MRC Biostatistics Unit, Institute of Public Health, Robinson Way, University Forvie Site, CB2 0SR, Cambridge, UK
Department of Public Health and Primary Care, Institute of Public Health, University of Cambridge, University Forvie Site, CB2 0SR, Cambridge, UK
Abstract
Background
Cognitive decline is a major threat to well being in later life. Change scores and regression based models have often been used for its investigation. Most methods used to describe cognitive decline assume individuals lose their cognitive abilities at a constant rate with time. The investigation of the parametric curve that best describes the process has been prevented by restrictions imposed by study design limitations and methodological considerations. We propose a comparison of parametric shapes that could be considered to describe the process of cognitive decline in late life.
Attrition plays a key role in the generation of missing observations in longitudinal studies of older persons. As ignoring missing observations will produce biased results and previous studies point to the important effect of the last observed cognitive score on the probability of dropout, we propose modelling both mechanisms jointly to account for these two considerations in the model likelihood.
Methods
Data from four interview waves of a population based longitudinal study of the older population, the Cambridge City over 75 Cohort Study were used. Within a selection model process, latent growth models combined with a logistic regression model for the missing data mechanism were fitted. To illustrate advantages of the model proposed, a sensitivity analysis of the missing data assumptions was conducted.
Results
Results showed that a quadratic curve describes cognitive decline best. Significant heterogeneity between individuals about mean curve parameters was identified. At all interviews, MMSE scores before dropout were significantly lower than those who remained in the study. Individuals with good functional ability were found to be less likely to dropout, as were women and younger persons in later stages of the study.
Conclusion
The combination of a latent growth model with a model for the missing data has permitted to make use of all available data and quantify the effect of significant predictors of dropout on the dropout and observational processes. Cognitive decline over time in older persons is often modelled as a linear process, though we have presented other parametric curves that may be considered.
Background
Severe changes in the population pyramid have been acknowledged by international organisations such as the United Nations
At the individual level, the loss of physical and mental health are two main threats to the enjoyment of wellbeing in older ages and to the costs and health of carers. Age associated diseases such as dementia including Alzheimer's disease represent serious threats to healthy life of older persons and their carers. These diseases already affect over half a million people in England and Wales
As a central feature of the ageing process, understanding and measuring changes in cognitive function has therefore become a key issue for society and researchers.
Many tests have been used to measure cognition, although the Mini Mental Status Examination
The MMSE can detect decline and correlates with biological and neuropathological markers associated with the dementias
Historically, change has been regarded as an incremental or as a continuous process
There is large literature on agebased models fitted to examine change in specific cognitive abilities
Limitations for further exploration of parametric curves that could fit the data were sometimes imposed by the study design. For instance, Backman et al.
The effect of attrition is a major issue in longitudinal studies of the older population as individuals are more likely to die during the duration of the study or are too frail to be interviewed. If inferences about population decline are made without accounting for them results are likely to be biased towards less observed decline
In this paper, we fitted a linear, a quadratic and two piecewise latent growth models with the aim of investigating further possible functional shapes to describe cognitive decline in older age. These latent growth models included a model to describe the missing data mechanism and were parameterised within the selection models framework defined by Little and Rubin where the joint distribution of the complete data and a model for the missing data are factorised, conditional on the complete data model
All analyses were carried out in Mplus, Version 4
Methods
The data
Data from the Cambridge City over 75 Cohort Study were used to perform the analysis proposed. The main aims of the study were to establish the incidence and prevalence of dementia. Initially, people aged at least 75 years old in 1985 who were registered at five primary care practices in the Cambridge City area were invited to participate in the Cambridge City over 75 Cohort study
This screening interview was followed by a more detailed clinical interview of all participants suspected of having dementia, identified by a cut point of 23 on the MMSE, and a third of those with scores of 24 and 25 points on the MMSE.
Further waves of interviews were carried out to establish incidence of dementia
Characteristics of individuals at baseline are shown in Table
Baseline characteristics of the sample analysed.
Variable
N (%)
Non manual profession
755 (38%)
Left school aged <= 14 years old
1582 (70%)
Female
1266 (63%)
Walks unaided around town or block
1582 (73%)
Married
773 (33%)
Mean ± st. dev. age at baseline
81 ± 5 yrs.
Mean ± st. dev. MMSE (median)
Baseline
24.5 ± 5.2 (26)
T2
23.0 ± 7.2 (25)
T7
24.1 ± 5.3 (25)
T9
26.3 ± 4.5 (25)
A small number of participants did not take part in some study waves but rejoined the study later. Data from them were only considered up to their first dropout to produce a non intermittent data set. All participants were observed at baseline, but 43.4, 24.5 and 16.6% of the sample dropped out for the first time at T2, T7 and T9 respectively. These include drop out due to death.
Analysis
MMSE scores from the first four interview waves of the study were analysed using latent growth analysis where the latent variables (mean curve parameters) were adjusted for risk factors for cognitive decline such as education, gender and age cohort at baseline.
Four models were fitted: a linear, a quadratic and two piecewise models with change points at T2 and T7 respectively. The parameterisation of the models was such that the models' mean intercept represent mean cognitive status at baseline and the mean slope, average rate of change in cognition between T0 and T2. In the quadratic model, the quadratic term represents the rate of change in the linear term. In the piecewise models, the slope of the second piece represents the change in the linear term of the first piece after the change point. Correlations between latent variables were estimated unless fixed to zero to assure model fit. Figure
Diagram representing the latent growth linear model fitted
Diagram representing the latent growth linear model fitted.
A missing at random mechanism
Numerical integration is needed to estimate maximum likelihood parameters in latent growth curve models with random effects and missing data. As standard fit indices are not appropriate for these types of models, in order to identify the model that best fits the data, a Bayesian Information Criteria (BIC) based selection was undertaken. As the BIC is a measure of fit that considers the likelihood with a penalisation for the number of parameters, the most parsimonious model would be the one with the lowest BIC and therefore, that model should be chosen as the one with best fit.
Sensitivity of models to normality assumption
Latent growth models require normality of residuals. On visual exploration of this assumption, some departure of normality was identified. To improve the normality of the data, MMSE scores were transformed using the transformation
Figure
The plot on the left shows the mean curve of scores obtained after applying the transformation
The plot on the left shows the mean curve of scores obtained after applying the transformation
The results obtained from fitting the models to raw data are presented furtehr for two main reasons. First, the main aim of our investigation was to show a range of parametric models that can be used to model cognitive change taking into account the missing data. And second, the interpretation of the back transformed models is complex when nonlinear or nonisometric transformations are applied in these models.
Results
Table
Mean estimates and standard errors of latent variables of growth models fitted
Model with explicit formulation of missing data model
Parameter
Linear
Quadratic
Piecewise with change point at T2
Piecewise with change point at T7
Intercept
25.4 (0.2)*
25.4 (0.2)*
25.4 (0.3)*
25.3 (0.3)*
Slope1
0.9 (0.1)*
1.14 (0.2)
1.13 (0.2)
0.8 (0.1)
Quadratic term

0.07 (0.02)


Slope2

0.2 (0.1)
0.4 (0.4)
* significant residual variance
In the linear and quadratic models, there was significant variability about the models' intercept and slope, an indication of heterogeneity of individuals about these parameters. The correlation between the intercept and slope in the linear model was estimated as 0.64 (st. error = 0.08), and in the quadratic, at 0.47 (1.79). The correlation of the quadratic term with the intercept was estimated at 0.11 (st. error = 0.07) in the quadratic model, and with the slope at 0.03 (st. error = 0.02). In the two piecewise models the variances of the two slopes were fixed at zero to achieve better model fit.
At all interviews, MMSE scores before dropout of those who dropped out were significantly lower than for those who remained in the study (ttests, df = 1957, level 5%, p < 0.01 for all interviews).
All models consistently showed those with lower previous scores and lower functional ability as more likely to miss an interview. At the second and third interviews, women were less likely to dropout. Those in the youngest cohort interview were also found to be more likely to be observed in the fourth interview.
BIC values of the four models fitted were estimated at 27956.1, 27890.1, 28217.2 and 28179.3 for the linear, quadratic and the piecewise models with change points at T2 and T7 respectively. These BIC values identified the quadratic model as the model with best fit.
The same models were run without an explicit formulation of a model for the missing data. In all models, estimated slopes indicated less decline. For instance, in the case of the linear model, the mean slope was estimated at 0.7 (st. error = 0.09), a value that suggests a smoother decline when compared to the slope estimated by the linear model formulated within the selection modelling framework (estimate = 0.9, st error = 0.1). Factor scores were also estimated at different values, in particular, individual specific slopes were consistently estimated at higher values than the slopes estimated by the model with a formultation of the missing data model.
One of the main advantages of our proposed models is that they can be easily extended to model a nonignorable missing data mechanism. This could be used to perform a sensitivity analysis of the missing data assumptions. We extended the work of Dufouil et al.
To illustrate the method, results obtained from the quadratic model are presented in Tables
Odds ratio (95% C.I.) missing data model results.
Interview
Risk Factor
Second
Third
Fourth
Previous MMSE
0.82 (0.79–0.85)
0.81 (0.78–0.84)
0.79 (0.80–0.85)
Education
0.85 (0.64–1.13)
0.99 (0.75–1.54)
0.84 (0.55–1.27)
Marital status
0.84 (0.65–1.08)
0.82 (0.59–1.54)
0.84 (0.56–1.24)
Social class
0.93 (0.72–1.20)
0.90 (0.63–1.27)
1.04 (0.70–1.55)
Functional ability
0.52 (0.38–0.71)
0.43 (0.26–0.68)
0.30 (0.19–0.62)
Gender
0.88 (0.69–1.10)
0.60 (0.46–0.86)
0.51 (0.33–0.77)
Younger cohort
0.73 (0.54–1.01)
0.43 (0.27–0.71)
0.27 (0.13–0.56)
Medium cohort
0.84 (0.61–1.14)
0.58 (0.35–0.94)
0.47 (0.22–1.05)
Odds ratio (95% C.I.) nonignorable missing data model results.
Interview
Risk Factor
Second (assumed MMSE_{2 }= 0.10)
Third (assumed MMSE_{7 }= 0.15)
Fourth (assumed MMSE_{9 }= 0.20)
Unobserved MMSE
0.90
0.86
0.81
Previous MMSE
0.87 (0.84–0.90)
0.90 (0.87–0.94)
0.84 (0.81–0.88)
Education
0.79 (0.62–0.99)
0.85 (0.66–1.10)
0.61 (0.39–0.96)
Marital status
0.83 (0.67–1.03)
0.81 (0.63–1.04)
0.77 (0.52–1.13)
Social class
0.89 (0.72–1.11)
0.82 (0.64–1.05)
0.88 (0.59–1.30)
Functional ability
0.55 (0.41–0.72)
0.46 (0.30–0.70)
0.34 (0.15–0.76)
Gender
0.79 (0.64–0.98)
0.55 (0.43–0.72)
0.51 (0.33–0.79)
Younger cohort
0.88 (0.66–1.18)
0.65 (0.43–0.98)
0.62 (0.26–1.48)
Medium cohort
0.94 (0.70–1.27)
0.72 (0.47–1.16)
0.93 (0.38–2.31)
Discussion
Overall, we propose that the use of latent growth models to investigate cognitive decline at the marginal and individual level provides researchers with good opportunities that represent advancement with respect to methods previously used in the field. Change scores are known to be severely affected by reliability issues
On the other hand, some regression based analysis are not suitable for the correlated measures that are observed in longitudinal data. Despite generalised estimating equations being more flexible in this regard, estimates of change at the individual level are not easily obtained. When mixed effects models are used, random effects are often not reported
Latent growth analysis is flexible enough to estimate the full trajectory of change at the mean level also producing individual's growth factors that permit the reconstruction of individual's trajectories that are of interest to clinicians
To improve the fulfilment of this assumption, we transformed scores and ran all models using the resulting scores. Results obtained from transformed data showed some improvement in the satisfaction of the normality assumption and identified the quadratic as the model that fits the data best in the transformed scale. However, thoughtful consideration needs to be exercised in the interpretation of backtransformed results as parameter estimates apply to the transformed scale not the original MMSE scale. As the purpose of the paper is the illustration of the method proposed, and the interpretation of results is natural in the original MMSE scale, we opted for the presentation of all other results in the MMSE scale.
Our analysis of cognitive decline in older persons over a 9 year longitudinal has identified the quadratic curve as the one that best fits the data. The quadratic model suggests an initial drop followed by a deceleration in the rate of decline expressed by a positive quadratic term. A relatively small annualised rate of decline was estimated (around 1.14 points) in our study, a finding that agrees with other studies. For instance, JacqminGadda
Attrition has severe effects on studies of older people. In the literature different levels of analysis of the missing data mechanism present in the samples can be found. For instance, in studies where change scores were used to model change
Instead, we have opted for the actual modelling of the mechanism using Diggle and Kenward's logistic regression model. This was important as by explicitly modelling the missing data, we were able to quantify the effect of the last observed score on probability of dropout at each interview and to fully account for the effect of previous MMSE scores within the full model likelihood. We believe that the explicit formulation of a model for the missing data mechanism is a novel practise that is necessary to fully inform the measurement model in order to provide an accurate description of the cognitive change experimented by older individuals.
Furthermore, it raises further opportunities for conducting sensitivity analyses of the models with respect to the missing data as shown.
Our results agree with previous studies in which a missing at random mechanism was present in the data. Yet, we found that women are less likely to drop out than men at later interviews, as opposed to the findings by Van Beijsterveldt et al.
Perhaps one of the disadvantages of our study is the large amount of missing data after the second interview. This is common in most studies of the older population and the fact that a missing data model was implemented should smooth its consequences regarding parameter estimates. Our models could benefit from the inclusion of further variables such as the person's cardiovascular disease history, genetic data and social network, for instance. They would not only inform the measurement model, but also the missing data model. Another key variable that would benefit the understanding of the process is the reason for dropout. If information was available about the cause of drop out, then it would be plausible to perform a competing risk analysis or a multiple group analysis discriminating by reason for drop out. This is under investigation at the moment.
Another possible limitation of the study is the fact that we considered a growth model based on 'time in study' rather than on the individual's age at each interview, an approach that facilitated the fit of the missing data model based on fixed times of observation. The discussion about the most suitable temporal matrix to describe cognitive change is contentious
Our research has proved that making parametric assumptions when investigating a process as complex as the loss of cognitive function as measured by the MMSE in older ages deserves thorough consideration.
Conclusion
Our study has showed that cognitive decline in older persons is not a linear process. The quadratic growth model, identified as the one with best fit, permitted the estimation of initial mean cognitive status, rate of decline and of its change. Estimates of person specific parameters were produced.
Significant heterogeneity between individuals in initial cognitive status and rate of decline was identified.
By fitting a latent growth model, we maximised the use of all data available to investigate cognitive change in older persons. Individuals with lower previous cognitive scores have been identified as more likely to drop out.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
GM ran the analysis and produced the manuscript, FM contributed with comments on analysis and manuscript and CB contributed with comments on the manuscript. CB is a principal investigator of the CC75C study. All authors read and approved the final manuscript.
Appendix 1
The latent linear growth model represented in Figure
where
λ_{0t }and λ_{1t }are factor loadings with λ_{0t }fixed to one, and
The missing data mechanism was modelled using a logistic regression model, where the probability of individual
where ν_{i1 }are model covariates and ε_{1τ }are residuals that follow a logistic distribution.
Appendix 2
Basic Mplus code for linear model:
TITLE:
DATA: (data file location)
VARIABLE: NAMES ARE (name declaration);
CLASSES = c(1);
USEVARIABLES = sc1 sc2 sc3 sc4 married sclass mob
educ miss2 miss3 miss4 women young interm;
MISSING = ALL(999);
CATEGORICAL ARE miss2 miss3 miss4;
ANALYSIS: TYPE = MIXTURE MISSING RANDOM;
algorithm = integration;
integration = montecarlo;
MODEL:
%OVERALL%
i s  sc1@0 sc2@2 sc3@7 sc4@9;
i s ON educ women young interm;
miss2 ON sc1 educ married sclass mob women young interm;
miss3 On sc2 educ married sclass mob women young interm;
miss4 On sc3 educ married sclass mob women young interm;
sc3; sc2;
%c#1%
i s ON educ women young interm;
OUTPUT: cinterval;
SAVEDATA:
RESULTS ARE (file location);
SAVE = FSCORES;
Acknowledgements
We thank all study participants, their relatives and carers for their invaluable help. We also thank all our past sponsors for financial support spanning two decades (project grants from the Charles Wolfson Charitable Trust, Medical Research Council, Leopold Muller Foundation, East Anglian Regional Health Authority, Anglia and Oxford Regional Health Authority, British Council Joint Research Programme in Progressive Degenerative Diseases, Edward Storey Foundation, Research into Ageing and Addenbrooke's Hospital Alzheimer's Disease Research Fund and fellowship awards including Wellcome Trust Prize PhD Studentship, Isaac Newton Trust Research Fellowship, Taiwan Government PhD Studentship, Anglia and Oxford Regional Health Authority Training Fellowship and the National Health Service Executive Research and Development Unit Health Services Research Fellowships) and are grateful to the BUPA Foundation for ensuring the study continues today. FM is fundeed by MRC. U. 1052.00.004
Prepublication history
The prepublication history for this paper can be accessed here: