A Random-Effects Pattern-Mixture Model with Application to Comorbidity

Abstract

French

This study aims to better understand the contribution of potential non-ignorable nonresponse associated with attrition and wave-nonresponse in race/ethnicity disparities in health trajectories. The empirical work of this study is based on the 1992-2010 Health and Retirement Study (HRS). Both growth curve models (direct likelihood maximization and pattern mixture) have very similar results, but the standard errors tended to be slightly underestimated in the former. Results from the growth curve models suggest that with age, racial/ethnic disparities in health decrease for low educated individuals, persist for those with at least a high school education and for Hispanic elderly with a highs school or GED education, and increase among the lower educated. The study concludes that any possible non-ignorable differences between models are not large enough to affect inferences drawn from the data analysis.

Keywords

Modèles de courbe de croissance Modélisation motif mélangé Trajectoires de santé Personnes âgées Growth Curve Models Pattern-mixture Modeling Health Trajectories Elderly

Introduction

With improvements in statistical methods and increasing availability of longitudinal data, research on racial/ethnic health disparities using hierarchical modeling has burgeoned since 2000 (Lynch, 2008). A distinguished legacy of research has demonstrated that education exerts a long-term and cumulative effect on health by conferring vital resources such as income, wealth, occupational status, knowledge of health risks, and other social-psychological resources (e.g. Mirowsky and Ross, 1998; Ross and Wu, 1995; Rousset, Giret and Grelet, 2012). The review of empirical evidence by Jackson, Govia and Sellers (2010) and focus group research by Mudege and Ezeh (2009) implies that inequalities among ethnic groups are relevant to health outcomes. Specifically, poverty experienced earlier in life such as labor market disadvantage, low rates of pension plan participation and other forms of economic insecurities, can predispose individuals to certain health disadvantages as they grow older (Jackson et al., 2010). In addition, exposure to toxic substances and environments earlier during working lives could predispose older people to diseases such as cancer or blindness (Mudege and Ezeh, 2009).

A few of these researchers have also acknowledged the importance of properly addressing and understanding the effect of nonrandom non-response due to loss to follow-up, refusal, institutionalization, and death in longitudinal studies (Beckett, 2000; Kelley-Moore and Ferraro, 2004; Lillard and Panis, 1998). Even though pattern mixture models have been developed to address the potential non-ignorability of missing data, there is a lack of empirical analysis about their effect on health changes over time among the elderly population. To my knowledge, Xiao Xu and his colleagues incorporated pattern-mixture approach in their attempted to examine trajectories of depressive symptoms among elderly Whites, Blacks, and Hispanics (Xu et al., 2010). In order to correct these deficiencies of previous research, this study will utilize the 1992-2008 Health and Retirement Study (HRS) to better understand the contribution of potential non-ignorable nonresponse associated with attrition and wave-nonresponse in race/ethnicity disparities in health trajectories.

Method

Data and Design

The empirical work of this study is based on the 1992-2010 Health and Retirement Study (HRS). The HRS, which is a collaborative effort between the Social Security Administration (SSA) and the National Institute on Aging (NIA) at the National Institutes of Health (NIH), has been the principal source of health data on the advanced age population in America. The main purpose of HRS is to paint an emerging portrait of an aging America by assessing the amount and distribution of illness, its effects in terms of disability and chronic impairments, and the kinds of health services people receive.

The HRS has several analytic strengths. It is an ongoing longitudinal survey with a high re-interview rate. In addition to providing current information on gender, age, race, chronic health conditions, marital status, income, and education, individuals participated in in-depth interviews about health behaviors, health care utilization and costs, housing, income, assets, employment status and history, physical and mental health, insurance coverage, financial status, family structure, family support systems, labor market status, and retirement planning.

Measures

Dependent variable

Comorbidity was used to represent the health status of elderly men and women at each wave. It is a continuous variable measuring the co-occurrence (sum of) of multiple diagnosed health conditions in an individual within a year. The illnesses included are hypertension, diabetes, cancer, lung disease, heart disease stroke, psychiatric problems, and arthritis.

Independent variables

Age is treated as a continuous variable. Education is indicated by dummy variables for less than high school, high school or GED (reference), some college and college and beyond. Respondent’s race is indicated by dummy variables for White/Caucasian (reference), Black / African-American, Hispanics, and other racial groups. Following Vandecasteele and Debels (2007), the patterns of attrition will be classified into Always Participating, Monotone Attrition, and Variable Participation. I will follow Vandecasteele and Debels (2007) to define Monotone Attrition as a form of permanent attrition where the individual drops out and never returns to the study and Variable Participation as a form of temporary attrition, equivalent to wave non-response, where the individual drops out but returns to the study at least once.

Data Analysis

A growth curve model, using PROC MIXED in SAS, is used to estimate the distinctive health trajectories. Growth curve models belong to a general class of mixed models that take into consideration variables measured at different levels of aggregation. Many problems inherent to the traditional longitudinal approach can be overcomed using the generalized mixed modeling approach (Dupéré, Lacourse, Vitaro and Tremblay, 2007). Complete case approach removes the problem of missing data by removing these observations (Horton and Kleinman, 2007). Omitting participants with missing data assumes data are missing completely at random (MCAR¹) under the assumed analysis model (Little, 1995). Growth curve models are appropriate when the MAR² assumption holds and they have the ability to partition the contribution of between- and within-subject influences on the dependent variables, to incorporate time-varying covariates, and to handle unbalanced designs by taking into account certain types of missing data.

Growth curve modeling begins by describing change over time for each individual in the study. In addition to the simple descriptions of change, growth curve modeling allows us to explain systematic inter-individual differences in both the level and rate of change for a variable such as household income. Conceptually, a regression line (a growth curve) linking the outcome variable to time is fit for each individual in the study using an intercept or initial level ( $π_{0}$ ) and a slope or rate of change ( $π_{1}$ ). Individual trajectories (level and rate of change) are expected to differ across individuals. For a sample of n individuals, there are potentially n different initial levels and n different rates of change over time. The direct likelihood maximization growth curve model can be formulated as:

\begin{array}{ll} H e a l t h_{i j} = β_{0} + β_{1} (A g e_{i j}) + β_{2} (R a c e_{i}) + β_{3} (G e n d e r_{i}) + β_{4} ({Race}_{i}^{*} A g e_{i j}) \\ + β_{5} ({Gender}_{i}^{*} A g e_{i j}) + v_{0 i} + v_{1 i} (A g e_{i j}) + ∊_{i j} \end{array}

where i refers to the ith wave and j refers to the jth individual.

However, both the MCAR and MAR assume that the reasons for the missing data are “ignorable” (Little, 1995) – in short, that the analyses can proceed without explicitly jointly modeling the missing data and outcomes distributions. In addition, since the growth curve models does not jointly model the outcomes of interest ( $Y_{i}$ ) and the missing data indicator which indexes patterns of missing data variable ( $R_{i}$ ), the growth curve parameter estimates would likely to be biased if the possibility of non-ignorable non-response is not adequately accounted for.

To address this issue, Little (1995) has advocated the use of patter-mixture models (PMMs) for taking account of non-ignorable non-response. Thus, PMMs can be used to determine if non-ignorable non-response is present in longitudinal data (Little, 1993 and 1995). As such, when non-ignorable non-response in longitudinal studies is a concern (Fitzmaurice et al., 2001; Hedeker and Gibbons, 1997; Pauler et al., 2003), this approach can help clarify the gaps between missing-at-random based analyses and the range of plausible biases induced by possibly non-ignorable missing data (Paddock et al., 2006). In contrast, PMMs “re-expresses” the joint model of $Y_{i}$ and $R_{i}$ by a model of $Y_{i}$ conditional $R_{i}$ and accounts for non-ignorable non-response by allowing the distribution of $Y_{i}$ to differ by the patterns of missing data $f (y_{i} | R_{i}) \neq f (y_{i})$ (Paddock et al., 2006). The pattern-mixture approach augments the random effects models by including variables defined by a subject’s pattern of missing data (Hedeker and Gibbons, 1997; Paddock et al., 2006). This allows a researcher to specify a model that does not require the missing-data mechanism to be ignorable (Hedeker and Gibbons, 1997). Under this approach, the researcher first identifies the different patterns of missing data in attempt to explicitly model the missing data distribution (Paddock et al., 2006). He or she then proceeds to include the different patterns of missing data as model covariates (Paddock et al., 2006). The idea behind PMMs is that models are built for $Y_{i} | R_{i}$ but inferences are based on $f (y) = Σ_{R} f (y | R) p (R)$ .

The pattern-mixture growth curve model can be formulated as:

\begin{array}{l} H e a l t h_{i j} = β_{0} + β_{1} (A g e_{i j}) + β_{2} (R a c e_{i}) + β_{3} (G e n d e r_{i}) + β_{4} (A t t r i t i o n_t y p e_{i}) \\ + β_{5} ({Race}_{i}^{*} A g e_{i j}) + β_{6} ({Gender}_{i}^{*} A g e_{i j}) + β_{7} ({Attrition_type}_{i}^{*} A g e_{i j}) \\ + β_{8} ({Attrition_type}_{i}^{*} R a c e_{i}) + β_{9} ({Attrition_type}_{i}^{*} G e n d e r_{i}) \\ + β_{10} ({Attrition_type}_{i}^{*} {Race}_{i}^{*} A g e_{i j}) + β_{11} ({Attrition_type}_{i}^{*} {Gender}_{i}^{*} A g e_{i j}) \\ + v_{0 i} + v_{1 i} (A g e_{i j}) + ∊_{i j} \end{array}

where i refers to the ith wave and j refers to the jth individual.

Procedures

The methods and procedures utilized in this study followed that of Hedeker and Gibbons (1997). However, since their model was developed for the analysis of Inpatient Multidimensional Psychiatric Scale (IMPS) in mind, some aspects of this study and the variables selected will deviate from their models. Since prior researchers (i.e. Binstock and George, 2006; Mudege and Ezeh, 2009) have established that educational inequalities are reflected in trajectories of health across the life course, the analyses will be disaggregated by the education levels.

Results

Regardless of which educational group is being examined, all models suggest that elderly men and elderly women experience health deterioration as they age.

Less than High School

Contrary to findings from direct likelihood maximization and pattern mixture models, findings from complete case analysis suggest that Hispanics and other ethnic groups and Whites did not differ in initial health and rate of health deterioration (see Table 1). Findings from the complete case analysis also suggest that even though elderly men have better health conditions in 1992 than elderly women, the rate of health deterioration across age is more pronounced among them. However, analyses from the direct likelihood maximization growth curve models reveal no gender differences in the rate of health deterioration across age is more pronounced among these individuals.

Table 1.
Growth Curve Analysis of Health Trajectories (Elderly Men and Women with Less Than a High School Education), HRS 1992-2010

Variables Complete case analysis Direct likelihood maximization Pattern Mixture

Estimate SE Estimate SE Estimate SE

Intercept −5.4949* 0.173 −6.260* 0.132 −7.269* 0.129

Age 0.113* 0.003 0.117* 0.002 0.139* 0.002

Race (Ref. = White)

Black 0.808* 0.27 1.527* 0.201 2.259* 0.262

Hispanic 0.464 0.268 1.199* 0.210 1.992* 0.259

Other 0.319 0.73 0.545* 0.539 1.972** 0.748

BlackAge −0.007 0.004 −0.015 0.003 −0.292*** 0.004

HispanicAge −0.005 0.004 −0.012 0.003 −0.290* 0.004

OtherAge −0.008 0.011 −0.018 0.008 −0.244 0.011

Male −0.912* 0.219 −0.494 0.164 0.373 0.210

MaleAge 0.009 0.003 0.004 0.002 −0.108 0.003

Type of Attrition (Ref. = Always participate)

Monotone attrition 10.500* 0.212

Variable participation 9.235*** 0.498

Monotone attrition*Age −0.133*** 0.002

Variable participation*Age −0.121*** 0.006

Monotone attrition*Black −1.684*** 0.351

Variable participationBlack −1.394 0.689

Monotone attrition*Hispanic −1.838*** 0.375

Variable participation*Hispanic −2.117** 0.740

Monotone attrition*Other −2.527*** 1.089

Variable participationOther −3.451 1.509

Monotone attrition*Male −1.463*** 0.277

Variable participation*Male −2.226*** 0.540

Monotone attritionBlackAge 0.232*** 0.005

Variable participationBlackAge 0.169 0.010

Monotone attritionHispanicAge 0.217*** 0.005

Variable participationHispanicAge 0.270* 0.011

Monotone attritionOtherAge 0.032* 0.002

Variable participationOtherAge 0.036 0.022

Monotone attritionMaleAge 0.024*** 0.004

Variable participationMaleAge 0.033* 0.008

Variance Components

Level 1

Within-person 0.1864* 0.002 0.2186* 0.002 0.2144* 0.002

Level 2

Initial Status 27.014* 0.893 32.9382* 0.866 29.1421* 0.757

Rate of Change 0.006* 0.000212 0.007* 2E-04 0.006* 2E-04

Covariance −0.396* 0.014 −0.448* 0.012 −0.408* 0.011

−2LL 45,781.80 104,185.90 100,882.00

N 3,070 8,741 8,741

Cases 22,552 44,198 44,198

* p<0.0001, ** p<0.01, * p<0.05

The likelihood ratio test for the joint significance of the attrition-related model terms ( $β_{7}$ through $β_{12}$ ) yields $L R_{χ^{2}}$ = 104,185.9 - 100,882.0 = 3,303.9, which on 20 degrees of freedom is highly significant (p < .001). This test then confirms that type of attrition and its interaction with socio-demographic characteristics are significantly related to health, over and above the influences of socio-demographic characteristics, age, and socio-demographic characteristics. Attritors have worse health in 1992 than non-attritors but the rates of health deterioration across age were less pronounced for attritors than for non-attritors.

The positive and significant three-way interaction indicates that the Race × Age interaction (which indicates more dramatic health deterioration across age for Blacks, Hispanics, and other racial groups than for Whites) is significantly more pronounced for permanent (monotone) attritors than for non attritors. The positive and significant three-way interaction indicates that the Gender × Age interaction (which indicates more dramatic health deterioration across age for males) is significantly more pronounced for attritors than for non attritors.

High School or GED

As illustrated in Table 2, all models suggest that Blacks and Hispanics have worse health in 1992 than Whites but the rate of health deterioration across age is less pronounced among these minority groups when compared to Whites. Elderly men have better health in 1992 than elderly women. However, with the inclusion of patterns of missing data and its interaction with socio-demographic characteristics, elderly men have worse health than elderly women in 1992 and the rate of health deterioration across age is less pronounced for elderly men.

Table 2.
Growth Curve Analysis of Health Trajectories (Elderly Men and Women with a High School or GED Education), HRS 1992-2010

Variables Complete case analysis Direct likelihood maximization Pattern Mixture

Estimate SE Estimate SE Estimate SE

Intercept −5.000* 0.100 −5.120* 0.084 −5.379* 0.085

Age 0.102* 0.002 0.111* 0.001 0.114* 0.001

Race (Ref. = White)

Black 0.866 0.241 0.834* 0.188 1.470*** 0.245

Hispanic 0.782* 0.329 1.019 0.278 1.343* 0.337

Other 0.309 0.623 1.174* 0.489 0.925 0.641

Black*Age −0.004 0.004 −0.004 0.003 −0.014** 0.004

HispanicAge −0.011 0.005 −0.013* 0.004 −0.021 0.005

OtherAge 0.003 0.010 −0.012 0.008 −0.008 0.010

Male −0.329 0.157 −0.477*** 0.127 0.389* 0.153

MaleAge 0.005 0.003 0.008* 0.002 −0.006 0.002

Type of Attrition (Ref. = Always participate)

Monotone attrition 7.882* 0.189

Variable participation 6.467* 0.356

Monotone attrition*Age −0.105*** 0.002

Variable participation*Age −0.090*** 0.005

Monotone attrition*Black −1.234** 0.375

Variable participation*Black −2.688*** 0.682

Monotone attritionHispanic −0.682 0.601

Variable participationHispanic −2.492* 1.099

Monotone attritionOther −0.201 1.036

Variable participationOther −0.660 1.628

Monotone attrition*Male −1.970*** 0.221

Variable participation*Male −1.861*** 0.453

Monotone attritionBlackAge 0.015* 0.006

Variable participationBlackAge 0.039** 0.011

Monotone attritionHispanicAge 0.006 0.010

Variable participationHispanicAge 0.034* 0.018

Monotone attritionOtherAge −0.007 0.017

Variable participationOtherAge 0.008 0.026

Monotone attritionMaleAge 0.033*** 0.003

Variable participationMaleAge 0.030* 0.007

Variance Components

Level 1

Within−person 0.1643* 0.001436 0.183* 0.001 0.180* 0.001

Level 2

Initial Status 22.6866* 0.5793 25.408* 0.549 24.420* 0.524

Rate of Change 0.006* 0.000146 0.006* 0.000124 0.005* 0.000123

Covariance −0.345* 0.009 −0.037* 0.008 −0.366* 0.008

−2LL 71,581.60 122,948.50 120,642.70

N 5,316 10,426 10,426

Cases 37,921 58,592 58,592

* p<0.0001, ** p<0.01, * p<0.05

The likelihood ratio test for the joint significance of the attrition-related model terms ( $β_{7}$ through $β_{12}$ ) yields $L R_{χ^{2}}$ = 122,948.5 - 120,642.7 = 2,305.8, which on 20 degrees of freedom is highly significant (p < .001). This test then confirms that type of attrition and its interaction with socio-demographic characteristics are significantly related to health, over and above the influences of socio-demographic characteristics, age, and socio-demographic characteristics.

Attritors have worse health in 1992 than non-attritors but the rates of health deterioration across age were less pronounced for attritors than for non-attritors.

The positive and significant three-way interaction indicates that the Race × Age interaction (which indicates more dramatic health deterioration across age for Blacks than for Whites) is significantly more pronounced for attritors than for non attritors. Similarly, a more dramatic health deterioration across age for Hispanics than for Whites is significantly more pronounced for temporary attritors (wave-nonresponse) than for non attritors. The positive and significant three-way interaction indicates that the Gender × Age interaction (which indicates more dramatic health deterioration across age for males) is significantly more pronounced for attritors than for non attritors.

Beyond High School

Among elderly men and women with education beyond high school, race was not a significant predictor of initial health and the rate of health deterioration across age in the complete case model but was a significant predictor in each of the direct likelihood maximization and pattern mixture models (see Table 3). Results from growth curve models using direct likelihood maximization and pattern mixture suggest that Blacks and other ethnic groups have worse health than Whites in 1992. Results from the pattern mixture model also suggest that the rate of health deterioration across age is less pronounced among Blacks than Whites.

Table 3.
Growth Curve Analysis of Health Trajectories (Elderly Men and Women with Education beyond High School), HRS 1992-2010

Variables Complete case analysis Direct likelihood maximization Pattern Mixture

Estimate SE Estimate SE Estimate SE

Intercept −4.513* 0.092 −4.651* 0.080 −5.229* 0.080

Age 0.094* 0.002 0.094* 0.001 0.106* 0.001

Race (Ref. = White)

Black 0.376 0.227 0.587 0.184 0.947* 0.23

Hispanic −0.021 0.355 0.398 0.278 0.487 0.363

Other 0.467 0.420 0.661* 0.340 0.949* 0.431

BlackAge 0.001 0.004 −0.001 0.003 −0.009 0.004

HispanicAge 0.008 0.006 0.001 0.005 −0.001 0.006

OtherAge −0.004 0.007 −0.005 0.006 −0.012 0.007

Male −0.469* 0.132 −0.518* 0.112 0.237 0.128

MaleAge 0.007 0.002 0.008 0.002 −0.005 0.002

Type of Attrition (Ref. = Always participate)

Monotone attrition 6.984* 0.181

Variable participation 5.935*** 0.300

Monotone attrition*Age −0.097*** 0.002

Variable participation*Age −0.089*** 0.004

Monotone attritionBlack −0.548 0.385

Variable participationBlack −1.467* 0.648

Monotone attritionHispanic −0.200 0.582

Variable participationHispanic −1.270 1.038

Monotone attritionOther −1.209 0.719

Variable participationOther −1.115 1.373

Monotone attrition*Male −1.779*** 0.193

Variable participation*Male −1.991*** 0.385

Monotone attritionBlackAge 0.007 0.006

Variable participationBlackAge 0.022* 0.011

Monotone attritionHispanicAge −0.005 0.010

Variable participationHispanicAge 0.015 0.018

Monotone attritionOtherAge 0.017 0.012

Variable participationOtherAge 0.020 0.023

Monotone attritionMaleAge 0.033*** 0.003

Variable participationMaleAge 0.034* 0.006

Variance Components

Level 1

Within-person 0.1510* 0.001 0.1672* 0.001 0.1640* 0.001

Level 2

Initial Status 19.885* 0.478 21.827* 0.450 21.123* 0.434

Rate of Change 0.006* 0 0.006 0 0.006 0

Covariance −0.319* 0.008 −0.337* 0.007 −0.332* 0.007

−2LL 79,310.10 124,851.00 122,638.40

N 6,657 11,334 11,334

Cases 43,485 62,208 62,208

* p<0.0001, ** p<0.01, * p<0.05

Results from the complete case analysis and direct likelihood maximization reveal that elderly men have worse health than elderly women in 1992 and the rate of health deterioration across age is less pronounced for elderly men. However, no gender differences in initial health were found in the pattern mixture model. In addition, with the inclusion of type of attrition and its interaction with socio-demographic characteristics in model 3, the interaction coefficient between gender and age becomes negative, suggesting that the rate of health deterioration across age is significantly less pronounced among elderly men.

The likelihood ratio test for the joint significance of the attrition-related model terms ( $β_{7}$ through $β_{12}$ ) yields $L R_{χ^{2}}$ = 124,851.0 - 122,638.4 = 2,212.6, which on 20 degrees of freedom is highly significant (p < .001). This test then confirms that type of attrition and its interaction with socio-demographic characteristics are significantly related to health, over and above the influences of socio-demographic characteristics, age, and socio-demographic characteristics.

Attritors have worse health in 1992 than non-attritors but the rates of health deterioration across age were less pronounced for attritors than for non-attritors.

The positive and significant three-way interaction indicates that the Black × Age interaction (which indicates more dramatic health deterioration across for Blacks than for Whites) is significantly more pronounced for temporary attritors (wave-nonresponse) than for non attritors. The positive and significant three-way interaction indicates that the Male × Age interaction (which indicates more dramatic health deterioration across age for males) is significantly more pronounced for attritors than for non attritors.

Discussion and Conclusion

Analyses of the 1992-2010 HRS suggest that the type of health trajectory that characterizes the elderly men and women is largely depended upon the characteristics of the individuals (e.g. race and gender) and education. The findings of this study suggest that with age, racial/ethnic disparities in health decreased among Blacks regardless of the level of education and among Hispanic elderly with a highs school or GED education, and among other racial groups with less than a high school education.

While African Americans began the study in poorer health, they manifest less steep declines in health than Whites over time. Part of the reason for these findings hinges on the influences of mortality selection where frail members of the Black population experience mortality at younger ages (Ferraro and Farmer, 1996). This selection may lead to the patterns of survival as evidenced by a crossover effect (Markides et al., 1990), whereby at some age, the successfully aging Blacks become healthier than their White counterparts. Another plausible explanation is that at older ages, health differentials are reduced because everyone, regardless of race and class, eventually succumbs to biological forces that produce poor health and mortality (Lynch, 2008).

Consistent with previous research documenting the ‘Hispanic paradox’ (e.g. Franzini et al., 2001; Palloni and Arias, 2004), elderly Hispanics with at most a high school education (Tables 1 and 2) have worse health but a lower rate of health deterioration as they age than Whites. The converging trajectories that characterize health over time between Hispanic and White older adults with at most a high school or GED education as shown in Figures 2e and 2f seem to indicate the aging-as-leveler explanation best describes the experiences of these individuals. These researchers have identified the social buffering (more favorable health behaviors - less likely to smoke and drink), healthy migrant (those who migrate to the U.S. are healthier than their native-born counterparts), and salmon bias (return migration of migrants in poor health) effects as the contributing factors to Hispanics’ better than expected health outcomes despite their structural and socioeconomic disadvantages compared to whites.

When the patterns of missing data and its interaction with socio-demographic characteristics are controlled, elderly men have worse health than elderly women in 1992 and the rate of health deterioration across age is less pronounced for elderly men. This is in part due to the fact that women developed more serious illness while they were younger and their health declined more rapidly than their male counterparts over time. This finding is consistent with previous research that finds that gender gaps in chronic health problems and comorbidities increases with age even though women tend to live longer than men (Laditka and Laditka, 2002; Newman and Brach, 2001; Verbrugge, 1989).

Consistent with Horton and Kleinman (2007), findings from growth curve modeling suggest that the direct likelihood maximization and pattern mixture approaches are superior to the complete case analysis. The standard error estimates from the complete case analysis are as much as 34% larger than those of the missing data models. This is partly attributable to its smaller sample size, and may also indicate a selection bias due to discarding all the partially observed observations if data are assumed to be MCAR (Carpenter and Kenward, 2007).

It is also worth noting that the findings reveal that the any possible non-ignorable differences between growth curve models using direct likelihood maximization and pattern mixture are not large enough to affect inferences drawn from the data analysis. Both growth curve models using direct likelihood maximization and pattern mixture have very similar results, but the standard errors tended to be slightly underestimated in the former. Comparisons of these two approaches suggest that the robustness of the latter to the possibility that persons who completed 1992-2010 surveys were non-ignorably different from others.

Variables	Complete case analysis	Direct likelihood maximization	Pattern Mixture
Intercept	−5.4949***	0.173	−6.260***	0.132	−7.269***	0.129
Age	0.113***	0.003	0.117***	0.002	0.139***	0.002
Race (Ref. = White)
Black	0.808***	0.27	1.527***	0.201	2.259***	0.262
Hispanic	0.464	0.268	1.199***	0.210	1.992***	0.259
Other	0.319	0.73	0.545***	0.539	1.972**	0.748
Black*Age	−0.007***	0.004	−0.015**	0.003	−0.292***	0.004
Hispanic*Age	−0.005	0.004	−0.012***	0.003	−0.290***	0.004
Other*Age	−0.008	0.011	−0.018	0.008	−0.244*	0.011
Male	−0.912***	0.219	−0.494**	0.164	0.373	0.210
Male*Age	0.009*	0.003	0.004	0.002	−0.108**	0.003
Type of Attrition (Ref. = Always participate)
Monotone attrition					10.500***	0.212
Variable participation					9.235***	0.498
Monotone attrition*Age					−0.133***	0.002
Variable participation*Age					−0.121***	0.006
Monotone attrition*Black					−1.684***	0.351
Variable participation*Black					−1.394*	0.689
Monotone attrition*Hispanic					−1.838***	0.375
Variable participation*Hispanic					−2.117**	0.740
Monotone attrition*Other					−2.527***	1.089
Variable participation*Other					−3.451*	1.509
Monotone attrition*Male					−1.463***	0.277
Variable participation*Male					−2.226***	0.540
Monotone attritionBlackAge					0.232***	0.005
Variable participationBlackAge					0.169	0.010
Monotone attritionHispanicAge					0.217***	0.005
Variable participationHispanicAge					0.270*	0.011
Monotone attritionOtherAge					0.032*	0.002
Variable participationOtherAge					0.036	0.022
Monotone attritionMaleAge					0.024***	0.004
Variable participationMaleAge					0.033***	0.008
Variance Components
Level 1
Within-person	0.1864***	0.002	0.2186***	0.002	0.2144***	0.002
Level 2
Initial Status	27.014***	0.893	32.9382***	0.866	29.1421***	0.757
Rate of Change	0.006***	0.000212	0.007***	2E-04	0.006***	2E-04
Covariance	−0.396***	0.014	−0.448***	0.012	−0.408***	0.011
−2LL	45,781.80	104,185.90	100,882.00
N	3,070	8,741	8,741
Cases	22,552	44,198	44,198

Variables	Complete case analysis	Direct likelihood maximization	Pattern Mixture
Intercept	−5.000***	0.100	−5.120***	0.084	−5.379***	0.085
Age	0.102***	0.002	0.111***	0.001	0.114***	0.001
Race (Ref. = White)
Black	0.866**	0.241	0.834***	0.188	1.470***	0.245
Hispanic	0.782*	0.329	1.019**	0.278	1.343***	0.337
Other	0.309	0.623	1.174*	0.489	0.925	0.641
Black*Age	−0.004	0.004	−0.004	0.003	−0.014**	0.004
Hispanic*Age	−0.011*	0.005	−0.013***	0.004	−0.021**	0.005
Other*Age	0.003	0.010	−0.012	0.008	−0.008	0.010
Male	−0.329*	0.157	−0.477***	0.127	0.389*	0.153
Male*Age	0.005*	0.003	0.008***	0.002	−0.006**	0.002
Type of Attrition (Ref. = Always participate)
Monotone attrition					7.882***	0.189
Variable participation					6.467***	0.356
Monotone attrition*Age					−0.105***	0.002
Variable participation*Age					−0.090***	0.005
Monotone attrition*Black					−1.234**	0.375
Variable participation*Black					−2.688***	0.682
Monotone attrition*Hispanic					−0.682	0.601
Variable participation*Hispanic					−2.492*	1.099
Monotone attrition*Other					−0.201	1.036
Variable participation*Other					−0.660	1.628
Monotone attrition*Male					−1.970***	0.221
Variable participation*Male					−1.861***	0.453
Monotone attritionBlackAge					0.015*	0.006
Variable participationBlackAge					0.039**	0.011
Monotone attritionHispanicAge					0.006	0.010
Variable participationHispanicAge					0.034*	0.018
Monotone attritionOtherAge					−0.007	0.017
Variable participationOtherAge					0.008	0.026
Monotone attritionMaleAge					0.033***	0.003
Variable participationMaleAge					0.030***	0.007
Variance Components
Level 1
Within−person	0.1643***	0.001436	0.183***	0.001	0.180***	0.001
Level 2
Initial Status	22.6866***	0.5793	25.408***	0.549	24.420***	0.524
Rate of Change	0.006***	0.000146	0.006***	0.000124	0.005***	0.000123
Covariance	−0.345***	0.009	−0.037***	0.008	−0.366***	0.008
−2LL	71,581.60	122,948.50	120,642.70
N	5,316	10,426	10,426
Cases	37,921	58,592	58,592

Variables	Complete case analysis	Direct likelihood maximization	Pattern Mixture
Intercept	−4.513***	0.092	−4.651***	0.080	−5.229***	0.080
Age	0.094***	0.002	0.094***	0.001	0.106***	0.001
Race (Ref. = White)
Black	0.376	0.227	0.587**	0.184	0.947***	0.23
Hispanic	−0.021	0.355	0.398	0.278	0.487	0.363
Other	0.467	0.420	0.661*	0.340	0.949*	0.431
Black*Age	0.001	0.004	−0.001	0.003	−0.009*	0.004
Hispanic*Age	0.008	0.006	0.001	0.005	−0.001	0.006
Other*Age	−0.004	0.007	−0.005	0.006	−0.012	0.007
Male	−0.469***	0.132	−0.518***	0.112	0.237	0.128
Male*Age	0.007**	0.002	0.008***	0.002	−0.005**	0.002
Type of Attrition (Ref. = Always participate)
Monotone attrition					6.984***	0.181
Variable participation					5.935***	0.300
Monotone attrition*Age					−0.097***	0.002
Variable participation*Age					−0.089***	0.004
Monotone attrition*Black					−0.548	0.385
Variable participation*Black					−1.467*	0.648
Monotone attrition*Hispanic					−0.200	0.582
Variable participation*Hispanic					−1.270	1.038
Monotone attrition*Other					−1.209	0.719
Variable participation*Other					−1.115	1.373
Monotone attrition*Male					−1.779***	0.193
Variable participation*Male					−1.991***	0.385
Monotone attritionBlackAge					0.007	0.006
Variable participationBlackAge					0.022*	0.011
Monotone attritionHispanicAge					−0.005	0.010
Variable participationHispanicAge					0.015	0.018
Monotone attritionOtherAge					0.017	0.012
Variable participationOtherAge					0.020	0.023
Monotone attritionMaleAge					0.033***	0.003
Variable participationMaleAge					0.034***	0.006
Variance Components
Level 1
Within-person	0.1510***	0.001	0.1672***	0.001	0.1640***	0.001
Level 2
Initial Status	19.885***	0.478	21.827***	0.450	21.123***	0.434
Rate of Change	0.006***	0	0.006	0	0.006	0
Covariance	−0.319***	0.008	−0.337***	0.007	−0.332***	0.007
−2LL	79,310.10	124,851.00	122,638.40
N	6,657	11,334	11,334
Cases	43,485	62,208	62,208

Footnotes

Notes

References

Beckett

(2000) Converging Health Inequalities in Later Life - An Artifact of Mortality Selection? Journal of Health and Social Behavior 41: 106–19.

Binstock

George

(eds) (2006) Handbook of Aging and the Social Sciences, 6rd Ed. San Diego: Academy Press, Inc.

Carpenter

Kenward

(2007) Missing Data in Randomised Controlled Trails - A Practical Guide. http://www.haps.bham.ac.uk/publichealth/methodology/docs/invitations/Final_Report_RM04_JH17_mk.pdf.

Dupéré

Lacourse

Vitaro

Tremblay

(2007) Méthodes d’analyse du changement fondées sur les trajectoires de développement individuelle : Modèles de régression mixtes paramétriques et nonparamétriques. Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique 95(1): 26–57.

Ferraro

Farmer

(1996) Double Jeopardy, Aging as Leveler, or Persistent Health Inequality? A Longitudinal Analysis of White and Black Americans. Journal of Gerontology. Series B, Psychology sciences and social sciences 51(6): S319–28.

Fitzmaurice

Laird

Shneyer

(2001) An Alternative Parameterization of the General Linear Mixture Model for Longitudinal Data with Non-ignorable Dropouts. Statistics in Medicine 20: 1009–21.

Franzini

Ribble

Keddie

(2001) Understanding the Hispanic Paradox. Ethnicity and Disease 11(3): 496–518.

Hedeker

Gibbons

(1997) Application of Random-effects Pattern-mixture Models for Missing Data in Longitudinal Studies. Psychological Methods 2(1): 64–78.

Horton

Kleinman

(2007) Much Ado About Nothing: A Comparison of Missing Data Methods and Software to Fit Incomplete Data Regression Models. American Statistician 61(1): 79–90.

10.

Jackson

Govia

Sellers

(2010) Race and Ethnic Influences Over the Life Course. In: Binstock

George

(eds) Handbook of Aging and the Social Sciences, 7rd Ed. San Diego: Academy Press, Inc., 91–120.

11.

Kelley-Moore

Ferraro

(2004) The Black/White Disability Gap: Persistent Inequality in Later Life? Journal of Gerontology: Social Sciences 59B: S34–S43.

12.

Laditka

(2002) Recent Perspectives on Active Life Expectancy for Older Women. Journal of Women & Aging 14(1-2): 163–84.

13.

Lillard

Panis

CWA

(1998) Panel Attrition from the Panel Study of Income Dynamics. Journal of Human Resources 33(2): 437–57.

14.

Little

RJA

(1993). Pattern-mixture Models for Multivariate Incomplete Data. Journal of American Statistical Association 88(421): 125–33.

15.

Little

RJA

(1995) Modeling the Drop-out Mechanism in Repeated-measures Studies. Journal of the American Statistical Association 90: 1112–21.

16.

Lynch

(2008) Race, Socioeconomic Status, and Health in Life-Course Perspective: Introduction to the Special Issue. Research on Aging 30(2): 127–36.

17.

Markides

Liang

Jackson

(1990) Race, Ethnicity, and Aging: Conceptual and Methodological Issues. In: Binstock

George

(eds) Handbook of Aging and the Social Sciences, 3rd Ed. San Diego: Academy Press, Inc., 112–29.

18.

Mirowsky

Ross

(1998) Education, Personal Control, Lifestyle and Health: A Human Capital Hypothesis. Research on Aging 20(4): 415–49.

19.

Mudege

Ezeh

(2009) Gender, Aging, Poverty and Health: Survival Strategies of Older Men and Women in Nairobi Slums. Journal of Aging Studies 23(4): 245–57.

20.

Newman

Brach

(2001) Gender Gap in Longevity and Disability in Older Persons. Epidemiologic Reviews 23(2): 343–50.

21.

Paddock

Edelen

Wenzel

Ebener

Mandell

(2006) Pattern-Mixture Models for Addressing Nonignorable Nonresponse in Longitudinal Substance Abuse Treatment Studies. http://www.rand.org/content/dam/rand/pubs/working_papers/2007/RAND_WR441.pdf.

22.

Palloni

Arias

(2004) Paradox Lost: Explaining the Hispanic Adult Mortality Advantage. Demography 41(3): 385–415.

23.

Pauler

McCoy

Moinpour

(2003) Pattern Mixture Models for Longitudinal Quality of Life Studies in Advanced Stage Disease. Statistics in Medicine 22(5): 795–809.

24.

Rousset

Giret

Grelet

(2012) Typologies de parcours et dynamique longitudinale. Bulletin of Sociological Methodology/Bulletin de Méthodologie Sociologique 114(1): 5–34.

25.

Ross

(1995) The Links between Education and Health. American Sociological Review 60(5): 719–45.

26.

Vandecasteele

Debels

(2007) Modelling Attrition in Panel Data: The Effectiveness of Weighting. European Sociological Review (23)1: 81–97.

27.

Verbrugge

(1989) The Twain Meet: Empirical Explanations of Sex Differences in Health and Mortality. Journal of Health and Social Behavior 30(3): 282–304.

28.

Liang

Bennett

Quiñones Ye

(2010) Ethnic Differences in the Dynamics of Depressive Symptoms in Middle-Aged and Older Americans. Journal of Aging and Health 22(5): 631–652.

Variables	Complete case analysis		Direct likelihood maximization		Pattern Mixture
Variables	Estimate	SE	Estimate	SE	Estimate	SE
Intercept	−5.4949***	0.173	−6.260***	0.132	−7.269***	0.129
Age	0.113***	0.003	0.117***	0.002	0.139***	0.002
Race (Ref. = White)
Black	0.808***	0.27	1.527***	0.201	2.259***	0.262
Hispanic	0.464	0.268	1.199***	0.210	1.992***	0.259
Other	0.319	0.73	0.545***	0.539	1.972**	0.748
Black*Age	−0.007***	0.004	−0.015**	0.003	−0.292***	0.004
Hispanic*Age	−0.005	0.004	−0.012***	0.003	−0.290***	0.004
Other*Age	−0.008	0.011	−0.018	0.008	−0.244*	0.011
Male	−0.912***	0.219	−0.494**	0.164	0.373	0.210
Male*Age	0.009*	0.003	0.004	0.002	−0.108**	0.003
Type of Attrition (Ref. = Always participate)
Monotone attrition					10.500***	0.212
Variable participation					9.235***	0.498
Monotone attrition*Age					−0.133***	0.002
Variable participation*Age					−0.121***	0.006
Monotone attrition*Black					−1.684***	0.351
Variable participation*Black					−1.394*	0.689
Monotone attrition*Hispanic					−1.838***	0.375
Variable participation*Hispanic					−2.117**	0.740
Monotone attrition*Other					−2.527***	1.089
Variable participation*Other					−3.451*	1.509
Monotone attrition*Male					−1.463***	0.277
Variable participation*Male					−2.226***	0.540
Monotone attritionBlackAge					0.232***	0.005
Variable participationBlackAge					0.169	0.010
Monotone attritionHispanicAge					0.217***	0.005
Variable participationHispanicAge					0.270*	0.011
Monotone attritionOtherAge					0.032*	0.002
Variable participationOtherAge					0.036	0.022
Monotone attritionMaleAge					0.024***	0.004
Variable participationMaleAge					0.033***	0.008
Variance Components
Level 1
Within-person	0.1864***	0.002	0.2186***	0.002	0.2144***	0.002
Level 2
Initial Status	27.014***	0.893	32.9382***	0.866	29.1421***	0.757
Rate of Change	0.006***	0.000212	0.007***	2E-04	0.006***	2E-04
Covariance	−0.396***	0.014	−0.448***	0.012	−0.408***	0.011
−2LL	45,781.80		104,185.90		100,882.00
N	3,070		8,741		8,741
Cases	22,552		44,198		44,198