## Abstract

This study examines the co-development of cognitive and physical function in older Americans using an age-heterogeneous sample drawn from the Health and Retirement Study (1998–2008). We used multiple-group parallel process latent growth models to estimate the association between trajectories of cognitive function as measured by immediate word recall scores, and limitations in physical function as measured as an index of functional mobility limitations. Nested model fit testing was used to assess model fit for the separate trajectories followed by estimation of an unconditional parallel process model. Controls for demographic characteristics, socioeconomic status, and chronic health conditions were added to the best-fitting parallel process model. Pattern mixture models were used to assess the sensitivity of the parameter estimates to the effect of selective attrition. Results indicated that favorable cognitive health and mobility at initial measurement were associated with faster decline in the alternate functional domain. The cross-process associations remained significant when we adjusted estimates for the influence of covariates and selective attrition. Demographic and socioeconomic characteristics were consistently associated with initial cognitive and physical health but had few relations with change in these measures.

## Introduction

Like much of the world, the United States is experiencing significant population aging. Estimates project that between 2012 and 2050, the percentage of the U.S. population age 65 and older will increase from 13.7 % to nearly 20.9 %, with the percentage of adults aged 85 or older more than doubling from 1.9 % to 4.5 % (Ortman et al. 2014). The cognitive and physical functioning of aging adults represents a significant input to overall population health and wellness and has been the focus of extensive research. From 2000 to 2050, the percentage of adults age 65 or older with low cognitive function is projected to more than double from current estimates (Alwin et al. 2008), as is the number of adults age 65 and older experiencing some form of physical limitation (Waidmann and Liu 2000). Despite debate about current trends in the prevalence of cognitive and physical limitations (Langa et al. 2008; Martin et al. 2010; Rocca et al. 2011; Seeman et al. 2010), the number of adults experiencing these limitations will certainly increase. Accordingly, there is growing recognition that late-age cognitive and physical health are likely themselves interrelated, with declining cognitive abilities increasing the risk of later physical limitations (Kim et al. 2013; McGuire et al. 2006; Moody-Ayers et al. 2005; Stuck et al. 1999) and increasing physical limitations negatively influencing later cognitive health (Alfaro-Acha et al. 2006; Rajan et al. 2013; Wang et al. 2006). For example, initial cognitive and physical health has been shown to be associated with subsequent cognitive and physical decline (Atkinson et al. 2005; Black and Rush 2002; Krall et al. 2014). Both cognitive decline and increasing physical limitations have also been shown to increase the risk of disability and mortality (Gallacher et al. 2009; McGuire et al. 2006; Scott et al. 1997; Wang et al. 2006). As such, understanding the association between cognitive and physical function is vital to the development of successful policies to reduce the impact of disabilities on aging populations.

Existing research provides support for concomitant cognitive and physical development among older adults, but these studies have important limitations. Many studies in this area have used categorical measures of combined cognitive and physical outcomes (Atkinson et al. 2005; Black and Rush 2002), potentially obscuring or conflating the association between the two. Recent longitudinal work has focused on the short-term dynamic relation between cognitive and physical function (Krall et al. 2014), although studies of concomitant developmental trajectories estimated over longer periods are presently absent. Also, the generalizability of available studies is limited by samples drawn from localized geographic regions, focusing only on females (Atkinson et al. 2005; Krall et al. 2014) or adults at later stages of the aging process (Black and Rush 2002; Krall et al. 2014), and by relatively few statistical controls for socioeconomic status (SES). The current study attempts to address these limitations.

We investigate the association between cognitive and physical function, using parallel process latent growth models to examine the co-development of fluid memory measured as immediate word recall scores, and limitations in physical function measured as an index of physical mobility limitations. The Health and Retirement Study (HRS) (Juster and Suzman 1995) offers an ideal source of data for these analyses, providing observations of multiple cohorts representing adults from late middle age to late old age over a 10-year period (1998–2008). In addition, we estimate models controlling for a robust set of socioeconomic measures and conduct a sensitivity analysis to assess the influence of nonrandom missing data as a result of selective attrition on model development and interpretation. By analyzing a rich source of data using methods capable of addressing complications associated with the longitudinal study of health and aging, our work represents a valuable contribution to the study of concurrent cognitive and physical development among older adults.

### Associations Between Cognitive and Physical Function

The disablement process provides a useful framework to discuss the relationship between trajectories of cognitive and physical function. Conceptually, the disablement process describes the pathway between disease pathology and disablement, with pathologies creating impairments in specific body systems that produce restrictions in basic abilities and actions (functional limitations), in turn producing disability, or a restricted capacity to complete activities essential to survival and social functioning (Verbrugge and Jette 1994). Demographic, social, and behavioral risk factors contribute to the development of disability and represent potential points of intervention to reduce the prevalence and effect of disability. Health problems develop along this pathway directly, can be reinforced through feedback effects, and can initiate and continue the development and progression of other ailments. Declines in cognitive and physical function are important precursors to disability that restrict the capacity to effectively carry out activities (Verbrugge and Jette 1994). The current study is specifically interested in how cognitive and physical function influence each other’s development and how social risk factors relate to the progress of these functional limitations.

The connection between cognitive and physical function has long been a focus in studies of health and aging. A large body of research supports the conclusion that cognitive functioning is associated with, and influences the development of, limitations in activities of daily living (ADLs) and instrumental activities of daily living (IADLs) (Gill et al. 1996; Kim et al. 2013; McGuire et al. 2006; Moody-Ayers et al. 2005; Moritz et al. 1995; Spiers et al. 2005; Stuck et al. 1999; Wang et al. 2002). Studies investigating cognitive health as an outcome have found support for the assertion that ADL/IADL limitations and mobility limitations are associated with later cognitive status (Alfaro-Acha et al. 2006; Rajan et al. 2013; Wang et al. 2006). Research on mobility limitations and cognitive decline is less common than work using ADL/IADLs as indicators of physical health, although evidence has suggested that decline in physical mobility indicators (such as hand-grip strength and timed walking) precede cognitive impairment (Alfaro-Acha et al. 2006; Wang et al. 2006). These studies have specified cognitive and physical function as unique outcomes, not considering the potential interrelation of these health processes.

In contrast to the amount of research investigating cognitive and physical function as separate domains, relatively few studies have considered the bidirectional association between cognitive and physical health. Recent work using cross-lagged structural equation models found positive bidirectional associations between memory and physical function among a sample of high-functioning females ages 70–79 years at baseline, observing change over intervals averaging 1.5 years (Krall et al. 2014). Atkinson and colleagues (2005) examined predictors of combined cognitive and physical decline in a sample of females with moderate to severe disability, finding that impairments in either cognition or IADLs predicted future decline in both processes. Similarly, Black and Rush (2002) found that when controlling for sociodemographic variables and chronic health conditions in a sample of adults aged 75 and older at initial measurement, ADLs and cognitive performance predicted subsequent decline in both dimensions of health. Our work adds to existing research on concomitant cognitive and physical development by using statistical methods and data sources that overcome many of the limitations of available studies, as well as by focusing on the role of SES in the development of cognitive and physical health among older adults.

## Socioeconomic Determinants of Cognitive and Physical Function

That those with lower levels of socioeconomic resources have worse health outcomes than their higher-SES peers is among the most documented associations in the social and health sciences (Lynch and Kaplan 2000; Robert and House 2000). Differential access to health-enabling resources is considered a fundamental cause of health disparities (Link and Phelan 1995, 2000, 2005), and the resources to which Link and Phelan refer are various forms of economic, social, and cultural capital that embed individual health within the sociocultural context.

Substantial evidence has linked SES to cognitive and physical function among the aging. Previous studies of the relation between SES and cognitive decline have found a consistent association between education and cognitive health outcomes among the elderly (Alley et al. 2007; Cagney and Lauderdale 2002; Lee et al. 2003). In studies with robust controls for SES including lifetime wealth and current income, the association between education and cognitive function remained strong (Cagney and Lauderdale 2002; Lee et al. 2003). Both education and income have been positively associated with physical function (Berkman et al. 1993; Guralnik et al. 1993; Kaplan et al. 1993; Seeman et al. 1994), with disparities in disabilities related to education and income levels increasing between 1982 and 2002 (Schoeni et al. 2005). Studies of physical health trajectories have found that education was more strongly related to the onset of limitations in physical function, where income was more closely tied to change in physical health (House et al. 2005; Zimmer and House 2003). Unfortunately, little research has examined how socioeconomic characteristics simultaneously influence trajectories of both cognitive and physical health. The current study fills this gap by modeling the co-development of cognitive and physical trajectories and their socioeconomic antecedents.

## Mortality Selection in Longitudinal Studies of Health

Estimates of change in the health of older adults may be influenced by selective loss to the sample being analyzed. In panel studies of older adults, mortality selection is an important source of nonrandom dropout, with individuals of poorer cognitive and physical functioning at increased risk of mortality (Lavery et al. 2009; Scott et al. 1997; van Gelder et al. 2007). When the outcome variable(s) of interest are related to the likelihood of dropout, missing data are considered not missing at random (NMAR) (Enders 2011; Little and Rubin 2002; Rubin 1976). To test the sensitivity of our primary results to selective attrition, we implemented pattern mixture models that produce weighted average trajectories based on unique patterns of missing data. These supplementary analyses allowed us to assess whether the estimates of the cognitive and physical trajectories, the association between these measures, and the influence of covariates on these trajectories were sensitive to the influence of selective dropout.

## Data and Methods

Created as a national panel survey with biennial assessment of America’s aging population, the HRS began in 1992 as a representative sample of the U.S. population born between 1931 and 1941 (including spouses regardless of age; Hauser and Weir 2010). To supplement the original HRS cohort, the Asset and Health Dynamics among the Oldest Old (AHEAD) cohort was included in 1993 as a spinoff of the HRS, focusing on Americans considered to be the oldest-old: birth cohorts born in 1923 or earlier (Juster and Suzman 1995). In 1998, two additional cohorts were brought into the study, representing “Children of the Depression” (CODA, born in 1924–1930), and “War Babies” (WB, born in 1942–1947). We examined all four cohorts described, using measurements of immediate word recall and mobility limitations taken biennially from 1998 to 2008, with all predictor variables measured in 1998. In association with the National Institute of Aging and the Social Security Administration, the RAND Corporation developed a cleaned data file containing imputed wealth and income measures for all cohorts analyzed here (version J; St. Clair et al. 2010). Immediate word recall scores were taken from the HRS final core files.

Restricting the analytic sample to white, black, or Hispanic individuals who had nonmissing immediate word recall and mobility limitations measures in 1998 resulted in an overall sample of 17,713 persons with the following cohort-specific sample sizes1: WB n = 2,803; HRS n = 7,345; CODA n = 3,298; and AHEAD n = 4,267.

### Dependent Measures

Cognitive function represents a combination of fluid abilities related to complex problem solving and crystallized abilities developed by experience and culturally defined tasks (Alwin and Hofer 2008). The present analysis focuses on the former. We use immediate word recall scores, hereafter referred to as “word recall.” Word recall has been shown to be a measure sensitive to cognitive change in the domain of fluid abilities (Small et al. 1999) and is an example of a free recall task measuring the construct of episodic verbal memory (Lachman and Tun 2008). The word recall task asked respondents to recall a list of 10 common nouns (e.g., lake, car, army) in any order. The interviewer read from one of four possible lists of 10 nouns, with the initial list being randomly assigned and subsequent measurements using a different set of words across three successive waves (Ofstedal et al. 2005). To adjust for potential practice effects resulting from multiple exposures to the word recall task, a count of previous exposures at each wave was included as a time-varying covariate in all models estimating the trajectory of word recall.

The measure of physical function analyzed was a summation of 11 indicators of limitation in physical mobility (Nagi 1969; Rosow and Breslau 1966), hereafter referred to as “mobility limitations.” Respondents were asked whether they had difficulty in each of the following activities: stooping or crouching, climbing one flight of stairs without resting, climbing several flight of stairs without resting, moving large objects, sitting in a chair for two hours, getting up from a chair after sitting for long periods, lifting weights more than 10 pounds, raising arms above shoulder level, walking one block, walking several blocks, and picking up a dime from a table (1 = yes; 0 = no). The original HRS measures contained skip patterns such that respondents with severe limitations in simple functions were not asked questions about related and more complex physical functions. The RAND HRS version of these measures recoded missing values resulting from the skip pattern to indicate difficulty with the item. Proxy responses to the mobility limitations measures were coded as missing. In the summation of the variable, respondents with seven or more missing values on these 11 items were coded as missing. Although questions have been raised about the use of a single additive scale to represent mobility limitations, the items have been shown to load together with a Kuder-Richardson Formula 20 of .85 (Fonda and Herzog 2004; Haas and Rohlfsen 2010), largely supporting the use of a single additive scale.

Mobility limitations are an important indicator of general physical ability. In addition, compared with other measures of functional health, such as ADLs and IADLs, mobility limitations are less contingent on the social environment, the use of assistive technology, and the adoption of specific social roles; thus, they are more closely related to underlying physical capacity (Haas 2008; Martin et al. 2010). Further, because ADLs and IADLs represent a much more severe level of disability, they are less able to capture early indications of physical limitation and the healthiest end of the functional limitation distribution. Finally, IADLs represent cognitively demanding tasks and are therefore likely to be conflated with measures of cognitive functioning. For these reasons, mobility limitations represent the best available indicator of functional health as conceptualized in the disablement process.2

### Independent Measures

A number of measures were included to control for the association between individual characteristics and cognitive and physical function. Analyses controlled for gender and respondents’ age at initial interview centered on the mean age of each cohort. Respondents who were white, black, or Hispanic were included in analysis, with white individuals serving as the reference group. Marital status at time of initial interview was included to control for possible associations between health and marital status. Those who reported being married, married with spouse absent, or partnered were defined as married; and those who reported being separated, divorced, widowed, or never married were defined as unmarried (1 = married, 0 = unmarried).

The amount of education reported by the respondent was measured as a set of three dummy variables that represented less than high school completion (0–11 years), high school graduates (12 years, reference group), and those with more than 12 years of education. Occupation was measured as the job with longest reported tenure. Individuals who reported their longest employment tenure as professional or technical workers, managers, officials or proprietors, or clerical or kindred workers, and those in sales were defined as white-collar workers. Individuals who reported longest employment tenure as craftsmen, foremen, operators, laborers, service workers, or farmers were labeled as blue-collar workers. A dummy variable was included to capture the work experience of females who reported fewer than four years of lifetime work experience and who self-identified as a homemaker. A final category was created for individuals whose work experience did not fall within the previously defined categories. In all analyses, white-collar workers were used as the occupational reference group. Household income and household assets were summary measures of numerous sources of income and asset holdings included in the RAND HRS data. Both household income and household assets were transformed on a log scale to reduce the right skew of these variables and were centered on the cohort mean. Finally, a dichotomous measure indicating whether the respondent had any of the following chronic conditions was used to account for the association between cognitive and physical health and these common chronic diseases: high blood pressure, diabetes, cancer, lung disease, heart disease, stroke, psychiatric problems, and arthritis.

### Methods

We used Mplus version 7.3 (Muthén and Muthén 2012) to estimate multiple group parallel process latent growth models. Latent growth modeling (LGM) is considered an extension of structural equation modeling (SEM) in which the intercept and slope that create estimated trajectories are treated as latent variables (for technical description, see Bollen and Curran 2006).

Latent growth models are able to account for characteristics of data from longitudinal studies including unbalanced data structures resulting from differing numbers of interview waves and time intervals between assessments. Using a multiple group design, we specified a separate model for each cohort, allowing for invariance in model estimates to be formally tested across cohorts. For the mean and variance of each latent measure, all covariance estimates, and the hypothesized regression paths, we tested a model allowing the parameter to be estimated but held equal across cohorts (Model 2) against a model constraining the estimate to zero (Model 1). If Model 2 provided better model fit than Model 1, we tested a subsequent model allowing the parameter to be estimated separately for each cohort (Model 3) against Model 2. Hypothesis testing was conducted using the Wald chi-square test to assess the fit of Model 2 versus Model 1, as well as Model 3 versus Model 2. Household identification number was used to adjust standard errors for the potential nesting of observations within household.

Preliminary analysis of the dependent variables indicated that the distribution of the mobility limitations measure was right-skewed. The mobility measure was modeled as a count variable using a Poisson distribution. A negative binomial distribution was tested as a potential alternate specification for mobility limitations but resulted in model nonconvergence. The distribution of the word recall measure was adequately normal and modeled as a continuous variable.

We specified latent trajectories for each growth process using individually varying time scores based on age. Specifically, the time scores used were participants’ age at each measurement centered on the mean age of their cohort at initial measurement, divided by 10. The average latent intercept for each outcome represented the estimated value of the outcome at the mean age of the cohort at initial measurement, and the latent slope for each cohort represented the estimated amount of change in the outcome over a 10-year interval.

Analyzing the cognitive and physical trajectories of the four cohorts as described speaks primarily to the effects of age differences on cognitive and physical development, but also adjusts results for potential cohort effects. Individuals of similar ages experiencing the unique circumstances of a specific historical period are identified as a cohort (Ryder 1965), resulting in the potential for birth cohort to influence health outcomes net of age differences between the groups. For example, variation in access to education across cohort has been linked to cohort differences in cognitive and physical function (Alwin et al. 2008; Martin et al. 2010). Some studies have suggested increased physical limitations among younger cohorts (Lin et al. 2012; Seeman et al. 2010), and others have found no significant change in physical function in similarly aged cohorts (Martin et al. 2009, 2010). Also, studies of cognitive outcomes among comparably aged cohorts have suggested decreases in poor cognitive function (Freedman et al. 2002), with others finding stable patterns of cognitive function across time (Rodgers et al. 2003). As such, our analysis of the cohorts observed in the HRS speaks largely to the effects of age on cognitive and physical development but also accounts for the potential effects of cohort experiences.3

#### Parallel Process Latent Growth Modeling

Parallel process latent growth modeling, also referred to as a multivariate growth model (Bollen and Curran 2006), is an extension of LGM where an intercept and slope are estimated for two or more concurrent outcome variables of interest (Muthén and Curran 1997). The benefits of using a parallel process model to examine the co-development of cognitive and physical function include (1) the ability to estimate the dual trajectories of word recall and mobility limitations in a single statistical model, allowing for covariance estimates between the latent growth variables within and between the estimated trajectories; and (2) formal testing of hypothesized associations between the two growth processes. For example, the covariance between the intercept of word recall and intercept of mobility limitations adjusts the model for unobserved causes of common variation in baseline estimates of these measures unaccounted for by covariates. In a multiple group setting, the parallel process model allows for covariances and regression coefficients between growth parameters to be constrained to zero, to be estimated equally across groups, or to be freely estimated for each group. Also, an essential goal was to assess whether initial word recall and mobility limitations were associated with variation in the slope of the alternate growth process. The regression paths that we investigated extended from the latent intercept of each trajectory to the latent slope of the alternate trajectory. This specification assured that both latent and observed predictor variables temporally preceded the central outcomes of interest: namely, change in word recall and mobility limitations with increasing age. Figure 1 presents the four structural models tested.

#### Sensitivity Analysis

To control for potential biases in longitudinal studies introduced by selective dropout, results from the parallel process models developed under the assumption of data missing at random were compared against estimates adjusted for the effects of data NMAR on the outcomes of interest. Full information maximum likelihood is the default estimator used by Mplus when estimating outcome trajectories and assumes that missingness is not related to the underlying values of the outcome variables. In longitudinal studies where the outcome of interest is associated with the propensity for missing data, an NMAR mechanism is at work and may produce biased parameter estimates (Enders 2011; Muthén et al. 2011).

A pattern mixture model was used to test for potential biases resulting from selective dropout in the primary results. The pattern mixture model decomposes the average latent trajectory for each pattern of missing data and produces a weighted average trajectory based on the number of waves each individual was observed. For example, the average intercept $β¯^0$ is determined by the number of missing data patterns (numeric superscript), the proportion of cases in missing data pattern $pπ^p$, and the pattern-specific intercept estimate ($β^0p$ ), here including three missing data patterns for simplicity:
$β¯^0=π^1β^01+π^2β^02+π^3β^03.$
The slope for the pattern mixture models was constrained to be equal for the second and third observations because of convergence issues. A full description of the pattern mixture model is offered by Enders (2011).

## Results

Table 1 provides descriptive statistics for word recall and mobility limitations by cohort and year, including the mean and standard deviation of each measure as well as the number of observations and the percentage of original respondents providing information at each measurement occasion. Individuals in older cohorts generally had lower word recall scores, more mobility limitations, and greater amounts of missing data at later waves than those in younger cohorts.

### Model Fit Testing of Unconditional Single Trajectory Models

Table 3 presents Wald test results for the mean and variance estimates of the latent intercepts and slopes, parallel process covariance estimates, and the cross-process regression path estimates. For sake of organization, model fit results from the pattern mixture models are presented in Table 3 but are discussed following complete description of the primary results. Wald tests were used to determine whether the mean and variance estimates of the intercept and slope were significantly different from 0 and whether the estimates varied across cohorts. Model comparisons were conducted separately for each growth process. All estimates tested were significantly different from 0, and allowing the estimates to be specified separately for each cohort provided better model fit than having the estimate held equal across cohorts.

For covariance estimates, the covariance between the intercept and slope within each separate growth process was examined first, testing whether the estimate differed from 0 and next whether the estimate differed across cohorts. The intercept-slope covariance estimate for word recall was significantly different from 0 and varied across cohorts. For mobility limitations, this estimate was nonzero but did not differ across cohorts. Next, the best-fitting word recall and mobility limitation LGMs were combined into a single parallel process LGM. The covariance between the intercept of word recall and intercept of mobility limitations and the covariance between the slope of word recall and slope of mobility limitations were tested separately. Both cross-process covariance estimates were significantly different from 0 and varied across cohorts.

### Model Fit Testing of Parallel Process Structural Model

Using the unconditional parallel process model identified in the previous step as the comparison model (Fig. 1, panel A), we next tested the cross-trajectory regression paths. First, we regressed the slope of mobility limitations on the intercept of word recall (Fig. 1, panel B). When held equal across cohorts and compared with a model with the path constrained to 0, the model allowing the path to be estimated provided significantly better model fit (Wald χ2 = 78.64, df = 1, p < .001). The second path tested regressed the slope of word recall on the intercept of mobility limitations (panel C). Allowing the path to be estimated contributed to a significantly better fitting model (Wald χ2 = 4.69, df = 1, p = .03). Neither of the hypothesized paths were found to vary across cohorts. Next, we estimated both regression paths in the same model (panel D). The inclusion of both regression paths with estimates constrained across cohorts provided better model fit than either of the single-path models (Intercept IWR → Slope ML constrained to 0, Wald χ2 = 91.30, df = 1, p = < .001; Intercept ML → Slope IWR constrained to 0, Wald χ2 = 17.71, df = 1, p < .001). Testing the focal paths under an unconditional parallel process model indicated that initial word recall and mobility limitations were both associated with change in the alternate process consistently across cohorts.

### Interpretation of Latent Means, Cross-Process Regression Paths, and Independent Variables

Table 4 presents the estimated latent parameter means from the separate unconditional growth models as well as the covariance and cross-process regression estimates from the final conditional model. The latent intercept and slope estimates varied substantially across cohorts. For older cohorts compared with younger ones, the mean intercept for word recall was lower, and the mean slope showed a steeper rate of decline. For mobility limitations, older cohorts had more initial limitations than younger cohorts, with a slightly greater rate of increase in limitations with age. For example, the estimated number of initial mobility limitations for the HRS cohort at the mean age of 61 was 1.11 (e0.10 = 1.11), with an estimated 1.73-unit (e0.55 = 1.73) increase in limitations over 10 years of age. In the AHEAD cohort, the estimated number of initial mobility limitations at the mean age of 80 was 2.15 (e0.76 = 2.15), with an estimated 2.11-unit increase in mobility limitations over 10 years of age (e0.75 = 2.11).

The covariance estimates provided in Table 4 come from the best-fitting parallel process model conditional on covariates. Greater initial word recall scores were associated with greater decreases in word recall with age, excluding the WB cohort. Greater initial mobility limitations were associated with slower increases in mobility limitations with age. For the cross-process covariance parameters, greater initial word recall was associated with fewer initial mobility limitations; and as decline in word recall grew more rapid, the rate of increase in mobility limitations also accelerated.

When including all control variables in the best-fitting model, both structural regression paths remained significant. With a one-unit increase in initial word recall, the predicted number of mobility limitations developed over 10 years of age increased 0.07 units (b = 0.07, SE = 0.01, p < .001). With a one-unit increase in mobility limitations at baseline, the predicted word recall score after a decade of age increased 0.05 units (b = 0.05, SE = 0.01, p < .001), indicating that having fewer initial mobility limitations was associated with faster decline in word recall with increasing age.

### Covariate Estimates

Table 5 contains estimates from the final conditional model for the word recall trajectory. More exposures to the word recall task were associated with greater word recall scores, appearing most consistent for the AHEAD cohort. For all cohorts, being male, being black, having less than 12 years of education, and reporting occupational tenure other than white collar were associated with lower initial word recall than their respective reference groups. Household income was positively associated with initial word recall for all cohorts, while household assets were positively associated with word recall for all excluding the WB cohort. Reporting any chronic disease in 1998 was negatively associated with initial word recall for the HRS cohort.

As presented in Table 6, increasing age was associated with slightly decreased initial mobility limitations for all but the WB cohort. Females had more initial mobility limitations than did males. Those with 12 years of education had fewer initial limitations than those who completed less than 12 years of education, but they also tended to have more initial mobility limitations than those with greater education. Reporting blue-collar or other occupational tenure was associated with more initial mobility limitations relative to those reporting white-collar occupational tenure in the HRS and AHEAD cohorts. Both household income and household assets were negatively associated with initial mobility limitations, excluding household income for the AHEAD cohort. Finally, the expected number of initial mobility limitations for respondents reporting any chronic disease increased 283.44 % for the HRS cohort (100(e1.34 – 1) = 283.44) and 182.92 % for the AHEAD cohort (100(e1.04 – 1) = 182.92).

Few covariates were associated with change in either outcome. Decline in word recall scores was associated with greater age in the HRS, CODA, and AHEAD cohorts, and the word recall scores of females in the AHEAD cohort declined faster than for their male counterparts. For change in mobility limitations over time, reporting any chronic condition at baseline was the only covariate associated with change in mobility limitations for all cohorts. Over the 10-year change in age measured, reporting a chronic disease at baseline was associated with a 35.85 % decrease in the expected number of mobility limitations developed for the HRS cohort (100(e–0.44 – 1) = –35.85) and a 43.67 % decrease in expected mobility limitations for the AHEAD cohort (100(e–0.57 – 1) = –43.67). Although not consistent across cohorts, the increase in mobility limitations was generally slower for females than for males. Compared with white-collar workers, females reporting occupational tenure as homemaker had somewhat greater increases in limitations in the HRS cohort, and those reporting other occupational tenure had slower increases in limitations in the HRS and AHEAD cohorts.

### Sensitivity Analysis

Returning to Table 3 to describe development of the pattern mixture model, results were similar to the primary models with the exception that the slope variance of mobility limitations did not significantly differ across cohorts. All covariance estimates were nonzero and differed significantly across cohorts. Results from model tests of the cross-process regression paths supported the primary findings—namely, that initial word recall and mobility limitations were associated with change in the alternate process similarly across cohorts.

Table 7 presents the estimated latent parameter means, covariance estimates, and dropout indicator estimates for the final parallel process pattern mixture model. Compared with the primary results, the estimated growth parameter means for word recall from the pattern mixture model indicated greater initial word recall and slower decline, with differences appearing more pronounced for older cohorts. For mobility limitations, the pattern mixture model tended to estimate fewer initial limitations and greater increases than those produced by the traditional models, with differences again being greater for the older cohorts.

In terms of covariance estimates, results from the pattern mixture model generally aligned with results from the primary models. The dropout path estimates indicate how dropout in a given interval between observations was associated with the intercept and slope of each growth process. Dropout was consistently associated with lower initial word recall and faster decline in word recall with age, appearing more influential when dropout occurred earlier and for the older cohorts. For mobility limitations, dropout predicted greater initial mobility limitations for all excluding the WB cohort, with the effect again appearing to weaken with dropout at later waves of observation. Dropout was consistently associated with change in mobility limitations only for the HRS cohort, indicating greater increases in mobility limitations than those not dropping out.

Tables S1 and S2, provided as an online supplement (Online Resource 1), contain estimates from the conditional pattern mixture models for word recall and mobility limitations. Again, results generally aligned with those from the primary analyses.

## Discussion

Our study examined the co-development of cognitive and physical function in age-heterogeneous cohorts of older adults. Initial cognitive and physical health were associated with development of the alternate process for all cohorts, with better initial health in one domain having an adverse association with subsequent change in the alternate domain. The intercept and slope covariance within each distinct trajectory suggested the presence of floor and ceiling effects that likely contributed to the cross-process effects observed. Demographic and socioeconomic characteristics were consistently associated with initial cognitive and physical health but had few associations with change in these measures. The sensitivity analysis controlling for selective attrition supported results from the principal analyses.

Over the decade observed and without variation across cohort, initial word recall was positively associated with the rate of increase in mobility limitations, and fewer initial mobility limitations were associated with faster cognitive decline. For the separate outcome trajectories, better initial health was associated with faster progression of limitations with age. The covariances linking the separate growth models indicated that at initial measurement, favorable cognitive health was related to fewer mobility limitations, and faster decline in word recall was associated with more rapid increase in mobility limitations over the 10 years of aging observed. In sum, favorable cognitive and physical function at initial measurement was associated with less favorable change in both of these functional domains with increasing age, individuals with good cognitive health at the beginning of the study also tended to be more physically healthy, and those experiencing rapid cognitive decline also tended to experience rapid progression of mobility limitations.

Taken together, these findings suggest the presence of floor and ceiling effects influencing the development of the health trajectories observed: poor initial health in one domain associated with lower initial health in the alternate domain, which in turn is less susceptible to deterioration with age. The magnitude of the cross-process regression path estimates was small but consistent across cohorts and remained relatively unchanged when controlling for covariates and nonrandom attrition as tested in the sensitivity analysis. The cross-process regression paths also remained significant when we accounted for within-process covariances, indicating that in addition to each health trajectory’s unique development, initial function in one domain was associated with development of function in the alternate domain. The cross-process covariance and regression path estimates suggested not only that high cognitive or physical function at younger ages will be met with faster decline in each domain as age increases but also that initially good health in either domain may be associated with more rapid decline in the alternate health process as age progresses. These results may be partially explained by the process of biological senescence, with high functioning older adults having cognitive and physical reserves that will necessarily diminish with increasing age and frailty. Based on these findings, a challenge for those tasked with preserving the cognitive and physical health of our aging population will be protecting high performing older adults from experiencing concomitant decline in their cognitive and physical function with increasing age.

That initially poor cognitive and physical performance were not associated with more rapid functional decline in the alternate process as reported by a number of relevant studies highlights the complexities of measuring the relation between these outcomes. A primary difference between our approach and previous studies of concomitant decline in cognitive and physical function was the extended period of observation used to estimate change in cognitive and physical outcomes. Related work using cross-lagged path analysis found initially good cognitive or physical health to be positively associated with better health in the alternate domain (Krall et al. 2014), but estimates of change were restricted to an average of 1.5-year intervals. Our work specified change in the health processes using continuous trajectories based on measurements taken at six time points over a 10-year period, resulting in developmental trajectories that are more likely to reflect the floor and ceiling effects observed.

A number of other factors may contribute to the divergence of our results from other studies of concurrent cognitive and physical development of aging adults. Measures of cognition tend to differ across studies, including various versions of the Mini-Mental State Examination (MMSE) (Atkinson et al. 2005; Krall et al. 2014; Moody-Ayers et al. 2005) and the Short Portable Mental Status Questionnaire (SPMSQ) (Black and Rush 2002). Our use of Nagi physical mobility limitations differs from other analyses using ADL/IADL measures given that we desired to specify cognitive and physical decline as underlying abilities preceding disability rather than as disabilities reflecting the interaction between underlying abilities and the surrounding environment. In addition, our results reflect the association between cognitive and physical decline for an age-heterogeneous group of older individuals. The results and interpretation of the cross-process regression paths were the same for all cohorts observed, but the majority of remaining parameters were uniquely estimated for each cohort. Finally, the HRS provides a large sample size from which to draw conclusions—a luxury that other studies of concomitant cognitive and physical decline did not have. Although population weighting adjustments available in the HRS were not included, our results can be generalized to a larger population than other relevant studies.

The socioeconomic measures included in our study were more consistently associated with initial cognitive and physical function than change in either of these measures. For example, greater household income and household assets were associated with better word recall and fewer mobility limitations at initial measurement in 1998 but were generally not associated with change in these outcomes over the decade of follow-up. More socioeconomic indicators were associated with changes in mobility limitations than with changes in word recall, but these were not consistent across cohorts. These results align with those of others who have found SES to be differentially related to the onset and progression of functional limitations (House et al. 2005; Zimmer and House 2003). Disparities observed in these health processes seem to be largely developed before the latest years of life, indicating that interventions to reduce socioeconomic disparities in cognitive and physical function among older adults are likely more valuable when occurring at earlier stages of the life course.

The main findings were supported by sensitivity analysis using pattern mixture modeling to address potential biases caused by selective attrition. Results of model building in the sensitivity analysis produced a final pattern mixture model similar to the final primary model, with the covariance between growth estimates and structural paths of interest remaining significant and in the same direction as the primary models. The variation in parameter estimates across missing data treatments was itself interesting: compared with estimates produced by models assuming data missing at random, the pattern mixture model produced initial estimates of word recall and functional limitations that were more advantageous, but estimates of change that were slower for word recall and slightly faster for functional limitations. Pattern mixture models are not a complete solution to the potential influence of selective attrition in analyses of longitudinal data, but they do offer a conservative point of comparison to support our primary findings.

### Limitations and Future Directions

Our research introduced a novel approach to examining the interdependence of cognitive and physical function among aging Americans, but the chosen statistical design resulted in limitations to the estimation and interpretation of the trajectories of interest.

Foremost, the complexity of our model specification did not allow quadratic effects in the slopes of word recall or mobility limitations to be estimated. When using accelerated longitudinal designs, others have shown quadratic effects in trajectories of word recall using the HRS (Alwin et al. 2008). When estimating each growth trajectory separately, testing of quadratic parameters in the mobility limitations model resulted in fatal memory errors. Excluding potential quadratic effects in the trajectories being explored may have introduced bias in our estimates, but the use of a multiple group design allowing the intercepts and slopes to be freely estimated for each cohort would limit these biases to the cohort-specific estimated trajectories rather than influence the entire developmental trajectory that would result from using an accelerated approach.

Second, the large number of parameter estimates in the models may have introduced error into the model-building process due to the problem of multiple comparisons. Each model contained a large number of parameter estimates that were used to determine the specification of subsequent models. Not accounting for multiple testing may have influenced the model development process and subsequent interpretation of estimates. The model-building process was primarily exploratory, using nested model fit tests to guide the model specification process. To assess the results and conclusions of our work, further study should be done to confirm our findings using an independent sample.

Finally, the pattern mixture model used to evaluate the impact of selective attrition on the estimates of interest itself has limitations. While adjusting the growth trajectories of interest for the presence of NMAR data, this type of model introduces untestable and potentially biasing assumptions regarding values of unobserved values and model parameters. Specifically, the pattern mixture approach makes assumptions regarding model parameters that cannot be estimated because of missing data. Standard practice is to compare results from various types of NMAR models, each with their own set of untestable assumptions, in a combined sensitivity analysis. Attempts at specifying Diggle-Kenward and Wu-Carroll selection models were met with model nonconvergence. Fortunately, pattern mixture modeling stratifies the sample by missing data pattern rather than assuming that scores exist for the unobserved values, as does the selection modeling approach. Furthermore, the pattern mixture model does not incorporate a regression model to estimate missing data and thus does not introduce the possibility of bias caused by omitting an important predictor of missingness (Enders 2011). The sensitivity analysis presented here provides a test of the influence of selective attrition on our results, although these estimates should be viewed cautiously and only as a supplement to the primary findings.

In sum, our research provides an example of the complex and interdependent nature of health outcomes among aging adults. Future research should attempt to address the limitations described here as well as implement this type of model using other measures of cognitive and physical health. The measures that we analyzed represent two of the many indicators of cognitive and physical function, and further work is needed to assess whether our findings are robust to using alternative measures of these health processes. Generally, given the benefits of the temporal ordering of the intercept and slope of each growth process being examined, this form of research design should be considered by those interested in trajectories of associated health outcomes. We hope that our findings will inform substantive discussion on the topics of cognitive and physical function among aging adults, as well as provide methodological grounding for future studies of interdependent health trajectories.

## Notes

1

The cohort indicator RACOBYR based on birth year was used as the cohort identifier in comparison with the variable HACOHORT that identifies the cohort in which the household was originally sampled (St. Clair et al. 2010).

2

The primary analyses were replicated using ADLs in place of mobility limitations. In brief, greater initial word recall was associated with faster increases in ADLs with age, and this was consistent across cohorts. Initial ADLs were not associated with change in word recall. These results are available upon request.

3

For the sake of brevity, the term “cohort” was used to describe what would be more accurately defined as age-cohort throughout the manuscript.

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