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In longitudinal studies, measurements are taken repeatedly over time on the same experimental unit. These measurements are thus correlated. Missing data are very common in longitudinal studies. A lot of research has been going on ways to appropriately analyze such data set. Generalized Estimating Equations (GEE) is a popular method for the analysis of non-Gaussian longitudinal data. In the presence of missing data, GEE requires the strong assumption of missing completely at random (MCAR). Multiple Imputation Generalized Estimating Equations (MIGEE), Inverse Probability Weighted Generalized Estimating Equations (IPWGEE) and Double Robust Generalized Estimating Equations (DRGEE) have been proposed as elegant ways to ensure validity of the inference under missing at random (MAR). In this study, the three extensions of GEE are compared under various dropout rates and sample sizes through simulation studies. Under MAR and MCAR mechanism, the simulation results revealed better performance of DRGEE compared to IPWGEE and MIGEE. The optimum method was applied to real data set.

In the medical, epidemiological and social sciences, studies are often designed to investigate changes in the response of interest observed or measured over time on each subject. These are called repeated measures or longitudinal studies. Since the measurements are taken repeatedly over time on the same experimental unit, then the data are typically correlated. Ordinal responses are regularly experienced in these studies. It is exceptionally common for sets of longitudinal studies to be incomplete, in the sense that not all intended measurements of a subject outcome vector are actually observed. This turns the statistical analysis into a missing data problem. When data are incomplete, a number of issues arise in the analysis: 1) the issue of bias due to systematic differences between the observed measurements and unobserved data, 2) loss of efficiency and 3) complications in data handling and statistical inferences [

The issues of missing data are frequently encountered in longitudinal studies in the sense that nonresponse can happen any time from the beginning of the study. Two patterns of missing data can be observed for the response: 1) dropout (monotone pattern of nonresponse), in which an individual terminates the study prematurely from a scheduled sequence of visits for a number of reasons (both known and unknown), or 2) intermittent nonresponse, in which a subject returns to the study after occasions of nonresponse [

A lot of research has been going on ways to appropriately analyze longitudinal studies. When data is incomplete, rather than deleting missing values, it has been recommended to “impute” them [

Alternative solutions of handling longitudinal missing data have been explored, in particular, the Generalized Estimating Equations (GEE) method [

Literature of GEE for missing data for longitudinal ordinal response is comparatively scarce. In Toledano and Gatsonis [

In this paper, our main interest is the comparison of GEE methods in handling incomplete longitudinal ordinal outcomes when missing response is ignorable. This assumes the missing data are either MCAR or MAR. Comparisons are made by means of simulation study and the optimum model is applied to a real dataset. Through simulation study, the behavior of the methods in terms of mean squared error (MSE) and bias of the estimators are extensively studied, under correctly specified models.

This paper is organised as follows. Section 2 gives necessary notation and key definitions. Section 3 outlines the GEE, as well as IPWGEE, MIGEE and DRGEE approaches. A simulation study is presented in Section 4 followed by a simulation results and application in Section 5. Finally, discussion and concluding remarks are provided in Section 6.

Categorical variables occur frequently in many studies including but not limited to economic, health, education fields. In cases where the variables is categorical with only two levels, logistic regression take stage. However, in cases where there are more than two categories and the categories are ordered then polytomous ordinal regression come into play.

Ordinal outcomes are regularly experienced in longitudinal studies, particularly in randomized clinical trials. Apart from failing to meet the usual normality assumption for analysis and inference, these data are prone to missingness. Failure to deal with incomplete information jeopardizes the validity of inferences. Various authors [

Suppose that longitudinal data consists of N subjects and let Y i j be an ordered variable for subject i with C categories assessed at j^{th} occasion ( j = 1 , 2 , ⋯ , T ) . We define Y i j c = I ( Y i j = c ) for c = 1 , ⋯ , C , where I ( . ) is the indicator function equal to one when the argument is true and zero otherwise. Let Y i = ( Y i 1 , ⋯ , Y i T ) ′ denote the vector of repeated measurements of the i^{th} subject. Associated with each subject, there is a vector of covariates, say X i j , measured at time j. Let X i = ( X i 1 , ⋯ , X i T ) ′ be the covariates matrix for ith subject. The marginal distribution of Y i j will have a multinomial distribution such that:

f ( Y i j | X i j , β ) = ∏ c = 1 C μ i j c y i j c (1)

where μ i j c = μ i j c ( β ) = E ( Y i j c | X i , β ) = P ( Y i j = c | X i , β ) , is the probability of being at category c at time j given a set of covariates and β = ( β 0 c , β x ) is a vector of regression parameters. The cumulative proportional odds model is a popular choice to model μ i j c [

logit [ P r ( Y i j ≤ c | X i j ) ] = β 0 c + X ′ i j β x c = 1 , 2 , ⋯ , C − 1 (2)

where β 0 is the vector of intercept parameters and β x is the vector of coefficients and does not the depend on c.

Now if we let R i = ( R i 1 , ⋯ , R i T i ) be the indicator vector corresponding to Y i = ( Y i 1 , ⋯ , Y i T i ) and R i j = ( R i 1 , ⋯ , R i , j − 1 ) . Y i can be split into subvectors ( Y i 0 , Y i m ) where Y i 0 denotes the observed component and Y i m refers to the missing component. Now we let R i j = 0 if the outcome Y i j is missing and R i j = 1 if Y i j is observed. The joint distribution of the full data Y i and the indicator vector random variable R i can be factorised as

f ( Y i , R i | X i , θ , ψ ) = f ( Y i | X i , θ ) P ( R i = r i | Y i , X i , ψ ) , (3)

where f ( Y i | X i , θ ) denotes the marginal density of the measurement process, P ( R i = r i | Y i , X i , ψ ) denotes the missing data model whose parameter are contained in ψ . ψ is an unknown parameter governing the missing data mechanism and θ denotes the vector of parameters describing the response variable. The distribution of R i may depend on Y i . In terms of probability, we may define these distributions such that the data is said to be MAR if P r ( R i = r i | Y i 0 , Y i m , X i , ψ ) = P r ( R i = r i | Y i 0 , X i , ψ ) , MCAR if P r ( R i = r i | Y i 0 , Y i m , X i , ψ ) = P r ( R i = r i | X i , ψ ) and MNAR if P r ( R i = r i | Y i 0 , Y i m , X i , ψ ) = P r ( R i = r i | Y i 0 , Y i m , X i , ψ ) .

In this paper, our main interest is on missing data due to dropouts. For all components of Y i j that are not observed, the corresponding components of R i j will be 0. We can then replace the vector R i by a scalar variable D i , the drop out indicator, commonly defined as:

D i = 1 + ∑ j = 1 T R i j . (4)

D i denotes the time at which subject i dropped out. The model for drop outs process can therefore be written as

P ( R i = r i | Y i , X i , ψ ) = P ( D i = d i | R i , X i , ψ ) , (5)

where d i is the realisation of the variable D i . In Equation (4), it is assumed that all subjects are observed on the first occasion so that D i takes values between 2 and ( T + 1 ) . The maximum value ( T + 1 ) corresponds to a complete measurement sequence.

The GEE approach has its roots in the quasi-likelihood methods introduced by Wedderburn [

Suppose that longitudinal data consists of N subjects. For subject i , ( i = 1 , 2 , ⋯ , N ) , there are T observations and let Y i j denote the j^{th} response ( j = 1 , 2 , ⋯ , T ) , and let X i j denote the p × 1 vector of explanatory variables. Suppose Y i = ( Y i 1 , Y i 1 , ⋯ , Y i T ) ′ denote the corresponding column vector of response variable for the i^{th} subject with the mean vector μ i = ( μ i 1 , μ i 2 , ⋯ , μ i T ) where μ i j is the corresponding j^{th} mean. The marginal model specifies that a relationship between E ( Y i j ) = μ i j and the covariates X i j is as follows:

g ( μ i j ) = X ′ i j β , (6)

where g is a link function and β is the vector of regression parameters. On the other hand, the conditional variance of Y i j given X i j is given as V a r ( Y i j | X i j ) = ϕ ν ( μ i j ) , where ϕ is a scaling parameter and ν is a known variance function of μ i j . Based on Liang and Zeger [

U ( β ) = ∑ i = 1 N ∂ μ i ∂ β ′ V i − 1 ( Y i − μ i ) = 0 , (7)

where β ′ denotes a transpose vector of marginal regression parameters β ,

V i = A i 1 2 R i ( α ) A i 1 2 is a covariance matrix of Y i in which A i is a diagonal

matrix containing marginal variances. R i ( α ) is a “working” correlation matrix that expresses the marginal correlation between repeated measures and α is a vector of noises which may be handled by the introduction of the working correlation structure such as independence, autoregressive of the first order (AR(1)), exchangeable, or unstructured. For AR(1) the correlations decline exponentially between measures i.e. Corr ( Y i j , Y i h ) = ρ | j − h | . In the independence, the identity matrix serves as the working correlation matrix. On the other hand, for exchangeable structure the correlation between any two measures are assumed to be the same regardless of the time from one period to the next. Under unstructured case, every pair of measurements is given its own association parameter.

Under mild regularity conditions and correct specification of the marginal mean μ i , Liang and Zeger [

V β = lim N → ∞ N Σ 0 − 1 Σ 1 Σ 0 − 1 , (8)

where

Σ 0 = ∑ i = 1 N ∂ μ ′ i ∂ β V i − 1 ∂ μ i ∂ β ′ and Σ 1 = ∑ i = 1 N ∂ μ ′ i ∂ β V a r ( Y i ) V i − 1 ∂ μ i ∂ β ′ , (9)

where μ ′ i in Equation (9) denotes a transpose mean vector of μ i . In practice, the “sandwich” covariance matrix V β in Equation (8) is calculated by ignoring the limit and replacing β and α by their estimates, and also V a r ( Y i ) in expression Σ 1 by ( y i − μ ^ i ) ( y i − μ ^ i ) T [

This method is a simulation-based approach that imputes missing values multiple times [

Let β ^ k and U k be the estimate of a parameter of interest β and its covariance matrix from the k^{th} completed data set, ( k = 1 , 2 , ⋯ , M ) respectively. According to Little and Rubin [

β ^ ¯ = 1 M ∑ k = 1 M β ^ k (10)

and an estimate of the covariance matrix of β ^ ¯ is given by

V = W + ( M + 1 M ) B , (11)

where

W = 1 M ∑ k = 1 M U k and B = 1 M − 1 ∑ k = 1 M ( β ^ k − β ^ ¯ ) ( β ^ k − β ^ ¯ ) ′ .

here, W measures the within-imputation variability and B measures the between-imputation variability.

As Schafer [

When data are incomplete, GEE suffers bias from its frequentist nature and it is generally valid only under the strong assumption of MCAR [

w i j = P ( D i = j ) = ∏ t = 2 j − 1 [ 1 − P ( R i t = 0 | R i 2 = ⋯ = R i , t − 1 = 1 ) ] × P [ R i j = 0 | R i 2 = ⋯ = R i , j − 1 ] I { j ≤ T i } , (12)

where j = 2 , 3 , ⋯ , T + 1 , I { } is an indicator variable and D i is a dropout indicator for the subject i, where D i = 1 + ∑ j = 1 T R i j . The first visit Y i 1 is assumed to be always observed with R i 1 = 1 so that 2 ≤ D i ≤ T + 1 . Hence D i = T + 1 represents that subject i completes all the T visits, which were set prior by design. In the IPWGEE approach, GEE estimator for β is based on solving the equation:

U ( β ) = ∑ i = 1 N W i − 1 ∂ μ i ∂ β ′ ( A i 1 2 R i A i 1 2 ) − 1 ( y i − μ i ) = 0 , (13)

where W i = diag { R i 1 w i 1 , ⋯ , R i T w i T } is a diagonal matrix of event specific weights. A consistent estimator for β can be obtained by solving Equation (13), under the correct specification of the missing data model. Following [

U ( β ) = ∑ i = 1 N ∑ d = 2 T + 1 I ( D i = d ) w i d ∂ μ i ∂ β ′ ( d ) ( A i 1 2 R i A i 1 2 ) − 1 ( d ) { y i ( d ) − μ i ( d ) } = 0 , (14)

where y i ( d ) and μ i ( d ) are the first d − 1 elements of y i and μ i respectively. Provided that the w i d are correctly specified, IPWGEE provides consistent estimates of the model parameters under a MAR mechanism. Estimators from IPWGEE enjoy robustness properties similar to the ones from ordinary GEE, that is., the correlation structure does not need to be correctly specified.

The doubly robust method is an alternative approach that uses the inverse probability weights (IPW) to refine estimates of the model parameters [

Tsiatis [

γ opt = E ( Y i m | Y i 0 , X i , R i ) { ∂ μ i ∂ β ′ ( 1 1 ′ − W i − 1 ) ( A i 1 2 R i A i 1 2 ) − 1 ( y i − μ i ) } , (15)

where 1 and 1’ is a vector of 1’s of length T i and its transpose respectively, and Y i m denote the missing component of Y i . Undefined variables and parameters in Equation (15) are as defined before in section 0. The parameters β are estimated by solving the estimating equations,

U 1 ( θ ) = ∑ i = 1 N U 1 i ( θ ) = ∑ i = 1 N [ W i − 1 ∂ μ i ∂ β ′ ( A i 1 2 R i A i 1 2 ) − 1 ( y i − μ i ) + γ opt ] = 0. (16)

The estimator for β in Equation (16) is doubly-robust in the sense that it is consistent if at least one of the missing data models is correctly specified. In current application, we combine inverse probability weighting (IPW) with MI and the GEE as the analysis to construct DRGEE. The robustness of the imputation model is enhanced by ensuring necessary information is included in the model, while avoiding the bias from the final inference.

The aim of the DRGEE estimation; is to estimate the propensities for each incomplete variable conditional on the other variables, and impute the missing values on that variable by the inclusion of propensity functions (i.e. IPW) into the imputation model. Finally, the results of the analysis from M completed (imputed) data are combined into a single inference using Rubin [

We simulated data in order to mimic an ordinal longitudinal clinical trial data. We simulated 1000 datasets based on the marginal model (17) for random sample sizes N = 100 , 300 and 500. We consider a study with T i = 4 repeated ordinal measures (with four categories) and two covariates (one binary and the other continuous). For binary covariate ( x 1 ) individuals were assumed to have been assigned to two treatment arms (Higher dose = 1 and Mild dose = 0) and x 2 represent exposure period. The true marginal model is ( c = 1 , ⋯ , C − 1 ; i = 1 , ⋯ , N ) :

logit [ P r ( Y i j ≤ c | x ) ] = β 0 c + β x x ′ , for j = 1,2,3,4 (17)

where the model parameters are β = ( β 0 c , β x ) . Here x ′ = ( x 1 , x 2 ) is a vector of predictor variables. The parameter values used in the simulations are β 01 = − 0.4 , β 02 = 0.2 , β 03 = 0.5 , β 1 = 0.5 and β 2 = − 0.1 . The correlated ordinal response were generated using the NORTA method [

For comparison purposes, standard GEE was considered to analyse the full datasets. Each estimate is an average of 1000 estimates from the different simulated datasets. After analysing the full data set we then create the dropouts. Dropouts were created on the complete simulated datasets using different settings of missingness rate on response variable Y i j and according to the MCAR or MAR missing mechanism.

The dropout model is based on a logistic regression for the probability of dropout at occasion j, given that the individual was in the study up to occasion j − 1 . This probability is denoted by P ( h i j ; y i j ) , and the outcome history h i j is expressed as h i j = ( y i 1 , ⋯ , y i , j − 1 ) . In this study, the assumption is that dropout depends only on the current observed measurement y i j and the immediately preceding measurement y i , j − 1 . We therefore assume that dropout process is modelled by a logistic regression of the form

logit [ P r ( h i j , y i j ) ] = logit [ P r ( D i = j | D i ≥ j , h i j , y i j ) ] = ψ 0 + ψ 1 y i , j − 1 + ψ 2 y i j , (18)

with ψ 0 denoting the intercept of regression, ψ 1 and ψ 1 are respectively the coefficients of y i , j − 1 and y i j . The model (18) reduces to a MAR if ψ 2 = 0 (i.e. the missingness process is related to the observed outcome prior to dropout) and MCAR if ψ 1 = ψ 2 = 0 . In both MAR and MCAR settings, after simulating a data set without missing data, we adopted the following strategy. We assume that dropout can occur after the first time point. Thus in this study, four dropout patterns are possible, i.e., 1) dropout at the second point time, 2) dropout at the third time point, 3) dropout at the fourth time point, 4) no dropout.

According to Satty [

logit [ P r ( D i = j | D i ≥ j , y i , j − 1 ) ] = ψ 0 + ψ p r e v y i , j − 1 (19)

where j = 2 , 3 , 4 , ψ 0 = ( 2 , 2.3 , 2.3 ) and ψ p r e v = ( 0.3 , − 0.2 , − 0.37 ) . The values for ψ 0 and ψ p r e v were used to generate different dropout rates. The combination of this MAR logistic dropout model with the measurement model (18) defines our data generating model, which is hereinafter referred to as GM I.

We further consider a second data generating model, GM II, in which the outcomes are generated based on model (18) and random missingness is induced via the following MCAR logistic regression model:

logit [ P r ( D i = j | D i ≥ j , y i , j − 1 ) ] = ψ 0 + ψ p r e v y i , j − 1 (20)

where j = 2 , 3 , 4 , ψ 0 = ( 3.2 , 1.5 , 1.2 ) and ψ p r e v = 0 .

After creating the dropouts, the incomplete data sets were analysed using the three (3) extensions of GEE namely; MIGEE, IPWGEE and DRGEE. The performances of these methods were assessed in terms of mean squared error (MSE) and bias.

In the evaluation, inferences are drawn on the complete data before the dropouts are created. Complete-data results are used as the standard against which those obtained from applying IPWGEE, MIGEE and DRGEE approaches are compared. R software [

The performance of the three methods were evaluated using bias and mean squared error(MSE). These criteria were recommended in [

Bias = | β ^ ¯ − β | , (21)

where β is the true value for the estimate of interest, β ^ ¯ = ∑ s = 1 S β ^ s S is the

average estimate of interest, S is the number of simulation replications performed, and β ^ s is the estimate of interest within each of the s = 1 , ⋯ , S simulations. The mean squared error (MSE) was given by

MSE = | β ^ ¯ − β | 2 + SE ( β ^ ) 2 , (22)

where SE ( β ^ ) denotes the empirical standard error (SE) of the estimate over all simulations [

interest from all simulations [ 1 / ( S − 1 ) ] ∑ s = 1 S ( β ^ s − β ^ ¯ ) 2 . Alternatively, the average of the estimated within simulation SE for the estimate of interest ∑ s = 1 S SE ( β ^ s ) S could be used, where SE ( β ^ s ) denotes the standard error of the

estimate of interest within each simulation. Normally, small values of MSE are desirable [

In this section, we discuss the result of simulation study that compares the three techniques namely; MIGEE, IPWGEE and DRGEE for different sample size and different missingness rates on the response variable. The measurement at first time point were assumed to be observed for each individual. Note that the primary focus was to compare MIGEE, IPWGEE and DRGEE, but we extend the results to include those obtained from full datasets using standard GEE. The imputation model considered here is the imputation using chained equations [

A better method is expected to produce parameter estimates closer or similar to the true values, hence yielding small bias. Likewise, a small MSE denotes a better or precise method. Results are presented in Tables 1-3 for 8%, 25% and 33% dropout rates respectively, under MAR mechanism. For MCAR mechanism, results are presented in

Bias | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

β 01 | β 02 | β 03 | β 1 | β 2 | β 01 | β 02 | β 03 | β 1 | β 2 | |

N = 100 | ||||||||||

GEE | 0.0237 | 0.0063 | 0.0017 | 0.0118 | 0.0027 | 0.2505 | 0.2412 | 0.2465 | 0.2127 | 0.0006 |

MIGEE | 0.4757 | 0.4950 | 0.5602 | 0.0415 | 0.0144 | 0.3143 | 0.2752 | 0.3229 | 0.0154 | 0.0003 |

IPWGEE | 0.8251 | 0.5441 | 1.2058 | 0.7192 | 0.1923 | 0.6808 | 0.2961 | 1.4540 | 0.5173 | 0.0370 |

DRGEE | 0.0142 | 0.0441 | 0.3703 | 0.0862 | 0.0160 | 0.0407 | 0.0673 | 0.2068 | 0.1723 | 0.0005 |

N = 300 | ||||||||||

GEE | 0.0078 | 0.0123 | 0.0154 | 0.0020 | 0.0014 | 0.0685 | 0.0664 | 0.0664 | 0.0691 | 0.0002 |

MIGEE | 0.0567 | 0.0256 | 0.0548 | 0.0639 | 0.0187 | 0.0035 | 0.0009 | 0.0032 | 0.0043 | 0.0004 |

IPWGEE | 0.7325 | 0.4520 | 1.0329 | 1.0138 | 0.1948 | 0.5588 | 0.2286 | 1.0990 | 1.0484 | 0.0382 |
---|---|---|---|---|---|---|---|---|---|---|

DRGEE | 0.2769 | 0.1347 | 0.0645 | 0.0945 | 0.0121 | 0.0779 | 0.0189 | 0.0044 | 0.0092 | 0.0001 |

N = 500 | ||||||||||

GEE | 0.0025 | 0.0068 | 0.0100 | 0.0057 | 0.0006 | 0.0390 | 0.0384 | 0.0388 | 0.0385 | 0.0000 |

MIGEE | 0.0461 | 0.1154 | 0.1126 | 0.2283 | 0.0067 | 0.0114 | 0.0195 | 0.0177 | 0.0679 | 0.0000 |

IPWGEE | 0.6949 | 0.5496 | 1.1140 | 1.0700 | 0.2281 | 0.5166 | 0.3445 | 1.2860 | 1.1615 | 0.0520 |

DRGEE | 0.1641 | 0.2591 | 0.0189 | 0.1784 | 0.0012 | 0.0276 | 0.0716 | 0.0173 | 0.0627 | 0.0000 |

Notes: Also estimates from full datasets (GEE). Approximately (8%) missing values on the outcome variable.

Bias | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

β 01 | β 02 | β 03 | β 1 | β 2 | β 01 | β 02 | β 03 | β 1 | β 2 | |

N = 100 | ||||||||||

GEE | 0.0237 | 0.0063 | 0.0017 | 0.0118 | 0.0027 | 0.2505 | 0.2412 | 0.2465 | 0.2127 | 0.0006 |

MIGEE | 0.3192 | 0.3373 | 0.2830 | 0.7887 | 0.0284 | 0.1025 | 0.1143 | 0.0805 | 0.6221 | 0.0008 |

IPWGEE | 0.9322 | 0.1136 | 0.7884 | 1.4709 | 0.2109 | 0.8694 | 0.0134 | 0.6220 | 2.1653 | 0.0445 |

DRGEE | 0.4259 | 0.3067 | 0.1236 | 0.0054 | 0.0082 | 0.1822 | 0.0945 | 0.0156 | 0.0195 | 0.0007 |

N = 300 | ||||||||||

GEE | 0.0078 | 0.0123 | 0.0154 | 0.0020 | 0.0014 | 0.0685 | 0.0664 | 0.0664 | 0.0691 | 0.0002 |

MIGEE | 0.0757 | 0.0244 | 0.0159 | 0.1344 | 0.0054 | 0.0644 | 0.0573 | 0.0674 | 0.0445 | 0.0007 |

IPWGEE | 0.3794 | 0.8610 | 1.4330 | 0.8438 | 0.2269 | 0.1447 | 0.7420 | 2.0543 | 0.7120 | 0.0515 |

DRGEE | 0.4050 | 0.2510 | 0.3149 | 0.3018 | 0.0289 | 0.3161 | 0.1624 | 0.1461 | 0.1089 | 0.0014 |

N = 500 | ||||||||||

GEE | 0.0025 | 0.0068 | 0.0100 | 0.0057 | 0.0006 | 0.0390 | 0.0384 | 0.0388 | 0.0385 | 0.0000 |

MIGEE | 0.0080 | 0.0220 | 0.0347 | 0.2257 | 0.0063 | 0.0006 | 0.0012 | 0.0036 | 0.0613 | 0.0000 |

IPWGEE | 0.6705 | 0.4881 | 1.1094 | 0.9383 | 0.2109 | 0.4500 | 0.2385 | 1.2310 | 0.8807 | 0.0445 |

DRGEE | 0.1210 | 0.0257 | 0.0535 | 0.0751 | 0.0178 | 0.2981 | 0.2068 | 0.1534 | 0.0451 | 0.0003 |

Notes: Also estimates from full datasets (GEE). Approximately (25%) missing values on the outcome variable.

Bias | MSE | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

β 01 | β 02 | β 03 | β 1 | β 2 | β 01 | β 02 | β 03 | β 1 | β 2 | |

N = 100 | ||||||||||

GEE | 0.0237 | 0.0063 | 0.0017 | 0.0118 | 0.0027 | 0.2505 | 0.2412 | 0.2465 | 0.2127 | 0.0006 |

MIGEE | 0.6596 | 0.3532 | 0.3445 | 0.4913 | 0.0171 | 0.4358 | 0.1253 | 0.1192 | 0.2425 | 0.0003 |

IPWGEE | 0.5056 | 0.7440 | 1.2986 | 0.7512 | 0.2035 | 0.2556 | 0.5535 | 1.6863 | 0.5644 | 0.0414 |
---|---|---|---|---|---|---|---|---|---|---|

DRGEE | 0.1678 | 0.0255 | 0.1731 | 0.1542 | 0.0156 | 0.1385 | 0.0898 | 0.1133 | 0.0803 | 0.0011 |

N = 300 | ||||||||||

GEE | 0.0078 | 0.0123 | 0.0154 | 0.0020 | 0.0014 | 0.0685 | 0.0664 | 0.0664 | 0.0691 | 0.0002 |

MIGEE | 0.2929 | 0.2695 | 0.2568 | 0.1196 | 0.0208 | 0.0893 | 0.0880 | 0.0705 | 0.0201 | 0.0004 |

IPWGEE | 0.7047 | 0.5080 | 1.1535 | 0.8065 | 0.2020 | 0.4972 | 0.2586 | 1.3312 | 0.6507 | 0.0408 |

DRGEE | 0.7620 | 0.4790 | 0.3708 | 0.2186 | 0.0555 | 0.5837 | 0.2300 | 0.1400 | 0.0489 | 0.0031 |

N = 500 | ||||||||||

GEE | 0.0025 | 0.0068 | 0.0100 | 0.0057 | 0.0006 | 0.0390 | 0.0384 | 0.0388 | 0.0385 | 0.0000 |

MIGEE | 0.2859 | 0.1804 | 0.1412 | 0.4514 | 0.0011 | 0.0819 | 0.0326 | 0.0200 | 0.2043 | 0.0000 |

IPWGEE | 0.9661 | 0.2342 | 0.8597 | 1.1017 | 0.2041 | 0.9395 | 0.0603 | 0.7457 | 1.2159 | 0.0417 |

DRGEE | 0.1506 | 0.1027 | 0.1002 | 0.3607 | 0.0074 | 0.0229 | 0.0109 | 0.0103 | 0.1303 | 0.0000 |

IPWGEE | MIGEE | DRGEE | ||||||
---|---|---|---|---|---|---|---|---|

Sample | Drp | Par | Bias | MSE | Bias | MSE | Bias | MSE |

N= 100 | 8% | β 01 | 0.6987 | 0.4882 | 0.3781 | 0.1430 | 0.1259 | 0.0159 |

β 02 | 0.4009 | 0.1607 | 0.5997 | 0.3597 | 0.0607 | 0.0037 | ||

β 03 | 0.9872 | 0.9746 | 0.7049 | 0.4972 | 0.3725 | 0.1387 | ||

β 1 | 1.3864 | 1.9230 | 0.6884 | 0.4753 | 0.0129 | 0.0004 | ||

β 2 | 0.2362 | 0.0558 | 0.0716 | 0.0051 | 0.0345 | 0.0012 | ||

25% | β 01 | 0.9630 | 0.9275 | 0.5147 | 0.2650 | 0.4494 | 0.2020 | |

β 02 | 0.2084 | 0.0436 | 0.6363 | 0.4051 | 0.2857 | 0.0817 | ||

β 03 | 0.8412 | 0.7079 | 0.7259 | 0.5272 | 0.3867 | 0.1496 | ||

β 1 | 1.0229 | 1.0467 | 0.2274 | 0.0517 | 0.1807 | 0.0329 | ||

β 2 | 0.2000 | 0.0400 | 0.0226 | 0.0005 | 0.0265 | 0.0007 | ||

33% | β 01 | 0.9673 | 0.9358 | 0.1610 | 0.0259 | 0.0966 | 0.0094 | |

β 02 | 0.0706 | 0.0051 | 0.2584 | 0.0668 | 0.2068 | 0.0428 | ||

β 03 | 0.6288 | 0.3956 | 0.3695 | 0.1367 | 0.7956 | 0.6331 | ||

β 1 | 0.7844 | 0.6153 | 0.7775 | 0.6055 | 0.0237 | 0.0007 | ||

β 2 | 0.1883 | 0.0354 | 0.0204 | 0.0004 | 0.0121 | 0.0001 | ||

N= 300 | 8% | β 01 | 0.8206 | 0.6936 | 02656 | 0.0706 | 0.2841 | 0.0807 |

β 02 | 0.3443 | 0.1187 | 0.2328 | 0.0542 | 0.2636 | 0.0695 | ||

β 03 | 0.9451 | 0.8935 | 0.2469 | 0.0610 | 0.0177 | 0.0005 | ||

β 1 | 0.8985 | 0.8073 | 0.3749 | 0.1406 | 0.2187 | 0.0479 | ||

β 2 | 0.2005 | 0.0402 | 0.0077 | 0.0000 | 0.0091 | 0.0000 | ||

25% | β 01 | 0.7498 | 0.5626 | 0.1263 | 0.0164 | 0.2924 | 0.0856 |

β 02 | 0.5852 | 0.3429 | 0.1906 | 0.0367 | 0.0910 | 0.0084 | ||
---|---|---|---|---|---|---|---|---|

β 03 | 1.1789 | 1.3903 | 0.1987 | 0.0399 | 0.4986 | 0.2486 | ||

β 1 | 0.6393 | 0.4089 | 0.1273 | 0.0162 | 0.2175 | 0.0473 | ||

β 2 | 0.2032 | 0.0413 | 0.0110 | 0.0001 | 0.0193 | 0.0004 | ||

33% | β 01 | 0.8742 | 0.7645 | 0.1347 | 0.0188 | 0.3772 | 0.1424 | |

β 02 | 0.3449 | 0.1192 | 0.0192 | 0.0006 | 0.0978 | 0.0097 | ||

β 03 | 0.9912 | 0.9828 | 0.0345 | 0.0014 | 0.7059 | 0.4985 | ||

β 1 | 1.0100 | 1.0193 | 0.0371 | 0.0014 | 0.3549 | 0.1259 | ||

β 2 | 0.2045 | 0.0418 | 0.0044 | 0.0000 | 0.0156 | 0.0002 |

Notes: Also estimates from full datasets (GEE). Approximately (33%) missing values on the outcome variable.

IPWGEE | MIGEE | DRGEE | ||||||
---|---|---|---|---|---|---|---|---|

Sample | Drp | Par | Bias | MSE | Bias | MSE | Bias | MSE |

N= 500 | 8% | β 01 | 0.7128 | 0.5085 | 0.1724 | 0.0298 | 0.2176 | 0.0475 |

β 02 | 0.5105 | 0.2610 | 0.1008 | 0.0103 | 0.2106 | 0.0455 | ||

β 03 | 1.1048 | 1.2210 | 0.0681 | 0.0048 | 0.0163 | 0.0004 | ||

β 1 | 0.8703 | 0.7578 | 0.2369 | 0.0561 | 0.1542 | 0.0238 | ||

β 2 | 0.1939 | 0.0376 | 0.0109 | 0.0001 | 0.0055 | 0.0000 | ||

25% | β 01 | 0.8487 | 0.7205 | 0.1448 | 0.0210 | 0.2573 | 0.0664 | |

β 02 | 0.3432 | 0.1179 | 0.1438 | 0.0207 | 0.1793 | 0.0323 | ||

β 03 | 0.9585 | 0.9188 | 0.0893 | 0.0080 | 0.5664 | 0.3210 | ||

β 1 | 1.1562 | 1.3340 | 0.0406 | 0.0017 | 0.0363 | 0.0013 | ||

β 2 | 0.2000 | 0.0400 | 0.0036 | 0.0000 | 0.0104 | 0.0001 | ||

33% | β 01 | 0.5026 | 0.2533 | 0.0499 | 0.0026 | 0.0967 | 0.0096 | |

β 02 | 0.7618 | 0.5810 | 0.0736 | 0.0055 | 0.0519 | 0.0028 | ||

β 03 | 1.3373 | 1.7890 | 0.1357 | 0.0184 | 0.0776 | 0.0062 | ||

β 1 | 0.7810 | 0.6104 | 0.1566 | 0.0245 | 0.2350 | 0.0552 | ||

β 2 | 0.2187 | 0.0478 | 0.0069 | 0.0000 | 0.0063 | 0.0000 |

Examining

Shifting focus to

In

As expected, it can be seen that in most cases IPWGEE was more biased compared to the MIGEE and DRGEE. In addition, IPWGEE has larger MSE values than the other methods. It can be seen that for sample size 300, MIGEE performed better than DRGEE for different dropout rates, except for 25% dropout rate where MIGEE was better than DRGEE for sample size 300 and 500. Generally, the bias was negligible for all methods showing asymptotically parameter estimates. In sum, although all methods performed equally well in terms of bias and MSE, DRGEE provided better parameter estimates than the single robust counterparts.

In

Now shifting focus to

The dataset used is from a homoeopathic clinic in Dublin, made available in [

We think the MAR mechanism may be reasonable because, for instance, a patient’s visit to a clinic may depend on his/her previous observed pain score: if s/he scored a high pain score on his/her last visit, s/he may be likely to attend the next visit to treat the disease efficiently. Both monotone dropouts pattern and nonmonotone missingness were observed in the data. The amount of monotone dropouts was considerable (33.8%), while that of nonmonotone missigness was much smaller (1.8%). Overall, approximately 36% of the pain score data were missing/not observed. Some descriptive statistics of the dataset are summarized in

For the ordinal response scale, we used the following proportional odds model

logit [ P r ( Y i j ≤ c | x i j ) ] = β 0 c + β x ′ i j , c = 1 , ⋯ , 5 , j = 1 , ⋯ , 12, (23)

where Y i j is the pain score status of the i^{th} patient at j^{th} visit, x i j is the covariate vector at time j. Here, the covariate vector is formed by Sex, Age, Time, Type and Years.

DRGEE was applied to the real dataset. The reason why we chose DRGEE as an optimum method was: 1) simulation results showed that it performed better than MIGEE and IPWGEE under MAR and 2) MAR mechanism was observed in the arthritis data. When dealing with DRGEE it is necessary to correctly specified inverse probability weighting and imputation model, in order to obtain consistent estimates of β . The weights were based on a logistic regression model for dropout:

logit [ P ( D i = j | D i ≥ j , v i j ) ] = ψ 0 j + ψ v ′ i j , j = 1, ⋯ ,12, (24)

where v i j include sex, age, type, history of observed pain scores. Here, D i = 1 if the pain score was observed and 0 otherwise. We incorporate weights obtained in Equation (24) in the imputation model, in order to get double robust estimates. Available data was analysed without alteration or any attempts to impute data missing on the response variable. This was under ordinary GEE. Results from the two approaches are shown below in

Variable name | Description | Range | % miss | Mean | Mode |
---|---|---|---|---|---|

Sex | 0 = Female, 1 = Male | 0/1 | 0 | ||

Age | Patient age in years | 18 - 88 | 0 | 59.5 | 57 |

Time | Number of patient visits | 1 - 12 | 0 | ||

Type | Type of arthritis (RA = 1, OA = 0) | 1/0 | 0 | ||

Years | Number of years with symptom | 0 - 57 | 0 | 10.7 | 1 |

Response variable | |||||

pain scores | scores on the arthritis pain | 1 - 6 | 35.6% | 4 |

Note: Missing values on the response variable. Type of arthritis (RA = rheumathoid arthritis, OA = ostheo-arthritis).

GEE | DRGEE | |||||
---|---|---|---|---|---|---|

Parameter | Est | SE | P r > | t | | Est | SE | P r > | t | |

Sex | −0.3366 | 0.5073 | 0.5070 | 0.5627 | 0.2780 | 0.0429 |

Age | −0.0253 | 0.0139 | 0.0708 | −0.0278 | 0.0087 | 0.0007 |

Time | 0.2115 | 0.0422 | <0.0001 | 0.2719 | 0.0249 | <0.0001 |

Type | −0.8775 | 0.3372 | 0.0093 | −0.6069 | 0.2288 | 0.0079 |

Years | −0.0048 | 0.0149 | 0.7454 | 0.0026 | 0.0094 | 0.7768 |

Note: Approximately (36%) data missing on the response variable. Available data analyzed using GEE.

The results showed that Time effect was significant and the variable Years was non significant for both methods. It can be noticed that p-value for Age goes from non-significant (0.07) in the ordinary GEE to a significant one in DRGEE. Similar trend was observed for the variable Sex. Both methods provide the same conclusion for effects of type of arthritis a patient is diagnose with. The negative effect for Type means that the chance of a patient to feel/record minimal pain is lower among the patients who had rheumathoid arthritis type compared to those who had ostheo-arthritis (the estimated odds e − 0.6069 = 0.5450 in the DRGEE method). Both methods provided the same conclusion for the effect of Age. That is, each unit increase in Age, the odds of feeling mild pain or minimal pain decreases by 3% (for instance, in DRGEE it is e − 0.0278 = 0.9725 ). Furthermore, the standard error produced by DRGEE are marginally smaller than one produced by usual GEE. Overall, it can be seen that there is gain in using DRGEE method due to its doubly robust property.

In this paper, the focus was to compare three techniques for handling incomplete ordinal outcome based on GEE under MCAR and MAR dropouts in longitudinal data. Three methodologies were used, namely: multiple imputation, inverse probability weighting and its doubly robustness counterpart. First, dropouts were created at different rates on simulated datasets of various sample sizes and the three methods were applied to these incomplete datasets. Then the optimum method was used on the Arthritis data as an application to real data. The dropout rates in simulated data were diverse, ranging from 8% to 33% with the aim to investigate the performance of the approaches when different amount of data are missing. The sample sizes were varied to see how these methods will behave. The performances of the three approaches were evaluated in terms of mean squared error and bias.

For multiple imputation, we make sure that the imputed values bore the structure of the data, uncertainty about the structure and included any knowledge about the process that led to the data missing [

Generally, the results from simulation study showed that all the methods can be satisfactorily used for incomplete ordinal outcomes with the assumption of MAR and MCAR mechanism. It is worth mentioning that almost all methods that are valid under MAR hold under MCAR. This is because MCAR is a special case of MAR. Consequently, ignoring missigness under MCAR will not introduce systematic bias, but will increase the standard error of the sample estimates due to the reduced sample size [

Specifically, when we consider both bias and MSE, a better performance was observed for DRGEE over single robustness alternatives MIGEE and IPWGEE in the simulation study. This is consistent with the results reported in [

In this study, missing values were only on the response variable. However, this does not limit the applicability of DRGEE, MIGEE and IPWGEE to that case only. These methods can be extended to situation where missing values are on the response and covariates variables. It is also important to note that DRGEE, MIGEE and IPWGEE all rely on the assumption that the missingness is MAR (and hence necessarily under MCAR). Typically, the possibility that the missing mechanism is MNAR cannot be ruled out. Whence, caution should be exercised in interpreting results from any of these procedures. Under MNAR, researchers are always encouraged to do sensitivity analysis [

In conclusion, based on the results of this simulation, the DRGEE is recommended because consistency is guaranteed under the MAR (and hence necessarily under MCAR) if at least one of the missing data models is correctly specified. It became clear that the IPWGEE method does not always yield the best results, even if the MAR mechanism holds. In addition, it is advisable to include few and necessary auxiliary variables when constructing weights for individuals, while too many variables can be harmful. For instance, when the number of individuals is small, we run the risk of giving too much weight to one specific subject.

Sincere acknowledgements to the African Union for giving me the opportunity to do this research.

The authors declare no conflicts of interest regarding the publication of this paper.

Ditlhong, K.E., Ngesa, O.O. and Kombo, A.Y. (2018) A Comparative Analysis of Generalized Estimating Equations Methods for Incomplete Longitudinal Ordinal Data with Ignorable Dropouts. Open Journal of Statistics, 8, 770-792. https://doi.org/10.4236/ojs.2018.85051