Visuospatial Reasoning Trajectories and Death in a Study of the Oldest Old: A Formal Evaluation of Their Association

Abstract

Objectives: To model trajectories of visuospatial reasoning measured using Kohs Block Design test under realistic missing data assumptions and evaluate their association with hazard of death. Methods: A joint longitudinal-survival model was used to estimate trajectories of visuospatial reasoning under a missing not at random assumption of participants from the Origins of Variance in the Old–Old: Octogenarian Twins study. Sensitivity analyses to missing data assumptions were conducted. Results: Visuospatial reasoning declined at constant rate. Baseline age, dementia status, education, and history of stroke were associated with visuospatial reasoning performance, but only dementia was associated with its rate of decline. Importantly, our results demonstrated an association between poorer visuospatial reasoning and increased hazard of death. Baseline age and sex were associated with risk of death. Discussion: We confirmed an association between visuospatial reasoning and death under plausible missing data assumptions.

Keywords

joint longitudinal-survival missing not at random cognitive decline

Introduction

Age effects in cognitive function are well documented, and it is generally accepted that individuals tend to perform more poorly in cognitive tests with increasing age (Barnes et al., 2003; De Vries, 2004). Researchers have postulated that fluid abilities, that is, abilities that reflect individuals’ capacity for insight into complex problem-solving tasks, independent of modalities and cultural settings are more susceptible to aging declines, whereas crystallized abilities, that is, abilities that result from the investment of fluid abilities in experience and culturally defined tasks (Catell, 1971) tend to be better preserved with age. In longitudinal studies of older adults, mortality plays an important role with extensive literature indicating poorer performance in individuals who die during a study follow-up time than in survivors (Deeg, 2002; Johansson & Berg, 1989; Rabbitt et al., 2002).

It has also been postulated that as individuals approach death, rate of cognitive decline accelerates. This hypothesis, known as the terminal decline hypothesis (Riegel & Riegel, 1972), has been evaluated in several publications (Gerstorf et al., 2008; Muniz-Terrera, van den Hout, Piccinin, Matthews, & Hofer, 2013; Wilson, Beckett, Bienias, Evans, & Bennett, 2003; Wilson, Segawa, Buchman, Boyle, Hizel, & Bennett, 2012). Yet, results are inconsistent and findings vary across populations and cognitive abilities, with differences partially explained by the use of different methodological tools in the various publications. For example, Piccinin et al (Piccinin, Muniz, Matthews, & Johansson, 2011) evaluated the terminal decline hypothesis modeling Information Scores, a measure of crystallized ability and also modeling Kohs Block Design scores, a measure of fluid ability, using data from a sample of deceased participants from a study of the oldest old in Sweden, the OCTO Twin study (McClearn, 1997). They reported that, in individuals who die during the study follow-up, Information scores followed a curvilinear trajectory as individuals approach death, which is consistent with the terminal decline hypothesis. However, in this study, they also reported that Block Design scores declined at a constant rate, failing to confirm the hypothesis.

Although this finding may initially appear to be in contradiction with the theory of fluid abilities being more sensitive to aging-related changes, it is important to bear in mind several factors when interpreting this finding. To begin with, the process-based approach, employed in Piccinin et al. (2011) rests on the idea that if intraindividual change in cognition is driven by the process of dying, then the effect of chronological time or age will be minimized by accounting for the dying process that is inducing change in cognitive function. Second, as the analytical sample used was comprised only of deceased study participants, who are likely to be different from those who remain alive during the study follow-up time, findings are not generalizable to the entire sample. Third, questions regarding an association between aging-related changes in visuospatial ability and mortality remained unanswered.

Hence, the question of how this specific measure of fluid ability, Block Design, changes as individuals age remains unanswered in this sample of the oldest old, and importantly, a new question regarding what features of the Block Design trajectory (i.e., is it the level of performance or its change?) are associated with mortality in the whole sample, not only in the sample of deceased individuals, arises.

Sex differences in cognitive function have been largely discussed in the literature (Agneta & Johanna, 2009; Barnes et al., 2003; De Vries, 2004; Finkel, Reynolds, Berg, & Pedersen, 2006; McCarrey, An, Kitner-Triolo, Ferrucci, & Resnick, 2016; Miller & Halpern, 2014), although it is not yet clear whether these differences generalize to the oldest old (Hassing, Wahlin, & Bäckman, 1998) and whether they exist for all or only for some cognitive abilities. Particularly, sex differences in level and rate of change of visuospatial abilities have been questioned (Mitolo et al., 2015; Palmiero, Nori, Rogolino, D’amico, & Piccardi, 2016; Voyer, Voyer, & Bryden, 1995; Waschl, Nettelbeck, Jackson, & Burns, 2016). In this article, we examine whether sex differences in visuospatial abilities exist in a cohort of the oldest old.

A better understanding of the relationship between visuospatial ability and mortality will inform clinicians and researchers as to whether or not this cognitive ability, in addition to other cognitive functions, should be the target of interventions and training programs designed to improve performance, and hence reduce the risk of death.

Method

Data

Participants of the OCTO Twin study (McClearn, 1997) were sampled from 737 pairs aged 80 years or older from the Swedish Twin Registry. The pairwise response rate, apart from nonresponse due to death of one or both twins in a pair (188 pairs), was 65%, resulting in 351 intact twin pairs aged 80 years or older (702 individuals: 149 monozygotic and 202 same-sex dizygotic pairs). These individuals were first interviewed between 1991 and 1993 and then at 2-year intervals for four further occasions. After accounting for mortality, only 10% of missing occasions were due to refusal to participate. The sample eligible for analysis after elimination of individuals who had dementia at baseline comprised of 358 women and 193 men, with an average age at study entry of 82.8 (SD = 2.5).

Measures

Visuospatial reasoning was measured using Kohs Block Design test (Dureman & Salde, 1959). In this task, participants are shown a set of cards displaying red and white blocks arranged in a pattern that they have to reproduce. The maximum score in this test is 42. Figure 1 depicts observed trajectories of Block Design scores of individuals in the sample analyses plotted as a function of time in study.

Figure 1.

Observed Kohs Block Design scores plotted as a function of time in study.

In the OCTO Twin study, dementia was diagnosed by consensus according to the revised third edition of the Diagnostic and Statistical Manual of Mental Disorders (DSM-III-R). To identify cases of dementia at each new wave of testing, a multidisciplinary team consisting of a physician, a registered nurse, and two neuropsychologists reviewed cognitive test results and medical records, including reported medical history, medicine use, and self-reported information about diseases. The diagnoses made at this consensus conference were based on all available information.

Using the dementia diagnosis information, we derived a binary indicator that took the value of 1 if the individual was ever diagnosed with dementia and 0 otherwise. Self-reports of smoking habits were employed to create an indicator for smoking status (1 = ever smoked, 0 = never smoked). Information from medical records and self-reports of medication usage were used to create indicator variables for stroke (1 = ever had a stroke, 0 = never), diabetes (1 = ever diagnosed with Diabetes Mellitus, 0 = never diagnosed), and hypertension (1 = yes, 0 = no). About 26% of the sample had been diagnosed with dementia, 39% had a history of smoking, 15% had a stroke, 14% had diabetes, and 43% had a history of hypertension. Education was measured as the number of years spent in formal education. Only individuals with valid sociodemographic and health conditions information were included in the data analysis. About 5% of the sample lived beyond the study follow-up time, and the average age at death was 90.22 (SD = 4.24) years. Table 1 shows descriptive characteristics of the analytical sample examined.

Table 1.

Mean and Standard Deviation of Block Design Test Scores and Age at Each Assessment of the Subsample of OCTO Twin Study Participants.

	Assessments
	Baseline	1st follow-up	2nd follow-up	3rd follow-up	4th follow-up
Deceased
N	478	359	267	191	117
M Block Design (SD)	11.09 (7.18)	11.15 (7.16)	11.17 (6.93)	10.82 (6.95)	10.04 (6.99)
M Age (SD)	83.30 (0.12)	84.22 (2.33)	86.26 (2.29)	88.23 (2.35)	90.20 (2.38)
Survivors
N	29	28	28	28	27
M Block Design (SD)	15.24 (6.40)	16.07 (5.60)	16.44 (6.22)	15.53 (6.24)	15.00 (6.33)
M Age (SD)	82.12 (0.43)	84.81 (2.39)	86.84 (2.43)	88.55 (2.21)	90.59 (2.20)

Note. N = sample size at each follow-up time.

Statistical Analyses

We fitted a series of joint longitudinal survival models, formulated under a nonignorable missing data assumption, to visuospatial reasoning scores modeled as a function of time in study and to mortality data. For a mathematical formualtion of the model, see Appendix A. As the shape of the longitudinal trajectory is unknown, we estimated models describing linear and accelerating change to identify the parametric shape that best fitted the data. To select the best fitting model, we compared Bayesian information criterion (BIC) indices (Raftery, 1996) from each model. The BIC is an index that permits identification of the best fitting and most parsimonious model. Models with lowest BIC values are preferred.

The intercept and linear and quadratic slopes were adjusted for sociodemographic and physical health-related variables. Specifically, we adjusted the intercept and slopes for age at study entry, sex (0 = male, 1 = female), education, dementia diagnosis, smoking status, and health condition indicators (stroke, diabetes, hypertension). Continuous covariates were centered at their mean value at study entry (7.13 years for education and 83.3 years for age at study entry).

Mortality data were modeled using a Weibull proportional hazard model that included a parameter (α) representing the association between visuospatial reasoning scores at time t with hazard of death at that time and adjusted for the same set of covariates considered in the longitudinal model. The linking parameter α is interpreted as a regression parameter in the survival submodel and represents the expected value of the effect of the longitudinal process on the hazard of death for fixed values of the covariates. Importantly, if α = 0, the longitudinal and survival processes are independent and the missing mechanism ignorable.

Following recommended practice (Verbeke & Molenberghs, 2000), we conducted sensitivity analyses to missing data assumptions. Specifically, we compared raw estimates, z values and confidence intervals from JM and linear mixed models, to evaluate their differences. For completeness, we also fitted a survival model to death times excluding the link with the trajectory of visuospatial ability.

All models were fitted using the R package JM (Rizopoulos, 2010) using maximum likelihood estimation. The linear mixed effects model was fitted using the lme package and the survival packages, ignoring the association between both processes, was fitted using the survreg function. The joint model using the Joint Model function that uses both functions jointly.

Results

At each study wave, we compared visuospatial reasoning scores of participants who died between consecutive waves and participants who survived and consistently found statistically significant differences between these groups (t tests, all p values <.05). Deceased individuals consistently had worse performance than survivors (Table 1). On average, participants who died were older at study entry than those who survived (83.30 [SE = 0.43] vs. 82.13[SE = 0.13]).

A series of Fisher’s exact tests conducted to evaluate the association of history of stroke, hypertension, diabetes, dementia, and smoking with survival status implied no significant association between these variables and survival status.

Joint Longitudinal-Survival Model

BIC values from the JM indicated that a linear trajectory fitted the data best (BIC_linear = 12283 vs. BIC_quad = 12317), so we report results from the JM that models visuospatial reasoning scores as declining at constant rate (linear trajectory).

Longitudinal submodel

Results from the longitudinal submodel are reported in Table 2, under the Joint Longitudinal-Survival Model columns. These results indicate that performance at study entry on the Block Design test for a reference man (aged 83.3 years at study entry, with 7.13 years of education and no history of stroke, hypertension, diabetes, smoking, or dementia) was estimated at 13.10 (SE = 0.76) points in the Block Design test. Average rate of change was estimated as –0.22 (SE = 0.12) points per year, although the estimate did not reach conventional statistical significance level.

Table 2.

Results From the Joint Model Fitted to Block Design Scores of a Sample of OCTO Twin Study Participants.

	Joint longitudinal-survival			Linear mixed effects
	Est. (SE)	95% CI	z value	Est. (SE)	95% CI	z value
Fixed effects
Level	13.10 (0.76)*	[11.59, 14.58]	17.07	13.09 (0.76)*	[11.6, 14.58]	17.04
Baseline age	−0.49 (0.10)*	[–0.71, –0.27]	−4.56	−0.48 (0.10)*	[–0.07, –0.27]	−4.47
Dementia status	−3.60 (0.67)*	[–4.91, –2.27]	−5.33	−3.63(0.67)*	[–4.94, –2.31]	−5.39
Education	0.77 (0.13)*	[0.50, 1.04]	5.68	0.77 (0.13)*	[0.50, 1.04]	5.67
Sex	0.24 (0.69)	[–1.12, 1.61]	0.34	0.23 (0.69)	[–1.12, 1.60]	0.33
Stroke	−1.95 (0.80)*	[–3.51, –0.37]	−2.42	−1.95 (0.80)*	[–3.52, –0.39]	−2.44
Diabetes	−0.25 (0.83)	[–1.82, 1.42]	−0.30	−0.19 (0.83)	[–1.82, 1.42]	−0.23
Hypertension	−0.48 (0.58)	[–1.68, 0.62]	−0.81	−0.53 (0.59)	[–1.68, 0.62]	−0.89
Smoking status	−1.05 (0.69)	[–2.41, 0.31]	−1.50	−1.04 (0.69)	[–2.41, 0.31]	−1.49
Time	−0.22 (0.12)	[–0.46, 0.02]	−1.78	−0.21 (0.11)	[–0.42, 0.06]	−1.86
Age at Entry × Time	−0.01 (0.02)	[–0.05, 0.02]	−0.69	−0.01 (0.02)	[–0.05, 0.02]	−0.69
Dementia × Time	−0.58 (0.11)*	[–0.79, –0.36]	−4.92	−0.55 (0.11)*	[–0.77, –0.33]	−4.92
Education × Time	0.008 (0.02)	[–0.03, 0.05]	0.41	0.008 (0.02)	[–0.03, 0.05]	0.41
Sex × Time	−0.02 (0.11)	[–0.23, 0.19]	−0.16	−0.02 (0.11)	[–0.24, 0.19]	−0.22
Stroke × Time	0.20 (0.12)	[–0.03, 0.43]	1.66	0.20 (0.12)	[–0.03, 0.43]	1.63
Diabetes × Time	−0.09 (0.13)	[–0.34, 0.16]	−0.69	−0.10 (0.13)	[–0.36, 0.15]	−0.78
Hypertension × Time	−0.10 (0.09)	[–0.27, 0.07]	−1.16	−0.09 (0.09)	[–0.26, 0.08]	−1.04
Smoking × Time	−0.06 (0.05)	[–0.41, 0.01]	−1.81	−0.06 (0.05)	[–0.42, 0.009]	−1.19
Random effects
Level	5.68	[5.21, 6.12]		5.64	[5.21, 6.12]
Time	0.35	[0.23, 0.54]		0.37	[0.23, 0.54]
Residual	3.32	[3.15, 3.51]		3.31	[3.15, 3.51]
R(Level, Time)	−0.29	[–0.48, –0.02]		−0.32	[–0.48, –0.02]

Note. Est. = unstandardized estimate; CI = confidence interval.

*significant at 5% level

Baseline age, dementia status, education, and having suffered a stroke were found to be associated with performance at study entry in the Block Design test, but only dementia was associated with its rate of decline. Specifically, per extra year of older age at study entry, performance at study entry decreased by almost half a point in the Block Design test scale (–0.49 [SE = 0.10]); per extra year of education, performance improved by 0.77 (SE = 0.13) points and individuals with a history of stroke scored about 2 points lower at study entry (–1.95 [SE = 0.80]) than individuals who had not had a stroke. Also at study entry, individuals with dementia scored 3.60 (SE = 0.67) points lower than individuals without dementia, and their decline was 0.58 (SE = 0.10) points faster than the decline of individuals without dementia.

Survival submodel

The association parameter, α, that links the longitudinal and survival models, was estimated at −0.05 (SE = 0.00), a result that indicates that the hazard of death dropped by 5.13% per unit increase in Block Design scores (see Table 3, right column). This nonnull value of the association parameter demonstrates an association between poorer visuospatial reasoning and mortality, as individuals with poorer visuospatial reasoning performance in the Block Design test were found to be more likely to die. The scale parameter of the survival submodel did not provide support for an exponential model (scale = 1.91), indicating that the hazard of death is not flat. After controlling for visuospatial reasoning performance, only baseline age and sex were associated with hazard of death. Specifically, the hazard of death increased by 5.13% per year of older age at study entry, and women’s hazard of death was 30% lower than men’s. Q-Q plots of subject-specific residuals versus the corresponding fitted values depicted in the top panel of Figure 2 suggest good fit of the model to the observed data. Marginal survival and cumulative risk function for death are displayed in the bottom panel of Figure 2.

Table 3.

Results From Independent Survival Analysis and From Survival Submodel of Joint Longitudinal-Survival Analysis.

	Independent model		Joint model
Survival model	Est. (SE)	p value	Est. (SE)	p value
Intercept	2.02 (0.06)	<.000	−3.08 (0.23)	<.000
Age at study entry	−0.04 (0.008)	<.000	0.05 (0.02)	.00
Dementia status	−0.11 (0.05)	.02	−0.15 (0.12)	.21
Education	−0.002 (0.00)	.77	0.04 (0.02)	.06
Sex	0.14 (0.06)	.01	−0.26 (0.11)	.00
Stroke	0.00 (0.064)	.99	0.00 (0.13)	.94
Diabetes	−0.05 (0.06)	.42	0.09 (0.13)	.46
Hypertension	0.005(0.05)	.22	−0.18 (0.09)	.07
Smoking status	−0.05 (0.03)	.06	0.04 (0.05)	.41
Association			−0.05 (0.00)	<.000
Scale	0.52		1.91	.000

Note. Est. = unstandardized estimate.

Figure 2.

Top panel: subject specific residuals plotted versus the corresponding fitted values and their QQ plots. Bottom panel: Marginal survival and cumulative risk function for death.

Sensitivity Analyses

Linear mixed effects model

Average visuospatial reasoning at baseline for a reference individual was estimated at 13.09 (SE = 0.76) points in the Block Design test. Although there was some indication of a decrease in cognition over time, similarly to the JM, the rate of change for the LME did not reach conventional significance levels (see results in Table 3). All other fixed effects estimates were consistent in direction and statistical significance with those obtained from the JM.

The comparison of JM and LME fixed effects estimates of trajectory parameters of visuospatial reasoning scores identified some minor differences between estimates produced by both models. Differences in estimates were easier to evaluate inspecting z scores, which identified various degrees of sensitivity. Some z values did not differ across models, and where differences were found, they were consistently small (differences ranged from –0.62 to 0.09). There was no consistency in the sign of the z values’ differences, as positive values suggested that z values of JM estimates were larger than the corresponding LME z values and negative differences indicated the reverse. The inspection of confidence intervals about the estimates is also informative of differences between estimates and suggests no real differences between fixed effects estimates obtained from each model.

Survival model (independent of visuospatial reasoning scores)

The scale parameter of the Weibull model was estimated at 0.52, a value that indicates no support for an exponential model. Only sex, diagnosis of dementia, and age at study entry were significantly associated with hazard of death (see Table 2, left column). Specifically, after controlling for history of hypertension, diabetes, stroke, and smoking status, the hazard of death decreased slightly (4.4%) per year of older age at study entry, women had 15% higher hazard of death than men, and the hazard of death of demented individuals was 11.6% lower than the hazard of individuals without dementia.

Discussion

We evaluated the association between repeated measurements of visuospatial reasoning and death in a study of the oldest old fitting a JM formulated under a not at random missing data assumption. Results revealed an association between visuospatial reasoning and death such that poorer performers had an increased hazard of death compared with individuals with better performance on the Block Design test. In addition, the joint analysis of longitudinal and mortality data demonstrated that, after accounting for visuospatial reasoning and other sociodemographic and health-related factors, older individuals at study entry and women have a lower hazard of death than men and individuals who are younger when joining the study. Dementia was associated with faster decline and poorer performance, and history of stroke, older age at study entry were negatively associated with performance at study entry.

These results are in general agreement with the literature. For example, older age and poorer education have been reported to be associated with poorer performance but not rate of change of global cognition scores (Piccinin et al., 2013). Previous investigations have also reported a negative effect of stroke on cognitive decline (Gottesman & Hillis, 2010; Rist et al., 2014). Although previous reports identified health conditions and health detrimental behaviors to be associated with mortality, in our study, we did not find evidence of an association between hazard of death and stroke, diabetes, smoking status, or hypertension. Yet, the lack of an association between these variables and increased hazard of death may be the result of a selection bias, as the OCTO Twin study is a study of the oldest old; therefore, only individuals who survived up to the age of 80 years took part. Our results are complementary to previous examinations of change in visuospatial ability conducted using data from this same study. For instance, in an evaluation of the terminal decline hypothesis, Piccinin et al. (2011) failed to identify accelerating rate of change prior to death in the sample of deceased study participants and reported that in this study of the oldest old, individuals’ visuospatial reasoning skills declined linearly before death. Taken together with our results, which were generated in an analysis of the entire sample and where visuospatial reasoning changes were examined as a function of chronological time, a consistent picture emerges regarding the linearity of change in visuospatial ability in this study of the oldest old. Importantly, our analyses further add to Piccinin et al.’s work by formally testing the association between mortality and visuospatial ability.

Our analysis has some limitations. From a methodological perspective, although we fitted a model that enabled us to evaluate the association between survival and visuospatial reasoning, because of the lack of information about cause of death, we could not distinguish between potentially different associations of cognitive functioning and mortality due to different causes. We adjusted our models for dementia status to understand differences in visuospatial trajectories and survival between individuals diagnosed with dementia and individuals who were dementia-free. Yet, it is possible that individuals, with different dementia types, differ in their visuospatial ability trajectories and survival. Our ability to understand these differences was hampered by the low number of individuals in our sample with different types of dementia.

In addition, the sample we analyzed was comprised of individuals who were aged 80 years and older at study entry. Individuals who participated in the study are likely to be highly selected, as they are possibly healthier and with better cognitive functioning than individuals of the same age who did not participate in the study, and, therefore, our results may be susceptible to a healthy survivor effect (left truncation). This would explain, for instance, the relatively flat trajectory estimated and the fact that health conditions and health detriment behaviors did not emerge as associated with survival. A further limitation of our work is the lack of consideration of mental health conditions such as depression that has been previously reported to be associated with poor cognition in some studies of older adults. In the sample we analyzed, the average CES-D score at study entry was 8.23 (SD = 7.91) in individuals free of dementia and 8.71 (SD = 7.45) in individuals with dementia, and only 20% of the sample scored above the CES-D cutoff point of 16 for clinical depression at study entry and the majority of study participants remained below the clinical depression cutoff point over time (Haynie, Berg, Johansson, Gatz, & Zarit, 2001). This may be the result of a healthy survivor effect, although similarly low rates of depressive symptoms have been documented in other studies of Swedish older adults (Kiljunen et al., 1997).

Importantly, our analyses have strong methodological underpinnings. To begin with, we employed joint modeling, an efficient methodology specifically formulated to improve knowledge about the association between longitudinal outcomes and an event of interest. In our context, we confirmed an association between visuospatial ability and death. Second, as JM are formulated under a not at random missing data mechanism, our approach reflects more plausible missing data assumptions than the missing at random assumption made by commonly used methods for longitudinal data. Finally, we followed recommended practice conducting thorough sensitivity analyses to these assumptions. Although we present results from an independent survival model, as well as the survival submodel fitted as part of the JM, their comparison is not part of the sensitivity analyses in a strict sense, as the sensitivity analyses relate specifically to the missing data assumptions. Furthermore, if results were to be compared, it would be necessary to bear in mind that the two models are conditioned on different covariates, so estimates from each model have different interpretations. Results obtained from the JM survival submodel show a drop in significance levels for dementia status compared with the independent survival model results, and a change in the direction of the association between sex and mortality. The drop in significance levels of dementia status can be explained by the inclusion of visuospatial ability in the model, as it is likely that both variables share the variance to be explained. The change in direction of the effect of sex on survival opens interesting questions that need further exploration. Previously reported sex differences in dementia prevalence may explain this result. For instance, women have been reported to be more likely to develop AD than men, and visuospatial abilities are more affected in AD and Lewy Body dementias than in other dementia types (Podcasy & Epperson, 2016), so to fully understand sex differences once visuospatial abilities are accounted for in the survival submodel, it would be necessary to know the distribution of dementia types in this sample by sex and mortality status. This would also contribute to the literature about survival and dementia type, which reported mixed findings (Lee & Chodosh, 2009). Unfortunately, as previously mentioned, we were unable to investigate this due to the low number of cases in some dementia types.

Global sensitivity of results to missing data assumptions was evaluated by inspection of raw and z scores of fixed effects estimates and confidence intervals of estimates. Inspection of confidence intervals and of z values suggested negligible differences between fixed effects estimates obtained from each model. Other studies have also reported similar findings when comparing joint model estimates with naïve analyses (Graham, Ryan, & Luszcz, 2011). Furthermore, these results are in agreement with our expectations as in the JM, the survival model is conditional on the random effects, but the longitudinal submodel is independent of death. Indeed, the JM still models an immortal cohort (Jones, Mishra, & Dobson, 2015), but better reflects the missing not at random assumption.

The use of a sophisticated analytical technique improved our understanding of the association between decline in visuospatial abilities and mortality while modeling visuospatial ability decline under more realistic missing data assumptions. This is in contrast with independent analysis of survival data that often ignores features of longitudinal processes that may be associated with it, and that can be accounted for when jointly modeling both processes. As shown in our analyses, the JM approach has the capability of improving our knowledge about the survival process, and hence we recommend their use for improved prediction of survival.

Footnotes

Appendix A

A general formulation of a shared parameters model, also known as joint longitudinal-survival model, can be expressed by the two equations presented below.

\begin{array}{l} Y_{i} (t_{i j}) = m_{i} (t_{i j}) + ε_{i} (t_{i j}), j = 1 \dots, r_{i}, i = 1, \dots, n \\ h (t_{i j}) (| M_{i} (t_{i j}), W_{i}) = h_{0} (t_{i j}) \exp {γ^{T} W_{i} + α m_{t} (t_{i j})} . \end{array}

Where in the top equation, $Y_{i} (t_{i j})$ represents the observed value of the longitudinal outcome for person i at time t_ij, $m_{i} (t_{i j}) = β^{T} X_{i} (t_{i j}) + b_{i}^{T} Z_{i} (t_{i j})$ represents the expected value of the longitudinal process at the jth occasion for person I; with errors $ε_{i} (t_{i j})$ _~ $N (0, σ^{2})$ . Random coefficients are represented by $b_{i}$ _~MVN (0,D) and $X_{i} (t_{i j})$ and $Z_{i} (t_{i j})$ are the rows of design matrices corresponding to fixed and random effects, respectively (for subject i at time t_ij).

The bottom equation represents the hazard of an event (death in the context of the article), as a function of $h_{0} (t_{i j})$ the baseline risk function and the true and unobserved value of the longitudinal process at time t, $m_{t} (t)$ , and is conditional on $M_{i} (t_{i j}) = {m i (u), 0 < = u < = t_{i j}}$ , the history of the process up to time t_ij_, $W_{i}$ is a row vector of baseline covariates associated with the survival process.

Authors’ Note

The code used for data analysis can be obtained from the corresponding author. Annie Robitaiile is presently affiliated to Department of Psychology Faculty of Social Sciences Université du Québec à Montréal Montreal, QC.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by NIH/NIA Program Project Grant (P01AG043362; 2013-2018).

References

Agneta

Johanna

(2009). Sex differences in cognitive functions. Acta Psychologica Sinica, 41, 1081-1090. doi:10.3724/SP.J.1041.2009.01081

Barnes

L. L.

Wilson

R. S.

Schneider

J. A.

Bienias

J. L.

Evans

D. A.

Bennett

D. A.

(2003). Gender, cognitive decline, and risk of AD in older persons. Neurology, 60, 1777-1781. doi:10.1212/WNL.59.3.364

Catell

(1971). Abilities: Their structure, growth, and action. New York, NY: Houghton Mifflin.

De Vries

G. J

. (2004). Minireview: Sex differences in adult and developing brains: Compensation, compensation, compensation. Endocrinology, 145, 1063-1068. doi:10.1210/en.2003-1504

Deeg

D. J. H.

(2002). Attrition in longitudinal population studies: Does it affect the generalizability of the findings? An introduction to the series. Journal of Clinical Epidemiology, 55, 213-215. doi:10.1016/S0895-4356(01)00472-3

Dureman

Salde

(1959). Psykometriska och experimental-psykologiska metoderfor klinisk tillampning. Uppsala, Sweden: Almqvist & Wiksell.

Finkel

Reynolds

C. A.

Berg

Pedersen

N. L.

(2006). Surprising lack of sex differences in normal cognitive aging in twins. International Journal of Aging & Human Development, 62, 335-357. doi:10.2190/C39X-9QHY-49DM-X9GJ

Gerstorf

Ram

Estabrook

Schupp

Wagner

G. G.

Lindenberger

(2008). Life satisfaction shows terminal decline in old age: Longitudinal evidence from the German Socio-Economic Panel Study (SOEP). Developmental Psychology, 44, 1148-1159. doi:10.1037/0012-1649.44.4.1148

Gottesman

R. F.

Hillis

A. E.

(2010). Predictors and assessment of cognitive dysfunction resulting from ischaemic stroke. The Lancet Neurology, 9, 895-905. doi:10.1016/S1474-4422(10)70164-2

10.

Graham

P. L.

Ryan

L. M.

Luszcz

M. A.

(2011). Joint modelling of survival and cognitive decline in the Australian longitudinal study of ageing. Journal of the Royal Statistical Society. Series C: Applied Statistics, 60, 221-238. doi:10.1111/j.1467-9876.2010.00737.x

11.

Hassing

Wahlin

Bäckman

(1998). Minimal influence of age, education, and gender on episodic memory functioning in very old age: A population-based study of nonagenarians. Archives of Gerontology and Geriatrics, 27, 75-87. Retrieved from http://www.ncbi.nlm.nih.gov/pubmed/18653153

12.

Haynie

D. A.

Berg

Johansson

Gatz

Zarit

S. H.

(2001). Symptoms of depression in the oldest old: A longitudinal study. The Journals of Gerontology. Series B, Psychological Sciences & Social Sciences, 56, P111-118. Retrieved from http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=11245358

13.

Johansson

Berg

(1989). The robustness of the terminal decline phenomenon: longitudinal data from the Digit-Span Memory Test 197. Journal of Gerontology, 44, 184-186. doi:10.1093/geronj/44.6.P184

14.

Jones

Mishra

G. D.

Dobson

(2015). Analytical results in longitudinal studies depended on target of inference and assumed mechanism of attrition. Journal of Clinical Epidemiology, 68, 1165-1175. doi:10.1016/j.jclinepi.2015.03.011

15.

Kiljunen

Sulkava

Niinisto

Polvikoski

Verkkoniemi

Halonen

(1997). Depression measured by the zung depression status inventory is very rare in a Finnish population aged 85 years and over. International Psychogeriatrics, 9, 359-368.doi:10.1017/S1041610297004493

16.

Lee

Chodosh

(2009). Dementia and life expectancy: What do we know? Journal of the American Medical Directors Association, 10, 466-471.doi:10.1016/j.jamda.2009.03.014

17.

McCarrey

A. C.

Kitner-Triolo

M. H.

Ferrucci

Resnick

S. M.

(2016). Sex differences in cognitive trajectories in clinically normal older adults. Psychology and Aging, 31, 166-175. doi:10.1037/pag0000070

18.

McClearn

G. E.

(1997). Substantial genetic influence on cognitive abilities in twins 80 or more years old. Science, 276, 1560-1563. doi:10.1126/science.276.5318.1560

19.

Miller

D. I.

Halpern

D. F.

(2014). The new science of cognitive sex differences. Trends in Cognitive Sciences, 18, 37-45. doi:10.1016/j.tics.2013.10.011

20.

Mitolo

Gardini

Caffarra

Ronconi

Venneri

Pazzaglia

(2015). Relationship between spatial ability, visuospatial working memory and self-assessed spatial orientation ability: A study in older adults. Cognitive Processing, 16, 165-176. doi:10.1007/s10339-015-0647-3

21.

Muniz-Terrera

van den Hout

Piccinin

A. M.

Matthews

F. E.

Hofer

S. M.

(2013). Investigating terminal decline: Results from a UK population-based study of aging. Psychology and Aging, 28, 377-385. doi:10.1037/a0031000

22.

Palmiero

Nori

Rogolino

D’amico

Piccardi

(2016). Sex differences in visuospatial and navigational working memory: The role of mood induced by background music. Experimental Brain Research, 234, 2381-2389. doi:10.1007/s00221-016-4643-3

23.

Piccinin

A. M.

Muniz

Matthews

F. E.

Johansson

(2011). Terminal decline from within- and between-person perspectives, accounting for incident dementia. Journals of Gerontology Series B: Psychological Sciences & Social Sciences, 66, 391-401. doi:10.1093/geronb/gbr010

24.

Piccinin

A. M.

Muniz-Terrera

Clouston

Reynolds

C. A.

Thorvaldsson

Deary

I. J.

. . . Hofer

S. M.

(2013). Coordinated analysis of age, sex, and education effects on change in MMSE scores. The Journals of Gerontology. Series B: Psychological Sciences & Social Sciences, 68, 374-390. doi:10.1093/geronb/gbs077

25.

Podcasy

J. L.

Epperson

C. N.

(2016). Considering sex and gender in Alzheimer disease and other dementias. Dialogues in Clinical Neuroscience, 18, 437-446. doi:10.1016/j.jcin.2015.10.034

26.

Rabbitt

Watson

Donlan

Mc Innes

Horan

Pendleton

Clague

(2002). Effects of death within 11 years on cognitive performance in old age. Psychology and Aging, 17, 468-481. doi:10.1037//0882-7974.17.3.468

27.

Raftery

A. E.

(1996). Approximate Bayes factors and accounting for model uncertainty in generalised linear models. Biometrika, 83, 251-266. doi:10.1093/biomet/83.2.251

28.

Riegel

K. F.

Riegel

R. M.

(1972). Development, drop, and death. Developmental Psychology, 6, 306-319. doi:10.1037/h0032104

29.

Rist

P. M.

Chalmers

Arima

Anderson

Macmahon

Woodward

. . . Tzourio

(2014). Baseline cognitive function, recurrent stroke, and risk of dementia in patients with stroke. Stroke, 44, 1795-1790. doi:10.1161/STROKEAHA.111.680728

30.

Rizopoulos

(2010). JM: An R package for the joint modelling of longitudinal and time-to-event data. Journal of Statistical Software, 35(9). Retrieved from http://www.jstatsoft.org/v35/i09

31.

Verbeke

Molenberghs

(2000). Linear mixed models for longitudinal data. Berlin, Germany: Springer Series in Statistics.

32.

Voyer

Bryden

M. P.

(1995). Magnitude of sex differences in spatial abilities: A meta-analysis and consideration of critical variables. Psychological Bulletin, 117, 250-270. doi:10.1037/0033-2909.117.2.250

33.

Waschl

N. A.

Nettelbeck

Jackson

S. A.

Burns

N. R.

(2016). Dimensionality of the Raven’s advanced progressive matrices: Sex differences and visuospatial ability. Personality and Individual Differences, 100, 157-166. doi:10.1016/j.paid.2015.12.008

34.

Wilson

R. S.

Beckett

L. A.

Bienias

J. L.

Evans

D. A.

Bennett

D. A.

(2003). Terminal decline in cognitive function. Neurology, 60, 1782-1787. doi:10.1212/01.WNL.0000068019.60901.C1

35.

Wilson

R. S.

Segawa

Buchman

A. S.

Boyle

P. A.

Hizel

L. P.

Bennett

D. A.

(2012). Terminal decline in motor function. Psychology and Aging, 27, 998-1007. doi:10.1037/a0028182