 Research article
 Open Access
 Published:
Joint modelling rationale for chained equations
BMC Medical Research Methodology volume 14, Article number: 28 (2014)
Abstract
Background
Chained equations imputation is widely used in medical research. It uses a set of conditional models, so is more flexible than joint modelling imputation for the imputation of different types of variables (e.g. binary, ordinal or unordered categorical). However, chained equations imputation does not correspond to drawing from a joint distribution when the conditional models are incompatible. Concurrently with our work, other authors have shown the equivalence of the two imputation methods in finite samples.
Methods
Taking a different approach, we prove, in finite samples, sufficient conditions for chained equations and joint modelling to yield imputations from the same predictive distribution. Further, we apply this proof in four specific cases and conduct a simulation study which explores the consequences when the conditional models are compatible but the conditions otherwise are not satisfied.
Results
We provide an additional “noninformative margins” condition which, together with compatibility, is sufficient. We show that the noninformative margins condition is not satisfied, despite compatible conditional models, in a situation as simple as two continuous variables and one binary variable. Our simulation study demonstrates that as a consequence of this violation order effects can occur; that is, systematic differences depending upon the ordering of the variables in the chained equations algorithm. However, the order effects appear to be small, especially when associations between variables are weak.
Conclusions
Since chained equations is typically used in medical research for datasets with different types of variables, researchers must be aware that order effects are likely to be ubiquitous, but our results suggest they may be small enough to be negligible.
Background
Multiple imputation [1] has become a popular approach for the analysis of incomplete data, with several mainstream statistical packages now incorporating multiple imputation tools. It involves making several draws of the missing data from their posterior predictive distribution given the observed data and an imputation model. For multivariate, nonmonotone missing data there are two main approaches for constructing an imputation model: joint modelling and chained equations. Joint modelling imputation requires the specification of a parametric joint model for the complete data: current implementations impute under the multivariate normal model, the log linear model and the general location model [2]. However for datasets containing different types of variables the current classes of joint models [3–5] may not be appropriate for the joint distribution of the data. The alternative method, chained equations imputation [4, 6], is more flexible as it specifies a separate imputation model, typically a univariate regression model, for each incomplete variable and updates the missing data for each variable in turn.
Chained equations imputation has been proposed under several different names including: fully conditional specification, stochastic relaxation, variablebyvariable imputation, regression switching, sequential regressions, ordered pseudoGibbs sampler, partially incompatible MCMC and iterated univariate imputation [3]. In addition to handling variables of varying types, the chained equations approach has other flexible features such as incorporating restrictions, logistical and consistency bounds (for example, to handle imputation of gender specific variables or impute only questions that were not intentionally skipped in a questionnaire [4]). van Buuren and GroothuisOudshoorn [7] discuss the wide range of medical fields that have used chained equations imputation (e.g. addiction [8], epidemiology [9], infectious diseases [10], genetics [11], cancer [12], obesity and physical activity [13]), and a brief review of available software that have implemented chained equations imputation is given by [14]. Given the popularity of chained equations, among users of varying degrees of expertise, there is now guidance in its practical use (e.g. [14] and [15]).
Despite its widespread use, a known theoretical weakness of the chained equations method is that the implicit joint distribution underlying the separate models may not always exist: that is, the conditional models may be incompatible [4, 5, 16–18]. In such situations, the results after chained equations imputation may systematically differ according to the order in which the missing variables are updated in the chained equations algorithm. We shall refer to this phenomenon as an “order effect”.
Previous authors [3–5] have stated that chained equations imputation under a set of normal linear regression models, with all other variables as covariates and no interactions, is equivalent to a Gibbs sampler that draws from a multivariate normal distribution. van Buuren [3] also states for a dataset of three partially observed binary variables that chained equations under a set of logistic regression models, with all other variables included as main effects only, is equivalent to a joint modelling imputation under a log linear model with the threeway factor term set to zero. However, none of these papers provides a proof beyond stating that the set of conditional models is compatible [16] and are all derived from the specified joint distribution.
Independently and concurrently with our work, Liu et al. [19] have given sufficient conditions (which include compatibility of the conditionals) under which, as the sample size tends to infinity, the stationary distribution of the Markov chain generated by the chained equations algorithm (assuming that this stationary distribution exists and that the chain converges to it) converges to the posterior predictive distribution of the missing data implied by a joint Bayesian model. That is, under these sufficient conditions, the total variation of the distance between the chained equations stationary distribution and the posterior predictive distribution tends to zero as the sample size tends to infinity. As a corollary, Liu et al show the equivalence of the two imputation methods in finite samples under a condition we have independently identified and named the “noninformative margins” condition.
Our work is complementary to that of Liu et al. Firstly, we have taken a different approach to prove the equivalence of the two imputation methods in finite samples. Additionally, in specific examples, we prove whether the noninformative margins condition is satisfied or not, and in a simulation study we demonstrate the consequences when the conditional models are compatible but do not satisfy the noninformative margins condition.
In this paper, we provide a “noninformative margins” condition that, together with compatibility of the conditionals (and assuming that the Markov chain generated by the chained equations converges to a stationary distribution), guarantees that the imputed values obtained using chained equations (at convergence) are drawn from the posterior predictive distribution of the missing data implied by a Bayesian joint model. We give examples of chained equations algorithms that satisfy the noninformative margins condition when the joint model is the multivariate normal model and the saturated multinomial model, and examples where this condition is not satisfied when the joint model is an unsaturated multinomial model and the general location model. A simulation study considers a simple chained equations algorithm in which the conditional models are compatible but do not satisfy the noninformative margins condition, and shows that it is not equivalent to any joint model procedure.
Methods
Notation
Suppose K random variables $X={({X}_{1},\dots ,{X}_{K})}^{\u22ba}$ are intended to be observed on N subjects. We use subscripts i and j to index subjects and variables respectively (i = 1,…,N; j = 1,…,K). Let x = (x _{ ij }) denote an (N × K) matrix, whose i,j element is x _{ ij }. Column j of matrix x is denoted by ${x}_{j}={({x}_{1j},\dots ,{x}_{\mathit{\text{Nj}}})}^{\u22ba}$. It is assumed that the rows of matrix x are independent and identically distributed draws from a probability distribution with probability distribution function p (X ∣ θ), where θ is an unknown parameter.
In practice some subjects have missing observations on up to K  1 variables and we write x _{ j }= (xjobs,xjmis) for any j, x = (x ^{obs},x ^{mis}) and p (x ∣ θ) = p(x ^{obs},x ^{mis} ∣ θ), with superscript obs and mis denoting the observed and missing data respectively. In keeping with the assumptions of joint modelling imputation and chained equations imputation, the missing data mechanism is assumed to be ignorable for Bayesian inference [20] p. 120, so that inferences about θ can be based on the marginal observed data posterior p (θ ∣ x ^{obs}).
Joint modelling imputation
Joint modelling imputation requires the specification of a parametric joint model p(x ^{obs},x ^{mis }∣ θ) for the complete data and a prior distribution p(θ) for parameter θ. Imputations are independent draws from the posterior predictive distribution of the missing data given the observed data p(x ^{mis }∣ x ^{obs}) [2] p. 105, which under the ignorability assumption is
Therefore, to draw from this posterior predictive distribution, first draw θ ^{∗} ∼ p (θ ∣ x ^{obs}) followed by x ^{mis∗} ∼ p (x ^{mis} ∣ x ^{obs},θ ^{∗}) [2] p. 105. When it is difficult to draw from the observed data posterior p(θ∣x ^{obs}), Markov chain Monte Carlo methods can be used. For example, the data augmentation algorithm of Tanner and Wong [21] draws missing values from the posterior predictive distribution x ^{mis∗} ∼ p (x ^{mis }∣ x ^{obs},θ ^{∗}) and then draws θ from the complete data posterior θ ^{∗} ∼ p (θ ∣ x ^{obs},x ^{mis∗}), where ∗ denotes the last drawn values of θ or x ^{mis}. Upon convergence this produces a draw from the joint posterior distribution p(θ,x ^{mis} ∣ x ^{obs}).
Chained equations imputation
For every incomplete variable the chained equations algorithm requires an imputation model, typically a univariate regression model, and an accompanying prior distribution for the model’s parameter. Let X _{j } = (X _{1},…,X _{ j1},X _{ j+1},…,X _{ K })^{T} denote the vector of random variables excluding variable X _{ j } and ${x}_{j}=({x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\mathit{\text{mis}}})$ the submatrix of x corresponding to variables X _{j }. We write p (x _{ j } ∣ x _{j },ψ _{ j }) for the probability distribution function of the imputation model for variable X _{ j }and p (ψ _{ j }) for the prior distribution of the unknown parameter ψ _{ j }.
Chained equations draws the imputations using an iterative algorithm, typically with 10 to 20 iterations [15]. To start off, the missing values of each incomplete variable are replaced by its mean or a random sample of its observed values. Suppose, without loss of generality, that variables X _{1},…,X _{ R } (R ≤ K) are incomplete and variables X _{ R+1},…,X _{ K } are fully observed. Given the imputations from the last iteration $({x}_{1}^{(t1)},\dots ,{x}_{R}^{(t1)})$, iteration t of the chained equations algorithm consists of the following draws [18]
During each iteration the following two steps are applied to each incomplete variable X _{ j } in turn: ${\psi}_{j}^{\ast}$ is drawn from the posterior distribution proportional to $p\left({\psi}_{j}\right)p({x}_{j}^{\mathit{\text{obs}}}\mid {x}_{j}^{\ast},{\psi}_{j})$ and missing values ${x}_{j}^{\mathit{\text{mis}}\ast}$ are drawn from the predictive posterior $p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\ast},{\psi}_{j}^{\ast})$. The imputations from the last iteration form the imputed dataset. The whole iterative algorithm is repeated to obtain further imputed datasets.
Equivalence of joint modelling and chained equations imputation
We investigated, in finite samples, sufficient conditions under which a chained equations algorithm with compatible conditional models imputes missing data from the predictive distribution of the missing data implied by the joint model and its accompanying prior. We provide examples of chained equations algorithms (with compatible conditional models) where our identified condition is satisfied and examples where it is not satisfied.
Simulation study
We conducted a simulation study to explore the consequences for chained equations imputation when the conditional models were compatible with the same joint model but the noninformative margins condition of Proposition 1 was not satisfied. In particular, we looked for evidence of “order effects”, where the distribution from which the final imputed values of the variables were drawn differed according to the order in which the variables were updated in the chained equations sampler. If the chained equations algorithm imputes all variables from the predictive distribution of the missing data implied by a specific joint model, then order effects cannot occur [22]. Thus, the existence of order effects implies that the chained equations algorithm is not equivalent to imputing from any joint model.
The simulation study was based on a general location model, discussed in the Theoretical results section below, with one incomplete binary variable Y and two continuous variables W _{1} and W _{2}, where W _{1} was also incompletely observed. We compared joint modelling imputation under the general location model, considered as a gold standard, with the chained equations algorithm that imputes the binary variable Y under a logistic regression model and the continuous variable W _{1} under a normal linear regression model.
We generated 500 datasets, each with a sample size of 100. For each dataset, the rows were independent, identically distributed realizations of the general location model Y ∼ Bernoulli (3/10), W _{1} ∣ Y ∼ N (10 + β Y,9) and W _{2} ∣ W _{1},Y ∼ N (9 + 8/9 + 1/9W _{1} + β Y,8 + 8/9). The data model was a simplified version of data that can occur in the medical literature [23]. The simulation study was repeated when β, the regression coefficient for covariate Y, was set to 1 and 3. The analysis of interest was the normal linear regression of W _{2} on W _{1} and Y. To ensure that any observed order effects could only be due to the failure of the noninformative margins condition we considered the simplest setting, that of data missing completely at random [20] p. 16, and set the values of Y and W _{1} to be missing for the first 50 individuals in the dataset. Below we describe the joint modelling imputation procedure and the chained equations algorithm that were separately applied to the same 500 datasets.
We used the data augmentation algorithm (as described under the heading “Joint modelling imputation”) to perform joint modelling imputation under the general location model and the joint prior given in the general location example (see example 4 of the Results), setting hyperparameters τ = ν = 1/2 and κ = 3/2. The number of imputed datasets generated, the burnin period and the number of iterations between imputed datasets was 100. The analysis model was applied to each dataset separately and the mean of the multiple estimates of β, the coefficient for Y, was calculated.
In the (standard) chained equations algorithm, a logistic regression model for Y given W _{1} and W _{2} was first fitted to those rows of the dataset in which Y was observed. Let ${\widehat{\psi}}_{Y}$ denote the maximum likelihood estimate of the parameters of this model and $\widehat{V}$ denote its associated estimated variancecovariance matrix. A draw ${\psi}_{Y}^{\ast}$ was then made from the multivariate normal approximation $N({\widehat{\psi}}_{Y},\widehat{V})$ and used to impute the missing Y values. The continuous variable W _{1} was imputed using the linear regression model W _{1} ∣ Y,W _{2} ∼ N (λ + ξ Y + ϕ W _{2},ω) and prior distribution p (λ,ξ,ϕ,ω) ∝ ω ^{3/2}.
To start off the chained equations algorithm the missing values of Y and W _{1} were replaced with a random sample of their observed values. We augmented the chained equations algorithm such that, within each iteration we fitted the analysis model immediately after updating the binary variable Y and also immediately after updating the continuous variable W _{1}. The simulation study focused on systematic differences between the two resulting estimates of β. Given the imputations from the last iteration $({y}^{(t1)},{w}_{1}^{(t1)})$, iteration t of the augmented chained equations algorithm consisted of the following steps:

1.
Generate y ^{(t)} by imputing values for the missing binary observations, conditioning on ${w}_{1}^{(t1)}$ and w _{2}.

2.
Linearly regress w _{2} on ${w}_{1}^{(t1)}$ and y ^{(t)} and store the estimate for the coefficient of Y, denoted by ${\widehat{\beta}}^{b}$.

3.
Generate ${w}_{1}^{\left(t\right)}$ by imputing values for the missing continuous observations, conditioning on y ^{(t)} and w _{2}.

4.
Linearly regress w _{2} on ${w}_{1}^{\left(t\right)}$ and y ^{(t)} and store the estimate for the coefficient of Y, denoted by ${\widehat{\beta}}^{c}$.
The chained equations algorithm was implemented with 10010 iterations. The first 10 iterations were regarded as burnin and the estimates from these iterations discarded. The remaining 10000 estimates of ${\widehat{\beta}}^{b}$ were averaged, and likewise for ${\widehat{\beta}}^{c}$. We denote these means as ${\stackrel{\u0304}{\beta}}^{b}$ and ${\stackrel{\u0304}{\beta}}^{c}$ and their difference by ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$. The quantity ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ can be interpreted as an estimate of the order effect for imputation in one dataset. We estimated the (Monte Carlo) standard error of ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ using the batchmeans method, a method for computing standard errors for correlated output [24] p. 124, and calculated a 95% confidence interval from this.
Linear discriminant analysis is an alternative way to estimate a logistic regression [25, 26]. A modified chained equations algorithm using linear discriminant analysis on all individuals with observed Y has been proposed as an alternative way to impute the binary variable Y[27]. Because the linear discriminant likelihood is the joint distribution of Y, W _{1} and W _{2}, this model has the advantage of recovering some information about ψ _{ Y } in the W margin. We repeated the simulation study using this modified chained equations algorithm.
We assessed the sensitivity of our results by repeating the simulation study using different specifications. For joint modelling imputation we increased the number of imputed datasets generated, the burnin period and the number of iterations between imputed datasets to 250. For the standard and modified chained equations procedures we (1) increased the burnin period of the chained equations sampler to 1000 iterations and (2) sampled every 50^{th} iteration thereby reducing serial correlation (with a burnin period of 10 iterations). To check that our results were not dependent upon our choice of prior distributions we repeated the simulation study with improper imputation procedures; that is, using maximum likelihood estimates of ψ _{ j } instead of Bayesian draws ψ _{ j } from its posterior distribution $p\left({\psi}_{j}\right)p\left({x}_{j}^{\mathit{\text{obs}}}\right{x}_{j}^{\ast},{\psi}_{j})$. Lastly, we also repeated the simulation study with a sample size of 1000 observations.
Results
Theoretical results
Equivalence of joint modelling and chained equations imputation
In this section, we give our key result Proposition 1, which shows that, in finite samples, compatibility of the conditionals and our proposed noninformative margins condition are sufficient for chained equations and joint modelling to yield imputations from the same predictive distribution.
Consider a joint model p (x ∣ θ) and prior p (θ). From here onwards we shall use p (.∣.) to refer to any probability distribution derived from this joint model. In particular, for each j = 1,…,R, p (x _{ j }∣ x _{j },θ) is the conditional distribution of x _{ j }given x _{j }and θ implied by the joint model, and p(x _{j }∣ θ) is the conditional distribution of x _{j }given θ. The distribution of x given θ factorizes as
Let ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ be functions of θ such that p (x _{ j }∣ x _{j },θ) = p(x _{ j }∣ x _{j },ψ _{ j }) and $p({x}_{j}\mid \theta )=p({x}_{j}\mid {\stackrel{~}{\psi}}_{j})$. That is, the distribution of x _{ j }given x _{j }and the distribution of x _{j }depend on θ only through the functions ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$, respectively, of θ. Let $p({\psi}_{j},{\stackrel{~}{\psi}}_{j})$ denote the joint prior for ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ implied by p (θ), and let p (ψ _{ j }) and $p\left({\stackrel{~}{\psi}}_{j}\right)$ denote the corresponding marginal priors.
The chained equations algorithm applies the following two steps for each x _{ j }in turn: Step CE1Draw ${\psi}_{j}^{\ast}$ from the distribution proportional to $p\left({\psi}_{j}\right)p({x}_{j}^{\mathit{\text{obs}}}\mid {x}_{j}^{\ast},{\psi}_{j})$. Step CE2 Draw ${x}_{j}^{\mathit{\text{mis}}\ast}$ from $p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\ast},{\psi}_{j}^{\ast})$.
The choice of the parameterizations ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ does not affect the output of step CE2, but a parsimonious choice will help to make the condition of Proposition 1 hold.
Proposition 1.
> Upon convergence, the chained equations algorithm defined by CE1 and CE2 and the joint model p (x ∣ θ) and prior p (θ) draw from the same predictive distribution of x ^{mis}if, for each incomplete variable x _{ j }, $p({\psi}_{j},{\stackrel{~}{\psi}}_{j})=p\left({\psi}_{j}\right)p\left({\stackrel{~}{\psi}}_{j}\right)$, i.e. if the joint prior distribution for ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ factorizes into independent priors for ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$.
Proof.
Using the condition of Proposition 1,
Now, integrating out ${\stackrel{~}{\psi}}_{j}$,
Therefore, step CE1 yields a draw from $p({\psi}_{j}\mid {x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\ast})$.
Next, ${x}_{j}^{\mathit{\text{mis}}}$ and ${x}_{j}^{\mathit{\text{obs}}}$ are conditionally independent given ${x}_{j}^{\ast}$ and ${\psi}_{j}^{\ast}$ so $p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\ast},{\psi}_{j}^{\ast})=p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\ast},{\psi}_{j}^{\ast})$ and step CE2 yields a draw from $p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\ast},{\psi}_{j}^{\ast})$.
Hence steps CE1 and CE2 together yield a draw from $p({\psi}_{j},{x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\ast})$, and in particular they draw ${x}_{j}^{\mathit{\text{mis}}\ast}$ from $p({x}_{j}^{\mathit{\text{mis}}}\mid {x}_{j}^{\mathit{\text{obs}}},{x}_{j}^{\ast})$. The latter is a fullconditional distribution corresponding to the joint density p (x) implied by the joint model. Once ${x}_{j}^{\mathit{\text{mis}}\ast}$ has been sampled, ${\psi}_{j}^{\ast}$, the sampled value of ψ _{ j }, is not used again in the chained equations algorithm. So, the application of steps CE1 and CE2 to each j in turn and then iterating is a Gibbs sampler which, at convergence, yields a draw from p(x ^{mis}∣ x ^{obs}), the predictive distribution implied by the joint model and its accompanying prior.
Comment 1. The condition of Proposition 1 does not hold if the conditional and marginal parameters ψ _{ j } and ${\stackrel{~}{\psi}}_{j}$ are not distinct (i.e. if their joint parameter space is not the product of their separate parameter spaces), and in particular if the combined dimension of ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ is greater than that of θ. Distinctness of parameter spaces is a property of the model p (x ∣ θ) and not of the prior p (θ). It will be used in the examples of the unsaturated multinomial distribution and the general location model to identify joint models where the condition of Proposition 1 does not hold. Comment 2. Heuristically, the condition of Proposition 1 says that there is no information about ψ _{ j } in the marginal distribution of x _{j }, so we call it the “noninformative margins” condition. When such information does exist, it is used by the joint modelling sampler but not by the chained equations algorithm, so they may draw from different distributions. Our simulation study will show that this occurs in an example by demonstrating order effects. Comment 3. As the noninformative margins condition has only been shown to be sufficient, then potentially if this condition is not satisfied the chained equations algorithm defined by CE1 and CE2 and the joint model p (x∣θ) and prior p (θ) could still draw from the same predictive distribution of x ^{mis}. Comment 4. This proof holds for improper prior distributions provided the posterior distributions are proper. Comment 5. When the noninformative margins condition holds true the chained equations algorithm is a Gibbs sampler, and so order effects cannot occur [22].
Example 1: Multivariate normal
Consider the multivariate normal joint model X ∼ N (μ,Σ) with parameter θ = (μ,Σ) and prior distribution p (θ) ∝ Σ  ^{κ} $(\kappa \in \mathbb{Q})$. We show that the corresponding chained equations algorithm, which imputes under a set of normal linear regression models, satisfies the noninformative margins condition of Proposition 1 (and hence draws from the same joint model as joint modelling imputation).
For each j we partition the mean vector μ as ${({\mu}_{j},{\stackrel{~}{\mu}}_{j})}^{\text{T}}$ and the covariance matrix Σ as
such that X _{ j }∼ N (μ _{ j },σ _{ j }) and ${X}_{j}\sim N({\stackrel{~}{\mu}}_{j},{\stackrel{~}{\Sigma}}_{j})$. The conditional distribution of X _{ j }given X _{j }is the normal linear regression model ${X}_{j}\mid {X}_{j}\sim N({\alpha}_{j}+{\beta}_{j}^{\text{T}}{X}_{j},{\omega}_{j})$ where ${\beta}_{j}^{\text{T}}={\varsigma}_{j}^{\text{T}}{\stackrel{~}{\Sigma}}_{j}^{1}$, ${\alpha}_{j}={\mu}_{j}{\beta}_{j}^{\text{T}}{\stackrel{~}{\mu}}_{j}$ and ${\omega}_{j}={\sigma}_{j}{\varsigma}_{j}^{\text{T}}{\stackrel{~}{\Sigma}}_{j}^{1}{\varsigma}_{j}$[2] p. 157. Using our notation for chained equations imputation (see under the subsection “Chained equations imputation” of the Methods section) ψ _{ j }= (α _{ j },β _{ j },ω _{ j }) and $\stackrel{~}{{\psi}_{j}}=({\stackrel{~}{\mu}}_{j},{\stackrel{~}{\Sigma}}_{j})$.
The joint prior for ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ derived from p (θ) is $p({\psi}_{j},{\stackrel{~}{\psi}}_{j})=\Sigma {}^{\kappa}\times {\stackrel{~}{\Sigma}}_{j}$[2] p. 158159. So, using a standard result from matrix algebra for the determinant of a partitioned matrix,
Therefore, the noninformative margins condition of Proposition 1 is satisfied.
Example 2: Saturated multinomial distribution
We now show for joint modelling imputation under a saturated multinomial model and a Dirichlet prior for θ, that the corresponding chained equations algorithm satisfies the noninformative margins condition of Proposition 1.
Consider K categorical variables X = (X _{1},…,X _{ K }), where each X _{ j }takes possible values 1,…,I _{ j }(j = 1,…,K). Variables X define a Kway contingency table. Let c = (c _{1},…,c _{ K }) denote a generic cell of the contingency table, θ _{ c }denote the cell probability pr (X = c) and ∂ denote the set of all cells of the contingency table. The joint distribution of X is a multinomial distribution with parameter θ = (θ _{ c } : c ∈ ∂) and index equal to 1.
Summing the table counts over variable X _{ j }produces a collapsed contingency table defined by variables X _{j }. Let c _{j }= (c _{1},…,c _{ j1},c _{ j+1},…,c _{ K }) denote a generic cell of the collapsed table, ${\stackrel{~}{\psi}}_{{c}_{j}}$ denote the cell probability pr (X _{j }= c _{j }) and ${\stackrel{~}{\partial}}_{j}$ denote the set of all cells of the collapsed table. The marginal distribution of X _{j } is multinomial with parameter ${\stackrel{~}{\psi}}_{j}=\{{\stackrel{~}{\psi}}_{{c}_{j}}:{c}_{j}\in {\stackrel{~}{\partial}}_{j}\}$, where ${\stackrel{~}{\psi}}_{{c}_{j}}=\sum _{{c}_{j}=1}^{{I}_{j}}{\theta}_{c}$. The conditional distribution of X _{ j }given X _{j }= c _{j } is the multinomial distribution with parameters ${\psi}_{{c}_{j}}=\left\{\mathit{\text{pr}}\right({X}_{j}={c}_{j}\mid {X}_{j}={c}_{j}):{c}_{j}=1,\dots ,{I}_{j}\}$. So, the full set of parameters for the conditional distribution of X _{ j }given X _{j } is ${\psi}_{j}=\{{\psi}_{{c}_{j}}:{c}_{j}\in {\stackrel{~}{\partial}}_{j}\}$.
If the prior distribution for θ is Dirichlet with hyperparameter α = {α _{ c } : c ∈ ∂}, then the implied prior distributions for ${\stackrel{~}{\psi}}_{j}$ and ψ _{ j }are independent: the prior for ${\stackrel{~}{\psi}}_{j}$ is Dirichlet with hyperparameter ${\stackrel{~}{\alpha}}_{j}=\{{\stackrel{~}{\alpha}}_{{c}_{j}}:{c}_{j}\in {\stackrel{~}{\partial}}_{j}\}$, where ${\stackrel{~}{\alpha}}_{{c}_{j}}=\sum _{{c}_{j}=1}^{{I}_{j}}{\alpha}_{c}$, and the prior distribution for ψ _{ j }is the product of the set of independent Dirichlet distributions $\{{\psi}_{{c}_{j}}\sim D({\alpha}_{{c}_{j}}):{c}_{j}\in {\stackrel{~}{\partial}}_{j}\}$, where ${\alpha}_{{c}_{j}}=\{{\alpha}_{c}:{c}_{j}=1,\dots ,{I}_{j}\}$ is a subset of α[2] p. 256. Since the prior for $({\psi}_{j},{\stackrel{~}{\psi}}_{j})$ can be factored into independent distributions for ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$, the noninformative margins condition of Proposition 1 is satisfied.
Example 3: Unsaturated multinomial distribution
When the joint model is an unsaturated multinomial model, we give an example where the conditional and marginal parameters ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ are not distinct (see comment 1 of Proposition 1). Consequently the noninformative margins condition of Proposition 1 is not satisfied.
Consider K categorical variables X = (X _{1},…,X _{ K }) as described in the saturated multinomial example. Assume that all cell probabilities are positive, θ (c) gt; 0 for all c, to ensure that every multinomial distribution considered can be written as a log linear model and that all possible conditional distributions exist [28] p. 202. Let the joint model be the hierarchical log linear model that contains all twoway factors between the K variables and no higher order factors. We shall refer to this as the all twoway factor hierarchical model. Under this model, for any j the conditional distribution of X _{ j }given X _{j }follows a multinomial logistic regression model (or a logistic regression model when X _{ j }is binary) where the regression model includes variables X _{j }as main effects only (i.e. no interaction terms).
In generating class notation [29] the all twoway factor hierarchical model is written as [ 1 2]…[ 1 K]… [ (K  1) K]. Any hierarchical log linear model for X can be represented by an undirected graph in which the graph vertices are the variables X _{1},…,X _{ K }and two vertices X _{ g }and X _{ h }are connected by an edge if and only if the log linear model contains twofactor or higher order terms involving variables X _{ g }and X _{ h }. Any model containing all twofactor terms is therefore represented by the complete graph. Asmussen and Edwards [29] state that two vertices in a graph are adjacent if there is an edge between them. For any subset a of the vertices, Asmussen and Edwards define the boundary of a to be the set of vertices that are not in a but that are adjacent to one or more vertices in a. So, for the complete graph, the boundary of X _{ j }is X _{j }. Theorem 2.3 of [29] states that a hierarchical log linear model is collapsible onto X _{j }if and only if the boundary of X _{ j }is contained in a generator of the hierarchical log linear model. Further, Theorem 4.1 of [29] states that if a hierarchical log linear model is not collapsible onto X _{j }, then parameters ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ are not distinct. For any j, X _{j }, the boundary of X _{ j }in the complete graph, contains all of the remaining K  1 variables. When K ≥ 4, X _{j } is not contained in any of the generators, [ 1 2]…[ 1 K]… [ (K  1) K], of the all twoway factor hierarchical model, and hence the log linear model is not collapsible onto X _{j }. Therefore, parameters ψ _{ j }and ${\stackrel{~}{\psi}}_{j}$ are not distinct, and so the noninformative margins condition of Proposition 1 is not satisfied when K ≥ 4.
Example 4: General location model
We give an example of a chained equations algorithm derived from joint modelling under a general location model that does not satisfy the noninformative margins condition of Proposition 1. Our simulation study is based on this example.
Suppose that the data on each individual consist of one incomplete binary variable Y and K  1 continuous variables W = (W _{1},…,W _{ K  1})^{T}, where one or more of the continuous variables are also incomplete. Let the joint distribution of X = (Y,W)^{T} be the general location model Y ∼ Bernoulli (γ) and W ∣ Y ∼ N (μ _{0} + μ _{1} Y,Σ), for unknown parameters θ = (γ,μ _{0},μ _{1},Σ) [30]. Let the joint prior for θ be p (θ) = γ ^{τ1} (1γ)^{ν1} ∣ Σ ∣ ^{κ} with hyperparameters τ,ν gt; 0 and $\kappa \in \mathbb{Q}$.
From the multivariate normal example above it is straightforward to show that the noninformative margins condition of Proposition 1 holds for imputing any W _{ j }.
The conditional distribution of Y given W, p(Y ∣ W, ψ _{ Y }), is the logistic regression model with covariates W[25]. The marginal distribution of W, $p(W\mid {\stackrel{~}{\psi}}_{Y})$, can be written as a mixture of normal distributions
This cannot be parameterized more parsimoniously than ${\stackrel{~}{\psi}}_{Y}=(\gamma ,{\mu}_{0},{\mu}_{1},\Sigma )=\theta $. As ψ _{ Y } is a function of θ, it is determined by $\theta ={\stackrel{~}{\psi}}_{Y}$.
Consequently, given ${\stackrel{~}{\psi}}_{Y}$ the parameters of the logistic regression model ψ _{ Y }are fully determined, and so ${\stackrel{~}{\psi}}_{Y}$ and ψ _{ Y } are not distinct. Therefore, as discussed in comment 1 above, the noninformative margins condition of Proposition 1 does not hold.
Using the same argument, the noninformative margins condition of Proposition 1 does not hold for a chained equations algorithm derived from joint modelling under the restricted general location model, with cell probabilities restricted by the all twofactor hierarchical model (discussed in the unsaturated multinomial example) and cell means restricted to be a linear function of the categorical variables.
Simulation study results
This section reports the results of the simulation study, where the chained equations conditional models were compatible with the same joint model but the noninformative margins condition of Proposition 1 was not satisfied.
Figures 1a and 1b show, for the first 30 of the 500 datasets, the value of ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ (estimate of the order effect in one dataset) along with its 95% confidence interval, for the (standard) chained equations procedure (i.e., binary variable Y imputed under the logistic regression model). In a number of the datasets the 95% confidence intervals did not cross zero. Thus, there was clear evidence of order effects, with the magnitude of such effects varying between datasets. Such statistically significant evidence of an order effect occurred in 164 and 386 of the 500 datasets for β = 1 and β = 3 respectively. The range of absolute values of ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ was larger for β = 3 than for β = 1. These results confirm that the chained equations procedure was not equivalent to any joint model procedure.
The average magnitudes of the order effects ${\stackrel{\u0304}{\beta}}^{b}$, ${\stackrel{\u0304}{\beta}}^{c}$ and ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ over the 500 datasets are shown in Table 1. Consider the results for the chained equations algorithm (labelled LR). In keeping with Figure 1, the average magnitude of the order effect was larger for β = 3 than β = 1. The means of ${\stackrel{\u0304}{\beta}}^{b}$ and ${\stackrel{\u0304}{\beta}}^{c}$ did not differ systematically, consistent with the direction of the order effect being arbitrary and dataset dependent. Estimates ${\stackrel{\u0304}{\beta}}^{b}$ and ${\stackrel{\u0304}{\beta}}^{c}$ appeared to be equally good estimates of β.
The forest plots of ${\stackrel{\u0304}{\beta}}^{b}{\stackrel{\u0304}{\beta}}^{c}$ with 95% confidence intervals corresponding to the modified chained equations algorithm (i.e Y imputed under the linear discriminant model) are shown in Figures 1c and 1d. Order effects were smaller than in Figures 1a and b, but were still present because the modified algorithm did not use information about W _{1} and W _{2} when Y was missing. Out of the 500 datasets the number of datasets that showed statistically significant evidence of an order effect was 113 for β = 1 and 330 for β = 3.
From Table 1, the complete case estimates of β were unbiased. When β = 3 the linear discriminant analysis values ${\stackrel{\u0304}{\beta}}^{c}$ and ${\stackrel{\u0304}{\beta}}^{b}$, and the joint modelling imputation estimate ${\widehat{\beta}}_{\mathit{\text{JM}}}$ were slightly biased towards the null. This bias was due to the prior (which was the same for imputation under the linear discriminant analysis model and joint modelling imputation under the general location model) and it disappeared when the sample size was increased to 1000 observations and when the simulation study was conducted using improper imputation; i.e using maximum likelihood estimates instead of Bayesian draws of parameters. For joint modelling imputation, and the standard and modified chained equations procedures changing their specifications (e.g., larger number of iterations, burnin period) gave the same pattern of results as above (results not shown but available on request from the authors).
In preliminary simulation studies, when the noninformative margins condition was satisfied the results were consistent with a zero order effect (results not shown but available on request from the authors).
Discussion
We have defined a noninformative margins condition which, together with compatibility of the conditional models, we have proved is sufficient for a chained equations algorithm to impute missing data from the predictive distribution of the missing data implied by the joint model and its prior distribution. Also, we have shown that compatibility of the conditional models is not alone a sufficient condition. In a scenario where the conditionals models were compatible but the noninformative margins condition failed, our simulation study showed that the distribution from which the final imputed values of the variables were drawn differed, in a datasetdependent manner, according to the order in which the variables were updated in the chained equations sampler.
In work that is complementary to the finitesample results presented in this paper, Liu et al. [19] identified sufficient conditions for chained equations imputation and imputation under a fully Bayesian model to be asymptotically equivalent; that is, for the supremum of the difference between the two imputation distributions to converge to zero as the sample size tends to infinity. This implies that when the noninformative margins condition is not satisfied but the conditional models are compatible with the same joint model, the order effects identified in our simulation study will disappear as the sample size tends to infinity.
In our simulation study the average magnitude of the order effects was small and did not induce bias. Given that chained equations imputation is a widely used approach to imputation, these results are somewhat reassuring. However, the scope of these simulations was limited and it remains possible that chained equations imputation could lead to more substantial bias in different situations; for example, when there are many partially observed variables.
When the noninformative margins condition does not hold we expect some loss of efficiency in general (because some information is discarded). However, in our simulation study we did not detect any sizable loss of efficiency. The issue of variance estimation for chained equations imputation is beyond the scope of this paper.
The advantage of chained equations imputation is that we do not need to specify the joint distribution of the variables. In cases where it is not known that there is a joint distribution, several methods for checking compatibility have been proposed (e.g., [16, 31–35]). In practice, these methods are either limited to discrete distributions or are difficult to apply for multivariate distributions of more than 2 or 3 dimensions. This means that it may not be possible to check that the conditionals are compatible with the same joint model or that our noninformative margins condition holds true. van Buuren and other authors [3, 18], in the examples they considered, concluded that chained equations is a robust approach even when the set of conditionals are not compatible with the same joint model. The findings of our simulation study support this body of work. Other studies [3, 4, 36, 37] have compared chained equations and joint modelling, when missingness is multivariate, nonmonotone and ignorable, in settings which reflect real data (e.g. mixture of different types of variables, nonlinear relationships and interactions between variables, semicontinuous variables). None of these studies has reported substantial differences in the performances of joint modelling imputation and chained equations imputation. Nonetheless many authors emphasize the need for further understanding of the theoretical underpinnings of the chained equations approach and the establishment of the robustness of the chained equations method (e.g. [3, 7, 14, 38]).
Conclusions
In finite samples, compatibility of the conditionals and our noninformative margins condition are sufficient for chained equations and joint modelling to yield imputations from the same predictive distribution. Furthermore, our simulation study demonstrated that, even in a simple setting, a chained equations procedure that does not satisfy the noninformative margins condition is not necessarily equivalent to a joint model procedure, even though when its conditional models are compatible.
When conditionals are incompatible or the noninformative margins condition is not satisfied, the distribution from which the imputed values are drawn can differ according to the order in which the variables are updated in the chained equations sampler, thereby introducing order effects.
Given the widespread use of chained equations imputation in medical research for datasets with different types of variables, researchers must be aware that order effects are likely to be ubiquitous. As noted by van Buuren [3], further work is needed to verify the robustness of chained equations to incompatibility of the conditional models in more general and realistic settings. Equally, future work could evaluate the robustness of chained equations imputation when the sample size is smalltomoderate, the conditionals are compatible and the noninformative margins condition is not satisfied.
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Acknowledgements
This work was supported by the Medical Research Council [Grant G0900724, Unit Programme number U105260558].
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IRW and JRC proposed the project. All authors contributed to the examples and/or the simulation study. RAH carried out the programming for the simulation study. RAH drafted the manuscript, and RAH, IRW and SRS edited the manuscript. All authors read and approved the final manuscript.
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Hughes, R.A., White, I.R., Seaman, S.R. et al. Joint modelling rationale for chained equations. BMC Med Res Methodol 14, 28 (2014). https://doi.org/10.1186/147122881428
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Keywords
 Chained equations imputation
 Gibbs sampling
 Joint modelling imputation
 Multiple imputation
 Multivariate missing data