Progression of Physical Function Decline and Gender Difference Among the Elderly in China: A Multiple-Group Latent Transition Analysis

Background: Previous research has shown the physiological basis and empirical evidence for judging representative functional statuses and the functional decline pattern characterized by the specic combination and coevolution of muti-limitations among the elderly. However, as for health measures, under the conventional variable-centered view, studies mostly outlined physical function conditions by quantized severity levels rather than the qualitative prole and possessed the status change just uptrend or downside. This paper's central issue is to identify the constructed latent status of physical function among the Chinese elderly, explore the law of the coexistence and progress of functional limitations, also rene the comprehension of gender inequality with age. Methods: We selected 1578 valid elderly samples who participated in the Chinese Longitudinal Healthy Longevity Survey in 2002, 2008, and 2014, and undertook latent transition analysis to construct the latent status, conduct the membership change overtimes, and examine gender disparity by introducing the multiple-grouplatent model. Results: Chinese elders could be divided into ve typical physical functional latent statues.Upper extremity mobility was a distinguishing factor with unimanual dexterity or grip strength. The functional decline started with highly strength-required actions, then, the risk of mobility dysfunction and unimanual indexterity released predominantly, companied by visual and hearing problems. Upper limb disorders rarely occurred alone. The gender differences in distribution and transition of functional status were both distinguishable(ΔG 2 =74.59,df=4,p<0.000;ΔG 2 =110.86,df=40, p<0.000). Male elderly preferred maintaining extreme states with a lower risk of deterioration and a higher chance of recovery; females tended toward an unhealthier state and sank into the middle state. Whether gender inequality within physical health expands or shrinks with aging was related closely to the initial status. Conclusion: This study bridges the gap between individualized studies and health measures by typology reference, differentiating the key functional stage and persons at risk. Medical treatments could prioritize the upper limb mobility recure and the strength enhance. Health equity promotion should consider gender typing and original status. Further, individual behaviors under diverse health types would be compared to demonstrate the heterogeneity in real needs and formulate targeted health interventions. the categories into a binary outcome: 2 for response 'can, without help' (unlimited), and 1 for other answers (limited).

Thus, as the direct manifestation of impairments, reasonably, it is believed that physical function status can be de ned quintessentially, and its decline advanced gradually shaped by each given disparate types of coexistence and accumulation of diverse but interlinked underlying physical dysfunction. Advancers, Visual Problem Advancer using ve functional limitation category and stated that further decline was characterized by the addition of either visual problems or hearing problems, which were both associated with an increased risk of limitations in upper extremities.
The previous studies indeed provided physiological basis and classi cation evidence to realize the judge and explain the representative physical functional status characterized by a speci c combination and coevolution of muti-limitations. Besides, with the expanding scope and structuration of well-being, identifying the pro le of individuals' status such as Wolinsky's three-dimensional con guration (1980) and ICF (WHO, 2001), has been brought out, which indicates a new idea for assessing physical function. In China, the physical function has been extensively explored in clinical medicine or nested in disability study in sociology and demography. In either case, under the conventional variable-centered investigation by score or count of the scale (Xu et al. This paper's central issue is to utilize latent transition analysis to structure the latent status, conduct the membership change overtimes, and examine gender disparity by introducing the group variable, to identify the typical physical function status of the Chinese elderly, explore the law of the coexistence and progress of functional limitations, and re ne the comprehension of gender inequality with age. The signi cances achieved in this work are: in terms of methodology, to bridge the gap between individualized studies and health measure by typology reference, which is more conducive to grasp the abstract property of subgroups than taking index-quantitative level as a criterion; theoretically, deeply into variables mining, to establish a nite and speci c correlation among functional components, rewarding to proof and develop the rationale of the formation and progress of functional limitations; as for the practical sense, to re ne the comprehension of health states change rules of the aging in China.

Methods
Aim and Design Firstly, we proposed typical functional status featured by the particular performance of multi-limits coexistence and accumulation to subdivide the elderly. Then, each status's prevalence and transition probabilities were computed to probe into the relationships within each physical functioning and its development. Lastly, health disparity between genders over time was ulteriorly demonstrated in a distinctive angle of the old folks' classi cation and physical status transfer. crouch and stand up three times, walk continuously for 1 km, pick up a book from the oor by standing, sit to stand without using hands, turn around within ten steps. We recoded the categories variables into a binary outcome: 2 for response 'can, without help' (unlimited), and 1 for other answers (limited).
Models and Processes We used latent transition models (LTM) by SAS9.4 software PROC LTA to estimated parameters by the maximum likelihood of the Expectation Maximum algorithm and Newton-Raphson (Collins and Lanza, 2010). General LTM revealed the typical functional state of the elderly, and the proportion of each state tracked the membership change over time. Multiple-group LTM was used to test gender inequivalent on the distribution and transformation of functional state among Chinese elders. The latent status analysis identi ed latent statuses of physical function by response patterns of elderly on 11 observed indicators, divided individuals into distinct subgroups, and calculated latent status prevalences at baseline survey. Then we gained the latent status prevalences at multiple time points and transition probabilities of the elderly from one status to another within the longitudinal time intervals through transformation matrixes established by a hidden Markov progress. Before modeling, Harman's One-factor Test was used to examine the common method biases of the sample. The rst common factor variance contribution rates were 28.84%, 31.04%, 29.22% at three-time points, all below the critical value of 40%. This inferred that common attributes of each functional status rather than the certain self-reported bias, environment, context, or the project itself, determined the elderly's response pattern. Model t and model selection As an exploratory analytical approach, we used the Bayesian information criterion (Schwartz, 1978) and Akaike information criterion (Akaike, 1973) to choose the optimal number of latent statuses, the smaller which were, the better competing LTM tted. There might be complex situations that each criterion did not identify the consistent model as optimal. Parsimony and explicable of latent status were two primary principles to choose a single best model. This study computed models with 2-7 latent status preset through PROC LTA syntaxes.
Hypothesis test of measurement invariance LTMs is generally premised on measurement invariance hypothesis to avoid excess and confusing latent status with inconsistent meaning across times and groups created by additional information derived from multiple measurements. Thus, latent status change can be measured straightforwardly, and group equivalence tests make the most sense. The hypothesis is tested statistically by comparing two LTMs, one with item-response probabilities constrained into parallel parameters by de ning equivalence sets, another with them varied across times or groups. In this study, the difference G 2 test provided a formal test of the null hypothesis that these two models tted equally well; the better model was with lower AIC and BIC values. In practice, particularly in multiple-group LTM, even statistically signi cant, it is indispensable to make an ultimate judgment after combining research background and theoretical knowledge. In this paper, we further conducted the heterogeneous unrestricted model with entirely un xed response probabilities. If latent statuses of each elderly subgroup were still basically consistent in structure and connotation, we assumed measurement equivalence con dently and believed the discrepancy of type partition on the physical function in distinct groups did not considerably impact model tting and interpretation conceptually.
Hypothesis test of gender differences in status distribution and transformation If latent status prevalence at start time and transition probability are equivalent, we deem that no gender differences in the whole distribution and change of physical function status. This paper compared the model with latent status prevalences in 2002 and transition probabilities all xed to the model with only latent status prevalence in 2002 constrained for testing gender differences of distribution, and to the one where only transition probabilities equaled for testing gender differences of transformation. Group differences were examed as the same strategy to compare nested models t by difference G 2 test and information criteria.

Results
Descriptive physical function characteristics of the study sample Physical function was declining with age. Statistical description showed that the sample's proportion without function limits decreased from 64.70% in 2002 to 24.74% in 2014 (Table 1). Speci cally, crouching and standing up for the elderly in China was the most arduous, followed by the outdoor walk and weight-bearing. Furthermore, the declining trend of visual function was evident with age; the completion and function decline on the actions as sitting to standing, picking up objects, and turning around were at a moderate level; general upper limb control was better and more stable. As for gender, female elderly were in more inferior function; explicit discrepancies were always observed in crouching and standing up, walking outdoors, carrying weights and vision, and gradually distinct in other functional indicators with age.
Item-response probabilities and latent statuses of physical function of the elderly Of the general LTA models with 2-7 latent statuses, a ve-status model had the best t in terms of the BIC and the principle of parsimony(i.e., BIC values were 11831.06, 10238.15, 9480.25, 9212.08, 9227.74, 9390.09 for models with 2,3,4,5,6,7 statuses, respectively). The ve statuses could be interpreted meaningfully and emerged with reasonable status membership probabilities(see Supplementary); no irrelevant outlier status appeared. The latent status of the Chinese elderly's physical function was stable across three times(i.e., BIC values were 9631.96,9212.08 for models with unconstrained and constrained item-response probabilities; ΔG2 = 108.05, df = 110,p = 0.53).
Five latent statuses with corresponding item-response probabilities of "limited" were de ned ( Fig. 1): Latent status 1: Completely Dysfunction. Characterized by more than 66% probabilities of limitations on almost all function indicators and a 54.32% risk of losing hearing, the elderly in this status were extremely poor in the strength, balance, gait, and exibility of upper limbs with a high risk of visual impairment and moderate risk of hearing trouble. Latent status 2: Health. Conversely, this latent status revealed no more than 10% risk of each function limit, where the elderly had intact physical function. Latent status 3: Lower extremity Dysfunction. Old folks in this status hardly completed "Crouch and stand up three times", "Lift the weight of 5kg", "Walk continuously for 1km" -the three most di cult for the elderly (see Table 1), which requires highly for strength, mobility, and balance. "Pick up a book from the oor by standing" is a crucial indicator to enter this state (82.19%). Additionally, they were at the upper-middle risk of trouble in "Sit to stand" (67.19%), "Turn around (≤ 10 steps)"(65.55%), and the increased rate of loss of vision and hearing (60.62%, 37.59%). These actions nearly involve lower extremities ability, and unimanual dexterity also loads much on body agility and single-leg balance by factor analysis (Greene et al.,1993).
Latent status 4: Limitation on High-intensity action. This status was featured by over 50% response probability for "limited" on three high-intensity actions as "Crouch and stand up three times"(64.64%), "Lift the weight of 5kg"(49.76%), "Walk continuously for 1km"(49.53%). The elderly in this status faced medium-low risk in visual (28.84%), and low possibility in other limits. This status re ected the low level of function deterioration. Latent status 5: Upper limb Mobility Dysfunction. In utterly contrary to latent status 3, the elderly in this status could not complete "hands behind the neck", "hands behind the lower back", "raise arms upright" independently( 90%), but performed well on strength/manipulating-required movements like carrying and picking up things, or other actions.  The latent status of Upper limb Mobility Dysfunction was unstable, and the elderly in this state were most likely to fully recover (59.10%, 33.22%) or get into trouble in high-intensity action (31.93%, 44.60%). With age, the recovery opportunity got much smaller; the worsening risk of Lower extremity Dysfunction (0.00-10.81%) and Completely Dysfunction (3.54-9.34%) got higher and higher.  heterogeneous unrestricted model demonstrated that the latent status composed by item-response probabilities was consistent in structure and connotation between male elderly and female elderly(see Supplementary). The ve-status multiple-group LTA model with item-response probabilities gender-constrained had the extremely similar items-response probabilities to the general LTA model above. These suggested that the latent status of the Chinese elderly's physical function was stable; there was no difference in the elderly's division by physical functional state across gender.These ve latent statuses"Completely Dysfunction", "Health", "Lower extremity Dysfunction", "Limitation on High-intensity action", "Upper limb Mobility Dysfunction" were typical.
The trend of latent status prevalences in each gender group was consistent with the overall sample. With age, the latent status prevalence of Health considerably decreased, but that of Lower extremity Dysfunction and Limitation on High-intensity action grew continually. The proportion of the elderly of Completely Dysfunction had a small rise, and which of Upper limb Mobility Dysfunction slightly uctuated (Fig. 2).
Generally, the male elderly's physical function was better than that of the female, and the difference was signi cant(ΔG 2 = 74. 59 The gender differences in the latent status transition were distinguishable (ΔG 2 = 110.86, df = 40, p < 0.000). As a whole, the chance of recovery of both genders reduced, and the risk of function deterioration was up, resulting in the change of latent status prevalences over three-time points. As Table 2 presented, for the elderly in the latent status of Complete Dysfunction, both genders' physical function tended to improve in the rst time interval, with over 55% chance into Lower extremity Dysfunction and around 35% chance into Limitation on High-intensity action.  , possibly due to their more professional working environment lled with risk and incidence in fatal diseases. A comprehensive health class study released an apparent paradox that more men were in the youngest and healthiest class, and women dominated in the oldest and sickest class (McClintock et al., 2016). Hence, the male survivors entering old age would be healthy and robust individuals; accordingly, living older men have an upraised opportunity to improve their physical function (Rogers, 2000; Huang and Wu, 2012). In addition to the inherent physiological variance, the cumulative effect of health inequality in gender in uence females' inferiority (Wei and Wang, 2017). Originating from distinct roles in family and society, especially in education and marital status, profound unfairness on social resources availability and effectivity exists between genders, triggering a disadvantaged position in females' health conditions later (Wheaton and Crimmins, 2016). The government should make the contrapuntal policy in response to older women's demographic, social, economic, and health characteristics.
The aging trend of health inequality has always being a complicated problem and research focus (Zheng and Zen, 2016). Two competitive hypothesescumulative disadvantage and age neutralization effect were formed by introducing the view of life course into social strati cation and inequality in health.
A few studies stated that health inequality caused by education and residence expanded with age (Hu, 2014; Li and Zhang, 2014), but others held opposite opinions (Christenson and Johnson, 1995). More precisely, the inequality was extensive from youth and middle age to the early stage of old age, but tended to shrink or converge later (House et al.,1994;Kim and Durden, 2007). Other researches emphasized that gender inequality in speci c health areas like depression symptoms had no suggestive change with age, but it was hardly judged to wide or narrow in self-rated health (Jiao, 2014).
Our study concludes whether gender inequality grew or shrunk with age are closely correlated to the initial health state. Firstly, among the elderly in the emergent period or extreme state of physical dysfunction, the cumulative advantage/disadvantage works, and female comparative disadvantage in keeping t is expanding. The transition probabilities proposed the female in the latent statuses of Limitation on High-intensity action, and Completely Dysfunction held a faster-growing possibility of deterioration and reduced chance of preserving or recovering with age. In contrast, men had a positive trend to maintain in the states of intact or sight decline on physical function. However, for the elderly with physical function intact, the gender differences change little with age, neither re ecting cumulative effect nor age neutralization effect. While Zeng believed among the undisabled elderly, women had an expanding health disadvantage comparing to men with age (Zeng et al., 2007).
Overall, from the new perspective of distribution and transformation of physical function state, the aging phenomenon is ineluctable and dominant in physiology health, which produces a marked effect at the beginning of the functional decline. Once started, the proceeding is irreversible. As for health inequality during the life course, when the elderly are at the emergent or later period of the physical dyfunction, socioeconomic factors dominated, and the cumulative disadvantage leads to the broad gender disparity. Arti cial intervention and economic support may bring more major utility to promote health and fairness among the elderly under these function states.
One major drawback of the approach is that data were from three-time nodes, not considering the annual change and complex situations. For individuals who had changed many times within the two periods, the explanation on status change was biased and rough. During data processing, this work also excluded the samples who dropped out, lost contact, and dead, considering the simplicity and low impact on classi cation results from extreme conditions or data missing. In the future, we would take more re ned health dimensions like cognition as the latent variable into the comprehensive health status construction; it worth to attention to choose manifest indicators and name the latent status based on differentiation and medical professionalism. It is also looking forward to continuing to study health inequality besides gender disparity.  Notes: Item-response probabilities constrained invariant across times and gender;row as the initial latent status of the time interval,column as the nal latent status of the time interval;bold part as the retention rate