IMPUTATION METHODS ON GENERAL LINEAR MIXED 

MODELS OF LONGITUDINAL STUDIES 

YANN-YANN SHIEH 

AMERICAN INSTITUTES FOR RESEARCH 

1000 THOMAS JEFFERSON ST. NW. WASHINGTON DC 20007. YSHIEH@AIR.ORG

KEY WORDS: Imputation, missing data, mixed model, longitudinal data. 

Abstract: Survey data collection is a very efficient way to gather information for research of interest. Many state and federal 

government agencies collect data through surveys. Long-term longitudinal studies are the most appropriate studies for the 

study of individual change over time. As a result, most longitudinal studies and surveys have non-responses and missing data 

issues. The problem of missing data arises frequently in practice in applied research settings. Imputation is a way to handle 

missing data. General linear mixed models are commonly used in the analysis of unbalanced repeated measures designs 

(Verbeke & Molenberghs, 2000). In this study, we simulate longitudinal data under a variety of missing data patterns that is 

contrasted with modeling using missing value imputation methods in the use of general linear mixed modeling of 

longitudinal surveys with the focus of documenting the characteristics of the model parameter estimates and their standard 

errors. 

1. INTRODUCTION 

Survey data collection is a very efficient way to gather information for research of interest. Many state and federal 

government agencies collect data through surveys. Some studies are often designed to investigate changes in a specific 

parameter, which is measure repeatedly over time. Long-term longitudinal studies are the most appropriate studies for the 

study of individual change over time and factors likely to influence change over time. As a result, most longitudinal studies 

and surveys have non-responses and missing data issues. Missing or incomplete data are a serious problem in many fields of 

research for data analysis, leaving the question: How to handle the missing values in such a way to make the result be as 

close to the truth as possible? 

One approach is to analyze cases with complete data. Another is to use of complete-data statistics on data sets with missing 

values filled-in. Alternatively, one can used maximum-likelihood approaches, such as that in general linear mixed modeling, 

to deal with missing data. Imputation is the name given to any method whereby missing values in a data set are filled-in with 

plausible estimates. There are several methods for imputing the missing values. In some cases, when auxiliary information is 

properly used, imputation increases statistical accuracy. A key point that is clear from the missing data literature is to choose 

a computational method or combination of methods based on the nature of the problem, the computational resources, the 

accuracy requirement, and the degree of difficulty of any required theoretical derivations. 

 

The purposes of this study are the followings. First, we simulate longitudinal data with different missing data patterns and 

examine the impact of missing data pattern on the quality of parameter estimates and their standard errors. Second, we select 

examples for some of these unbalanced missing data designs– but focus primarily on a case study where observation times 

are fixed but data are missing at some of the time points, to impute the missing data and compare the results with and without 

imputed values using general linear mixed modeling of data sets. 

2. GENERAL THEORY AND METHODOLOGY 

2.1 General linear mixed model 

The data analyses will be expressed in the general linear mixed model family: , Y X i i ii i = β + + Z b ε for i n = 1,...,

independent units, where ( , ), ib N ≈ 0 D ( , ), ε i ≈ N 0 Σ and bi and εi are statistically independent, Yi is the ( iτ × 1) response 

vector. Xi is a ( iτ × p) design matrix for the fixed effects. β is a (p× 1) vector of unknown fixed regression coefficients. Zi is 

a ( iτ × q) design matrix for the random effects. Though a number of estimation strategies are available, the current paper uses 

maximum likelihood estimation as implemented in the general linear mixed modeling procedure in HLM. 

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2.2 Missing value mechanisms and pattern 

There are a number of different ways to conceptualize how missing data arises. Little and Rubin (1987) introduced specific 

missing data terminology as a standard framework to deal with missing data mechanisms and their effect on data analysis. 

Little and Rubin (1987) found it useful to distinguish between data that are Missing Complete at Random, Missing at 

Random, and Non-ignorable Missing, where: 

(1) Missing Completely at Random (MCAR). If the probability that a response is missing is independent of both the 

observed data for that case and the unobserved responses are simple a random sample from the observed data. An 

example of MCAR missing data arises when investigators randomly assign research participants to complete two-thirds 

of a survey instrument. Graham, Hofer, MacKinnon (1996) illustrate the use of planned missing data patterns of this type

of gather responses to more survey items from fewer research participants than one ordinarily obtains from the standard 

survey completion paradigm where every research participant receives and answers each survey question. 

(2) Missing at Random (MAR). If the probability that a response is missing depends on the observed data, but not on the 

unobserved data. This assumes the parameters of the model for the data are distinct from the parameters of the 

missingness mechanism. The missingness mechanism is ignorable. For example, in a reading comprehension test at the 

beginning of a survey administration session, research participants with lower reading comprehension scores may be less 

likely to complete the entire survey. The missing data are due to some other external influence. 

(3) Non-ignorable Missing. When respondents and non-respondents, with the same values of some variables observed for 

both, differ systematically with respect to values of the variable missing for the non-respondent. In other words, the 

pattern of data missingness is non-random and it is not predictable from other variables in the database. For example, a 

participant in a weight-loss study does not attend a weigh-in due to concerns about his/her weight loss, his/her data are 

missing due to non-ignorable factors. 

In practice it is usually difficult to meet the MCAR assumption. MAR is an assumption that is more often but not always 

tenable. Ignorability is a judgment made by the data analyst. Rubin (1976) addressed the problem of missing data. He 

mentioned that when making sampling distribution inferences about the parameter of the data, it is appropriate to ignore the 

process that causes missing data if the missing data are missing at random and the observed data are observed at random, but 

the inferences are generally conditional on the observed pattern of missing data. Rubin (1976) farther suggested that when 

dealing with real data, data analyst or statistician should explicitly consider the process that causes missing data and needs 

models for the process. 

2.3 General theory for imputation 

Two principal approaches to estimation with missing data are weighting and imputation. Weighting typically is used in unit 

nonresponse which is viewed as the inverse of the response probabilities associated with the response mechanism. Imputation 

is used in item nonresponse. The imputed values are sample-based. There are a variety of imputation methods. The goal of 

any imputation technique is to produce a complete data set, which can then be analyzed using complete-data inferential 

method. The observed values are used to impute values for the missing observations. Two kinds of imputation methods are 

discussed: Single Imputation and Multiple Imputation. 

 

Single Imputation 

(1) Mean Imputation: The sample mean of a variable replaces any missing data for that variable. 

(2) Hot-deck Imputation: Missing values are replaced with values taken from matching respondents. 

(3) Last Value Carried Forward or LVCF: The last observed value is used to fill in missing values at subsequent points in a 

longitudinal study. 

(4) Predicted Mean: An ordinary least-squares multiple regression algorithm is used to impute the most likely value. In this 

method, researchers develop a regression equation based on complete case data for a given variable, treating it as the 

outcome and using all other relevant variables as predictors. 

Single imputation is easy to employ with a single value imputed for a missing value. However, there are disadvantages of 

single imputation since it does not reflect extra uncertainty and does not display variation due to missing data. Rubin (1986) 

sees a disadvantage of single imputation “...the one imputed value cannot in itself represent uncertainty about which value to 

impute: If one value were really adequate, then that value was never missing. Hence, analyses that treat imputed values just 

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like observed values generally systematically underestimate uncertainty, even assuming the precise reason for nonresponse 

are known.”

Multiple Imputation 

Multiple Imputation, first proposed by Rubin in the early 1970’s (Rubin, 1976) as a way to address survey non-response and 

issues associated with single imputation, involves replacing each missing values by M (M>=2) imputed values to create M

complete data sets. Multiple imputation carries out analysis under each set of imputation and combines analyses to reflect 

within-imputation and between-imputation variability. Several techniques involved in multiple imputation are mentioned by 

Rubin (1986), Little and Rubin (1987), Schafer and Olsen, (1998), and Schafter (1999). In the current study, we use two 

multiple imputation methods as follows. 

(1) Predictive Model Based Method: An ordinary least-squares regression method of imputation is used. Model is estimated 

from the observed data, then using this estimated model, a new linear regression parameters are randomly drawn from 

their Bayesian posterior distribution. 

(2) Propensity Score: An implicit model approach based on Propensity Scores and an Approximate Bayesian Bootstrap is 

used to generate the imputations. 

Multiple imputation has its advantage and disadvantage. The major advantages of multiple imputation as indicated by Rubin 

(1986) are that standard complete-data methods are used to analyze each completed data set; moreover, the ability to utilize 

data collector’s knowledge in handling the missing values is not only retained but actually enhanced. In addition, multiple 

imputations allow data collectors to reflect their uncertainty as to which values to impute. Disadvantages include the time 

intensiveness imputing five to ten data sets, testing models for each data set separately, and recombining the model results 

into one summary. 

3. SIMULATION STUDY– LONGITUDINAL GROWTH DATA WITH MISSING DATA PATTERNS 

The data sets under consideration for the present paper are derived from National Education Longitudinal Study of 1988 

(NELS:88) to obtain the parameters for simulation. NELS:88 is the most current and comprehensive source of information 

on personal and contextual factors in the educational life of U.S. adolescents over time. It began in 1988 with a cohort of 

25,000 eighth-graders and follow-up data were collected in 1990, 1992, 1994, and 2000. We used students math 

achievement scores to simulate missing data under different missing data mechanism. 

3.1 Simulated model 

Data were simulated from two-level models with either four or eight waves. These models describe an increasing linear trend 

in achievement for students being tested on a standard test, administered each semester, over a two- or four-year period. 

Each student (denoted by i) has an ‘ability’ latent variable written as ai which remains constant over the testing periods. On 

test period t (t=1, ..., T) the test score for the i-th student is yit.. The mean test score for student i is a linear function of time 

and student’s ability ai: it 0 1 ai μ = b + tb +

 

 

The actual test score on time period t has an random errors eit, so that it it it y = u + e

The ability distribution for the ai has a mean of 0 and variance of σ

2

a (i.e., N(0,σ

2

a) ) and the error distribution for the eit 

also has mean of 0 and variance of σ

2

e (i.e., N(0,σ

2

e)). These are standard assumptions for the two-level model level model. 

3.2 Time waves 

Two longitudinal time waves were examined in this study (T = 4 and T = 8). In addition, we modeled the linear trend 

regression over time as different for boys and girls. For T = 4 the regression for boy was 50+10t and for girls it was 35+15t, 

so that the boys started at score 50 at t = 1 and increased to score 90 at t = 4, while girls started at score 35 at t=1 and 

increased to score 95 at t = 4. For T = 8, the intercepts were assumed the same and the sloped were halved, giving the same 

initial and final means. Therefore, the data were generated from an interaction model including a dummy variable g for 

gender (g=0 for boys, g=1 for girls). it 50 10 15 i 5 i

*

i it.

y = + t − g + g t + a + e

3.3 Missing data mechanisms, and probabilities of missingness 

Two missing data patterns were used in this study: 

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(1) MCAR-missing completely at random. In this case, any test score is missing independently of the others with a constant 

probability p over the four or eight time periods of either 0.05 or 0.1. The proportions of complete observations with all 

scores are for T = 4, 0.81 for p = 0.05 and 0.66 for p = 0.1, and for T = 8, 0.66 for p = 0.05 and 0.43 for p = 0.1. 

(2) MAR-missing at random. Any test score is missing independently of the others, but with a probability pt

 which increases 

with time: pt

 = 0.0, 0.025, 0.05, and 0.075 for T = 4, and pt = 0.0, 0.0125, 0.025, 0.0375, 0.05, 0.0625, 0.075, and 0.0875 

for T = 8. 

3.4 Data generation procedures 

A two-stage sampling procedure for generating simulation data was used. Two types of data were examined: full data and 

complete data. We generated the T values of yit from the model for each student i , and generated a corresponding set of 

dummy indicators dit from the missingness model, where dit = 0 if the corresponding test score yit is to be missing (with 

probability 1-p or 1-pt) and dit = 1 if yit is to be observed (with probability 1-p or 1-pt). The ‘full data’ for case I consists of 

the set of ∑ responses. If all d

t

dit it for this case i is 1, the response is complete and the case is appended to the ‘complete 

case’ data set, otherwise, it is omitted as incomplete. Therefore, the ‘full data’ set consists of n strings of between 1 and 4 or 1 

and 8 responses. The ‘complete case’ data set consists of less than n strings of length 4 and 8. 

3.5 Parameter estimation 

To be able to make better comparisons between parameter values of different magnitude, the following quantities for each 

data set were computed: 

(1) The average values of the estimated parameters (both fixed –regression coefficient, and random-variance components). 

(2) The bias of estimated parameters, by subtracting the true values of the parameters. 

(3) The standard errors of the parameter estimates. 

4. STUDY EXAMPLES FOR IMPUTATION AND ANALYSIS STRATEGY 

For the current paper, based on different missing data mechanisms and patterns, four incomplete data sets are selected for 

imputation from the above incomplete data set. These incomplete data sets are described in the following section. 

4.1 Incomplete data sets 

Data Subset 1 (MCAR). We assume 500 students taking tests in the four continuous semesters. Some students missed test 

scores. 

Data Subset 2 (MAR). We assume 500 students taking tests in the four continuous semesters. Some students missed test 

scores. 

Data Subset 3 (MCAR). We assume 500 students taking tests in the eight continuous semesters. Some students missed test 

scores. 

Data Subset 4 (MAR). We assume 500 students taking tests in the eight continuous semesters. Some students missed test 

scores. 

4.2 Imputation data sets 

Currently a number of statistical software packages and procedures are available to impute missing values. In this paper, we 

use Solas (Software for Missing Data Analysis 3.2) software. The imputation data sets are labeled as a function of the data 

subset on which the imputations were made (data subset 1, 2, 3, or 4) and as a function of the imputation method used to fillin missing values: a) No mputation, b) Hot-deck Imputation , c) Group Mean Imputation, d) Last Value Carried Forward 

(LVCF), e) Predicted Mean f) Multiple imputation based on M imputations using multiple regression analysis, g)M=c

(Multiple imputation based on M imputations using Propensity Scores and an Approximate Bayesian Bootstrap approach). 

For each data set a range of M values were used. Due to space considerations results on only some of the M values are 

reported. 

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4.3 Parameter estimation 

A general linear mixed model approach using maximum likelihood estimation (HLM program) was used for all the data sets 

under consideration. Parameter estimates and standard errors were obtained for each study data set parameter. In the case of 

analyses of incomplete and single-value imputation method data sets, a single set of results were obtained. In the case of 

multiple imputation, where M=c, c sets of results were obtained and pooled as follows to generate the M=c results reported. 

The M within-imputation estimates for θ (the parameter of the interest) are pooled to give the multiple imputation estimates: 

. Now, suppose that complete data inference about θ would be made by (θ−θ 1 (

1

ˆ*

M m

θ M m

−

= ∑ =

) ˆθ

)

* '

]

∗

) ∼ Ν(0,Υ). Then, one can 

make normal based inferences for θ based upon (θ−θ∗

) ∼ Ν(0,V), where , such that ˆ ˆ 1 VWM M B ( 1)

−

=+ +

1 (

1

M m WM U m

∧

−

= ∑ =

is the average within-imputation variance, and 

( ) 1 * ( )

1

ˆ ˆ ˆˆ ˆ ( 1) [( )( ) M

m m B M m θ θθ θ

−

=− − − ∑ =

is the between-imputation variance. 

5. RESULTS AND SUMMARY 

Across the series of analyses conducted, the following results were observed: 

5.1 Parameter estimates for missing data set 

 

The average parameter estimates for complete data set and full data set across examined conditions are shown in the table 1. 

In looking at Table 1, both complete cases and full data estimates are consistent, as expected under theory, for both MCAR 

and MAR missingness patterns. For the bias parameter estimates and standard errors associated with the parameter estimates, 

when sample size increased, the bias and standard errors decreased as expected. However, biases were much smaller than 

standard errors (see Tables 2 and 3). In general, the biases of the complete cases estimates were larger and their standard 

errors were consistently larger than those of the full data estimates. 

Table 1 

Average Parameter Estimates by Examined Data Sets and Conditions 

Complete case Full case 

Condition N γ00 γ10 γ01 γ11 σ0

2

σε

2

γ00 γ10 γ01 γ11 σ0

2

σε

2

SIZE 

 

500 16 49.967 7.501 -14.939 3.746 59.169 40.013 50.016 7.502 -14.967 3.741 59.395 40.256

1000 16 49.986 7.497 -14.966 3.743 59.596 40.032 50.017 7.498 -15.012 3.743 59.703 40.126

2000 16 50.016 7.491 -14.977 3.743 59.907 40.134 50.032 7.497 -15.007 3.748 60.122 40.161

ICC 

0.3 12 50.006 7.499 -14.957 3.738 29.766 70.125 50.026 7.498 -14.992 3.742 29.805 70.281

0.5 12 49.962 7.490 -14.964 3.745 49.502 50.044 50.025 7.499 -14.995 3.743 49.675 50.309

0.7 12 49.998 7.498 -14.963 3.746 69.535 30.052 50.019 7.499 -14.994 3.744 69.756 30.111

0.9 12 49.992 7.499 -14.958 3.747 89.427 10.018 50.017 7.500 -15.002 3.747 89.725 10.022

TIME 

 

4 24 50.083 9.998 -15.012 4.987 60.131 40.152 50.073 9.998 -15.003 4.988 60.158 40.312

8 24 49.896 4.995 -14.909 2.501 58.984 39.967 49.971 5.000 -14.988 2.500 59.322 40.050

MISSING

NESS 

MACR 24 49.974 7.496 -14.945 3.741 59.417 40.049 50.019 7.500 -14.996 3.744 59.743 40.182

MAR 24 50.005 7.497 -14.976 3.748 59.698 40.071 50.024 7.498 -14.995 3.745 59.737 40.179

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Table 2 

Average Bias Parameter Estimates by Examined Data Sets and Conditions 

 Complete case Full case 

Condition N γ00 γ10 γ01 γ11 σ0

2 σε

2

γ00 γ10 γ01 γ11 σ0

2 σε

2

SIZE 

500 16 -0.033 0.001 0.061 -0.004 -0.831 0.013 

 

0.016 0.002 0.033 -0.009 -0.605 0.256

1000 16 -0.014 -0.003 0.034 -0.007 -0.404 0.032 

 

0.017 -0.002 -0.012 -0.007 -0.297 0.126

2000 16 0.016 -0.009 0.023 -0.007 -0.093 0.134 0.032 -0.003 -0.007 -0.002 0.122 0.161

ICC

0.3 12 0.006 -0.002 0.043 -0.012 -0.234 0.125 0.026 -0.002 0.008 -0.008 -0.195 0.281

0.5 12 -0.038 -0.010 0.036 -0.005 -0.498 0.044 0.025 -0.001 0.006 -0.007 -0.326 0.309

0.7 12 -0.002 -0.002 0.037 -0.004 -0.465 0.052 0.019 -0.001 0.006 -0.006 -0.244 0.111

0.9 12 -0.008 -0.001 0.042 -0.003 -0.573 0.018 

 

0.017 -0.001 -0.002 -0.003 -0.275 0.022

TIME

4 24 0.083 -0.002 -0.012 -0.013 0.131 0.152 0.073 -0.002 -0.003 -0.012 0.158 0.312

8 24 -0.104 -0.005 0.091 0.001 -1.016 -0.033 -0.030 0.000 0.012 0.000 -0.679 0.050

MISSING

NESS

MACR 24 -0.026 -0.004 0.055 -0.010 -0.583 0.049 0.019 0.000 0.004 -0.007 -0.257 0.182

MAR 24 0.005 -0.003 0.024 -0.003 -0.302 0.071 

 

 

0.024 -0.002 0.005 -0.006 -0.263 0.179

Table 3 

Average Standard Errors of Parameter Estimates by Examined Data Sets and Conditions 

Complete case Full case 

Condition N γ00 γ10 γ01 γ11 σ0

2 σε2

γ00 γ10 γ01 γ11 σ0

2 σε2

SIZE

500 16 0.796 0.135 1.048 0.186 4.829 1.518 0.672 0.124 0.901 0.178 4.111 1.433

1000 16 0.569 0.100 0.724 0.124 3.804 1.025 0.485 0.091 0.623 0.116 3.040 0.906

2000 16 0.398 0.068 0.520 0.114 2.574 0.717 0.338 0.063 0.434 0.084 2.231 0.616

ICC 

0.3 12 0.593 0.142 0.774 0.218 2.321 1.921 0.510 0.128 0.668 0.175 2.027 1.721

0.5 12 0.580 0.117 0.743 0.156 3.134 1.345 0.501 0.109 0.653 0.148 2.759 1.262

0.7 12 0.581 0.091 0.748 0.121 4.327 0.807 

 

 

0.493 0.084 0.646 0.115 3.473 0.717

0.9 12 0.596 0.054 0.791 0.071 5.161 0.274 0.487 0.049 0.645 0.066 4.249 0.240

TIME

4 24 0.595 0.142 0.761 0.205 3.573 1.165 0.552 0.137 0.701 0.187 3.281 1.181

8 24 0.580 0.060 0.767 0.078 3.899 1.009 0.445 0.048 0.606 0.065 2.973 0.789

MISSING

NESS

MACR 24 0.620 0.104 0.817 0.153 4.033 1.121 0.503 0.093 0.657 0.127 3.108 0.977

MAR 24 0.555 0.098 0.711 0.130 3.439 1.052 0.494 0.092 0.649 0.125 3.146 0.993

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5.2 Parameter estimates for imputed data sets 

Parameter estimates and standard errors were obtained for each example data set and each imputation data set. Tables 4, 5 

and 6 displayed the results. In looking at the Table 4, all the imputation data sets produced similar results. Compared the 

results from Table 4 to Tables 5 and 6, in general, the standard errors from the analysis of the imputation data sets were 

greater than from the no imputation data sets. For the parameter estimates, the intercepts for the imputation data sets were 

greater than those from the no imputation data sets; but the slopes were less than those from the no imputation data sets. For 

the variance component σε

2

, predicted mean, a single imputation method, tended to produce smaller values than those from 

other imputation methods and was less than those from no imputation data sets. 

Table 4. Parameter estimates and standard errors for underlying no imputation data sets 

Parameter estimates 

Missing 

Dataset SIZE ICC TIME MISSING γ00 γ10 γ01 γ11 σ0

2

σε

2

complete 500 0.5 4 MACR 50.147 10.012 -15.022 4.980 49.873 50.216

complete 500 0.5 4 MAR 50.161 10.002 -15.041 4.988 49.983 50.228

complete 500 0.5 8 MACR 49.354 4.994 -14.886 2.510 47.950 49.378

complete 500 0.5 8 MAR 49.854 4.994 -14.886 2.510 48.870 49.878

full 500 0.5 4 MACR 50.133 10.007 -15.025 4.978 49.839 50.626

full 500 0.5 4 MAR 50.157 9.999 -15.033 4.980 49.671 50.730

full 500 0.5 8 MACR 49.898 5.000 -14.914 2.501 48.839 50.276

full 500 0.5 8 MAR 49.898 5.000 -14.914 2.501 48.839 50.276

SE 

complete 500 0.5 4 MACR 0.855 0.220 1.119 0.324 4.262 2.103

complete 500 0.5 4 MAR 0.811 0.230 1.098 0.324 4.184 2.152

complete 500 0.5 8 MACR 0.709 0.086 0.888 0.105 3.956 1.589

complete 500 0.5 8 MAR 0.709 0.086 0.888 0.105 3.956 1.589

full 500 0.5 4 MACR 0.766 0.215 1.015 0.318 3.826 2.014

full 500 0.5 4 MAR 0.765 0.218 1.037 0.322 4.359 2.860

full 500 0.5 8 MACR 0.577 0.075 0.783 0.098 3.358 1.330

full 500 0.5 8 MAR 0.577 0.075 0.783 0.098 3.358 1.330

Table 5. Parameter estimates, underlying data subset, and imputation method 

SIZE ICC TIME MISSING Imputation γ00 γ10 γ01 γ11 σ0

2

σε

2

500 0.5 4 MACR Hot-deck 52.403 9.259 -16.384 5.149 61.456 53.224 

500 0.5 4 MAR Hot-deck 51.550 9.474 -17.366 5.435 47.334 52.145 

500 0.5 8 MACR Hot-deck 51.388 4.803 -16.396 2.617 56.704 52.081 

500 0.5 8 MAR Hot-deck 50.657 4.972 -16.171 2.446 49.819 50.701 

500 0.5 4 MACR Group mean 52.877 9.112 -16.110 5.163 52.638 55.286 

500 0.5 4 MAR Group mean 52.148 9.158 -17.529 5.589 44.384 56.971 

500 0.5 8 MACR Group mean 53.433 4.421 -15.503 2.478 43.631 68.896 

500 0.5 8 MAR Group mean 51.198 4.815 -15.974 2.379 45.153 58.984 

500 0.5 4 MACR LVCF 52.403 9.259 -16.384 5.149 61.456 53.224 

500 0.5 4 MAR LVCF 51.550 9.474 -17.366 5.435 47.334 52.145 

500 0.5 8 MACR LVCF 51.388 4.803 -16.396 2.617 56.704 52.081 

500 0.5 8 MAR LVCF 50.657 4.972 -16.171 2.446 49.819 50.701 

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Table 5 (continued). Parameter estimates, underlying data subset, and imputation method

500 0.5 4 MACR Predicted 51.263 9.813 -15.888 5.016 52.304 46.599 

500 0.5 4 MAR Predicted 51.407 9.589 -17.690 5.660 45.494 48.908 

500 0.5 8 MACR Predicted 51.212 4.899 -16.420 2.688 44.745 49.226 

500 0.5 8 MAR Predicted 50.572 5.028 -16.190 2.454 45.736 50.414 

500 0.5 4 MACR Multiple R 51.264 9.802 -15.889 4.997 59.280 45.841 

500 0.5 4 MAR Multiple R 51.274 9.655 -17.584 5.606 46.891 50.177 

500 0.5 8 MACR Multiple R 51.006 4.945 -16.111 2.618 55.544 49.221 

500 0.5 8 MAR Multiple R 50.541 5.035 -16.224 2.477 49.237 50.635 

500 0.5 4 MACR Propensity 51.271 9.762 -15.896 5.064 57.653 46.721 

500 0.5 4 MAR Propensity 51.125 9.740 -17.298 5.447 46.593 50.947 

500 0.5 8 MACR Propensity 50.958 4.947 -16.034 2.621 53.877 51.036 

500 0.5 8 MAR Propensity 50.507 5.047 -16.138 2.448 48.236 51.451 

Table 6. Standard errors of Parameter estimates, underlying data subset, and imputation method 

SIZE ICC TIME MISSING Imputation γ00 γ10 γ01 γ11 σ0

2

σε

2

500 0.5 4 MACR Hot-deck 0.782 0.215 1.066 0.293 7.839 7.295

500 0.5 4 MAR Hot-deck 0.687 0.198 1.004 0.289 6.880 7.221

500 0.5 8 MACR Hot-deck 0.618 0.073 0.843 0.100 7.530 7.217

500 0.5 8 MAR Hot-deck 0.550 0.067 0.805 0.098 7.058 7.120

500 0.5 4 MACR Group mean 0.766 0.219 1.044 0.298 7.255 7.435

500 0.5 4 MAR Group mean 0.699 0.207 1.021 0.303 6.662 7.548

500 0.5 8 MACR Group mean 0.608 0.084 0.829 0.115 6.605 8.300

500 0.5 8 MAR Group mean 0.552 0.073 0.806 0.106 6.720 7.680

500 0.5 4 MACR LVCF 0.791 0.215 1.109 0.293 7.839 7.295

500 0.5 4 MAR LVCF 0.687 0.198 1.004 0.289 6.880 7.221

500 0.5 8 MACR LVCF 0.618 0.073 0.843 0.100 7.530 7.217

500 0.5 8 MAR LVCF 0.550 0.067 0.805 0.098 7.058 7.120

500 0.5 4 MACR Predicted 0.727 0.201 0.992 0.274 7.232 6.826

500 0.5 4 MAR Predicted 0.668 0.192 0.977 0.280 6.745 6.993

500 0.5 8 MACR Predicted 0.568 0.071 0.775 0.097 6.689 7.016

500 0.5 8 MAR Predicted 0.536 0.067 0.783 0.098 6.763 7.100

500 0.5 4 MACR Multiple R 0.745 0.199 1.015 0.272 7.699 6.770

500 0.5 4 MAR Multiple R 0.669 0.192 0.987 0.281 6.848 7.083

500 0.5 8 MACR Multiple R 0.608 0.071 0.829 0.097 7.453 7.016

500 0.5 8 MAR Multiple R 0.548 0.067 0.802 0.098 7.017 7.116

500 0.5 4 MACR Propensity 0.744 0.201 1.014 0.274 7.593 6.835

500 0.5 4 MAR Propensity 0.680 0.196 0.994 0.286 6.826 7.138

500 0.5 8 MACR Propensity 0.606 0.073 0.826 0.099 7.340 7.144

500 0.5 8 MAR Propensity 0.547 0.068 0.799 0.099 6.945 7.173

6. CONCLUSIONS 

The procedures used fort he analysis of multilevel models are unusual in that they allow the same analysis for incomplete 

responses as for complete responses in longitudinal studies. This feature is not shared by other time series models like AR 

and MA, which require the same form of missing data analyses, as do general regression models with missing covariates. 

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In this study, since the analysis for incomplete response is the same as for complete responses there seems to be no reason for

restricting analysis to complete cases when multilevel modeling is used for analysis of longitudinal data, and the missingness 

process can be assumed to be MAR or MCAR. If the incomplete observations result from a non-random missingness process, 

that is, if the probability of being missing is related to the value which is missing, then both complete cases and full data 

parameter estimates will be biases, as is true in general for analysis of incomplete data. 

Multiple imputation is a valuable technique that allows the use of complete-data statistics on data sets with missing values. 

Comparing the incomplete and imputation data set analyses results using general linear mixed modeling procedures for the 

growth data, general linear mixed modeling of incomplete data sets with maximum likelihood method is an effective and 

flexible way of dealing with missing values. 

7. REFERENCES 

Graham, J. W., Hofer, S. M., & MacKinnon, D. P. (1996). Maximizing the usefulness of data obtained with planned 

missing value patterns: An application of maximum likelihood procedures. Multivariate Behavioral Research, 31(2), 197-

218. 

Hedejer, R., & Gibbons, R. D. (1997). Application of random-effects pattern-mixture models for missing data in 

longitudinal studies. Psychological Methods, 2, 64-78. 

Jennrich, R.I., and Schluchter, M.D. (1986). Unbalanced repeated measures with structured covariance matrices. 

Biometrics, 42, 805-820. 

Little, R., and Rubin, D. B. (1987). Statistical analysis with missing data. John Wiley & Sons, New York. 

Potthoff, R.F., and Roy, S.N. (1964). A generalized multivariate analysis of variance model useful expectially for growth 

curve problems. Biometrika, 51, 313-326. 

Rubin, R. (1976). Inference and missing data. Biometrika, 63, 581-592. 

Schafter, J. L. (1999). Multiple imputation: A primer. Statistical Methods in Medical Research, 8, 3-15. 

Schafer, J. L., & Olsen, M. K. (1998). Multiple imputation for multivariate missing data problems: A data analyst’s

perspective. Multivariate Behavioral Research, 33, 545-571. 

Verbeke, G., and Molenberghs, G. (2000). Linear mixed models for longitudinal data. Springer-Verlag, New York. 

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