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Forecasting dementia incidence
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\noindentKeywords: dementia incidence, time trends, forecasting
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More than 55 million people worldwide live with dementia, and this number is projected to increase over time (nichols2022estimation). Dementia is a multidimensional challenge with major implications for affected individuals, their families, social policy, and national economies. In England and Wales, the number of people living with dementia is predicted to increase substantially in the near future ahmadi2017temporal, causing significant increases in health and social care costs collins2022will,banks2025long. These forecasts, however, are sensitive to future trends in dementia incidence wolters2020twenty. Moreover, forecasts of trends in dementia incidence are sensitive to modeling assumptions CHEN2023e859. If the dementia incidence trend changes, the future burden of dementia and associated costs might differ substantially from current forecasts. Therefore, credible estimates of the dementia incidence trend and forecasts of its likely trajectory are important in shaping social care policy.
Nationally representative data that use consistent case definitions of dementia over time only exist since about 2000, affording only short sample periods to conduct inference. Consequently, epidemiological studies to date have extrapolated the observed dementia incidence rate trend without considering time-series uncertainty ahmadi2017temporal,CHEN2023e859,wolters2020twenty. We address this gap in methodology by formulating statistical methods that account for samples that are short in the time dimension and large in the number of cross sectional observations.
We estimate the model in two steps, using recent advances in estimation of multi-state models with many parameters and contribute our own time series model. Via the multi-state model, we estimate a time series of dementia incidence based on CHEN2023e859 which accounts for (i) individuals' dropping out (censoring) due to death which can lead to underestimating dementia incidence if censoring is not addressed, and (ii) potential misclassification of dementia. In a second step, we use the estimated incidence rate and its sampling uncertainty as inputs into a Kalman filter to estimate mean and variance of a dementia process free of cross-sectional noise.
This procedure allows us to recover and model a stochastic process for population level dementia incidence, while also addressing the aforementioned problems of dementia measurement and censoring. We incorporate this sampling uncertainty in dementia incidence by treating it as observational noise in a Kalman filter. Using this setup, we can compare different time series models of dementia incidence and use our preferred model for forecasting.
Our two-step estimation methodology extends to all contexts where some parameters in the main model are a time series that is estimated, rather than known with certainty, which arises frequently in models for panel data.
We estimate these trends in dementia incidence in England using data from the English Longitudinal Study of Aging over the 2002-2018 period. While we find some evidence that incidence fell in the early part of our sample period, there is little evidence of a long-term trend in dementia incidence. Our preferred model is a random walk with zero drift, meaning that while dementia incidence changes over time, the optimal forecast is its most recent value. We also produce confidence intervals for these dementia forecasts and find that we cannot reject significant increases or declines in incidence over the next decade.
To account for the short time dimension, we present two lines of argument to establish that the proposed Kalman filter performs well in small samples. First, we study the convergence properties in terms of the Kalman gain factor. Second, we study the sensitivity or power of the filter to detect changes in the dementia trend and its false positive rate as a function of the signal-to-noise ratio and the number of available time periods. We learn that even with our short sample, the filter's properties stabilize rapidly enough to warrant inference on the direction of the dementia trend.
The advantage of our procedure is that we are able to use the most recent developments in multi-state modeling, including misclassification models and interval-censoring, and add a time series method to this framework by modeling the underlying time trend that shifts the dementia hazard. The proposed methodology is very easy to implement and can be applied in any context where units move between different states over time.
aalen2008survival suggest joint modeling and estimation of the stochastic time trend along with demographic variables. However, they note that “{[}a{]} lot of work remains to be done on how best to implement the models.” Some studies engage in this joint modeling gjessinghort,timedepfrailty,unkel2014time,dynamicfrailty,ragni2025timedepfrail, but use simpler empirical frameworks than what we use. In contrast, our two-step method trades statistical efficiency for computational feasibility. Our methodology is deliberately kept simple and has potential for wide-spread adoption.
davis2016handbook,davis2021count suggest joint modeling and estimation of the stochastic time trend along with demographic variables using count time series models. These models perform well when the time series is long. In contrast, our approach works well when the number of cross-sectional units is large but there is only a modest time series dimension.
We used data from the English Longitudinal Study of Ageing (ELSA), a longitudinal panel study of a representative sample of people aged 50 years or more living in private households in England. We use the ELSA data spanning 17 years across wave 1 (2002-03) to wave 9 (2018-19), allowing us to measure transitions into dementia and death over eight two-year intervals and a total of 70,806 individual health transitions. Mortality data were linked to participants who had provided written consent for linkage to official records from the National Health Service central register.
We use the measure of dementia from ahmadi2017temporal, which uses an algorithmic case definition based on coexistence of both cognitive and functional impairment, or a report of a doctor's diagnosis of dementia by the participant or caregiver. Cognitive impairment is defined as impairment in two or more domains of cognitive function (orientation to time, immediate and delayed memory and verbal fluency). These measures are available for all nine waves except verbal fluency at wave six, for which we impute using information from waves five and seven. For individuals unable to take the cognitive function tests, the Informant Questionnaire on Cognitive Decline was administered to a proxy informant (usually an immediate family member), and a score higher than 3·6 is used to identify cognitive impairment. Functional impairment is defined as an inability to carry out one or more activities of daily living independently, which includes getting into or out of bed, walking across a room, bathing or showering, using the toilet, dressing, cutting food, and eating. This case definition is less likely to be affected by changes in diagnostic criteria and clinical practice over time. Appendix (ref) provides further details of the measurement or dementia and the sample we use.
We estimate the time series process of dementia incidence using a two step estimator. In the first step, we estimate dementia incidence as a function of demographics and time indicator variables. In the second step, we develop models of the underlying time series process given by the coefficients of the time indicators and their variance estimates.
We model mortality and dementia incidence jointly using a multi-state model cox1965theory,jackson2011multi where demographics and time affect transitions. The state space $S=\left\{ 1,2,3\right\} $, where the transition of interest is from no-dementia $\left(1\right)$ to having dementia $\left(2\right)$ while also allowing transitions to death $(3)$. Figure (ref) shows the state space.
Individuals may develop dementia and die between survey waves leading to interval censoring. Failure to account for censoring will underestimate dementia incidence rates since those succumbing to dementia between waves are more likely to die and thus drop out of the sample between waves. Even if dementia incidence is constant, changing mortality rates can produce spurious upward or downward trends if unmodeled. We account for censoring by following leffondre2013interval,binder2014missing,CHEN2023e859 and model mortality and health transitions explicitly.
The transition intensity or hazard from state $r$ to $s$ at time $t$ is the instantaneous probability
Transition-specific intensities depend on $t$ and the demographic variables age and sex $(\vect z_{i}\left(t\right),t)$ via
The parameters $q_{rs0}$ are baseline hazards and $g_{rs}\left(t\right)$ is a hazard-shifting time series. Time dependence of the transition intensity obtains exclusively via the time-dependent covariates. Equation ((ref)) does not include interactions between demographics and time; CHEN2023e859 test for these interactions and found them to be insignificant.
Time is continuous although mortality and dementia are observed only at discrete times. We have nine waves, giving us transitions for each individual at points $t=1,\dots,8$ . Using wave indicator variables, the process driving transitions from no dementia to dementia is $g_{12}\left(t\right)$ :
whereas the other transitions $g_{13}\left(t\right)$ and $g_{23}\left(t\right)$ are modeled as linear time trends. Tables (ref) and (ref) in Appendix (ref) detail the functions $f_{rs}\left(.\right)$ and $g_{rs}(t)$. The object of our study is $g_{12}(t)$, which is why we model it with maximal flexibility; the functions $g_{13}(t)$ and $g_{23}(t)$ are more parsimonious. Appendix (ref) discusses $q_{rs}$ and resulting probabilities in matrix notation. For individual $i$ observed at time $t_{ik-1}$, the entries of $P\left(w,\vect z_{i}\left(t\right),t\right)$ become
Equation ((ref)) shows the probability of transitioning from $s$ to $r$ between $t_{ik-1}$ and $t_{ik}$, and $S_{t_{ik}}\in S$ is individual $i$'s health state at time $t_{ik}$ for $k=1,\dots,m_{i}$ and $m_{i}$ is the number of $i$'s observed transitions.\footnote{Based on eight two-year waves of follow-up, if an individual is observed throughout the entire study period, they will contribute maximally eight observations.} jackson2011multi contains explicit expressions of $p_{rs}\left(\vect z_{i}\left(t_{ik-1}\right),t_{ik-1}\right)$ in terms of $q_{rs}\left(\vect z_{i}\left(t_{ik-1}\right),t_{ik-1}\right)$, which shows that the relationship between the two quantities accounts for unobserved transitions between waves. For example, it accounts for the fact that between any two waves an individual may develop dementia and then die, contributing to both dementia and death probabilities, although we would only observe the latter.
We now write the individual contribution to the likelihood in terms of transition probabilities for individual $i$, who is observed at times $t_{i1},...t_{im_{i}}$. The likelihood for one individual obtains from multiplying together individual transition probabilities along all possible transitions a person can undergo. The joint distribution for health states in all periods for an individual over the sample period, conditional on the initial state is
where we have suppressed dependence on $(\vect z_{i},w)$ for readability. The right-hand side of ((ref)) is a series of the conditional probabilities shown in ((ref)).
Our definition of dementia uses an algorithm as well as a doctor diagnosis to reduce bias. Inevitably, there is misclassification, which we account for via adjusting the likelihood as follows. Define measured health $S_{t_{ij}}^{*}$ as different from actual health. By assumption, misclassification probabilities at time $t_{ij}$ (the $j$th period $i$ is observed) only depend on the immediately preceding state, i.e. $\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}|S_{t_{i1}},\dots,S_{t_{ij}}\}=\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}|S_{t_{ij}}\}$. The probability of being wrongly classified as having dementia is $\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}=2|S_{t_{ij}}=1\}$ and the converse probability is $\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}=1|S_{t_{ij}}=2\}$. Death is measured with certainty so that $\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}=3|S_{t_{ij}}=3\}=1$ and $\ensuremath{\mathbb{P}}\{S_{t_{ij}}^{*}=3|S_{t_{ij}}\in\{1,2\}\}=0$. Parameter estimates of the transition intensity matrix (ref) in Appendix (ref), the misclassification matrix in Table (ref) in Appendix (ref), and of the initial state distributions are estimated by maximizing the log-likelihood based on jackson2002hidden. Appendix (ref) derives an expression for the full log-likelihood.
The relevant sample size obtains from the number of transitions for each $i$, $m_{i}-1$, so that $I$ individuals a total of $\sum_{i=1}^{I}(m_{i}-1)=n$ individual-transition observations. For each $i$, $L_{i}$ (ref) contributes $m_{i}-1$ transitions. Maximizing $\ell(\gamma)=\sum_{i=1}^{I}\log L_{i}$ yields vector $\gamma$ which includes the parameters of the expressions in ((ref)): the dummy variables for the time trend for dementia incidence $\{\{\beta_{k}\}_{k=1}^{T}\}$ as defined in ((ref)), the time trends for mortality as part of $g_{13}(.)$ and $g_{23}(.)$, and the parameters of the function $f_{rs}(.)$, which capture the role of demographics. The full vector satisfies $\sqrt{n}\left(\hat{\gamma}-\gamma\right)\overset{\text{a}}{\ensuremath{\sim}}N\left(0,\Sigma\right)$ under regularity conditions vandervaart1998. Define the portion of $\Sigma$ holding covariances of the estimates $\{\hat{\beta}_{k}\}_{1}^{T}$ as $\Sigma_{TT}$. Normality implies that $\hat{\beta}_{k}$ can be written as
where $\left\{ \varepsilon_{k}\right\} _{1}^{T}\overset{\text{a}}{\ensuremath{\sim}}N\left(0,\Sigma_{TT}\right)$ and $\varepsilon_{k}\overset{\text{a}}{\ensuremath{\sim}}N\left(0,\sigma_{kk}^{2}\right)$ and (ref) implies that \[ \operatorname{var}\left(\{\hat{\beta}_{k}\}_{k=1}^{T}|\left\{ \beta_{k}\right\} _{k=1}^{T}\right)=\Sigma_{TT}, \] i.e. conditional on knowing the true realization of the dementia process $\left\{ \beta_{k}\right\} _{k=1}^{T}$, all variability in $\{\hat{\beta}_{k}\}_{1}^{T}$ is due to cross-sectional sampling uncertainty. This insight is critical for the estimation of the time series models below.
This section uses the estimated dementia incidence trend $\{\hat{\beta}_{k}\}_{k=1}^{T}$ and its variance $\hat{\Sigma}_{TT}$ to estimate the underlying stochastic process for $\{\beta_{k}\}_{k=1}^{T}$. For our main specification, we model the process for $\{\beta_{k}\}_{k=1}^{T}$ as a random walk:
where the per period shock $\eta_{k}\ensuremath{\sim} N(0,\sigma_{\eta}^{2})$. Our goals in this section are to recover the time series $\{\beta_{k}\}$ and the shock variance $\sigma_{\eta}^{2}$. However, what we have is $\{\hat{\beta}_{k}\}_{k=1}^{T}$ and the time-varying measurement variance $\sigma_{\varepsilon}^{2}\left(k\right)\equiv\sigma_{kk}^{2}.$
The Kalman filter is a popular tool for estimating the time series properties of a variable that is measured with error harvey1990forecasting. The error-prone measurement in our case is the sequence of estimates $\hat{\beta}_{k}$ from which we estimate the stochastic process $\beta_{k}$ and its dynamic variance; measurement error in $\hat{\beta}_{k}$ arises from cross-sectional sampling uncertainty. Different than most filters where the measurement error variance is estimated jointly with remaining model parameters, we obtain measurement error variances in the first stage. In the second stage, variances $\sigma_{kk}^{2}$ are given, which is why we refer to our filter as “constrained.”
Using information until time $k-1$, the forecast $\beta_{k|k-1}$ is called the prior estimate at time $k$ and is available before $\hat{\beta}_{k}$ is. Once it is, the update $\beta_{k|k}$ is called the posterior estimate of $\beta_{k}$ at time $k$.
Given the random walk assumption, the formula of the Kalman filter for any period $k$ consists of the forecasting equations for $\beta$ and its variance:
The Kalman gain is
As the next $\hat{\beta}_{k}$ becomes available, the forecast error $v_{k}\equiv\hat{\beta}_{k}-\beta_{k|k-1}$ wherefrom the posterior $\beta_{k\mid k}$ and its variance obtain via the updating equations
for the posterior and $P_{k|k}=\left(1-K_{k}\right)P_{k|k-1}$ for the variance. The cross-sectional sampling uncertainty $\hat{\sigma}_{kk}^{2}$ enters the filter via the gain in ((ref)). A large $\hat{\sigma}_{kk}^{2}$ decreases $K_{k}$ so that the model tends to attach more weight to its prior $\beta_{k|k-1}$ in ((ref)) and less weight to the correction suggested by the observation $\hat{\beta}_{k}$. This downweighting is intuitive since a large cross-sectional sampling uncertainty implies the forecast error likely reflects this uncertainty and is thus decreased. Conversely, a small cross-sectional uncertainty $\sigma_{kk}^{2}$ implies a Kalman gain tending towards unity so that the observation $\hat{\beta}_{k}$ receives more weight than the prior $\beta_{k\mid k-1}$.
The solution to the Kalman filter depends on the parameters $\hat{\sigma}_{kk}^{2}$ and $\sigma_{\eta}^{2}$ with $\hat{\sigma}_{kk}^{2},k\in\{1,...,T\}$ given as explained in Section (ref). The parameter $\sigma_{\eta}^{2}$ must still be estimated by running the filter on the $\{\hat{\beta}_{k}\}$ and maximizing the likelihood. Denoting the variance of the forecast error by $F_{k}=P_{k|k-1}+\hat{\sigma}_{kk}^{2}$ , estimating $\sigma_{\eta}^{2}$ by ML is equivalent to finding the value that minimizes $\frac{1}{T}\sum_{k=1}^{T}\log(F_{k})+\frac{1}{T}\sum_{k=1}^{T}\frac{v_{k}^{2}}{F_{k}}$, harvey1990forecasting.
Initially, $\beta_{0|0}=0$ with posterior variance $P_{0|0}=\infty$ corresponding to a diffuse prior distribution of $\beta_{0|0}$ and Appendix (ref) contains a treatment that derives successive updating steps explicitly. While the random walk model in ((ref)) is parsimonious, it implies no drift in dementia incidence since in that model $E[\beta_{k}|\beta_{k-1}]=\beta_{k-1}$. To allow for more varied dynamics in dementia incidence, we augment the model with a drift term $\nu_{k}$,
The drift term $\nu_{k}$ may be stochastic or constant according to
where $\xi_{k}\ensuremath{\sim}\ensuremath{\mathcal{N}}\left(0,\sigma_{\xi}^{2}\right)$. The specification based on ((ref))-((ref)) becomes a random walk model with time-constant drift if $\sigma_{\xi}^{2}=0$ implying that $\nu_{k}=\nu_{k-1}=\nu$. Furthermore, if $\nu_{k}=0$, then ((ref)) reduces to ((ref)), implying no drift in dementia incidence. Empirically, a zero drift corresponds to a flat trend which is equivalent to no change in incidence. Not restricting $\hat{\sigma}_{kk}^{2}$ in ((ref)) yields the standard Kalman filter whose results we report, too.
While previous studies have presented evidence on whether dementia incidence has changed over time, they often do not show whether these are significant. Here, we test whether we can reject the hypothesis of no drift in dementia incidence; and, if there is a drift, whether we can reject whether this drift is constant.
We use three approaches to study the nature of the trend. First, an $F$-test checks whether to reject that all coefficients on the time dummies are identically zero. Second, a $t$-test checks for the presence of a non-zero deterministic drift. Third, another $t$-test checks for stochastic drifts.
Under $H_{0}:\beta_{k}=0\,\forall k$, the $F$-statistic asymptotically follows
In Section (ref), we assumed the covariance matrix $\Sigma_{TT}$ to be diagonal; here, we make use of all entries and thus do not need to adjust for auto-correlation.
To test whether the dementia incidence trend has changed over time, we formulate $H_{0}:\nu_{k}=0,\,\sigma_{\xi}^{2}=0$, i.e. a test of the hypothesis of zero deterministic drift against the alternative that $\nu_{k}=\nu\neq0$ and $\sigma_{\xi}^{2}=0$, which amounts to a zero deterministic drift. Appendix (ref) defines the test statistics and processes for the $t$-statistics and explains covariance estimation.
Figure (ref) shows the annual dementia incidence by age and sex using the multi-state model described in Section (ref). Dementia incidence increases rapidly with age and is higher for men. Rates at age 80 are $3.7\%$ and $3.8\%$ for women and men, respectively. By $90$, the rate rises to $12.3\%$ and $13.6\%$. The $90\%$ confidence intervals grow rapidly with age, especially for men, as mortality decreases sample size.
Figure (ref) shows the estimated annual rate over time. For 80-year-old women, it falls from 4.7% to 4.0% between 2004 and 2010 before falling further to 3.7% in 2018. These estimates are similar to those of CHEN2023e859, who use the same data and a similar model. They find that the annual rate for 80-year-old men decreases from 4.7% to 3.3% from 2004 to 2010 before rebounding to 4.2% in 2018. We adjust the years reported by CHEN2023e859 by two since we plot end of wave estimates. The key differences between our and their results arise from their use of polynomials for time trend estimates while we use wave dummies capturing more fine-grained variation. Although these results suggest declining population level dementia incidence, the large confidence intervals beg the question of whether the observed variation arises from sampling variability.
Next, we discuss results from the models using the filter explained in Section (ref). Table (ref) presents estimated coefficients of two model classes. The first (left panel) is where we constrain the measurement error variance $\sigma_{\varepsilon}^{2}=\sigma_{kk}^{2}$. Therefore, the estimated cross-sectional sampling variance from the multi-state model enters the time series model as measurement error variance. In the second class of models (right panel), measurement error variance is estimated jointly with other parameters.
The top panel shows parameter estimates. The first column presents estimates when the incidence process $\{\beta_{k}\}_{k=1}^{T}$ follows a random walk with zero drift. The estimate of $\hat{\sigma}_{\eta}=0.148$ is significant and its confidence interval shows that it is bounded away from zero and explains a substantial amount of variation.
Two reasons affect dementia incidence over time: (i) the underlying population level dementia incidence $\{\beta_{k}\}_{k=1}^{T}$ and (ii) sampling variability in the ELSA data producing measurement error. We reject that (ii) alone is responsible for changes displayed in Figure (ref) as the CI for $\hat{\sigma}_{\eta}$ excludes zero.
To see how a shock to the dementia process translates to a shift in incidence, recall that the model is estimated in logs: column (1) implies a one-SD shock increases incidence by $14.8\%$ in the RW model with zero drift. Allowing a drift (column 2), a one-SD shock increases dementia incidence by $13.7\%$, while the estimated drift implies a statistically insignificant $2.8\%$ average yearly reduction in incidence. Column 3 presents a RW model where the drift is itself a stochastic process. This least parsimonious model allows for shocks to $\{\beta_{k}\}$ as well as those to the stochastic drift process $\{\nu_{k}\}$. We do not find evidence of a stochastic drift as $\hat{\sigma}_{\xi}=0$. Moreover, this estimate occurs on the boundary of the parameter space and produced a zero eigenvalue in the Hessian. Switching to a model with a deterministic drift (column 2) we find that the 90% confidence interval contains zero. We thus find no evidence for any drift in dementia incidence over our sample period.
The bottom panel shows model fit and diagnostic tests. The first row displays the log-likelihood which is found based on one-step ahead prediction errors generated by the Kalman filter. Because out-of-sample errors determine model fit, there is no reason that adding more parameters should increase the log-likelihood. Indeed, adding parameters by modelling the drift as its own stochastic process decreases the likelihood, suggesting worse out-of-sample performance. While the likelihood is slightly higher in column 2 than in column 1, it is considerably lower in column 3, despite a more flexible specification.
The Bayesian Information Criterion (BIC)=$\big(-2\cdot(\widehat{\text{Log-likelihood}})+((\#\text{ of parameters})\cdot\ln T)\big)$ penalizes the likelihood for the number of parameters so that a higher BIC implies a worse model fit. Such accounting for the number of parameters shows the fit always deteriorating with more parameters.
The Ljung-Box ($\text{Q}\left(4\right)$) statistic measures the degree of autocorrelation in the residuals, here at lag four. Under the null of 0-autocorrelation, this statistic is a $\chi^{2}(4)$ variable. While no model exceeds the $95$th percentile ($9.49$ ), the random walk with constant drift model (column 2) minimizes this statistic. The Bowman-Shenton ($\text{BS}$) tests measure the prediction errors' deviations from normality via the skewness and excess kurtosis. Under the null of normal prediction errors, the $\text{BS}$ test statistic follows a $\chi^{2}(2)$ distribution. While again no model exceeds the $95$th percentile ($5.99$), this time the random walk with stochastic drift model minimizes this statistic. Finally, we include the estimated first and second order residual autocorrelations. The random walk model with stochastic drift induces significant negative autocorrelations, again highlighting its overfitting problems.
The right three columns show estimates when $\sigma_{\varepsilon}^{2}$ is freely-varying and estimated as a parameter within a Kalman filter procedure as opposed to a multi-state model one. Doing so produces a modestly larger estimate of $\sigma_{\varepsilon}^{2}$ than when using an estimated $\sigma_{\varepsilon}^{2}$ from the multi-state model. In the random walk with zero drift specification, $\hat{\sigma}_{\varepsilon}=0.179$. As a result, this approach interprets all estimated time series variation in $\hat{\beta}_{k}$ as measurement error arising from cross-sectional sampling variability and none of it as variability in the underlying dementia process, so that $\hat{\sigma}_{\eta}=0$. We do not report confidence intervals for the estimates of $\sigma_{\eta}$ in those columns because the Hessian is singular in the direction of this parameter for columns 4-6 making them difficult to justify or interpret.
The left panel of Table (ref), where $\sigma_{\varepsilon}$ is estimated using the multi-state model, is our preferred approach, because it uses additional information to discipline estimates of $\sigma_{\varepsilon}$ revealing that measurement error from sampling variability alone cannot fully explain variability in the estimated dementia series in Figure (ref). Instead, some of the variability in estimated dementia incidence represents time series variation in population level dementia incidence.
Both the model fit indicated by the likelihoods in Table (ref) and the evidence in Table (ref) point to our preference for a random walk model with zero drift. Although we reject the models with non-zero constant and stochastic drifts, they nevertheless help elucidate some of the possible dynamics.
Figure (ref) presents estimates of the time series of dementia incidence. Each panel shows the hazard shifter $\exp\{\hat{\beta}_{k}\}$ (red line) from (ref) with black error bars showing $90\%$ confidence intervals. Posterior process estimates from the Kalman filter, $\exp\left\{ \beta_{k|k}\right\} $ appear in blue with the ribbon showing a 90% confidence interval based on $P_{k\mid k}$. Forecasts along with a 90% confidence interval are green.
Panel (a) presents forecasts from the random walk model with no drift: Table (ref), col. 1 shows estimates of this preferred model. The graph shows that series from the filter and the multi-state model are close although filtered estimates are less variable than those from the multi-state model; our modified filter downweights observations $\hat{\beta}_{k}$ estimated from the multi-state model if they are either extreme and/or come with large confidence intervals. The 90% confidence intervals from the two models are also similar, corresponding to a roughly even split between cross-sectional and process noise variance. The model highlights that estimates for $\hat{\beta}_{k}$ for 2012 and 2016 are outliers. In 2012, the black error bar just about touches the Kalman estimate $\exp\left\{ \beta_{k\mid k}\right\} $ with its upper end and falls far outside the blue ribbon for its lower end, likely making this an outlier. For the year 2016, we see that the upward correction relative to 2014 is only moderate due to the large amount of uncertainty communicated by the black error bar, which per (ref) will result in a much lower gain thus downweighting that observation. This specification assumes zero drift implying a flat forecast.
Panel (b) shows estimates from the random walk with constant but potentially non-zero drift model. The corresponding Kalman filter parameter estimates are shown in column two of Table (ref). An estimated non-zero negative drift parameter implies that this model predicts a decline in dementia incidence. Relative to panel (a), the updated posterior process estimates track the estimated $\hat{\beta}_{k}$ slightly less well than the model with zero drift implying that the downward shift caused by $\nu<0$ is not justified. This worsening of model fit is even greater for the one-step ahead forecasts on which the likelihood is based but which are not pictured here.
Figure (ref) (c) graphs the model estimates corresponding to column three in Table (ref). This model includes a whole drift process in purple. The process $\exp\left\{ \nu_{k\mid k}\right\} $ approaches one throughout the sample period implying that the drift parameter in log incidence approaches zero, commensurate with a flattening trend process. However, based on Table (ref), this model performs even worse than the previous one depicted in panel (b). Figures (ref) (b) and (ref) (c) show some evidence of a negative drift but are outperformed by the zero drift model.
We formally test whether the wave-to-wave variation in measured dementia incidence is due to sampling variability, i.e. the hypothesis that $\beta_{k}=0$ for all $k$. To this end, we evaluate the $F$-test statistic shown in ((ref)): $\hat{\beta}^{\prime}\hat{\Sigma}^{-1}\hat{\beta}/T\approx3.298,$ which is distributed $\chi_{7}^{2}$ under the null that $\beta_{k}=0$ for all $k$. The test statistic has a $p$-value of $77.1\%$, meaning that we cannot reject that all $\beta_{k}$ are zero and thus cannot reject the hypothesis of no drift in dementia incidence.
Table (ref) presents specific tests for the nature of the dementia trend. In particular, we test for whether a non-zero drift in dementia incidence may be present. A zero drift parameter in ((ref)) corresponds to a model with no drift. The top row presents the test for whether a zero, deterministic drift, $\nu=0,\,\sigma_{\xi}^{2}=0$, can be rejected against a non-zero, deterministic drift, i.e. $\nu_{k}\neq0,\,\,\sigma_{\xi}^{2}=0$. The test statistic, shown in equation ((ref)), has a $t-$distribution (or normal distribution in large samples) under the null hypothesis. The test statistic is $-0.685$ with a $p-$value of 49.5%, meaning that we cannot reject a zero, deterministic drift. The second row presents a test that discriminates between a stochastic and deterministic drift $\nu_{k}$. In other words, we test whether $\nu_{k}\neq0$ and $\sigma_{\xi}^{2}=0$, i.e. if there is a non-zero drift and whether it is non-stochastic. The associated test statistic appears in ((ref)). We cannot reject this hypothesis at the 5% level. Finally, the third row presents a test for whether we can reject no drift in favor of a stochastic one where we impose $\nu_{k}=0$ in addition to $\sigma_{\xi}^{2}=0$ under the null. The relevant test statistic, defined in ((ref)), assumes no drift, making it inconsistent if one is present, which however buys more power if this assumption is correct. We note that the test based on $t_{\nu}$ in the first row of Table (ref) is evidence that a deterministic trend would be zero, which inspires confidence in the assumption that $\nu_{k}=0$, which we impose under both null and alternative hypotheses of the test in the third row. In summary, we do not reject the null hypothesis of no trend and we do not reject the null hypothesis of no stochastic drift. In sum, we do not reject that the true model has a zero, deterministic drift.
However, the inference appearing in the top row of Table (ref) originates with a normal approximation for $t_{\nu}$ in ((ref)), which is only true if the denominator $\sigma_{\text{HAC}/\text{L}}$ in ((ref)) or ((ref)) is known and not estimated. Because our test statistic is based on $\hat{\sigma}$, estimated on a short sample, $t_{\nu}$ will follow a $t$-distribution with $T-1$ degrees of freedom. Using such critical values, the $p$-value rises to $51.6\%$. Therefore, combining the conclusions of no stochastic drift in rows two and three with the evidence of $\nu=0$ in row one of Table (ref), we conclude that the drift parameter is a constant not different from zero. Figure (ref) in Appendix (ref) presents a sensitivity analysis and graphs $p$-values for different variance estimators and lag lengths, which all support this conclusion. Furthermore, the evidence of the tests for trend is consistent with those results presented in Table (ref), which implies that a model with no drift is preferred.
In this section, we show that the gain and the estimated process covariance converge rapidly to their steady state values after only a few time periods. Thus, the Kalman filter works well even with only eight periods of data.
Relative to joint parameter estimation, using the Kalman filter is inefficient because finding $\beta_{1|1},\dots,\beta_{T|T}$ involves repeated updating of the filter. Initialization using a diffuse prior does not use future information on dementia incidence and relies on convergence. Joint estimation of all parameters would use all information to inform incidence in the initial period. Hence, the gain may converge slowly leading to imprecise estimation of $\beta_{k|k}$ in a short sample.
Recall the gain from ((ref)) and note that $K_{k}$ is the share of variability in incidence attributable to the underlying process ($P_{k|k-1}$) relative to the sum of that and cross-sectional/noise variance ($\hat{\sigma}_{kk}^{2}$). The larger this share is, the more weight our noisy measurement $\hat{\beta}_{k}$ receives in the updating step.
In this section, we prove that the Kalman gain $(K_{k})$ converges to a fixed point, building on doi:10.1137/0331041, who proved that the posterior variance $P_{k\mid k}$ converges to a fixed point when $\hat{\sigma}_{kk}^{2}$ is constant. We extend the proof by both allowing for time-varying $\hat{\sigma}_{kk}^{2}$, and by showing explicitly that $K_{k}$ converges to a fixed point. More technical details appear in Appendix (ref).
In our context, the cross-sectional variance $\hat{\sigma}_{kk}^{2}$ is time-varying and given exogenously, which prevents $K_{k}$'s convergence to a constant. However, as long as $\hat{\sigma}_{kk}^{2}$ varies only modestly, we may still consider the filter elements $K_{k}$ and $P_{k\mid k}$ convergent. Following the assumptions in Section (ref), start the algorithm with a diffuse prior. Formally, we have
The infinite prior variance in Assumption (ref).(ref) formalizes our ignorance about the state of the system before measurements would begin. Its main consequence is $K_{1}=1$ meaning we give full weight to the first measurement $\hat{\beta}_{1}$ which becomes the process estimate for $k=1$. Assumption (ref).(ref) ensures we retain a meaningful concept of filter convergence even though a time-varying variance implies that fixed points will depend on $k$. The intuition is that the measurement variance ratio between successive $k$ is bounded above by the square of the ratio of total and noise variances. The RHS is always bigger than one, implying that this condition always implies a constant observational noise variance, i.e. $\frac{\hat{\sigma}_{kk}^{2}}{\hat{\sigma}_{k+1k+1}^{2}}=1$, as a special case. Generally, this condition requires that $\sigma_{\eta}^{2}$ is large enough relative to $\hat{\sigma}_{kk}^{2}$. The discussion following Lemma (ref) contains alternative expressions.
We define the signal-to-noise ratio as $s_{k}\equiv\sigma_{\eta}^{2}/\sigma_{kk}^{2}$, which measures the strength of the dementia process variance relative to the cross-sectional or noise variance $\sigma_{kk}^{2}$. Because $\sigma_{kk}^{2}$ varies over time, $s_{k}$ varies over time, as well, and is only constant if we set $\sigma_{\varepsilon}^{2}\equiv\sigma_{kk}^{2}$ for all $k$. The Kalman gain is given in
A proof and extended version of Lemma (ref) appears in Appendix (ref). Figure (ref) shows the speed of convergence of the Kalman filter using estimated parameter values of $s_{k}=\frac{\sigma_{\eta}^{2}}{\sigma_{kk}^{2}}$. The $y$-axis shows the values of the gain $K_{k}$ for different values of $s_{k}$ on the $x$-axis. To keep the analysis simple yet account for the time-varying nature of $s_{k}$, the figure evaluates convergence for a range of plausible values of $s$, with a CI drawn in by the black lines. The mean $\bar{s}=$ 1.26. Using this value, after the first iteration (i.e., after the first time period) the gain is $K_{1}=0.56$. After the second, the gain $K_{2}=0.65$. Moving from from the second iteration to further iterations provides little extra gain as in the limit, $K_{\infty}=0.66$. Appendix (ref) contains an extended figure.
A potential problem is that convergence may be slow if some realizations of $s_{k}$ are very small. However, the Kalman filter largely converges to its asymptotic value after two periods even at the bottom of the CI for $\bar{s}$ (denoted $s^{l}$). Put differently, the Kalman filter is inefficient, but after one period the additional potential efficiency gain from other estimators is negligible.
CHEN2023e859 show that a dementia incidence trend may have stopped declining over time. Our findings in Section (ref) lead to our preferred model being a random walk with no drift, meaning it is not possible to predict incidence fluctuations. We further find evidence that dementia incidence does fall in the first half of our sample, and rises in the second, which underscores the difficulty in predicting changes in dementia incidence.
Sampling variability possibly obscures falling dementia incidence at the population level. While the Kalman filter addresses this problem by down-weighting new realizations of estimated dementia incidence when estimating population-level dementia incidence, it may still errantly infer that population level dementia incidence has not fallen when in fact it has. Therefore, this section evaluates the probability that the Kalman filter correctly finds that a dementia shock $\eta_{k}$ is negative when it is in fact negative, under the assumption that the true process is a random walk.
We consider the null hypothesis that a shock at time $k$ to the dementia process $\eta_{k}$, defined in ((ref)), is not negative against the one-sided alternative that it is,
The false positive rate $\mathbb{P}\left\{ \text{reject}\,H_{0}|H_{0}\,\text{is true.}\right\} \equiv\alpha$ and the false negative rate $1-\mathbb{P}\left\{ \text{reject}\,H_{0}|H_{0}\,\text{is false.}\right\} \equiv1-\theta$ where $\theta$ is statistical power. Since the change in the measured dementia incidence using the Kalman filter is $\beta_{k|k}-\beta_{k-1|k-1}$, the power $\theta$ , or, equivalently, the probability that $\beta_{k|k}<\beta_{k-1|k-1}$ when a negative shock $\eta_{k}$ has occurred, is
To express this inequality in terms of the shocks $\eta_{k}$, $\varepsilon_{k}$, and initial conditions $\beta_{0}$ and $\beta_{1\mid0}$, we combine ((ref)) with the state updating equation ((ref)) to obtain $\beta_{k|k}-\beta_{k-1|k-1}=K_{k}v_{k}.$ Thus, ((ref)) becomes $\mathbb{P}\left\{ \beta_{k|k}<\beta_{k-1|k-1}|\eta_{k-1}<0\right\} =\mathbb{P}\left\{ K_{k}v_{k}<0|\eta_{k-1}<0\right\} $. Lemma (ref) below shows that $K_{k}v_{k}$ is a function of the history of $\eta_{k}$ (the “signal” of the process) as well as $\varepsilon_{k}$ (the “noise” of the process) and presents an asymptotic approximation which becomes accurate beyond $k=2$. Table (ref) shows the coefficients to second and third order.
Implementing the exact formulae provided by Lemma (ref) in Appendix (ref) presents a high computational burden for sample sizes beyond $T=10$ as the length of the products of Kalman gains $K_{1}\dots K_{k}$ doubles with each time step.\footnote{This result is established formally in Lemma (ref).(ref) in Appendix (ref).} Fortunately, because $K_{i}\in\left(0,1\right)$ by Lemma (ref),\footnote{A similar result holds in the multi-variate case.} the longer chains converge to zero while those of finite length approach a stable limit.
As can be seen in Table (ref), $c_{3}(3)\geq c_{2}(3)$ meaning that the most recent coefficients on $\eta_{i}$ outweigh the more distant ones. Likewise, $\abs{d_{3}(3)}\geq\abs{d_{2}(3)}$. This pattern where coefficients satisfy $\abs{d_{i}(k)}\geq\abs{d_{i-1}(k)}$ holds more generally which we show in Appendix (ref). As a result, the marginal contribution by more distant $c_{i}$ and $d_{i}$ decays to zero as $k$ grows meaning that more distant structural and noise shocks contribute less information. The formulae are given in
The approximation becomes accurate for $k\geq3$ as shown in Figure (ref), so that the simplified formula given in Lemma (ref) should be preferred almost always.
We use ((ref)) from Appendix (ref) to derive power for $k=2$: \[ \theta=\ensuremath{\mathbb{P}}\left\{ K_{2}v_{2}<0\mid\eta_{2}<0\right\} =\mathbb{P}\left\{ -\varepsilon_{1}+\varepsilon_{2}<-\eta_{2}\mid\eta_{2}<0\right\} . \] Assuming homoskedasticity of the shocks $\big(\eta_{k}\ensuremath{\sim} N\left(0,\sigma_{\eta}^{2}\right)$ and $\varepsilon_{k}\ensuremath{\sim} N\left(0,\sigma_{\varepsilon}^{2}\right)\big)$, $\frac{-\varepsilon_{1}+\varepsilon_{2}}{\sigma_{\eta}}$ has mean 0 and $\text{var}\left(\frac{-\varepsilon_{1}+\varepsilon_{2}}{\sigma_{\eta}}\right)=2\frac{\sigma_{\varepsilon}^{2}}{\sigma_{\eta}^{2}}=2/s$ so that
where $F\left(.\right)$ is the CDF of a normal random variable with variance $2/s$. Equation ((ref)) shows that $\theta$ increases in $-\eta_{2}/\sigma_{\eta}$. Because $\text{var}\left(\frac{-\varepsilon_{1}+\varepsilon_{2}}{\sigma_{\eta}}\right)$ decreases with $\sigma_{\eta}$ (and thus $s$), and $F\left(-\frac{\eta_{2}}{\sigma_{\eta}}\right)$ (and thus $\theta$) decreases in the variance when $\eta_{2}<0$, we have $\frac{\partial\theta}{\partial s}>0,$ i.e. power to detect a standard deviation shock in $\eta_{k}$ increases in the signal-to-noise ratio.
Using Lemma (ref) in Appendix (ref) or Lemma (ref) above allows us to derive an analytical expression for power $\theta$, which we use to construct Figure (ref). The top panel of Figure (ref) highlights how $\theta$ depends on the number of periods $k$ and the signal-to-noise ratio. In particular, the top left panel displays $\theta$ as a function of $\eta_{k}/\sigma_{\eta}$ for various values of $k$ for a signal-to-noise ratio equal to its sample average of $\bar{s}$=1.26 estimated from our data. Power increases with $k$ since as the number of time periods grows, more information is available for accurate measurement. Power converges quickly: it grows only slightly between $k=3$ and $k=\infty$. If the standardized shock to $\frac{\eta}{\sigma_{\eta}}=-1$, then the probability of measuring a fall in $\beta$ is 0.73. If the standardized value of the shock is $\frac{\eta}{\sigma_{\eta}}=-2$, then the probability of measuring a fall in $\beta$ is $0.98$. The right panel shows that power increases with $s$, holding constant $k=3$. For the extreme value of $s=10$, we see that the filtering algorithm is extremely sensitive.
Next we consider size, which is $\alpha=\mathbb{P}\left\{ K_{k}v_{k}<0\mid\eta_{k}\geq0\right\} $. Inspection of this expression and noting that $\eta_{k}$ is symmetric reveal that $\alpha=1-\theta$; we thus omit any derivations of size. Similar to the graphs for power, we plot $\alpha$ as functions of $\frac{\eta_{k}}{\sigma_{\eta}}$ in the bottom panels of Figure (ref) for various values of $k$ given a unit signal-to-noise ratio $s$ in panel (a) and for various $s$ given $k=4$ in panel (b). Panel (a) shows that size decreases with the shock $\eta$ and converges as fast as power does. Panel (b) shows that size decreases with $s$.
In summary, the Kalman filtering algorithm converges rapidly. As a result, it possesses good power to reject the null after only a few time periods.
We estimate time trends in dementia incidence in England over the 2002-2018 period and forecast future dementia incidence. While we find some evidence that incidence falls in the early part of our sample period, there is little evidence of a long-term trend in dementia incidence. Our preferred model is a random walk with zero drift, meaning that while dementia incidence changes over time, the optimal forecast is its most recent value. We also produce confidence intervals for these dementia forecasts and find that we cannot reject significant increases or declines in incidence over the next decade.
Our filtering algorthim allows us to decompose the incidence process variance into cross-sectional sampling variability and dementia incidence variability at the population level. The advantage of our method is that it is computationally very simple: the multi-state model is implemented in the R msm package which produces the trend as parameter estimates with associated variances. In the second step, we model the coefficients using a Kalman filter for which we provide Julia code available in the online supplement.
Data sets that contain internally consistent case definitions of dementia over time have only recently become available, meaning that they are relatively short in the time dimension. To assess the performance of the Kalman filter using short samples, we perform a rigorous analysis of its convergence properties. Its main element, the Kalman gain, distinguishes between signal, i.e. dementia process shocks, and cross-sectional sampling variability commonly referred to as measurement noise in the time series literature. We find that the sequence of gains converges so fast that only a small number of observations in the time domain do not hinder its ability to discriminate between signal and noise.
We also study the implied power and size properties of the filter. We define these probabilities with respect to the filter's ability to correctly detect a negative shock (power) or falsely discover a negative shock (size). We define our null hypothesis in terms of a positive shock, because the major question of our study is to find out whether the trend in dementia incidence has stopped declining. Using the available evidence, we can reject that a positive shock has occurred and conclude that we have a flat dementia trend.
We believe that the methods in this paper will prove useful in multiple contexts. First, the methods in this paper can be used to forecast dementia incidence using the newly available ELSA “sister studies” which are structured similarly to ELSA and contain many of the same variables that can be used to construct measures of dementia. Second, the problem of forecasting with samples that are of large size in the cross-sectional dimension and short in the time dimension is very common. Modeling and forecasting the dementia incidence process is one example, but there are many others in the fields of epidemiology, health, and the social sciences where the presented methodology can be adopted.
Another interesting extension of our current approach would be to add more health states. We model three health states and use age and sex as explanatory variables and include a stochastic time trend. While this approach may seem modest, it is computationally demanding. Implementing models with more health states would be challenging, since it would require joint modeling of many stochastic processes. The methodology proposed by katzmaus might be a useful way to construct a high-dimensional model that is computationally feasible and lets us estimate stochastic time trends. Building a richer model would also allow us to consider prevalence of multiple health states, i.e. the total number of people in each of those states. To achieve this end, we could combine our transition model with the methodologies in Bhadra01062011,Ionides03042023, who model multiple states and consider equilibrium impacts of transitions between those states.
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