Abstract
Studies indicate more rapid cognitive decline in patients with Lewy body dementia (LBD) compared to Alzheimer’s disease (AD). However, there has been less focus on any difference in the variability of cognitive decline. We assessed Mini-Mental State Examination (MMSE) test performance at baseline and annually for 5 years in 222 patients with mild dementia in the DemVest study who had either AD (137) or LBD (85). We used linear mixed models (LMMs) with random intercepts (variability in MMSE at baseline) and slopes (variability in MMSE decline), with years in study, age, gender, and diagnosis as independent variables. A non-linear (quadratic) trajectory was selected, interacting with age and diagnosis. To study differences in variance, we compared a regular LMM (i.e., a homoscedastic model), which assumes equal variance across groups, to a heteroscedastic model, assuming unequal intercept and slope variance based on diagnosis. The heteroscedastic model gave a better fit (Likelihood ratio test: χ2 = 30.3, p < 0.001), showing overall more variability in LBD. Further, the differences in intercept and slope variances were tested using a modified Wald test. The MMSE intercept variance (AD: 2.78, LBD: 7.75, difference: 4.97, p = 0.021) and slope variance (AD: 2.62, LBD 7.81, difference: 5.19, p = 0.004) were both higher in LBD. In conclusion, patients with LBD in the DemVest study have a higher variability in MMSE scores at study inclusion, and in MMSE decline compared to AD. Accordingly, clinical trials on LBD may need a larger sample size compared to AD.
Keywords
INTRODUCTION
Dementia is characterized by progressive cognitive decline, but there is huge variation in the rate of decline [1–3]. For example, patients with dementia with Lewy bodies (DLB) have more rapid cognitive decline compared to patients with Alzheimer’s disease (AD) [4–7]. This represents a potential testimony to more complex underlying DLB-pathology where α-synuclein and AD-pathology frequently co-occur [8]. In AD, there is considerable heterogeneity in cognitive trajectories, meaning that patients represent a mixed population who undergo both slow and rapid cognitive decline [9]. A longitudinal community based clinical-pathological study on 872 autopsied subjects found that neocortical Lewy bodies were associated with an increased risk of dementia and more rapid cognitive decline, independent of AD pathology. Further, neocortical Lewy bodies were associated with more fluctuations in individual cognitive trajectories, as estimated by an increased residual variance [10]. Another community-based autopsy study identified that patients with neuropathological AD had 70% more yearly variability in cognition compared to participants without AD pathology, and patients with neocortical Lewy bodies had almost twice the yearly variability [11]. DLB shares significant neuropathological, neurochemical, and clinical features with Parkinson’s disease dementia (PDD), including the dominant Lewy body pathology and the two are often combined as Lewy body dementia (LBD) [12]. In addition, there is considerable variation within these diseases; for example, DLB patients with low cerebrospinal fluid Aβ42 have a more rapid decline than those with normal cerebrospinal fluid values [13]. Knowing the variability in rates of decline is important for patient management, planning of care, and for planning clinical trials.
The estimation of cognitive decline in longitudinal studies is frequently performed using linear mixed-effects models (LMMs) on cognitive outcomes measured at specified times, for example, annual Mini-Mental State Examinations (MMSE). In addition to estimating the mean cognitive performance and rate of change during the study period using fixed effects, LMMs also estimate the variance by random effects which hierarchical structure must be determined by the investigator. Most clinical longitudinal studies will apply a straightforward model of the variance by specifying the random effects as the subject-specific intercept and the subject-specific slope. The remaining unexplained variance represents the residual variance. For the purposes of this study, we will refer to the random intercepts, slopes, and residual variance as variance components that make up the total variance. LMMs assumes homogeneity of variance, i.e., that the two groups compared have the same variance components. This is a relatively strong assumption to make about diseases with different underlying pathologies, as it pertains to both intercept, slope, and residual variance (for reviews of LMMs, see [14–16]). To our knowledge, the variance components in patients with a clinical diagnosis of AD and LBD have not been compared in longitudinal studies. This is relevant for sample size estimation for clinical trials, as variance components partly determine the statistical power in LMMs and thus sample size.
We therefore aimed to investigate differences in overall variance and variance components pertaining to cognitive trajectories over five years in patients with AD and LBD. Given the more pronounced pathological heterogeneity in LBD, we hypothesized that the variance would be greater in LBD compared to AD.
METHODS
Study participants
The dementia study of Western Norway (DemVest) is a longitudinal cohort study of patients with mild dementia. Briefly, patients with mild dementia, defined as a MMSE of at least 20 and a Clinical Dementia Rating scale (CDR) of at least one, were included. Dementia was diagnosed at study inclusion according to the Diagnostic and Statistical Manual of Mental Disorders (Fourth Edition: DSM-IV) criteria based on structured interviews, clinical examination, standardized neuropsychological tests and a neuropsychiatric assessment. Routine blood work, including assessment of thyroid function and Vitamin B12 status was performed. Structural magnetic resonance imaging of the brain was used to rule out other causes of dementia such as tumors. In a subgroup of patients with suspected DLB (n = 41), the diagnosis of DLB was supported by dopamine transport imaging (DAT scan). Exclusion criteria were no dementia, moderate or severe dementia, acute delirium, previous bipolar or psychotic disorder, terminal illness, or major somatic illness. AD was diagnosed according to the National Institute of Neurological and Communicative Disorders and Stroke and the Alzheimer’s Disease and Related Disorders Association (NINCDS-ADRDA) criteria. DLB was diagnosed according to the revised criteria from 2005 [17]. Further details of the study procedures have been described previously [18]. There were 137 participants with AD and 85 with LBD (69 DLB and 16 PDD). MMSE and CDR were scored at baseline and annually. In this study data from the first 5 consecutive years were used. This was done to avoid issues related to statistical estimation in the presence of floor-effects and a high mortality. From this cohort, 56 patients underwent autopsy. The sensitivity of a clinical diagnosis of DLB was 73% and the specificity was 93% using the neuropathological diagnosis as the gold standard. For LBD, the corresponding sensitivity and specificity was 80% and 92%. The sensitivity of a clinical diagnosis of AD was 81% and the specificity 88% [19].
Ethics
The Regional Committee for Medical and Health Research Ethics approved the study, and a later notification of change detailing the use of biomarkers (REC number: 2010/633). The participants provided written consent after the study had been explained in detail, in the presence of a caregiver.
Background on linear mixed-effects models
LMMs applied to repeated measures over time are often referred to as growth models. A basic model without predictors typically includes a fixed effect for time (linear or non-linear) with random intercepts and random slopes. The variance is not modelled as one, as is typically done in other models, but is partitioned into different piles of variance, typically random intercepts, random slopes and residual variance, and a covariance matrix needs to be specified by the investigator. Random intercepts allow the outcome to be higher and lower for each individual compared to the mean (i.e., the fixed intercept) level of the outcome, and are reported from baseline if this is zero on the time scale. However, in a random intercept model with a fixed effect of time, all participants are assumed to change at the same rate in the outcome variable (i.e., all have the same MMSE decline rates). This is most often unrealistic. Random slopes allow the outcome to change at different rates in each individual as a function of time (i.e., different MMSE decline rates) relative to the mean change. Notably, both random intercepts and slopes represent differences between persons whereas the residual variance represents changes within-persons. In a model with intercepts and slopes, this will typically pertain to individual fluctuations around their predicted trajectories according to their intercept and slope. The covariance between the intercepts and slopes can be positive, meaning for example that patients with lower initial MMSE scores tend to also have steeper slopes, or negative, indicating the opposite. Figure 1 illustrates some of these concepts. Please see Lesa Hoffman’s book for a review of LMMs [14].

Background on linear mixed effects models and variance components. Theoretical MMSE trajectories for four patients over five years illustrate variance components and covariance. All include a fixed, linear effect of time (indicated by “mean”) and random effects. A) A random intercept model where there is only deviance from the mean in the intercepts, random slopes being equal for all four patients. That there are no fluctuations, giving rise to no residual variance. B) Given that the time variable is set to zero at baseline, the random intercepts are equal and have no variance, whereas the four patients have different MMSE decline rates (i.e., random slope variance). There are no deviations from a linear trajectory, and no residual variance. C) The same trajectories as in A where the slopes fluctuate around a linear trajectory (gray lines). A random intercept model cannot explain these fluctuations, giving rise to residual variance. D-F) All pertain to random intercept and slope covariance. In D, patients who start off with lower MMSE scores (random intercepts), decline faster (random slopes), giving rise to a positive intercept-slope covariance. The opposite occurs in E, where the patients with higher initial MMSE scores decline faster, giving rise to a negative intercept-slope covariance. Finally, there is no discernable relationship between intercepts and slopes in F, giving rise to zero intercept-slope covariance. MMSE, Mini-Mental State Examination; Pt., patient.
Statistics
Descriptive statistics are reported as percentages, means, and standard deviations. MMSE performance over 5 years was assessed using linear mixed-effects models with random intercepts and slopes (random coefficient models). To avoid a high degree of model complexity, we included age and gender as covariates. Using LMM, a linear or non-linear term for time was selected based on the Bayesian Information Criterion (BIC) as misspecification of the fixed effect of time could introduce spurious variability in the random slopes. A simple random coefficient model with MMSE as the outcome and time as the predictor showed BIC improvement of –26 using a quadratic model of time compared to a linear model. Importantly for this study, the quadratic model gave a better fit compared to a linear model in both AD (BIC –16) and LBD (BIC –10). Thus, we proceeded using a quadratic effect for the fixed effect of time. A model with age, gender, LBD, time (years in study), quadratic time, and an interaction between LBD and linear time is thus used in the fixed part of a linear mixed-effects model. Further, we selected an unstructured variance-covariance matrix based on BIC and we did not include an interaction between LBD and quadratic time due to poorer BIC and lack of significance. Following the multilevel mixed-effects reference manual for Stata 13 [20], the aforementioned standard model where time (slopes) are nested in individuals (intercepts) was used as the homoscedastic model of reference using the Stata function “xtmixed”. A second identical model which allowed intercepts and slopes to vary according to diagnosis was used to model whether diagnosis was a source of heteroscedasticity, or difference in the random effects. The models were compared using the likelihood ratio test of nested models, Akaike’s Information Criteria (AIC) and BIC. The model with the lowest AIC and BIC indicates a better fit to the data, taking into account a penalty for more complex models. Using the obtained estimates from the random effects, the estimated variance and its significance was calculated for intercepts and slopes according to the presence of AD or LBD. Finally, to test our hypothesis, the differences in intercept and slope variance between LBD and AD were calculated using the approach described by UCLA: Statistical Consulting group, using a modified Wald’s test [21, 22]. Two sensitivity analyses were performed comparing the described homoscedastic and heteroscedastic models. The first with AD versus patients with DLB (PDD excluded) and the second using linear time instead of quadratic time. The analyses were conducted in Stata 15 (StataCorp. 2017. Stata Statistical Software: Release 15. StataCorp LLC, College Station, TX). The obtained intercept, slope, and residual variance were entered in a power analysis to show differences for a potential clinical trial had our study been a pilot for a trial on patients with AD or LBD. Further, a simple example of the relationship between variance components and statistical power is provided based on a worked example using the method proposed by Liu and Liang on a hypothetical clinical trial using the expected distribution of the Alzheimer’s Disease Assessment Scale- cognitive subscale (ADAS-cog) as the outcome, which is typically used in clinical trials [23]. We used the R package “longpower” for power calculations.
RESULTS
Study participants
Table 1 summarizes the study participants. The LBD group (n = 85) included more males, but the two groups did not differ in age or baseline MMSE score. The standard deviation of MMSE is higher in LBD patients, showing that there was somewhat more cognitive heterogeneity in these patients at baseline.
Baseline characteristics
AD, Alzheimer’s disease; DLB, dementia with Lewy bodies; LBD, Lewy body dementia; m, mean; PDD, Parkinson’s disease dementia; SD, standard deviation.
Overall higher variation in MMSE scores in LBD
As previously reported, patients with LBD have more MMSE decline over 5 years, seen in both the homoscedastic and heteroscedastic models (Table 2). AIC (–24.3) and BIC (–9.6) were lower in the heteroscedastic model compared to the homoscedastic model, indicating that the heteroscedastic model gave a better fit to the data than the homoscedastic model. This was further supported by a highly significant likelihood ratio test (χ2 30.3, p < 0.001) comparing the models. Patients with LBD had larger random intercepts and slopes. The intercept-slope covariance is also higher in LBD, indicating that the trajectories “fan out” more in LBD over time. The results are illustrated in Fig. 2, and a figure demonstrating the underlying raw MMSE data is available in Supplementary Figure 1. The models were also tested for differences in residual variance, or random effect, which was significantly higher in LBD using the homoscedastic model. However, this was reduced to non-significant in the heteroscedastic model, suggesting that the differences in residual variance were explained by differences in intercept and slope variance (data not shown).
MMSE and random effects over 5 years in Lewy body dementia versus Alzheimer’s disease (N = 222)
AIC, Akaike’s Information Criterion; AD, Alzheimer’s disease; BIC, Bayesian Information Criterion; 95% CI, 95% confidence interval; Est., estimate; LBD, Lewy body dementia; LR test, likelihood ratio test (nested models); MMSE, Mini-Mental State Examination; time, years in study. *p < 0.05, **p < 0.001.

Variability in MMSE trajectories. Each line represents the predicted MMSE trajectory of one patient, estimated from both the fixed and random effects of the heteroscedastic model.
Higher intercept and slope variance in LBD
The above model comparison does not directly compared intercepts in AD versus LBD, and slopes in AD versus LBD, but assesses overall heteroscedasticity. Thus, we estimated the variance in post-hoc analyses and found a significantly higher variance in both intercepts and slopes in LBD (intercept: 7.75, slope: 7.81) compared to AD (intercept: 2.71, slope: 2.61). The difference in the variance of the intercepts was 4.97, p = 0.021 and in slopes 5.19, p = 0.004 (please see Table 4 for a summary of the variances).
Sensitivity analyses
We performed an identical comparison of AD and DLB (PDD excluded). The results were highly consistent with the overall model (Table 3), and we similarly identified a better fit with the heteroscedastic model (AIC –25, BIC –11, likelihood ratio test: χ2 31.1, p < 0.001) indicating significant differences in variance also between DLB and AD. In DLB, the intercept (8.97) and slope variance (8.35) were higher than in AD (intercept 2.71, slope 2.61), with a difference of 6.27 in intercepts (p = 0.015) and 5.75 in slopes (p = 0.006), see Table 4. Further, we could also reproduce the findings using a linear time slope, as opposed to the better-fitting quadratic slope (Supplementary Table 1).
MMSE and random effects over 5 years in dementia with Lewy bodies versus Alzheimer’s disease (N = 206)
AIC, Akaike’s Information Criterion; AD, Alzheimer’s disease; BIC, Bayesian Information Criterion; 95% CI, 95% confidence interval; DLB, dementia with Lewy bodies; Est., estimate; LR test, likelihood ratio test (nested models); MMSE, Mini-Mental State Examination; time, years in study. * p < 0.05, ** p < 0.001.
Differences in intercept and slope variance between AD and LBD
Note: Variance of the random effects, not considering the residual variances. The significance of the variance and the hypothesis of differences in variance were both estimated using a modified Wald test. AD, Alzheimer’s disease; Coef., coefficient; DLB, dementia with Lewy bodies; LBD, Lewy body dementia; SE, standard error. *p < 0.05, **p < 0.001.
Variance and statistical power
The data from this study were entered as parameters in a power analysis. Given the intercept, slope, and residual variance, and intercept-slope covariance in this study, we set the premises to a 5-year study with 80% power, α= 0.05 in a two-sided test and two groups of 100 patients in each group. The minimal detectable change in MMSE points per year was 1.14 in LBD and 0.69 in AD. This represents a 65% increase in the minimally detectable difference between groups due to differences in slope-variance. However, estimates for power calculations will need to be derived from studies of shorter duration with shorter between-test intervals, using cognitive testing typically applied in clinical trials. Given a more realistic 18-month clinical trial scenario with 3-month follow up intervals, Fig. 3 demonstrates that higher slope variance results in a loss of power and the need for more participants to detect the same differences. The data were generated from a priori power analyses using theoretical data with ADAS-Cog, a test often used in clinical trials, as the outcome.

Slope and residual variance affect the statistical power of LMMs. Sample size refers to the total sample size divided over two equally large group which is necessary to detect a 1.5-point change on the ADAS-Cog scale. Increasing slope and residual variance necessitates a larger sample size to detect the same difference. Adapted from a worked example of ADAS-Cog, using the estimated intercept variance (55), slope variance (24), and residual variance (10) from the Alzheimer’s Disease Neuroimaging Initiative study. In the power calculation, the study design is assumed to be seven measurements at baseline and every three months over 18 months. The intercept-slope covariance is set to 0.8, power to 0.8 and α to 0.05. The correlation matrix is unstructured. ADAS-Cog, Alzheimer’s disease assessment scale, cognition (often used in clinical trials); LMMs, linear mixed-effects models.
DISCUSSION
We found that the LBD group had higher variance in MMSE test scores in both intercepts and slopes, i.e., the cognitive decline trajectories have a higher variability from the average decline in LBD patients compared to the AD group. Graphically, this looks as if the slopes fan out more from baseline, compared to AD. The differences in average decline persisted after adjusting for the increased variability in LBD. The same differences in the variability of cognitive decline trajectories were observed when only patients with DLB were compared to AD.
In our study, the heteroscedastic model provided a significantly better fit to the data than the homoscedastic model, supporting a higher overall variance in LBD compared to AD, where further analyses using a modified Wald test showed differences in both intercept and slope variance. The biological reasons for this difference cannot be ascertained in our study. One might speculate that this relates to the presence of multiple protein pathologies in LBD; α-synuclein, amyloid-β, and tau [24]. Concomitant TDP-43 pathology is also more frequent in patients with LBD as compared to AD [25]. From a clinical point of view, it is possible that the typical cognitive fluctuations observed in LBD may contribute to increased variability in cognitive performance. LBD patients have significantly more neuropsychiatric symptoms than AD patients [26], which may also contribute to the higher variability in cognitive trajectories in LBD. We did not assess explanatory variables of the differences in variance components related to diagnosis in this study as it would require complex statistical models which should preferably be used in larger studies.
LMMs are generally robust to misspecification of the covariance matrix and to heteroscedasticity. However, heteroscedasticity can be a source of bias if the heteroscedasticity is predicted by a covariate which also interacts with time in the model [27]. Thus, if the aim is to assess difference in rates of cognitive decline, the presence of heteroscedasticity should be checked and, if present, accounted for. Bias is also more likely to occur if the group with the higher variance also has a higher drop-out rate [28], which was also the case in this study due to higher mortality among LBD patients [7]. The estimated difference in MMSE decline per year between LBD and AD was higher in the heteroscedastic model (–0.96) as compared to the homoscedastic model (–0.81). However, these differences were minor and overall the fixed effect estimates in the homoscedastic model shows a relative robustness to the presence of heteroscedasticity in this study.
The increased slope-variance in LBD as compared to AD will result in relatively lower power to detect factors affecting cognitive decline in patients with LBD. Such information is key to properly calculate statistical power for clinical trials, as illustrated here by a general example of a simple power calculation which shows a linear increase in sample size as a function of increasing slope variance. Based on our study, we would expect a 65% increase in the minimally detectable effect size for decline per year in LBD compared to AD trials. However, trials are typically conducted with more frequent follow-up over shorter time periods with different cognitive tests than the MMSE that typically have a wider scoring range. Our power calculations did not take into account the increased mortality in LBD, which will further reduce statistical power, at least in studies of longer duration. However, due to the complexity of the individual time-to-time variation and drop-out in real studies, power calculations are likely best simulated based on existing data, preferentially with follow-up schedules and cognitive tests that are similar to clinical trials. In addition to an impact on statistical power, our findings highlight the need to identify clinical and biomarker predictors of the varied cognitive trajectories in LBD, which will guide management planning, and help understand the mechanistic underpinnings of the variation in clinical course.
There are limitations to our study. In particular, testing the significance of differences in random effects using a modified Wald test might have some weaknesses at this sample size [29]. The patients with LBD had a higher variance at study inclusion, which could represent a form of selection bias of patients which were more heterogenous early on. However, the LMM also adjusted for the differences in intercepts, as well as the covariance between intercepts and slopes which could have remedied this potential source of bias. Furthermore, the number of patients with LBD is relatively low, and thus replication in a larger sample would be very useful. Cognitive decline was based on MMSE only. Although MMSE is generally considered less sensitive than other screening scales to the cognitive impairment in early DLB due to the dominant memory and language items, the sensitivity to cognitive decline has been found to be similar to scales with more extensive executive and visuospatial items such as the Montreal Cognitive Assessment [30]. Finally, the diagnosis was mainly clinical, although the clinical diagnosis was confirmed in more than 80% of the cases in a subgroup coming to autopsy [19].
In conclusion, there is an increase in MMSE variance in LBD in our study, resulting in cognitive trajectories which fan out over time. This needs to be considered by investigators planning clinical trials in LBD.
Footnotes
ACKNOWLEDGMENTS
We want to thank all the participants, researchers, and technical staff that have made the DemVest study possible. This paper represents independent research partly funded by the National Institute for Health Research (NIHR) Biomedical Research Centre at South London and Maudsley NHS Foundation Trust and King’s College London. The views expressed are those of the author(s) and not necessarily those of the NHS, the NIHR or the Department of Health and Social Care.
