Antedependence Models for Nonstationary Categorical Longitudinal Data with Ignorable Missingness: Likelihood-Based Inference

Abstract

Time index-ordered random variables are said to be antedependent (AD) of order (p₁, p₂, …, p_n) if the kth variable, conditioned on the p_k immediately preceding variables, is independent of all further preceding variables. Inferential methods associated with AD models are well developed for continuous (primarily normal) longitudinal data, but not for categorical longitudinal data. In this article, we develop likelihood-based inferential procedures for unstructured AD models for categorical longitudinal data. Specifically, we derive maximum likelihood estimators (mles) of model parameters; penalized likelihood criteria and likelihood ratio tests for determining the order of antedependence; and likelihood ratio tests for homogeneity across groups, time-invariance of transition probabilities, and strict stationarity. Closed-form expressions for mles and test statistics, which allow for the possibility of empty cells and monotone missing data, are given for all cases save strict stationarity. For data with an arbitrary missingness pattern, we derive an efficient restricted EM algorithm for obtaining mles. The performance of the tests is evaluated by simulation. The methods are applied to longitudinal studies of toenail infection severity (measured on a binary scale) and Alzheimer’s disease severity (measured on an ordinal scale). The analysis of the toenail infection severity data reveals interesting nonstationary behavior of the transition probabilities and indicates that an unstructured first-order AD model is superior to stationary and other structured first-order AD models that have previously been fit to these data. The analysis of the Alzheimer’s severity data indicates that the antedependence is second-order with time-invariant transition probabilities, suggesting the use of a second-order autoregressive cumulative logit model.

1. Introduction

Longitudinal data are ubiquitous in biological research, hence a huge statistical literature exists on models and methods for their analysis. Modern parametric models for longitudinal data are of three main types ([1]): marginal, random-effects, and antedependence (also called Markov or transition) models. This article is concerned with models of the third type, by which the conditional distribution of the response variable at any time, given values of the response in the (recent) past and possibly values of explanatory variables in the present and (recent) past, is modeled in terms of the given quantities. Specifically, time index-ordered random variables Y₁, …, Y_n are said to be antedependent of order p₁, p₂, …, p_n, or AD(p₁, p₂, …, p_n), if Y_k, given at least p_k immediately preceding variables, is independent of all further preceding variables for k = 1, 2, …, n ([2], [3]). For example, if n = 4, if Y₃ is conditionally independent of Y₁ given Y₂, and if Y₄ is independent of (Y₁, Y₂, Y₃), then Y₁, Y₂, Y₃, Y₄ are AD(0,1,1,0). Antedependence models can be useful for nonstationary longitudinal data because they require nothing beyond the aforementioned conditional independencies; in fact, even the order of antedependence, p_k, let alone its strength, is allowed to vary over time. Note that 0 ≤ p_k ≤ k − 1 necessarily, and that AD(p₁, …, p_n) models are partially nested in the sense that AD(p₁, …, p_n) ⊂ AD(p₁ + q₁, …, p_n + q_n) if q_k ≥ 0 for all k. The special case in which p_k = min(k − 1, p) for some constant p is known as pth-order antedependence and denoted as AD(p). AD(p) models are completely nested; that is, AD(0)⊂AD(1)⊂ · · · ⊂AD(n − 1), with AD(0) being equivalent to mutual independence and AD(n − 1) being equivalent to completely general dependence. Antedependence (of specified order) is said to be structured if additional restrictions (such as stationarity) are imposed; otherwise it is unstructured.

Statistical inference procedures associated with fitting AD models to normally-distributed longitudinal data are well-developed; for a summary see [4]. A large variety of structured AD models, including stationary autoregressive models, are available, and likelihood-based inference has been widely studied for both structured models and the unstructured model. Likelihood-based inference is especially “nice” for the latter model, owing to the existence of closed-form expressions for maximum likelihood estimators (mles) of the parameters (from complete data) and the fact that various model selection and hypothesis testing procedures, such as those used to determine the order of antedependence, can be expressed simply in terms of residual sums of squares from regressions of response variables on their immediate predecessors. The procedures can be extended easily to accommodate ignorably missing data.

In contrast, inference procedures associated with fitting AD models to categorical longitudinal data are relatively underdeveloped. There are several structured AD models (most of which are stationary) available for such data, including binary Markov models ([5]), Markov generalized linear models ([6]; [1], pp. 190–207; [7], pp. 236–238), mixture transition models ([8]), and marginalized transition models ([9]; [10]; [11]; [12]). But missing from the existing toolkit are categorical-data analogues of the aforementioned inference procedures for unstructured AD models for normally-distributed longitudinal data. Although Anderson and Goodman [13] derived mles for the special case of an unstructured AD(1) model for complete categorical data with no empty cells and considered some related testing problems, general expressions for mles of unstructured AD model parameters in categorical settings do not yet exist, nor are methods available for determining the order of antedependence, for determining whether transition probabilities are time-invariant, or for dealing with missing data or empty cells. The goal of this article is to fill in these gaps, so as to help lift transition-model methodology for nonstationary categorical longitudinal data to the same developmental level as its continuous (normal) counterpart.

The remainder of the article is organized as follows. In Section 2, we set notation and define several parameterizations convenient for use with AD models of categorical longitudinal data. In Section 3, we derive maximum likelihood estimators of AD model parameters in various scenarios, assuming initially that the data are complete and then relaxing this to allow for missing data. Penalized likelihood-based model selection methods and likelihood ratio tests of various hypotheses of interest are presented in Sections 4 and 5, respectively. Section 6 describes simulation studies of two of the hypothesis tests. Section 7 presents two examples that highlight the usefulness of AD models for categorical longitudinal data. Section 8 is a brief discussion.

2. Notation and parameterizations

Let N denote a predetermined, nonrandom number of subjects on which a categorical characteristic of interest is observed repeatedly over time. We assume that n times of observation are intended to be common across subjects, but we allow for missing data, i.e. the possibility that some subjects are not actually observed at some of the appointed times. Let Y_i ≡ (Y_i1, …, Y_in)′ denote the vector of values of the characteristic at the n observation times for the ith subject, only some of which may actually be observed. Let 1, …, c ≥ 2 denote the characteristic’s categories (which are assumed not to change over time). Hence Y_i has cⁿ possible outcomes, each of which corresponds to a cell in a c × c × · · · × c (n times) contingency table. Let Y_k denote the characteristic’s value at time point k for a generic subject. For each possible outcome (y₁, …, y_n), let π_{y1 · · · yn} ≡ P(Y₁ = y₁, …, Y_n = y_n) denote the true cell probability with corresponding cell count N_{y1 · · · yn} (which may or may not be observed), and put π = (π_{y1 · · · yn}). Accordingly, $N={\displaystyle \sum _{(y_1,\dots ,y_n)\in C_n}}{N}_{y_1\cdots y_n}$, where C_n ≡ {1, …, c}ⁿ is the set of all cⁿ possible outcomes. We assume that the Y_i’s are independently and identically distributed as Multinomial(1, π) and that covariates, apart from indicator variables for groups, are either unavailable or not used in the analysis.

Since antedependence is defined in terms of certain conditional independencies, it is convenient to reparameterize in terms of certain conditional probabilities. Define π_{yk|y1 · · · yk−1} ≡ P(Y_k = y_k|Y₁ = y₁, …, Y_k−1 = y_k−1) for k = 2, …, n and (y₁, …, y_k) ∈ C_k. It is easily verified that the mapping from the nonredundant cell-probability parameterization ϒ ≡ {π_{y1 · · · yn} : (y₁, …, y_n) ∈ C_n \ {c, …, c}} to the nonredundant “sequential conditional-probability” parameterization Θ ≡ {π_{y1 + · · · +} : y₁ = 1, …, c − 1} ∪ {π_{yk|y1 · · · yk−1} : k = 2, …, n; y_k = 1, …, c − 1; (y₁, …, y_k−1) ∈ C_k−1} is one-to-one. (Here and subsequently, we indicate summation over a subscripted index by replacing that index with a “+.”) Moreover, under an AD(p₁, …, p_n) model, for each k such that p_k ≥ 1 and k − p_k ≥ 2 and each fixed (y_{k−pk, …, yk−1}) ∈ C_pk,

$$\{{\pi }_{y_k\mid y_1\cdots y_{k-p_k-1}{y}_{k-p_k}\cdots y_{k-1}}:(y_1,\dots ,y_{k-p_k-1})\in C_{k-p_k-1}\}\text{are}\text{equal};$$ (1)

hence we may represent their common value by transition probability parameter π_{yk | + · · · + yk−pk · · · yk−1}. Thus, the AD(p₁, …, p_n) model may be parameterized by the nonredundant set of parameters

$$\begin{array}{l}{\mathrm{\Theta }}^{(\mathbf{p})}\equiv \underset{k\ni p_k=0}{\cup }\{{\pi }_{+\cdots +y_k+\cdots +}:y_k=1,\dots ,c-1\}\\ \cup \underset{k\ni p_k\ge 1}{\cup }\{{\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}:y_k=1,\dots ,c-1;(y_{k-p_k},\dots ,y_{k-1})\in C_{p_k}\},\end{array}$$

which we call the transition-probability parameterization. (Here and throughout, we may sometimes write p for (p₁, …, p_n).) It is easily verified that the dimensionality of Θ^(p) is $(c-1){\sum }_{k=1}^n{c}^{p_k}$. Finally, let Θ̄^(p) denote the extended set of parameters obtained by adding to Θ^(p) the redundant probabilities, i.e. probabilities of the same form as those in Θ^(p) but with y_k = c.

To clarify the notation, consider a case in which binary (c = 2) longitudinal data are observed on 4 occasions. In this case,

$$\begin{array}{l}\mathrm{\Upsilon }=\{{\pi }_{1111},{\pi }_{1112},{\pi }_{1121},{\pi }_{1122},{\pi }_{1211},{\pi }_{1212},{\pi }_{1221},{\pi }_{1222},{\pi }_{2111},{\pi }_{2112},{\pi }_{2121},{\pi }_{2122},{\pi }_{2211},{\pi }_{2212},{\pi }_{2221}\}\text{and}\\ \mathrm{\Theta }=\{{\pi }_{1+++},{\pi }_{1\mid 1},{\pi }_{1\mid 2},{\pi }_{1\mid 11},{\pi }_{1\mid 12},{\pi }_{1\mid 21},{\pi }_{1\mid 22},{\pi }_{1\mid 111},{\pi }_{1\mid 112},{\pi }_{1\mid 121},{\pi }_{1\mid 122},{\pi }_{1\mid 211},{\pi }_{1\mid 212},{\pi }_{1\mid 221},{\pi }_{1\mid 222}\}.\end{array}$$

Furthermore, under an AD(0,1,1,1) [or equivalently AD(1)] model,

$$\begin{array}{l}{\pi }_{1\mid 11}={\pi }_{1\mid 21}(\equiv {\pi }_{1\mid +1}),{\pi }_{1\mid 12}={\pi }_{1\mid 22}(\equiv {\pi }_{1\mid +2}),{\pi }_{1\mid 111}={\pi }_{1\mid 121}={\pi }_{1\mid 211}={\pi }_{1\mid 221}(\equiv {\pi }_{1\mid ++1}),\text{and}\\ {\pi }_{1\mid 112}={\pi }_{1\mid 122}={\pi }_{1\mid 212}={\pi }_{1\mid 222}(\equiv {\pi }_{1\mid ++2}).\end{array}$$

Thus,

$$\begin{array}{l}{\mathrm{\Theta }}^{(0,1,1,1)}\{{\pi }_{1+++},{\pi }_{1\mid 1},{\pi }_{1\mid 2},{\pi }_{1\mid +1},{\pi }_{1\mid +2},{\pi }_{1\mid ++1},{\pi }_{1\mid ++2}\}\text{and}\\ {\overline{\mathrm{\Theta }}}^{(0,1,1,1)}=\{{\pi }_{1+++},{\pi }_{2+++},{\pi }_{1\mid 1},{\pi }_{2\mid 1},{\pi }_{1\mid 2},{\pi }_{2\mid 2},{\pi }_{1\mid +1},{\pi }_{2\mid +1},{\pi }_{1\mid +2},{\pi }_{2\mid +2},{\pi }_{1\mid ++1},{\pi }_{2\mid ++1},{\pi }_{1\mid ++2},{\pi }_{2\mid ++2}\}.\end{array}$$

3. Maximum likelihood estimation

In this section, we give several results pertaining to maximum likelihood estimation of the transition-probability parameterization of an AD(p₁, …, p_n) process under the multinomial sampling framework described previously. Should mles of the corresponding cell-probability parameterization be desired, they may be obtained using the relation ${\pi }_{y_1\cdots y_n}={\displaystyle \prod _{k=1}^n}\left(I(p_k=0){\pi }_{+\cdots +y_k+\cdots +}+I(p_k\ge 1){\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}\right)$ [where I(·) is an indicator function] and the parameterization-invariance of maximum likelihood estimation. Initially we assume that the data are complete, subsequently extending to permit missing data; throughout, however, we allow for empty cells.

3.1. Complete data

3.1.1. Unstructured antedependence

Theorem 1

Under AD(p₁, …, p_n), complete-data mles of the parameters of Θ^(p) are as follows: for k such that p_k = 0, ${\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(\mathbf{p})}=N_{+\cdots +y_k+\cdots +}/N$; for other k,

$${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}=\{\begin{array}{ll}0\hfill & \text{if}{N}_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}=0\hfill \\ \frac{N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}}{N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}}\hfill & \text{otherwise}.\hfill \end{array}$$

Proof

The likelihood function in terms of the parameters of Θ̄^{(p₁· · · p_n)} is proportional to

$$\begin{array}{l}\prod _{(y_1,\dots ,y_n)\in C_n}{\left(\prod _{k=1}^n\left[I(p_k=0){\pi }_{+\cdots +y_k+\cdots +}+I(p_k\ge 1){\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}\right]\right)}^{N_{y_1\cdots y_n}}\\ =\prod _{k=1}^n\left[\left(I(p_{k}0)\prod _{y_k=1}^c{\pi }_{+\cdots +y_k+\cdots +}^{N_{+\cdots +y_k+\cdots +}}\right)+\left(I(p_k\ge 1)\prod _{(y_{k-p_k},\dots ,y_k)\in C_{p_k+1}}{\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}}\right)\right].\end{array}$$ (2)

For k such that p_k = 0, the kth term of the outermost product in (2) is the kernel of the likelihood of a saturated c-nomial distribution with cell probabilities {π_{+· · ·+ yk+· · ·+} : y_k = 1, …, c}; for other k, the kth term is the product of c^p_k independent likelihood kernels, each corresponding to a saturated c-nomial distribution with cell probabilities {π_{yk|+· · · +yk−pk · · · yk−1} : y_k = 1, …, c}. The cell probabilities for each kernel sum to one and lie within [0, 1), but are not otherwise constrained. Thus, by well-known results on maximum likelihood estimation for saturated multinomial distributions, we have the following: (a) for those k such that p_k = 0, ${\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(\mathbf{p})}=N_{+\cdots +y_k+\cdots +}/N$; (b) for other k, if N_{+· · · + yk−pk · · · yk−1 + · · · +} = 0 then N_{+· · · + yk−pk · · · yk + · · · +} = 0, implying that ${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}=0$, and if N_{+· · · + yk−pk · · · yk−1 + · · · +} ≠ 0 then ${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}=N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}/N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}$. This completes the proof.

Observe that for each fixed k ∈ {1, …, n}, the mles of the marginal probabilities {π_{+· · · + yk+· · · +} : y_k = 1, …, c} and conditional probabilities {π_{yk|+· · · +yk−pk · · · yk−1} : (y_{k− pk, …, yk−1}, y_k) ∈ C_pk+1} under AD(p₁, …, p_n) depend on (p₁, …, p_n) only through the value of p_k. Also note that upon substituting min(k − 1, p) for p_k (k = 1, …, n) in Theorem 1 and writing Θ̄^(p) for Θ̄^{(0,1, …,p,p,…,p)}, we obtain the following corollary for AD(p) variables. The special case of this corollary when p = 1 and no cells are empty was given by [13].

Corollary 1

Under AD(p), complete-data mles of the parameters of Θ̄^(p) are as follows: if p = 0, ${\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(p)}=N_{+\cdots +y_k+\cdots +}/N$ for k = 1, …, n; if p ≥ 1, ${\widehat{\pi }}_{y_1+\cdots +}^{(p)}=N_{y_1+\cdots +}/N$,

$${\widehat{\pi }}_{y_k\mid y_1\cdots y_{k-1}}^{(p)}=\{\begin{array}{ll}0\hfill & \text{if}{N}_{y_1\cdots y_{k-1}+\cdots +}=0\hfill \\ \frac{N_{y_1\cdots y_k+\cdots +}}{N_{y_1\cdots y_{k-1}+\cdots +}}\hfill & \text{otherwise}\hfill \end{array}$$

for k = 2, …, p + 1, and

$${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p}\cdots y_{k-1}}^{(p)}=\{\begin{array}{ll}0\hfill & \text{if}{N}_{+\cdots +y_{k-p}\cdots y_{k-1}+\cdots +}=0\hfill \\ \frac{N_{+\cdots +y_{k-p}\cdots y_k+\cdots +}}{N_{+\cdots +y_{k-p}\cdots y_{k-1}+\cdots +}}\hfill & \text{otherwise}\hfill \end{array}$$

for k = p + 2, …, n.

3.1.2. Time-invariant transition probabilities

Consider a situation with a constant order of antedependence, p. If measurement times are equally spaced, it may be of interest to estimate parameters under AD(p) with a stationarity property imposed. One such property is that of time-invariant transition probabilities. If p ≥ 1 and ${\pi }_{y_{p+1}^{(k)}\mid y_1^{(k-p)}\cdots y_p^{(k-1)}}$ denotes P(Y_k = y_p+1|Y_k−p = y₁, …, Y_k−1 = y_p) for k = p + 1, …, n, the pth-order transition probabilities are said to be time-invariant if

$${\pi }_{y_{p+1}^{(p+1)}\mid y_1^{(1)}\cdots y_p^{(p)}}={\pi }_{y_{p+1}^{(p+2)}\mid y_1^{(2)}\cdots y_p^{(p+1)}}=\cdots ={\pi }_{y_{p+1}^{(n)}\mid y_1^{(n-p)}\cdots y_p^{(n-1)}}\text{for}\text{all}(y_1,\dots ,y_{p+1})\in C_{p+1}.$$ (3)

Theorem 2

Under AD(p) with p ≥ 1 and time-invariant pth-order transition probabilities, the complete-data mle of ${\pi }_{y_1\cdots y_p+\cdots +}^{(p)}$ is ${\widehat{\pi }}_{y_1\cdots y_p+\cdots +}^{(p)}=N_{y_1\cdots y_p+\cdots +}/N$, and the complete-data mle of the common pth-order transition probability, denoted by ${\widehat{\pi }}_{y_{p+1}^{+}\mid y_1\cdots y_p}^{(p)}$, is as follows:

$${\widehat{\pi }}_{y_{p+1}^{+}\mid y_1\cdots y_p}^{(p)}=\{\begin{array}{ll}0\hfill & \text{if}{\sum }_{k=p+1}^n{N}_{+\cdots +y_1^{(k-p)}\cdots y_p^{(k-1)}+\cdots +}=0\hfill \\ \frac{{\sum }_{k=p+1}^n{N}_{+\cdots +y_1^{(k-p)}\cdots y_{p+1}^{(k)}+\cdots +}}{{\sum }_{k=p+1}^n{N}_{+\cdots +y_1^{(k-p)}\cdots y_p^{(k-1)}+\cdots +}}\hfill & \text{otherwise}.\hfill \end{array}$$

Thus, when pth-order transition probabilities are time-invariant, they may be pooled over time to yield the mle of the common pth-order transition probability.

3.1.3. Strict stationarity

Variables Y₁, …, Y_n are said to be strictly stationary if the joint probabilities of all events are invariant to time shifts. The following lemma (proved in Web Appendix A) gives necessary and sufficient conditions for strict stationarity under AD(p) in terms of the transition-probability parameterization.

Lemma 1

Under AD(p) with p ≥ 1, the variables Y₁, …, Y_n are strictly stationary if and only if the transition probabilities are time-invariant and

$${\pi }_{y_1\cdots y_p+\cdots +}=\sum _{y_0=1}^c{\pi }_{y_p\mid y_0{y}_1\cdots y_{p-1}}{\pi }_{y_0{y}_1\cdots y_{p-1}+\cdots +}\text{for}\text{all}(y_1,\dots ,y_p)\in C_p\backslash \{c,\dots ,c\}.$$ (4)

Unfortunately, mles of neither the transition nor cell probabilities of a strictly stationary AD(p) model can be expressed in closed form. However, since the constraints imposed by (1), (3), and (4) may be written as nonredundant homogeneous smooth functions of the expected cell counts, the mles can be obtained numerically using the algorithm of [14].

3.2. Missing data

3.2.1. Monotone missingness

All of the expressions for complete-data mles given to this point can be extended easily to handle ignorable monotone missing data (“dropouts”), defined by the condition that Y_i,k+1 is missing whenever Y_i,k is missing (i = 1, …, N; k = 2, …, n − 1). Let N^(k)• denote the number of subjects having complete observations between time points 1 and k (inclusive), and let $N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(k)•}$ denote the number of these subjects for which Y_{k−pk =} y_k−pk, …, Y_k = y_k (regardless of whether Y_k+1, …, Y_n are observed or missing). Then, the observed-data likelihood is given by an expression identical to (2) except that $N_{+\cdots +y_k+\cdots +}^{(k)•}$ and $N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(k)•}$ are substituted for the corresponding complete-data counts. Hence, under AD(p₁, …, p_n), the monotone-missing-data mles (assuming ignorability) of the parameters of Θ̄^(p), denoted by ${\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(\mathbf{p})•}$ and ${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})•}$, are given by expressions identical to those in Theorem 1 except that N^(k)•, $N_{+\cdots +y_k+\cdots +}^{(k)•},N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(k)•}$, and $N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}^{(k)•}$ are substituted for the corresponding complete-data counts. Similarly, under AD(p), the monotone-missing-data mles of the parameters of Θ̄^(p) are obtained by substituting the analogous quantities into Corollary 1; furthermore, monotone-missing-data mles of time-invariant transition probabilities are obtained by substituting those same quantities into Theorem 2.

Mles under AD(p) may also be obtained easily for ignorably missing data with monotone drop-ins (also known as delayed or staggered entry), defined by the condition that Y_i,k+1 is observed whenever Y_i,k is observed (i = 1, …, N ; k = 2, …, n − 1). For such data, mles are as noted above (with p_k = min(k − 1, p)) but applied to the data in reverse time order. This follows from the fact that AD(p) random variables are also AD(p) when arranged in reverse time order ([4], p. 151). There is not an analogous reversibility result, however, for the more general case of AD(p₁, …, p_n).

3.2.2. Arbitrary missingness

When the data have an arbitrary ignorable pattern of missingness, a restricted EM algorithm [15] may be used to obtain mles under AD(p₁, …, p_n). Customized to the present setting, the algorithm exploits the fact that the AD(p₁, …, p_n) multinomial model can be regarded as a linearly constrained case of the saturated multinomial model, specifically one with linear equality constraints on the parameters of the sequential conditional-probability parameterization, as specified by (1). The constraints may be written as Aθ = 0, where θ is the vector of nonredundant sequential conditional probabilities and A is a matrix of ones, zeros, and minus ones corresponding to the prescribed order of antedependence; for example, for a binary AD(0,1,1) model, θ = (π₁₊₊, π_1|1, π_1|2, π_1|11, π_1|21, π_1|12, π_1|22)′ and

$$\mathbf{A}=\left[\begin{array}{ccccccc}0& 0& 0& 1& -1& 0& 0\\ 0& 0& 0& 0& 0& 1& -1\end{array}\right].$$

Partition the cell counts N = (N_{y1· · · yn}) into ${({\mathbf{N}}_{\mathit{obs}}^{\prime },{\mathbf{N}}_{un\mathit{obs}}^{\prime })}^{\prime }$, their observed and unobserved parts. Then the conditional expectation of the complete-data log-likelihood given the observed data and the current parameter estimate may be written as

$$Q(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}^{(m)})=\sum _{(y_1,\dots ,y_n)\in C_n}E(N_{y_1\cdots y_n}\mid {\mathbf{N}}_{\mathit{obs}},{\widehat{\mathit{\theta }}}^{(m)})(log{\pi }_{y_1+\cdots +}+log{\pi }_{y_2\mid y_1}+log{\pi }_{y_3\mid y_1{y}_2}+\cdots +log{\pi }_{y_n\mid y_1\cdots y_{n-1}}).$$

The restricted EM algorithm, in general form, is then as follows (a detailed illustration for a specific case may be found in Web Appendix B):

E step

Evaluate $Q(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}_R^{(m)})$. (The initial restricted estimate, ${\widehat{\mathit{\theta }}}_R^{(0)}$, may be taken to correspond to uniform cell probabilities, i.e. π_{y1 · · · yn} ≡ c⁻ⁿ.)

Restricted M step

1. Obtain the current unrestricted estimate, θ̂_U, by maximizing $Q(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}_R^{(m)})$ using explicit expressions given by Schafer ([16], sec. 7.3) for the estimated cell probabilities of a saturated multinomial distribution and the relationship between π and θ described in Section 2, and calculate ${\mathbf{I}}_U={-\frac{{\partial }^{2}Q(\mathit{\theta }\mid {\widehat{\mathit{\theta }}}_R^{(m)})}{\partial {\mathit{\theta }}^2}|}_{\mathit{\theta }={\widehat{\mathit{\theta }}}_R^{(m)}}$. It is worth noting that I_U is diagonal due to the linearity of Q(θ|θ̂^(m)) in the logs of the elements of θ.

2. Obtain a provisional restricted estimate, θ̂_R, via the equation

   $${\widehat{\mathit{\theta }}}_R={\widehat{\mathit{\theta }}}_U-{\mathbf{I}}_U^{-1}{\mathbf{A}}^{\prime }{({\mathbf{AI}}_U^{-1}{\mathbf{A}}^{\prime })}^{-1}\mathbf{A}{\widehat{\mathit{\theta }}}_U.$$

3. If $Q({\widehat{\mathit{\theta }}}_R\mid {\widehat{\mathit{\theta }}}_R^{(m)})>Q({\widehat{\mathit{\theta }}}_R^{(m)}\mid {\widehat{\mathit{\theta }}}_R^{(m)})$, then ${\widehat{\mathit{\theta }}}_R^{(m+1)}={\widehat{\mathit{\theta }}}_R$.

4. If not, then do step halving (and, if necessary, step quartering, etc.) on Δθ̂_U, where $\mathrm{\Delta }{\widehat{\mathit{\theta }}}_U={\widehat{\mathit{\theta }}}_U-{\widehat{\mathit{\theta }}}_R^{(m)}$.

The iteration between the E-step and restricted M-step is continued, and the estimate of θ updated as long as $Q({\widehat{\mathit{\theta }}}_R\mid {\widehat{\mathit{\theta }}}_R^{(m)})$ increases, until a convergence criterion is satisfied.

Restricted EM algorithms similar to the one just described may be devised without difficulty for use with the time-invariant transition probability AD(p) model as well, since for this model the constraints on the parameters likewise are linear. However, the strictly stationary AD(p₁, …, p_n) multinomial model imposes nonlinear constraints (4) in addition to linear constraints (1) and (3), so the same restricted EM algorithm is not applicable to it.

4. Penalized likelihood-based model selection

Consider the same data-model situation as that considered in the previous section, but now suppose that the order (p₁, …, p_n) of the AD model is not known and it is the goal of the investigator to determine the order that best fits the data, in some sense. Standard likelihood-based hypothesis testing is not entirely suitable for this purpose because AD(p₁, …, p_n) models are not completely nested. As an alternative, in this section we propose the use of penalized likelihood criteria; specifically, we consider information criteria of the form $IC(p_1,\dots ,p_n)=a(N)(c-1){\displaystyle \sum _{k=1}^n}{c}^{p_k}-2logL({\widehat{\mathit{\pi }}}^{(\mathbf{p})})$, where a(N) is a specified function of N. Many well-known penalized likelihood criteria, when applied to AD(p₁, …, p_n) models, are cases of this general form, including AIC, BIC, corrected AIC, quasi-AIC, and quasi-BIC. Note that the criteria are expressed in “smaller is better” form. In our example we feature AIC, for which a(N) = 2.

Since p_k can be any nonnegative integer less than or equal to k − 1, the number of AD models to compare is n!. Computing the information criterion for all n! models can be burdensome or even impractical when n is not small; for example, when n = 7 and the data are complete, about 24 hours of computing time on an Intel(R) Xeon(R) with W3520 processor (speed 2.67 GHz, memory 8158412 kB) are required to obtain AIC(p₁, …, p_n) for all 5040 models. However, it turns out that the AD model that minimizes IC(p₁, …, p_n) can be determined by optimizing p_k separately for each k, so that only [n(n + 1)=2] − 1 models must be fitted and much computational time can be saved. This is a consequence of the following theorem, which follows immediately from (2), Theorem 1, and results discussed in Section 3.2. In the theorem and subsequently, put $C_{u,k}^{+}=\{(y_u,y_{u+1},\dots ,y_k)\in C_{k-u+1}:N_{+\cdots +y_u\cdots y_k+\cdots +}>0\}$ and write $C_k^{+}$ for $C_{k,k}^{+}$.

Theorem 3

The minimizer of IC(p₁, …, p_n) is given by ${\widehat{p}}_k=\underset{p_k=0,\dots ,k-1}{\text{argmin}}IC(p_k)$ for k = 1, …, n, where, if the data are complete,

$$IC(p_k)=\frac{a(N)}{2}(c-1)c^{p_k}-[I(p_k=0)\sum _{y_k\in C_k^{+}}{N}_{+\cdots +y_k+\cdots +}log\left(\frac{N_{+\cdots +y_k+\cdots +}}{N}\right)$$ (5)

$$+I(p_k\ge 1)\sum _{(y_{k-p_k},\dots ,y_k)\in C_{k-p_k,k}^{+}}{N}_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}log\left(\frac{N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}}{N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}}\right)].$$ (6)

If the data are ignorably monotone missing, then N, N_{+ · · · + yk + · · · +}, N_{+ · · · + yk − pk · · · yk+ · · · +}, and N_{+ · · · + yk − pk · · · yk − 1 + · · · +} in (5) and (6) are replaced by N^(k)•, $N_{+\cdots +y_k+\cdots +}^{(k)•},N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(k)•}$, and $N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}^{(k)•}$; if the missingness is arbitrary, then those same quantities are replaced by their mles N̂_{+ · · · + yk+ · · · +}, N̂_{+ · · · + yk − pk · · · yk+ · · · +}, and N̂_{+ · · · + yk−pk · · · yk − 1+ · · · +} under AD(p₁, …, p_n), which may be obtained using the restricted EM algorithm as described in Section 3.2.2.

5. Hypothesis tests

Next we consider likelihood ratio tests of hypotheses concerning antedependence under the same data-model situation of the previous section. We first consider tests for the order of antedependence, beginning with the most general case of nested variable-order models and then specializing to constant-order models. Then we consider tests for homogeneity of antedependence across groups and for two forms of stationarity. For simplicity of presentation, we assume in this section that the data are complete, unless noted otherwise. For results on limiting distributions of test statistics, we assume standard multinomial sampling asymptotics, including fixed c and n as N → ∞, and all cell probabilities positive.

5.1. Order of antedependence

Consider testing AD(p₁, …, p_n) as a null hypothesis against AD(q₁, …, q_n) as the alternative hypothesis, where p_k ≤ q_k for all k and the inequality is strict for at least one k. Using (2) and Theorem 1, it is easily verified that the likelihood ratio test statistic is given by

$$G_{\mathbf{p},\mathbf{q}}^2\equiv 2\sum _{k=2}^{n}I(q_k>p_k)\sum _{(y_{k-q_k},\dots ,y_k)\in C_{k-q_k,k}^{+}}{N}_{+\cdots +y_k-q_k\cdots y_k+\cdots +}log\left(\frac{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(q)}}{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(p)}}\right)$$

Where ${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(\mathbf{p})}=I(p_k=0){\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(\mathbf{p})}+I(p_k\ge 1){\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}$. The limiting null distribution of $G_{\mathbf{p},\mathbf{q}}^2$ is chi-square with $d(\mathbf{p},\mathbf{q})\equiv \mathit{dim}({\mathrm{\Theta }}^{(\mathbf{q})})-\mathit{dim}({\mathrm{\Theta }}^{(\mathbf{p})})=(c-1){\sum }_{k=2}^n(c^{q_k}-c^{p_k})$ degrees of freedom. Hence the approximate size-α likelihood ratio test for AD(p₁, …, p_n) versus AD(q₁, …, q_n) rejects the former if $G_{\mathbf{p},\mathbf{q}}^2$ exceeds the 100(1 − α)th percentile of this chi-square distribution. However, the simulation study presented in the next section indicates that the Type I error rate of this test is somewhat larger than α. Aided by the following lemma (proved in Web Appendix B), we use Lawley’s [17] strategy of multiplying the likelihood ratio test statistic by a scale factor to obtain a modified version of the likelihood ratio test for which the Type I error rate is closer to its nominal level.

Lemma 2

$E(G_{\mathbf{p},\mathbf{q}}^2)=e(\mathbf{p},\mathbf{q})+O(N^{-2})$, where

$$\begin{array}{l}e(\mathbf{p},\mathbf{q})=2\sum _{k=2}^{n}I(q_k>p_k)[\sum _{(y_{k-q_k},\dots ,y_k)\in C_{k-q_k,k}^{+}}(f(N,{\pi }_{+\cdots +y_{k-q_k}\cdots y_k+\cdots +}^{(\mathbf{q})})\\ -{\pi }_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(\mathbf{q})}f(N,{\pi }_{+\cdots +y_{k-q_k}\cdots y_{k-1}+\cdots +}^{(\mathbf{q})}))-\sum _{(y_{k-p_k},\dots ,y_k)\in C_{k-p_k,k}^{+}}(f(N,{\pi }_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(\mathbf{p})})\\ -{\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}f(N,{\pi }_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}^{(\mathbf{p})})\left)\right]\end{array}$$

and

$$f(N,\pi )=N\pi log(N\pi )+\frac{1-\pi }{2}+\frac{1-{\pi }^2}{12N\pi }.$$

Based on Lemma 2, we define the modified likelihood ratio test statistic as

$${\overline{G}}_{\mathbf{p},\mathbf{q}}^2=\frac{d(\mathbf{p},\mathbf{q})}{\widehat{e}(\mathbf{p},\mathbf{q})}{G}_{\mathbf{p},\mathbf{q}}^2$$

where ê(p, q) is obtained by replacing the unknown probabilities in e(p, q) with their mles under AD(q₁, …, q_n).

In the special case of testing AD(p) versus AD(q), where 0 ≤ p < q ≤ n − 1, the likelihood ratio test statistic becomes

$$\begin{array}{l}{G}_{p,q}^2\equiv 2[\sum _{k=p+1}^q\sum _{(y_1,\dots ,y_k)\in C_{1,k}^{+}}{N}_{y_1\cdots y_k+\cdots +}log\left(\frac{{\widehat{\pi }}_{y_k\mid y_1\cdots y_{k-1}}^{(q)}}{{\widehat{\pi }}_{y_k\mid y_1\cdots y_{k-1}}^{(p)}}\right)\\ +\sum _{k=q+1}^n\sum _{(y_{k-q},\dots ,y_k)\in C_{k-q,k}^{+}}{N}_{+\cdots +y_{k-q}\cdots y_k+\cdots +}log\left(\frac{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q}\cdots y_{k-1}}^{(q)}}{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q}\cdots y_{k-1}}^{(p)}}\right)],\end{array}$$

for which the limiting null chi-square distribution has d(p, q) ≡ (n − p − 1)c^p − (n − p)c^p+1 − (n − q − 1)c^q + (n − q)c^q+1 degrees of freedom. The case q = p + 1 is worthy of special mention. It can be easily seen that the likelihood ratio test criteria for the sequence of tests of AD(p) versus AD(p + 1) (p = 0, 1, …, n − 2) sum to the likelihood ratio test criterion for AD(0) versus AD(n − 1). Thus, this sequence can be viewed as a decomposition of the well-known likelihood ratio test for complete independence (against the fully saturated alternative) into steps of order one. This suggests two practical strategies for selecting the order of antedependence. Analogously to the selection of order of a polynomial model for the mean structure of a regression model, one may use either a forward selection strategy [starting by testing AD(0) versus AD(1), and if AD(0) is rejected then testing AD(1) versus AD(2), etc.] or a backward elimination strategy [starting with a test of AD(n − 2) versus AD(n − 1), and if AD(n − 2) is not rejected then testing AD(n − 3) versus AD(n − 2), etc.]

If data are (ignorably) missing, $G_{\mathbf{p},\mathbf{q}}^2$ may be generalized as follows:

$$\begin{array}{l}{G}_{\mathbf{p},\mathbf{q}}^2=2\sum _{k=2}^{n}I(q_k>p_k)(\sum _{(y_{k-q_k},\dots ,y_k)\in C_{k-q_k,k}^{+}}{\widehat{N}}_{+\cdots +y_{k-q_k}\cdots y_k+\cdots +}^{(\mathbf{q})}log{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(\mathbf{q})}\\ -\sum _{(y_{k-p_k},\dots ,y_k)\in C_{k-p_k,k}^{+}}{\widehat{N}}_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{(\mathbf{p})}log{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-q_k}\cdots y_{k-1}}^{(\mathbf{p})})\end{array}$$

where the superscript on each estimated cell count and transition probability indicates the model under which the restricted EM algorithm is applied to obtain the estimate. Similar generalizations to accommodate missing data may be made to all likelihood ratio tests presented herein, so henceforth we display only complete-data versions of test statistics.

5.2. Homogeneity across groups

Suppose that the subjects can be classified into s distinct groups, and that the orders of antedependence are identically (p₁, …, p_n) across groups. Let N^g and $N_{y_1\cdots y_n}^g$ denote the sample size and the observed cell count for outcome (y₁, …, y_n), respectively, for group g, and consider testing whether the parameters are homogeneous across groups, i.e. testing H₀ : $H_0:{\pi }_{y_1+\cdots +}^{g,(\mathbf{p})}={\pi }_{y_1+\cdots +}^{(\mathbf{p})}$ and ${\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{g,(\mathbf{p})}={\pi }_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}(k=2,\dots ,n;g=1,\dots ,s)$ against H₁:not H₀. Under H₀, the mles are obtained merely by pooling across groups. Under H₁, the likelihood is obtained as the product of the s AD(p₁, …, p_n) multinomial likelihoods and is maximized by ${\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{g,(\mathbf{p})}=N_{+\cdots +y_k+\cdots +}^g/N^g$ and

$${\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{g,(\mathbf{p})}=\{\begin{array}{ll}0\hfill & \text{if}{N}_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}^g=0\hfill \\ \frac{N_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^g}{N_{+\cdots +y_{k-p_k}\cdots y_{k-1}+\cdots +}^g}\hfill & \text{otherwise}\hfill \end{array}$$

for g = 1, …, s. Thus the likelihood ratio statistic is

$$\begin{array}{l}{G}_{\mathit{hom},\mathbf{p}}^2=2\sum _{g=1}^s\sum _{k=1}^n\left[I(p_k=0)\sum _{y_k\in C_k^{g,+}}{N}_{+\cdots +y_k+\cdots +}^{g}log\left(\frac{{\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{g,(\mathbf{p})}}{{\widehat{\pi }}_{+\cdots +y_k+\cdots +}^{(\mathbf{p})}}\right)\right]\\ +I(p_k\ge 1)\sum _{(y_{k-p_k},\dots ,y_k)\in C_{k-p_k,k}^{g,+}}{N}_{+\cdots +y_{k-p_k}\cdots y_k+\cdots +}^{g}log\left(\frac{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{g,(\mathbf{p})}}{{\widehat{\pi }}_{y_k\mid +\cdots +y_{k-p_k}\cdots y_{k-1}}^{(\mathbf{p})}}\right)].\end{array}$$

Here $C_k^{g,+}$ and $C_{k-p_k,k}^{g,+}$ have similar meaning as $C_k^{+}$ and $C_{k-p_k,k}^{+}$ but for group g only. The limiting null distribution of both test statistics is chi-square with $(s-1)(c-1){\displaystyle \sum _{k=1}^n}{c}^{p_k}$ degrees of freedom. As with likelihood ratio tests for order of antedependence, Type I error rates associated with $G_{\mathit{hom},\mathbf{p}}^2$ are somewhat larger than nominal levels. A result similar to Lemma 2 can be obtained, leading to a modified likelihood ratio test for homogeneity whose actual and nominal sizes agree more closely.

5.3. Time-invariant transition probabilities

Consider testing the null hypothesis of time-invariant pth-order transition probabilities, as specified by (3), against the alternative of an unstructured AD(p) model. By Corollary 1, the unrestricted mle of a generic pth-order transition probability at time k under AD(p) is ${\widehat{\pi }}_{y_{p+1}^{(k)}\mid y_1^{(k-p)}\cdots y_p^{(k-1)}}^{(p)}=N_{+\cdots +y_1^{(k-p)}\cdots y_p^{(k-1)}{y}_{p+1}^{(k)}+\cdots +}/N_{+\cdots +y_1^{(k-p)}\cdots y_p^{(k-1)}+\cdots +}$ (unless the denominator is zero, in which case the mle is equal to 0). Thus, the likelihood ratio test statistic is

$$G_t^2=-2\sum _{(y_1,\dots ,y_{p+1})}\sum _k{N}_{+\cdots +y_1^{(k-p)}\cdots y_{p+1}^{(k)}+\cdots +}log\left(\frac{{\widehat{\pi }}_{y_{p+1}^{+}\mid y_1\cdots y_p}^{(p)}}{{\widehat{\pi }}_{y_{p+1}^{(k)}\mid y_1^{(k-p)}\cdots y_p^{(k-1)}}^{(p)}}\right)$$

where the summation is over those (y₁, …, y_p+1) ∈ C_p+1 and k = p + 1, …, n such that $N_{+\cdots +y_1^{(k-p)}\cdots y_{p+1}^{(k)}+\cdots +}>0$. The limiting null distribution of $G_t^2$ is chi-square with degrees of freedom equal to the number of nonredundant conditions in (3), or (c − 1)[n − (p + 1)]c^p.

5.4. Strict stationarity

Recall from Section 3.1.3 that the mles of cell probabilities of an AD(p) model under strict stationarity cannot be expressed in closed form, but they can be obtained numerically using Lang’s [14] algorithm. Let π̂^s,(p) be the vector of mles so obtained. Then, the likelihood ratio statistic for testing for strict stationarity under an unstructured AD(p) model is $G_s^2\equiv -2log\left(\frac{L({\widehat{\mathit{\pi }}}^{s,(p)})}{L({\widehat{\mathit{\pi }}}^{(p)})}\right)$. The limiting null distribution of $G_s^2$ is chi-square with (c − 1)[n − (p + 1)]c^p + (c^p − 1) degrees of freedom.

6. Simulation studies

Here we present simulation studies to evaluate the performance of the likelihood ratio test (and its modified version), with and without missing data, for two of the hypotheses described in Section 5. Studies of the tests for the other hypotheses yield similar conclusions and are not reported here.

6.1. Order of antedependence

Simulations were generated from two binary linear AD processes observed at four time points. The first process is defined as follows:

$$Y_1=\{\begin{array}{ll}1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2},\hfill \end{array}{Y}_2\mid Y_1=\{\begin{array}{ll}{Y}_1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{3}\hfill \\ 1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{3}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{3},\hfill \end{array}$$ (7)

$$Y_3\mid (Y_1,Y_2)=\{\begin{array}{ll}{Y}_1\hfill & \mathrm{w}.\mathrm{p}.\frac{\theta }{4}\hfill \\ Y_2\hfill & \mathrm{w}.\mathrm{p}.\frac{2-\theta }{4}\hfill \\ 1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4},\hfill \end{array}{Y}_4\mid (Y_1,Y_2,Y_3)=\{\begin{array}{ll}{Y}_1\hfill & \mathrm{w}.\mathrm{p}.\frac{\theta }{8}\hfill \\ Y_2\hfill & \mathrm{w}.\mathrm{p}.\frac{\theta }{8}\hfill \\ Y_3\hfill & \mathrm{w}.\mathrm{p}.\frac{2-\theta }{4}\hfill \\ 1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}.\hfill \end{array}$$ (8)

Here θ ∈ [0, 2] controls the degree of departure from first-order antedependence: when θ = 0 the process is AD(1), when θ > 0 the process is AD(3), and as θ increases, the departure from AD(1) is larger in the sense that the conditional (on intervening variables) dependence between variables lagged two or three time points apart increases. Note that the marginal distribution of Y_k − 1 is Bernoulli( $\frac{1}{2}$) for all k, regardless of θ. The second process differs from the first only by redefining Y₃ and Y₄ as follows:

$$Y_3\mid (Y_1,Y_2)=\{\begin{array}{ll}{Y}_1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ Y_2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ 1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4},\hfill \end{array}{Y}_4\mid (Y_1,Y_2,Y_3)=\{\begin{array}{ll}{Y}_1\hfill & \mathrm{w}.\mathrm{p}.\frac{\theta }{4}\hfill \\ Y_2\hfill & \mathrm{w}.\mathrm{p}.\frac{2-\theta }{8}\hfill \\ Y_3\hfill & \mathrm{w}.\mathrm{p}.\frac{2-\theta }{8}\hfill \\ 1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{4}.\hfill \end{array}$$ (9)

This difference renders the process AD(2) [rather than AD(1)] when θ = 0.

For each process, each of four sample sizes (N = 50, 100, 200, 1000), and each θ over a range of values, 10,000 samples were generated, and (nominal) size-0.05 likelihood ratio and modified likelihood ratio tests of AD(1) versus AD(3) for (7)&(8) and of AD(2) versus AD(3) for (7)&(9) were performed. In order to investigate the effect of missing data, we also performed the tests after deleting 25% of the observations in each sample, subject to the constraint that none of the subjects have more than two observations deleted. Empirical rejection rates are displayed in Table 1. These indicate that the actual size of the likelihood ratio test is significantly higher than the nominal size when N ≤ 200. In contrast, the actual size of the modified likelihood ratio test is consistently at or below the nominal size. As for power, it is perceptibly smaller for the modified test except when N = 1000. For both processes and both tests, the power increases with N and θ, as expected. Also as expected, there is a measurable loss of power when 25% of the observations are missing. Finally, a comparison of the results for the two processes suggests that the power for detecting a departure from a specified order of antedependence is higher when the difference in order for that departure is larger.

Table 1.

Empirical rejection rates of likelihood ratio test (LRT) and its modification (MLRT) for order of antedependence for data simulated from processes (7)&(8) and (7)&(9). Empirical sizes (rejection rates when θ = 0) more than two estimated standard errors from the nominal size (0.05) are set in bold type.

N	θ	(7)&(8)				(7)&(9)
		Complete		Missing		Complete		Missing
		LRT	MLRT	LRT	MLRT	LRT	MLRT	LRT	MLRT
50	0	0.073	0.029	0.066	0.029	0.097	0.038	0.088	0.032
	0.2	0.1	0.047	0.087	0.041	0.111	0.044	0.102	0.042
	0.4	0.138	0.068	0.122	0.058	0.14	0.062	0.122	0.052
	0.6	0.202	0.113	0.175	0.094	0.183	0.087	0.163	0.075
	0.8	0.285	0.181	0.248	0.147	0.24	0.123	0.205	0.103
	1	0.396	0.268	0.337	0.218	0.322	0.185	0.265	0.137
100	0	0.078	0.04	0.076	0.032	0.081	0.046	0.086	0.043
	0.2	0.103	0.057	0.100	0.049	0.099	0.06	0.104	0.052
	0.4	0.164	0.099	0.154	0.083	0.146	0.098	0.144	0.080
	0.6	0.267	0.183	0.247	0.150	0.221	0.159	0.204	0.121
	0.8	0.434	0.328	0.366	0.246	0.334	0.25	0.292	0.187
	1	0.628	0.518	0.512	0.389	0.477	0.385	0.402	0.280
200	0	0.068	0.044	0.072	0.043	0.063	0.049	0.069	0.049
	0.2	0.102	0.073	0.096	0.061	0.091	0.073	0.085	0.065
	0.4	0.231	0.184	0.193	0.137	0.186	0.16	0.158	0.129
	0.6	0.47	0.406	0.366	0.288	0.351	0.31	0.278	0.231
	0.8	0.745	0.687	0.603	0.518	0.565	0.524	0.443	0.390
	1	0.924	0.895	0.814	0.754	0.772	0.738	0.641	0.588
1000	0	0.053	0.051	0.057	0.051	0.055	0.051	0.057	0.050
	0.2	0.277	0.272	0.213	0.203	0.2	0.196	0.151	0.141
	0.4	0.865	0.862	0.731	0.714	0.682	0.675	0.544	0.527
	0.6	0.999	0.999	0.988	0.986	0.97	0.968	0.906	0.898
	0.8	1	1	1	1	0.999	0.999	0.994	0.993
	1	1	1	1	1	1	1	1	1

6.2. Time-invariant transition probabilities

Define binary AD(1) random variables Y₁, Y₂, Y₃ and Y₄ as follows:

$$Y_1=\{\begin{array}{ll}1\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2}\hfill \\ 2\hfill & \mathrm{w}.\mathrm{p}.\frac{1}{2},\hfill \end{array}P(Y_t=1\mid Y_1,\dots ,Y_{t-1})=\{\begin{array}{ll}\frac{1}{2}+\frac{(t-1)\lambda }{10}\hfill & \text{if}{Y}_{t-1}=1\hfill \\ \frac{1}{3}+\frac{(t-1)\lambda }{10}\hfill & \text{if}{Y}_{t-1}=2\hfill \end{array}$$ (10)

where 0 ≤ λ ≤ 5/3. For each fixed (t, λ), P(Y_t = 1|Y_{t − 1} = 1) ≠ P(Y_t = 1|Y_{t − 1} = 2); hence the variables are not AD(0) for any λ. When λ = 0, however, the transition probabilities are time-invariant; they become more variable over time as λ increases. Ten thousand realizations were simulated from (10) for λ = 0, 0.1, …, 1.0 and N = 50, 200, and 1000; and the likelihood ratio test for time-invariant transition probabilities was performed, both for complete data and for data with 25% of the observations deleted as before.

Empirical rejection rates for the tests are listed in Table 2. Results are as expected, with power increasing as either λ, N, or the proportion of non-missing data increases.

Table 2.

Empirical rejection rates for tests of time-invariant transition probabilities for data simulated from process (10). Empirical sizes more than two estimated standard errors from the nominal size (0.05) are set in bold type.

	N = 50		N = 200		N = 1000
λ	Complete	Missing	Complete	Missing	Complete	Missing
0	0.058	0.060	0.053	0.055	0.048	0.047
0.1	0.062	0.061	0.061	0.059	0.093	0.086
0.2	0.067	0.063	0.090	0.080	0.260	0.207
0.3	0.079	0.073	0.140	0.117	0.560	0.436
0.4	0.095	0.091	0.220	0.174	0.839	0.708
0.5	0.118	0.106	0.333	0.253	0.969	0.902
0.6	0.149	0.132	0.470	0.363	0.997	0.982
0.7	0.191	0.172	0.620	0.489	1	0.998
0.8	0.238	0.221	0.762	0.618	1	1
0.9	0.299	0.236	0.870	0.746	1	1
1	0.375	0.298	0.941	0.852	1	1

7. Examples

7.1. Toenail infection data

Molenberghs and Verbeke ([7], p. 8) report on a longitudinal study comparing two oral treatments (labeled here as A and B) for toenail dermatophyte onychomycosis. Severity of the infection (1 = severe, 2 = not severe) was scheduled to be observed at seven time points: baseline, at which time a three-month treatment period began; months 1, 2, and 3 during treatment; and months 6, 9, and 12 after initial treatment. The aims of the study were to compare the effectiveness of the treatments at reducing infection severity and to see how this evolves over time. For both treatment groups, all but 14 of the 2⁷ = 128 cells are empty. Furthermore, of the 146 and 148 people who received treatments A and B, respectively, only 107 and 117 provided their severity status at all time points (see Web Tables 1 and 2), so some data are missing. We assume that the missingness is ignorable.

The pattern of missingness is not monotone, so the restricted EM algorithm described in Section 3.2.2 was used to fit AD(p₁, …, p_n) models of all orders. The fitted models were then compared using the methods of Section 4. For treatment A, AIC selects AD(0, 1, 1, 1, 1, 1, 2); for treatment B it selects AD(0, 1, 2, 1, 1, 1, 1). Furthermore, forward selection and backward elimination strategies for building a constant-order AD model, based on size-0.05 modified likelihood ratio tests at each step, both select an AD(1) model for each group. In light of this, we perform the modified likelihood ratio test for homogeneity across groups assuming first-order antedependence, finding no evidence against it (P = 0.64) and concluding therefore that the two treatments do not differ significantly in effectiveness. The logical next step, namely testing for time-invariance of the transition probabilities under the homogeneous AD(1) model, is somewhat problematic due to the unequal spacing between measurements. Time-invariance can be tested straightforwardly over the first 4 measurements (which are equally spaced); for this test P = 0.026, suggesting that the lag-one transition probabilities up to the fourth measurement occasion are not time-invariant. Invariance with respect to time index rather than actual time, based on all 7 occasions, is rejected even more emphatically (P < 2.0 × 10⁻⁸) and an examination of the mles of the transition probabilities (Table 3) indicates why. The estimated transition probabilities for severe infection at time index t, given non-severe infection at the previous time, are small (less than 0.05) and do not change much over time. However, the estimated transition probabilities for severe infection at time index t, given severe infection at the previous time, vary substantially: they decrease over the first four time periods but increase considerably over the last two time periods, even though the last two periods are much longer. Thus, it appears that the “time-specific cure rate,” defined as 1.0 minus the probability of severe infection at time index t given that the infection was severe at the previous time, increases by approximately 8 to 12 percent each month during treatment and for three months thereafter, after which the curative effect of treatment rapidly wears off. Arguably, this phenomenon is more interesting than the behavior of the estimated marginal probabilities of severe infection, which decrease steadily over the first six months (including the first 3-month period after the cessation of treatment) before leveling off for the remaining six months.

Table 3.

Maximum likelihood estimates of marginal and transition probabilities under the homogeneous AD(1) model for the toenail infection data. Time is given in months after initial treatment.

Time	P(Yt = 1)	Transition probabilities
0	0.369	—	—
1	0.328	${\widehat{\pi }}_{1\mid 1}^{(1)}=0.869$	${\widehat{\pi }}_{1\mid 2}^{(1)}=0.022$
2	0.280	${\widehat{\pi }}_{1\mid +1}^{(1)}=0.792$	${\widehat{\pi }}_{1\mid +2}^{(1)}=0.042$
3	0.203	${\widehat{\pi }}_{1\mid ++1}^{(1)}=0.699$	${\widehat{\pi }}_{1\mid ++2}^{(1)}=0.015$
6	0.069	${\widehat{\pi }}_{1\mid +++1}^{(1)}=0.361$	${\widehat{\pi }}_{1\mid +++2}^{(1)}=0.009$
9	0.054	${\widehat{\pi }}_{1\mid ++++1}^{(1)}=0.625$	${\widehat{\pi }}_{1\mid ++++2}^{(1)}=0.016$
12	0.058	${\widehat{\pi }}_{1\mid +++++1}^{(1)}=0.722$	${\widehat{\pi }}_{1\mid +++++2}^{(1)}=0.028$

It is of interest to compare the homogeneous unstructured AD(1) model we’ve settled on to structured AD models that have been fit previously to these data. Molenberghs and Verbeke ([7], pp. 238–242) fitted three structured AD(1) logistic models, which are listed below. In these models, Y_ij represents infection severity (coded as 0 = not severe, 1 = severe) for subject i at time t_ij and is assumed to be Bernoulli with parameter μ_ij, and T_i is an indicator variable for Treatment B. The dependence on the previous observation is time-invariant for Model I (hence Molenberghs and Verbeke call it a stationary AR(1) model), but varies over time for Models II and III; in Model II this variation is attributed to the level of the previous outcome and in Model III it is attributed to the difference in spacing of measurements during the treatment and post-treatment periods.

1. logit(μ_ij) = β₀ + β₁T_i + β₂t_ij + β₃T_it_ij + α₀y_{i,j − 1}

2. logit(μ_ij) = (β₀₀ + β₁₀T_i + β₂₀t_ij + β₃₀T_it_ij)I(y_{i,j − 1} = 0) + (β₀₁ + β₁₁T_i + β₂₁t_ij + β₃₁T_it_ij)I(y_{i,j − 1} = 1)

   = β₀₀ + β₁₀T_i + β₂₀t_ij + β₃₀T_it_ij + (α₀ + α₁T_i + α₂t_ij + α₃T_it_ij)y_{i,j − 1}, where α_i = β_i1 − β_i0

3. logit(μ_ij) = β₀ + β₁T_i + β₂t_ij + β₃T_it_ij + α₀y_{i,j − 1} + α₁y_{i,j − 1}I(j ∈ {1, 2, 3, 4})

As it happens, the effects of treatment, time, or their interaction were not statistically significant in any of these models. However, α₀ was significant in all three models, and α₁ in Model III was significant also. Values of AIC for Models I–III and the homogeneous unstructured AD(1) model (based on the likelihood for the available data, conditional on baseline) are, respectively, 519.64, 524.15, 511.26, and 502.44, suggesting that the unstructured model fits best. Moreover, likelihood ratio tests of each of Models I–III against the unstructured model result in emphatic rejection of the structured models (P < 0.0001, P < 0.00001, and P < 0.002 respectively). The relatively superior fit of the unstructured model is likely due, as noted previously, to nonstationarity induced by an accumulating curative effect as treatment continues, which lingers for several months after the cessation of treatment before dissipating. Although Models II and III allow for time-varying dependence on the previous observation, they apparently are not flexible enough to adequately fit this type of nonstationary behavior.

7.2. Alzheimer’s disease data

The National Alzheimer’s Coordinating Center reposits global clinical dementia ratings (GCDR) data from patients at 29 Alzheimer’s Disease Centers throughout the United States. The GCDR is an overall staging of the severity of Alzheimer’s disease, computed based on the memory, orientation, judgment and problem solving, community affairs, home and hobbies and personal cares of the patients, and recorded at five levels of impairment (0 = no impairment, 0.5 = questionable impairment, 1.0 = mild impairment, 2.0 = moderate impairment and 3.0 = severe impairment). GCDR was scheduled to be observed annually from 2006 to 2012 for each patient. We are interested in examining how the GCDR for a subject at a given time depends on that subject’s GCDRs at previous times. Only 1097 out of 24787 patients had their GCDR observed at all time points, so much of the data is missing. We assume that the missingness is ignorable.

The missingness in these data is not monotone, so the restricted EM algorithm was used to fit AD(p₁, …, p_n) models of all orders, and the fitted models were compared using the methods of Section 4. AIC selects AD(0,1,2,2,2,3,2), and forward selection and backward elimination methods for constant-order AD models select an AD(2) model. Although strict stationarity under AD(2) is rejected (P < 10⁻⁵), time-invariance of the second-order transition probabilities is not rejected (P = 0.203). Accordingly, and with a view toward eventually modeling these data (which are ordinal) in terms of their cumulative logits, we examine the second-order cumulative transition probabilities P(Y_t ≤ j|Y_{t − 2} = y_{t − 2}, Y_{t − 1} = y_{t − 1}), pooled across time points t = 3, 4, 5, 6, 7 (see Table 4); for ease in interpretation and elimination of the arbitrariness of numerical labeling of impairment levels, we also dichotomize previous responses on the basis of whether there is no impairment or uncertain impairment (responses 0 or 0.5) or definitely some impairment (responses 1, 2, 3). The effect of GCDR in the immediately preceding year on GCDR in a given year is huge and positive, but it is interesting to note (by comparing the probabilities within rows 1 and 3 of the table, and also within rows 2 and 4) that the effect of GCDR two years prior, adjusted for the effect of GCDR in the immediately preceding year, though smaller than the first-order effect, is still quite substantial.

Table 4.

Second-order cumulative transition probabilities P(Y_t ≤ j|Y_{t − 2} = y_{t − 2}, Y_{t − 1} = y_{t − 1}) collapsed over time points t = 3, 4, 5, 6 and 7 and over dichotomized realizations of previous responses.

yt − 2	yt − 1	j = 0	j = 0.5	j = 1	j = 2
≤ 0.5	≤ 0.5	0.616	0.941	0.995	0.999
≤ 0.5	≥ 1	0.003	0.097	0.717	0.960
≥ 1	≤ 0.5	0.011	0.459	0.907	0.989
≥ 1	≥ 1	0.000	0.016	0.330	0.709

In light of these results, further model-building might reasonably adapt the well-known proportional odds, cumulative logit model (Agresti 2002, sec. 7.2) for use with a second-order autoregression on dichotomized previous responses, i.e.

$$\text{logit}[P(Y_t\le j\mid Y_{t-1},Y_{t-2})]=b_{j0}+b_{1}I(Y_{t-1}\le 0.5)+b_{2}I(Y_{t-2}\le 0.5)\text{for}j=0,0.5,1,2\text{and}t=3,4,5,6,7.$$ (11)

The mles of the coefficients (standard errors) in model (11) can be summarized as follows: b_j0 = −6.23(0.09), −3.94(0.08), −0.84(0.04) and 0.83(0.04) for j = 0, 0.5, 1 and 2 respectively; b₁ = 4.68(0.08) and b₂ = 2.01(0.07); and they are all overwhelmingly significant. Table 5 gives the fitted second-order cumulative transition probabilities corresponding to those in Table 4. The agreement between the entries in the two tables is not bad, especially in the first and last rows, and might be improved by adding an interaction effect or other terms to model (11).

Table 5.

Fitted [using model (11)] second-order cumulative transition probabilities P̂(Y_t ≤ j|Y_{t − 2} = y_{t − 2}, Y_{t − 1} = y_{t − 1}) collapsed over time points t = 3, 4, 5, 6 and 7 and over dichotomized realizations of previous responses.

yt − 2	yt − 1	j = 0	j = 0.5	j = 1	j = 2
≤ 0.5	≤ 0.5	0.612	0.939	0.997	0.999
≤ 0.5	≥ 1	0.015	0.126	0.762	0.945
≥ 1	≤ 0.5	0.175	0.675	0.979	0.996
≥ 1	≥ 1	0.002	0.019	0.301	0.695

8. Discussion

In this article, we have described how to estimate the parameters of unstructured antedependence models for categorical longitudinal data by the method of maximum likelihood. We have also presented likelihood-based methods for determining the order of antedependence by penalized likelihood criteria or formal hypothesis testing; for testing for homogeneity across groups; and for testing for time-invariance of transition probabilities and strict stationarity. Importantly, these methods allow for the possibility of empty cells and missing data. In particular, for data having an arbitrary missingness pattern, mles under unstructured antedependence of any order may be obtained using an efficient restricted EM algorithm. The methods are intended as a prefatory supplement to the common, but in our opinion unwise, practice of proceeding directly to the fitting of structured antedependence models to such data. They focus attention upon transition probabilities, which is often more appropriate than a focus on marginal probabilities. Moreover, using the methods to determine the order of antedependence and identify or refute the presence of other relevant structural features (e.g. time-invariant transition probabilities) may provide useful guidance in selecting (or discarding) a structured antedependence model. For example, our antedependence-based analysis of the toenail infection data, through its focus on the time-specific transition probabilities, led to an interesting interpretation of the curative effects of the treatments over time not noted by any previously published analysis, and indicated that an unstructured first-order AD model is superior to any structured antedependence model that has been fitted by previous authors. Our analysis of the Alzheimer’s disease data indicated the importance of using the data to select the order of antedependence rather than arbitrarily adopting a Markov (first-order antedependence) model; the analysis also justified the use of a structured antedependence model in which the transition probabilities are time-invariant.

We assumed throughout that covariates, other than previous responses and indicator variables for treatments or other groups, are unavailable or ignored in the analysis. If categorical covariates are available, they may be handled in the same way as treatments, by allowing for a different transition probability vector at each combination of levels of the covariates. Such models may have many parameters and may therefore require many observations to yield useful estimates. If, alternatively, the covariates are continuous, then generalized linear models with antedependence structure can be formulated. Research on these latter models is ongoing.

The models that are the subject of this paper, though they allow the order of antedependence to vary over time, are distinct from the variable-length Markov chains (VLMCs) of [18]. In a VLMC, the order of antedependence may depend on the specific realization, but not on time. The possibility of parsimoniously modeling categorical longitudinal data by a melded AD-VLMC model that allows for variation in order over time and realizations simultaneously is a topic for future research. Further work is also needed to extend the methods developed herein to multivariate categorical and mixed (some variables categorical, others continuous) data.