Sample and Definitions
According to the Diagnostic and Statistical Manuals of Mental Disorders (DSM-IV), the essential feature of dementia is the development of multiple cognitive deficits that include memory impairment and at least one of the following: aphasia (language deficit), apraxia (movement deficit), agnosia (deficit in recognition of objects or senses), or executive functioning deficit (American Psychiatric Association
2000). The cognitive deficits must represent a decline from past abilities and must be severe enough to cause impairment in occupational or social functioning (American Psychiatric Association
2000). The most common type of dementia is Alzheimer’s disease (AD), which accounts for 60 % to 80 % of dementia cases; the next most common type is vascular dementia, which alone accounts for approximately 10 % of cases but often accompanies AD (Alzheimer’s Association
2014). The estimated average age of onset of dementia in the United States is 83.7 years old (Plassman et al.
2011), and dementia is often accompanied by comorbidities, such as diabetes and a history of stroke (Langa et al.
2017). Dementia on its own is associated with significantly increased risk of death, and comorbidities such as diabetes have been shown to increase the risk of death further among those with dementia (Todd et al.
2013).
This study uses the Aging, Demographics, and Memory Study (ADAMS), a nationally representative, longitudinal study of cognitive health and dementia conducted in four waves from 2001 to 2009 (Langa et al.
2005). ADAMS, a probability subsample of the Health and Retirement Study (HRS), examined adults aged 70 and older with a series of cognitive, psychological, and neurological tests, and conducted an extensive medical history, an inventory of current prescription medications, a neurology-focused physical exam, and a family/caregiver questionnaire. The testing was conducted in person by trained technicians and nurses and was supervised by neuropsychologists (Langa et al.
2005). Diagnostic criteria were based on the DSM-III-R and the DSM-IV, and a consensus expert panel of physicians made the final diagnosis of dementia (Heeringa et al.
2009; Langa et al.
2005). The clinical diagnoses in ADAMS are especially valuable because unlike for certain other chronic diseases, such as diabetes, there is no single test or measure that can provide a “gold standard” ascertainment of dementia status in large samples (Lees et al.
2014).
Detailed descriptions of the ADAMS sample have been previously published (Heeringa et al.
2009; Langa et al.
2005; Plassman et al.
2007). Briefly, a stratified random subsample of 1,770 subjects in the Health and Retirement Study (HRS) were targeted for inclusion in ADAMS; of these, 227 died before they could be assessed, 687 refused or did not participate in ADAMS for other reasons, and 856 participated in ADAMS. The ADAMS subjects did not differ significantly from study nonparticipants in terms of age, sex, or education, but they were more likely to have scored in the cognitively normal range on cognitive screening tests in earlier waves of HRS (Langa et al.
2005). All ADAMS sampling weights incorporate statistical adjustment for differences in HRS cognitive scores between respondents and nonrespondents (Heeringa et al.
2009).
The initial wave of ADAMS, 2001–2003, examined 856 subjects to generate baseline estimates of dementia prevalence in the United States (Plassman et al.
2007). The subsequent waves followed 456 dementia-free individuals for dementia incidence (Plassman et al.
2011); longitudinal sampling weights adjust for differential attrition by baseline cognitive status (Heeringa et al.
2009). The second wave focused on subjects whose baseline status was
cognitively impaired, no dementia (CIND); this second wave assessed subjects 16 to 18 months after their baseline assessment. For the third and fourth waves, all living subjects who were dementia-free at baseline were in the sampling frame. Subjects in the third wave averaged 3.7 years since their most recent assessment, and subjects in the fourth wave averaged 1.8 years since their most recent assessment (Plassman et al.
2011). Despite the relatively long intervals between assessments, especially between Waves A and C for those not assessed in Wave B, ADAMS investigators could determine—based on informant reports, medical records, and clinical assessment—whether a subject experienced the onset of dementia at any time since the previous assessment. For example, if a 72-year-old subject was deemed dementia-free at baseline and then was assessed at age 76 and found to have dementia, investigators could determine that his age at the onset of dementia was 73. The assignment of ages at dementia onset during the interassessment interval allows for the estimation of dementia incidence rates rather than probabilities. Thus, the ADAMS data can be used to calculate age-specific incidence of dementia, an essential ingredient in making estimates of age-specific risk of developing dementia.
Mortality data come from ADAMS’ link to the HRS mortality tracking via the National Death Index (NDI), which provides vital status and, if deceased, month of death as of December 2011. The 856 ADAMS subjects constitute the individuals at risk of mortality. The ADAMS study team did not attempt to diagnose dementia posthumously in subjects who had not received a dementia diagnosis during their lifetimes. This study uses the mortality data to generate estimates of the age-specific ratio of mortality rates between those with and those without dementia. These estimated ratios reflect the distribution of comorbid conditions in the ADAMS subjects. They are thus nationally representative after the application of sampling weights, but they would not apply to future U.S. cohorts if the distribution of comorbidities among people with dementia changes. Mortality rates for the entire U.S. population come from the Social Security Administration (SSA) cohort life tables (Bell and Miller
2005).
Demographic Methods
The quantities estimated from the data are age-specific rates of dementia onset (incidence) and age-specific mortality rates by dementia status (with dementia vs. without). After age-specific incidence rates and mortality rates by dementia status have been estimated, multiple-decrement life table relations will be used to track exits from the dementia-free population via death or dementia onset (Preston et al.
2001: chapter 4). The number of exits via dementia onset features prominently in the estimates of risk of developing dementia.
The primary quantity of interest is the probability of developing dementia for a dementia-free person age
a (an age chosen by the investigator):
$$ {\Pi}_a=\frac{\left({\sum}_{x=a}^w{}_1{}^{dem}d_x^{DF}\right)}{l_a^{DF}}, $$
(1)
where Π
a
is the probability that a dementia-free person age a will develop dementia before death; w is the highest age interval; x indexes each age, at and above age a; \( {}_1{}^{dem}d_x^{DF} \) counts new cases of dementia (left superscripted dem) occurring in the previously dementia-free (right superscripted DF) population; the left subscripted 1 indicates that new cases of dementia are counted in single-year age intervals; and \( {l}_a^{DF} \) is the size of the dementia-free (DF) population at exact age a. In other words, Π
a
is the sum of all dementia cases occurring above age a, divided by the number of dementia-free persons at age a.
Also of interest is dementia-free life expectancy—that is, the average number of years a randomly chosen person age
a can expect to live free of dementia—under current rates:
$$ {DFLE}_a=\frac{\sum_{x=a}^w{}_1L_x^{DF}}{l_a}, $$
(2)
where DFLE is dementia-free life expectancy; a, x, and w are as previously defined; \( {}_1L_x^{DF} \) are person-years lived in a dementia-free (DF) state from age x to age x + 1; and l
a
is the size of the entire population aged a, obtained from a national life table.
One can also define
conditional dementia-free life expectancy as the average number of years that a
dementia-free person of a given age can expect to live free of dementia:
$$ DFL{E}_a^{\prime }=\frac{\sum_{x=a}^w{}_1L_x^{DF}}{l_a^{DF}}, $$
(3)
where
DFLE
′ indicates conditional dementia-free life expectancy; the numerator is the same as in Eq. (
2); and
\( {l}_a^{DF} \) is the size of the
dementia-free population aged
a. This quantity is valuable because the number of dementia-free person-years lived above age
a for someone who already has dementia at age
a is 0, but the people contributing 0 dementia-free person-years to the numerator in Eq. (
2) are still included in the denominator in Eq. (
2).
The quantities on the right side of Eqs. (
1), (
2), and (
3) are all multiple-decrement life table quantities, obtained using the approach in Preston et al. (
2001: chapter 4).
Age-specific incidence is estimated with the following equation:
$$ \mathrm{logit}\left({h}_x\right)=\upalpha +\upbeta x, $$
(4)
where
h is the incidence rate,
x is exact age, and α and β are parameters to be estimated. The model broadly conforms to the functional form of the age patterns of Alzheimer’s disease rates (Brookmeyer and Gray
2000; Brookmeyer et al.
2011; Ziegler-Graham et al.
2008).
The model is fit using a discrete-time logistic regression on a person-year data file (Allison
1984), using the 456 subjects followed longitudinally and the longitudinal survey weight provided by ADAMS. Each subject’s longitudinal survey weight is proportional to the subject’s predicted probability of attrition, which is calculated based on observed data in HRS and ADAMS on cognitive status, general health status, and sociodemographic information (Heeringa et al.
2009; Plassman et al.
2011). Age of dementia onset was reported in completed years; thus, for incident cases, the exact age at incidence was set at the reported age (last birthday) of onset plus 0.5. Subjects who never received a diagnosis of dementia from ADAMS investigators, including those who died without a dementia diagnosis, were censored. Among the censored subjects, those whose status at the end of the ADAMS study period was
alive, dementia-free contributed dementia-free person-years up to and including their exact age (in months) at their last assessment. A sensitivity analysis censored dementia-free survivors at the end of the ADAMS study period rather than at their last assessment.
Censored subjects whose status at the end of ADAMS was
died without dementia contributed dementia-free person years until their exact age at death. For example, if a subject’s status at the end of ADAMS was
died without dementia, and she died at age 78 and 5 months, then she contributed person-years of exposure until she was 78.41666. Her death would be assigned to the interval between exact ages 78.0 and 79.0. The approach of carrying the last assessment of deceased individuals forward until death is consistent with previous ADAMS reports (Plassman et al.
2011) and recommendations based on simulations of censored time-to-dementia data (Leffondré et al.
2013). It is based on the idea that if the deceased individuals had survived and developed dementia, the investigators could have been able to observe their dementia onset; decedents were therefore at risk of dementia onset until their deaths.
Considerable evidence in the literature suggests that age-specific incidence rates of dementia do not vary by sex (Chêne et al.
2015; Plassman et al.
2011; Ruitenberg et al.
2001). When a sex term was included in Eq. (
4), its coefficient was statistically insignificant (
p > .20). This insignificant result further justifies the pooling of males and females in the estimation of dementia incidence.
To estimate an age pattern of differential mortality, a Gompertz equation was fit with a Poisson regression on a person-year data file (Loomis et al.
2005), again using the longitudinal sampling weights provided by ADAMS investigators. Dementia status was modeled as a time-varying indicator to incorporate both baseline prevalent cases and incident cases (Palloni and Thomas
2013). The model is
$$ \ln \left({m}_{x,dem}\right)=\upalpha +{\upbeta}_1x+{\upbeta}_2dem+{\upbeta}_3x\times dem, $$
(5)
where m is the death rate, x is exact age, and dem is an indicator equal to 1 if the subject had dementia and equal to 0 otherwise. The parameters to be estimated from the data should be understood as follows: α represents the level of mortality in the entire population; β1 represents the age-pattern of mortality in the entire population; β2 represents the extent to which those with dementia die at a different (presumably higher) rate than those without dementia, regardless of age; and β3 is a parameter allowing differential mortality to vary by age.
As with the estimation of dementia incidence discussed earlier, subjects who died without a dementia diagnosis during the ADAMS study period contribute dementia-free person years until their exact age at death, and subjects who survived ADAMS without a dementia diagnosis contributed dementia-free person years until their last ADAMS assessment. A sensitivity analysis censored dementia-free survivors at the end of the ADAMS study period. Mortality data for the period after ADAMS (2009 to 2011) was used only for those with a dementia diagnosis whose state could not change until death. Not using mortality data from the post-ADAMS period for individuals without a dementia diagnosis avoids large misclassification errors whereby persons who develop dementia subsequent to ADAMS would wrongly contribute deaths without dementia and person-years without dementia to the calculations.
Based on Eq. (
5), the ratio of the mortality rate among persons with dementia to that among persons without dementia—also known as the mortality rate ratio (
RR)—is
$$ {RR}_x=\frac{ \exp \left(\upalpha +{\upbeta}_1x+{\upbeta}_2+{\upbeta}_3x\right)}{ \exp \left(\upalpha +{\upbeta}_1x\right)}= \exp \left({\upbeta}_2+{\upbeta}_3x\right), $$
(6)
where x is exact age. In this way, the ratio of the two mortality rates is estimated from the ADAMS sample, but the actual values of the mortality rates can be adjusted to match national data with many more deaths using national life tables.
Consistent with most of the literature, the ratio of mortality rates between those with and those without dementia were held constant across sex (Agüero-Torres et al.
1999; Garcia-Ptacek et al.
2014; Johnson et al.
2007; Lönnroos et al.
2013; Meller et al.
1999; Villarejo et al.
2011; Witthaus et al.
1999). When a sex term and an interaction term for sex by dementia status were included in Eq. (
5), the coefficient on interaction term was not statistically significant (
p > .30), providing additional justification for keeping differential mortality constant across sex. As with the modeling of incidence rates, pooling males and females to estimate differential mortality is useful with a small sample size, as in ADAMS. In this model, the only quantity that differed by sex was the overall level of age-specific mortality in the entire U.S. population.
For a given age, the mortality rate for the entire population can be decomposed into a weighted average of mortality rates of the diseased and disease-free populations, weighted by the age-specific prevalence of the disease:
$$ \begin{array}{l}{}_1m_x={}_1m_x^D\times {}_1P_x+{}_1{}^{death}m_x^{DF}\times \left(1-{}_1P_x\right)\\ {}={}_1{}^{death}m_x^{DF}{\times}_1{RR}_x\times {}_1P_x+{}_1{}^{death}m_x^{DF}\times \left(1-{P}_x\right),\end{array} $$
(7)
where 1
m
x
is the death rate in the entire population in the age interval x to x + 1; \( {}_1m_x^D \) is the death rate in the same age interval for those with dementia; 1
P
x
is the prevalence of dementia in that age interval; \( {}_1{}^{death}m_x^{DF} \) is the death rate in the age interval for the dementia-free population; and 1
RR
x
is the ratio of mortality rates (with-dementia vs. dementia-free) in the age interval.
The terms can be rearranged to solve for the mortality rate in the dementia-free population:
$$ {}_1{}^{death}m_x^{DF}=\frac{{}_1m_x}{\left({}_1P_x\times {}_1R{R}_x+1-{}_1P_x\right)} $$
(8)
and in the population with dementia:
$$ {}_1m_x^D={}_1{}^{death}m_x^{ND}\times {}_1R{R}_x, $$
(9)
where the overall mortality rate,
1
m
x
, comes from the national life table; the mortality rate ratio (
1
RR
x
) is from Eq. (
6); and the age-specific prevalence is the proportion of survivors to the middle of the age interval who have dementia, as detailed in the
appendix.
Using the incidence rates estimated in Eq. (
4) and the mortality rates found in Eqs. (
8) and (
9), a multiple-decrement life table is constructed for the population without dementia, including the crucial quantity of dementia-free person-years lived in each single-year age interval,
\( {}_1L_x^{DF} \). The approach incorporates elements of the increment-decrement life table to keep track of a model population with dementia. Single-year age groups are used, and no recovery from dementia is allowed. The life table relations used are developed in Preston et al. (
2001: chapter 4) and shown in detail in the
appendix.
After the multiple-decrement life table is completed, the summary quantities of interest—risk of developing dementia, unconditional expectancies, and conditional dementia-free life expectancy—can be calculated as in Eqs. (
1)–(
3).
Simulated Delays or Reductions in Dementia Incidence
A great deal of ongoing research, both privately and publicly funded, is developing treatments to delay AD and other dementias (Zissimopoulos et al.
2015), so the life cycle effects of different interventions that delay the onset of dementia are also estimated. A recent economic analysis of estimating a delayed onset of AD considered a five-year delay, which is treated here as the most optimistic of several scenarios of delayed dementia (Zissimopoulos et al.
2015). In the first scenario, the intervention delays dementia onset by one year and is effective for 50 % of the dementia-free population at age 70. In the second scenario, the same intervention affects 90 % of dementia-free 70-year-olds. In the third scenario, the intervention delays dementia onset by five years and is effective for 50 % of dementia-free 70-year-olds, and the fourth scenario delays dementia onset by five years for 90 % of dementia-free 70-year-olds. These interventions are modeled by splitting the model dementia-free population in half (or, for the second and fourth scenarios, into 10 %/90 % groups), subjecting the first group to the dementia incidence rates as modeled in Eq. (
4), and subjecting the second group to the dementia incidence rates as modeled by
$$ \mathrm{logit}\left({h}_x^{\prime}\right)=\upalpha +\upbeta \left(x-K\right), $$
(10)
where
K is the number of years of delay of dementia onset induced by the intervention, and the other quantities are as defined in Eq. (
4). This equation assigns what had been the age 70 incidence rate to age 70 +
K, what had been the age 71 incidence rate to age 71 +
K, and so forth.
Another type of intervention would reduce the risk of dementia at every age rather than delaying its onset. Such an intervention generates an incidence equation such as
$$ \mathrm{logit}\left({h}_x^{{\prime\prime}}\right)=\upalpha +\left(\upbeta k\right)x, $$
(11)
where
k is a value between 0 and 1 that represents the extent to which dementia incidence rises less steeply with age due to the intervention, and the other quantities are as in Eq. (
4). The closer
k is to 0, the more effective is the intervention in the sense of reducing the acceleration of dementia incidence. An intervention is simulated where
k = 0.9 in order to reduce the (logit of) acceleration of dementia incidence with age by 10 %.
Both the dementia-free and with-dementia populations are subject to the same mortality rates as in the original analysis, based on the 1920 cohort life tables and estimated mortality rate ratios. However, the changing sizes of these two populations resulting from the simulated intervention are assumed to change the overall mortality rate (Eq. (
7). These simulations illustrate the effects of possible future reductions in dementia incidence; they are not projected outcomes based on expected rates.
Estimation of Standard Errors and Confidence Intervals
To generate standard errors and confidence intervals around the lifetime probability and expectancy estimates, the parameter estimates generating the age-specific dementia incidence schedules (the fitted values of [α β] in Eq. (
4)) and differential mortality (the fitted values for [α β
1 β
2 β
3] in Eq. (
5)) are considered stochastic. Total mortality, derived from the SSA cohort life tables, was treated as deterministic (i.e., having 0 variance) (Abatih et al.
2008; Loukine et al.
2012); and the life table assumptions, such as linearity of survival within age intervals, were also considered not to contribute any additional variance.
For dementia incidence, the estimates of [α β] in Eq. (
4), along with their associated variance-covariance matrix, were used as the parameters of a bivariate normal distribution to draw 1,000 independent values of [α β], generating 1,000 incidence schedules. Separately, an analogous procedure with the estimated parameters and variance-covariance matrix from Eq. (
4) or Eq. (
5) was used to generate 1,000 age schedules of the mortality rate ratio between those with and those without dementia. Each incidence schedule was paired with one schedule of the mortality rate ratios and run through the life table operations, producing 1,000 dementia probability and expectancy estimates. Tables show the means and standard errors (square roots of variances) of the 1,000 estimates (Fishman
2015; Mooney
1997; Salomon et al.
2001).
Parameters from Eqs. (
4) and (
5) were estimated using Stata version 14 (StataCorp
2015), using first-order Taylor Series linearization for variance estimation with the
svy routine with longitudinal sampling weights provided by ADAMS (Heeringa et al.
2009). Random sampling for the estimation of standard errors was conducted in R using the
mvrnorm command in the MASS package (Venables and Ripley
2002), and life table operations were conducted using base R (R Core Team
2014). The HRS and ADAMS data are available to the public after a registration procedure (HRS
2013).