Proof

(a) Note first that π_i. > 0 so that

$$P(N_i>0)=1-{(1-{\pi }_{i.})}^n\to 1\text{as}n\to \infty .$$

Therefore,

$$E{\overline{Y}}_{i\mathrm{..}}=E(E{\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})={\overline{\mu }}_{i.}P(N_i>0)\to {\overline{\mu }}_{i.}\text{as}n\to \infty .$$

To show consistency we need to show that Var(Ȳ_i..) → 0. Recall that

$$\text{Var}({\overline{Y}}_{i\mathrm{..}})=\text{Var}(E({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N}))+E(\text{Var}({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N})).$$ (3.15)

Now,

$$\text{Var}(E({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N}))={\overline{\mu }}_{i.}^2(1-{(P(N_i>0))}^2)\to {\overline{\mu }}_{i.}^2(1-1^2)=0.$$

Finally,

$$\begin{array}{l}E(\text{Var}({\overline{Y}}_{i\mathrm{..}}\mid \overrightarrow{N}))=E\left({\delta }_i{\sigma }^2\frac{1}{N_i}P(N_i>0)\right)\\ =P(N_i>0){\delta }_i{\sigma }^2\sum _{m=1}^n{m}^{-1}P(N_i=m)\\ =P(N_i>0){\delta }_i{\sigma }^2\sum _{m=1}^n{m}^{-1}\left(\begin{array}{c}n\\ m\end{array}\right){\pi }_{i.}^m{(1-{\pi }_{i.})}^{n-m}\\ =P(N_i>0){\delta }_i{\sigma }^2\frac{1}{(n+1){\pi }_{i.}}\sum _{m=1}^n\left(1+\frac{1}{m}\right)\left(\begin{array}{c}n+1\\ m+1\end{array}\right){\pi }_{i.}^{m+1}{(1-{\pi }_{i.})}^{n-m}\\ \le P(N_i>0){\delta }_i{\sigma }^2\frac{2}{(n+1){\pi }_{i.}}\to 0\text{as}n\to \infty .\end{array}$$

Therefore, the Var(Ȳ_i..) and the bias of Ȳ_i.. converge to 0 and Ȳ_i.. is consistent. (b). By the asymptotic approximation of the multinomial proportions,

$$N_i^{1/2}({\widehat{\overrightarrow{\eta }}}_i-{\overrightarrow{\eta }}_i)\mid \overrightarrow{N}\stackrel{d}{\to }{\overrightarrow{W}}_i~N_c(\overrightarrow{0},V_i),$$

where V_i is given in (3.14). Therefore,

$$N_i^{1/2}({\stackrel{\sim }{\mu }}_i-{\overline{\mu }}_{i.})\mid \overrightarrow{N}=N_i^{1/2}{\overrightarrow{m}}_i^{\prime }({\widehat{\overrightarrow{\eta }}}_i-{\overrightarrow{\eta }}_i)\mid \overrightarrow{N}\stackrel{d}{\to }N(0,{\overrightarrow{m}}_i^{\prime }{V}_i{\overrightarrow{m}}_i).$$

Now, ē_i.. is the average of all the errors in the ith row, so that

$$N_i^{1/2}{\overline{e}}_{i\mathrm{..}}\mid \overrightarrow{N}~N(0,{\sigma }^2).$$

By (3.9) and the lemma above, conditionally on N, $N_i^{1/2}({\stackrel{\sim }{\mu }}_i-{\overline{\mu }}_i)$ and N^1/2ē_i.. are independent. Therefore,

$$N_i^{1/2}({\overline{Y}}_{i\mathrm{..}}-{\overline{\mu }}_{i.})\mid \overrightarrow{N}=(N_i^{1/2}{\overline{e}}_{i\mathrm{..}}+N_i^{1/2}({\stackrel{\sim }{\mu }}_i-{\overline{\mu }}_{i.}))\mid \overrightarrow{N}\stackrel{d}{\to }N(0,{\sigma }^2+{\overrightarrow{m}}_i^{\prime }{V}_i{\overrightarrow{m}}_i).$$

However,

$${\sigma }^2+{\overrightarrow{m}}_i^{\prime }{\overrightarrow{V}}_i{\overrightarrow{m}}_i={\sigma }^2{\delta }_i$$

and this proves the theorem.

Corollary

${\overrightarrow{Q}}_n=n^{1/2}{({\overline{Y}}_{\mathrm{1..}}-{\overline{\mu }}_{1.},\dots ,{\overline{Y}}_{r\mathrm{..}}-{\overline{\mu }}_{r.})}^{\prime }\stackrel{d}{\to }\overrightarrow{Q}~N_r(\overrightarrow{0},{\sigma }^{2}T)$, as n → ∞, where T is a diagonal matrix whose ith element is δ_i/π_i..

Proof

Note first that π̂_i. = N_i/n → π_i. a.s. (almost surely) Therefore, $N_i\stackrel{a.s}{\to }\infty$, so that we can use the previous theorem. From that theorem and the conditional independence of the Ȳ_i..,

$${\overrightarrow{P}}_n=(N_1^{1/2}({\overline{Y}}_{\mathrm{1..}}-{\overline{\mu }}_{1.}),\dots ,N_r^{1/2}{({\overline{Y}}_{r\mathrm{..}}-{\overline{\mu }}_{r.}))}^{\prime }\mid \overrightarrow{N}\stackrel{d}{\to }\overrightarrow{P}~N_r(0,{\sigma }^{2}M),$$

where M is a diagonal matrix whose i-th diagonal element is δ_i. Since this limiting distribution does not depend on N⃗, unconditionally we have

$${\overrightarrow{P}}_n\stackrel{d}{\to }\overrightarrow{P}.$$

Now, π̂_i is a consistent estimator of π_i.. Therefore, by Slutsky’s theorem,

$${\overrightarrow{Q}}_n=({\widehat{\pi }}_{1.}^{-1/2}{P}_{n1},\dots ,{\widehat{\pi }}_{r.}^{-1/2}{P}_{nr})\stackrel{d}{\to }{({\pi }_{1.}^{-1/2}{P}_1,\dots ,{\pi }_{r.}^{-1/2}{P}_r)}^{\prime }~N(0,T).$$

Now consider testing the hypothesis

$$H_o:{\overline{\mu }}_{i.}\text{equal}\text{for}\text{all}i=1,\dots ,r.$$ (3.16)

If the normal distribution in the corollary were exact and the δ_i/π_i. were known, then the model would be a generalized linear model, and this hypothesis would be tested using the following (see (3.3) and recall that q is the number of non-empty cells):

$$F=n\sum _i{\pi }_{i.}{({\overline{Y}}_{i\mathrm{..}}-\stackrel{\sim }{Y})}^2/{\delta }_i(r-1)S^2,$$ (3.17)

where

$$\stackrel{\sim }{Y}=\sum _i({\pi }_{i.}{\overline{Y}}_{i\mathrm{..}}/{\delta }_i)/\sum _i({\pi }_{i.}/{\delta }_i)$$

and under the null hypothesis

$$F~F_{r-1,n-q}.$$ (3.18)

In addition, Scheffe’ simultaneous confidence intervals for the set of contrasts between the μ̄_i. (functions Σc_iμ̄_i., Σc_i = 0) are given by

$$\sum _i{c}_i{\overline{\mu }}_{i.}\in \sum _i{c}_i{\overline{Y}}_{i\mathrm{..}}\pm S{[(r-1)F^{\alpha }]}^{1/2}{[\sum c_i^2{\delta }_i/n{\pi }_i]}^{1/2},$$ (3.19)

where F^α is the upper 100(1 − α)% point of the F-distribution defined in (3.18). Of course, the δ_i and π_i. would not be known in practice, so we estimate them in the obvious way. Let

$${\widehat{\pi }}_{i.}=N_i/n,{\widehat{\delta }}_i=1+\sum _j{\widehat{\eta }}_{ij}{({\overline{Y}}_{ij.}-{\overline{Y}}_{i\mathrm{..}})}^2/S^2.$$

(By the invariance principle, π̂_i. and δ̂_i are the MLE’s of π_i. and δ_i). We suggest using the test statistic:

$${\widehat{F}}_n=n\sum {\widehat{\pi }}_{i.}{({\overline{Y}}_{i\mathrm{..}}-\widehat{Y})}^2/{\widehat{\delta }}_i(r-1)S^2=\sum _i{N}_i{({\overline{Y}}_{i\mathrm{..}}-\widehat{Y})}^2/{\widehat{\delta }}_i(r-1)S^2,$$ (3.20)

where

$$\widehat{Y}=\sum _i({\widehat{\pi }}_i{\overline{Y}}_{i\mathrm{..}}/{\widehat{\delta }}_i)\sum _i({\widehat{\pi }}_{i.}/{\widehat{\delta }}_i).$$ (3.21)

Since π̂_i. and δ̂_i are consistent (see the proof below), F̂_n should have an approximate F-distribution F_{r − 1,n − q} under the null hypothesis. Similarly, we suggest using the following approximate confidence intervals for contrasts in the μ̄_i.:

$$\sum c_i{\overline{\mu }}_{i.}\in \sum c_i{\overline{Y}}_{i\mathrm{..}}\pm S{[(r-1)F^{\alpha }]}^{1/2}{\left[\sum c_i^2{\widehat{\delta }}_i/N_i\right]}^{1/2}.$$ (3.22)

We finish this section by showing that the test and confidence intervals given above are at least asymptotically correct.

Theorem 3

1. Let F̂_n be defined in Eq. (3.20). Then under the null hypothesis that μ_i.., i = 1, …, r are equal,

   $$(r-1){\widehat{F}}_n\stackrel{d}{\to }H~{\chi }_{r-1}^2,$$
   and hence the test which rejects the null hypothesis when

   $$F_n\ge F_{r-1,n-q}^{\alpha }$$

   is an asymptotic size α test.

2. The confidence intervals given in Eq. (3.19) are asymptotic 100(1 − α)% simultaneous confidence intervals for the set of contrasts given above.

Proof

1. Note first that π̂_ij = N_ij/n are known to be consistent estimators of the π_ij so that the π̂_i. and η̂_ij are consistent estimators of the π̂_i. and η_ij. We have shown that the Ȳ_i.. and $S_n^2$ are consistent estimators of μ_i. and σ². Therefore, the δ̂_i is consistent estimator of δ_i. Let Q⃗_n and Q⃗ be defined as in the corollary above. Let

   $$\begin{array}{l}\overrightarrow{\pi }=({\pi }_{1.},\dots {\pi }_{r.}),{\widehat{\overrightarrow{\pi }}}_n=({\widehat{\pi }}_{1.},\dots ,{\widehat{\pi }}_{r.}),\overrightarrow{\delta }=({\delta }_1,\dots ,{\delta }_r),\\ {\widehat{\overrightarrow{\delta }}}_n=({\widehat{\delta }}_1,\dots ,{\widehat{\delta }}_r),\\ h(\overrightarrow{Q},\overrightarrow{\pi },\overrightarrow{\delta },{\sigma }^2)=\sum _i{\pi }_{i.}{(Q_i-\stackrel{\sim }{Q})}^2/{\delta }_i{\sigma }^2,\stackrel{\sim }{Q}=\sum _i({\pi }_{i.}{Q}_i/{\delta }_i)/\sum _i({\pi }_{i.}/{\delta }_i).\end{array}$$
   By the usual results on the generalized linear model,

   $$H=h(\overrightarrow{Q},\overrightarrow{\pi },\overrightarrow{\delta },{\sigma }^2)~{\chi }_{r-1}^2.$$
   Therefore, by Slutsky’s theorem,

   $$h({\overrightarrow{Q}}_n,{\widehat{\overrightarrow{\pi }}}_n,{\widehat{\overrightarrow{\delta }}}_n,S_n^2)\stackrel{d}{\to }H~{\chi }_{r-1.}^2.$$
   If the μ̄_i. are all equal, then

   $$(r-1){\widehat{F}}_n=h({\overrightarrow{Q}}_n,{\widehat{\pi }}_n,{\widehat{\delta }}_n,S_n^2)\stackrel{d}{\to }H~{\chi }_{r-1}^2.$$
   Now, q ≤ rc, so that n − q→ ∞ and hence $(r-1)F_{r-1,n-q}^{\alpha }\to {\chi }_{r-1.}^2$. Therefore, the probability of rejecting under the null hypothesis is

   $$p(h({\overrightarrow{Q}}_n,{\widehat{\overrightarrow{\pi }}}_n,{\widehat{\overrightarrow{\delta }}}_n,S_n^2)-(r-1)F_{r-1,n-q}^{\alpha }>0)\to P(H-{\chi }_{r-1,\alpha }^2>0)=\alpha .$$
2. The simultaneous confidence intervals given in Eq. (3.22) are all satisfied if and only if

   $$h({\overrightarrow{Q}}_n,{\widehat{\overrightarrow{\pi }}}_n,{\widehat{\overrightarrow{\delta }}}_n{S}_n^2)\le (r-1)F_{r-1,n-q.}^{\alpha }.$$

   The remainder of the argument is similar to part (a).

Some additional comments

1. For ease in defining the model, we have assumed that the π_ij > 0. A careful reading of the paper, however, will show that we have only used π_i. > 0. If π_i. = 0, we can just drop the ith row from the design, so that this is really no assumption.

2. Again, for simplicity, we have assumed that the Y_ijk are normally distributed. However, the results are true under greater generality. In particular, the proof of Theorem 1 has only used that

   $$E{Y}_{\mathit{ijk}}={\mu }_{ij},var(Y_{\mathit{ijk}})={\sigma }^2.$$
   For Theorems 2 and 3, all we need is that

   $$Y_{\mathit{ijk}}={\mu }_{ij}+e_{\mathit{ijk}},$$
   where e_ijk are independently identically distributed with mean 0 and with finite variance σ². We have only used the normality to establish that

   $$N_i^{1/2}{\overline{e}}_{i\mathrm{..}}\stackrel{d}{\to }R~N(0,{\sigma }^2),$$

   which can be established under these more general conditions using the central limit theorem.

3. For simplicity, we have assumed that we were sampling from a very large population so that the distribution of the N_ij could be assumed multinomial. If we sample from a smaller population we should replace the multinomial distribution with a hypergeometric distribution. Theorem 1 follows equally well for a multivariate hypergeometric (with population size m) if we replace the covariance matrix V for the multinomial with the covariance for the hypergeometric, getting

   $${\delta }_i^{\ast }=1+(m-n)\sum {\eta }_{ij}{({\mu }_{ij}-{\overline{\mu }}_{i.})}^2/(m-1){\sigma }^2.$$

   Theorem 2 can be similarly modified.

4. As a final generalization, we mention a different sampling scheme. Suppose instead of getting a sample of size n from the whole population, we get independent samples of size N_i from each of the row classifications. As before, let N_ij be the number of observations in the (i,j) cell. We assume that

   $${\overrightarrow{N}}_i=(N_{i1},\dots ,N_{ic})~M_c(N_i,{\overrightarrow{\eta }}_i),{\overrightarrow{\eta }}_i=({\eta }_{i1},\dots ,{\eta }_{ic}),$$
   where η_ij is the proportion of the ith row class in column class j. Let Ȳ_i., $\widehat{{\eta }_{ij}}$, μ̄_i., μ̃_i, e_ijk and ē_i.. be defined as in Section 2. The lemma is still true for this model, since it is conditional on the N_i. Similarly, Theorem 1 is conditional on the N_i, so that we see that the Ȳ_i.. are independent (since they are from independent samples) and

   $$E({\overline{Y}}_{i\mathrm{..}})={\mu }_{i.},var({\overline{Y}}_{i\mathrm{..}})={\delta }_i{\sigma }^2/N_i.$$
   (Note that for this model, we do not have to worry about the case N_i = 0.) Hence Ȳ_i.. is unbiased and consistent for this model. Similarly Theorem 2 is derived conditionally on the N_i, so that we have

   $$N_i^{1/2}({\overline{Y}}_{i\mathrm{..}}-{\overline{\mu }}_{i.})\stackrel{d}{\to }{U}_i~N(0,{\delta }_i{\sigma }^2)\text{as}{N}_i\to \infty .$$
   Now suppose that

   $$n=\sum _i{N}_i,N_i/n\to {\pi }_{i.}\text{as}n\to \infty$$

   for some constants π_i.. Then $N_i/n\stackrel{P}{\to }{\pi }_{i.}$ as N_i→ ∞, so that the corollary and Theorem 3 follow also. Therefore, the procedure derived in Section 2 with π_i. replaced by lim_{n→ ∞}(n_i/n) applies equally well to the model considered in this paragraph.

4. Numerical evaluations of the test in (3.20)

The proposed test F_n in (3.20) was evaluated numerically under the null hypothesis of equality of row means in (3.16) for several models of 2 × 2, 2 × 3 and 3 × 4 designs with simulation size 10,000. Tables 1–3, give the models, the cell probabilities, the cell means and the equal row means. The assumed interactions were non-zero but non-significant and the means are plotted in the associated Figs. 1–3. Entries in the tables are the percent of rejections out of the 10,000 samples, for various sample sizes. The size of the entries should be compared to 5%. As seen from all three tables, although asymptotic, the proposed test performs well in maintaining the correct Type-I error.

Table 1.

Type_I error rates of the new test for row effect of 2 × 2 factorial designs, with non-significant interaction.

Model	Cell probabilities	Cell means, SD	Row means	n = 30	n = 50	n = 70	n = 100
Model I	(.3,.4,.1,.2)	(50,15,35,27.5) SD = 20	(30,30)	4.35	4.93	4.35	4.83
Model II	(.3,.4,.1,.2)	(50,30,5.71,55) SD = 50	(38.571,38.571)	4.90	5.62	5.59	6.34
Model III	(.3,.4,.2,.1)	(50,15,40,10) SD = 4	(30,30)	5.23	5.92	6.35	6.52
Model IV	(.2,.4,.3,.1)	(32,20,30,6) SD = 10	(24.24)	4.95	5.84	6.24	6.71

Table 3.

Type I error rates of the new test, for row effect of 3 × 4 factorial designs, with non-significant interaction.

Model	Cell probabilities	Cell means, SD	Row means	n = 70	n = 100	n = 200
Model I	(.8,.2,.15,.05,.1,.02,.04,.03,.05,.1,.1,.08)	(50,15,20,10,25,27.5,10,23.54,15,15,40,12.11) SD = 40	(21.88,21.88,21.88)	2.77	3.53	4.13
Model II	(.08,.2,.15,.05,.1,.02,.04,.03,.05,.1,.1,.08)	(40,30,20,50,35,27.5,10,45.63,30,15,20,63.83) SD = 40	(30.63,30.63,30.63)	2.81	3.46	4.07
Model III	(.08,.05,.15,.05,.1,.16,.04,.03,.05,.11,.1,.08)	(40,30,20,50,35,30,45,3.33,20,15,30,60.74) SD = 50	(30.91,30.91,30.91)	3.17	3.65	4.11
Model IV	(.08,.02,.1,.05,.1,.33,.04,.03,.05,.02,.1,.08)	(60,15,20,10,25,35,15,18.33,30,5,20,50) SD = 50	(30.4,30.4,30.4)	3.38	3.72	4.94

Fig. 1.

Graphs of means for 2 × 2 factorial designs.

Fig. 3.

Graphs of means for 3 × 4 factorial designs.

Fig. 2.

Graphs of means for 2 × 3 factorial designs.

Table 2.

Type I error rates of the new test, for row effect of 2 × 3 factorial designs, with non-significant interaction.

Model	Cell probabilities	Cell means, SD	Row means	n = 50	n = 70	n = 100
Model I	(.2,.3,.1,.1,.1,.2)	(50,15,20,35,27.5,23.75) SD = 15	(27.5,27.5)	4.22	4.37	5.03
Model II	(.2,.3,.1,.1,.1,.2)	(40,35,20,45,45,23.33) SD = 10	(34.17,34.17)	4.83	4.97	5.49
Model III	(.2,.3,.1,.1,.1,.2)	(65,5,60,30,45,30.83) SD = 50	(34.17,34.17)	4.08	4.30	4.95
Model IV	(.3,.1,.05,.05,.1,.4)	(30,40,60,30,45,33.89) SD = 30	(35.56,35.56)	3.98	4.23	4.38