Evaluation of Selective Survival and Sex/Gender Differences in Dementia Incidence Using a Simulation Model

Key Points Question Can selective survival plausibly explain reported sex/gender differences in dementia incidence? Findings In this decision analytical model of 100 000 simulated adults aged 50 years and without dementia at baseline, sex/gender differences in dementia incidence consistent with literature (ie, 15%-20% elevated risk for women aged ≥85 years) were only observed in the presence of moderate to strong effects of selective survival characteristics that differed by sex/gender. Meaning These findings suggest that selective survival may contribute to sex/gender differences in dementia incidence but do not preclude the potential for additional contributions from biological mechanisms.


Survival Function
To match survival rates in our simulations to US lifetable data, we calibrated the baseline mortality rate in men and the effect female sex/gender on mortality such that sex/gender-specific cumulative survival from age 50 matched lifetables. Here, we discuss the mortality hazard model and details of the calibration process.
For each individual in each five-year age band, [50][51][52][53][54][55], [55][56][57][58][59][60], …, [90][91][92][93][94][95], time to death was generated as a random variable drawn from an exponential survival distribution. In all simulation scenarios, we generated survival times for individual in age band as a random variable drawn from an exponential survival distribution based on the hazard function in Equation A.1. We assumed constant baseline mortality hazard within each age band : where ∼ (0, 1) and is the time from baseline study visit. Note that participants' ages are not included in the model because everyone is the same age at baseline in this simulation, thus there is no age effect to account for beyond the differences in between age bands. We calibrated survival in our simulations to the US 1919-1921 non-Latino white birth cohort. Calibrating survival in our simulations to US lifetable data involved choosing appropriate parameters for the hazard function so that the conditional probability of survival (survival to age + 5 conditional on survival to age ) for men and hazard ratios for mortality (women versus men) in our simulation closely matched those calculated in the lifetables. For each scenario, we fixed the effect of U on log hazard of death. The values chosen for these effects in each scenario are reported in eTable 1. We used R's optim function to solve for values of , the baseline mortality hazard for men in each 5-year age band, so that conditional probabilities of survival for men in each band closely matched those from US lifetables for each age band. We used the resulting optimized values of in the model and R's optim function to solve for values of 1 , the effect of female sex/gender on log hazard of death, so that mortality hazard ratios (women versus men) closely matched those calculated from US lifetables. eFigure 1 illustrates the success of our calibration for (a) survival probabilities and (b) mortality hazard ratios.

Cognitive Trajectories and Dementia
Calibrating dementia incidence rates in our simulation to real data involved choosing appropriate parameters in the model for cognitive trajectories, choosing an age-constant dementia cut-point (i.e., a threshold for cognitive function below which an individual would be classified as having dementia), and choosing a value for the constant rate of "random shock" dementia that together would produce reasonable cognitive trajectories (not too steep) and reasonable dementia incidence rates (reflective of real data). Here, we discuss the details for this calibration process, the resulting parameters, and the success of the calibration to dementia incidence rates for men (used as the reference group) reported in the Adult Changes in Thought study, which reported contemporary (1994-2010) age-and sex/gender-specific dementia incidence rates in a US population. 1,2 In all simulation scenarios, individuals could develop dementia in two ways: (1) their cognitive function fell below an age-constant dementia cut-point or (2) they experienced a "random shock" event (e.g. a serious stroke) that gave them dementia immediately. We used an age-constant rate for the "random shock" events. We generated person-specific cognitive trajectories from age 50 using a quadratic growth curve for cognitive decline with a random intercept, random linear slope, and random quadratic slope. Cognitive function for individual at time , where is the number of years from baseline, was determined by the quadratic mixed effects model defined in Equation A.2.
To develop dementia due to cognitive decline, an individual's cognitive trajectory had to fall below an ageconstant cut point for dementia. Parameters for the model were determined simultaneously with this age-constant dementia cut point so that together, trajectories would represent reasonable rates of decline and reasonable dementia incidence rates (reflective of reported rates). The age-constant cut point used across the simulation scenarios was -6.5. This cut-point was standardized to the distribution of cognitive function for 50-year-olds in our simulation which was roughly standard normal (mean = 0, SD = 1.0). Thus, an individual developed dementia due to cognitive decline when their cognitive function fell below 6.5 standard deviations below 0 in the distribution of cognitive function for 50-year-olds, regardless of their age. Because this cut-point is so extreme for individuals at younger ages, there were no incident dementia cases in our simulation until the [65, 70) age-band.
To ensure that individuals' cognitive trajectories declined continuously, we guaranteed a negative quadratic coefficient in our models for (Equation A.2) by taking the negative absolute value of the value drawn for 2 (the random quadratic slope) (i.e., by drawing 2 from a half-normal distribution). The cognitive intercept was set to 0 in all scenarios to obtain approximately standard normal distributions of baseline cognitive function. Values for the linear and quadratic coefficients in the model for in each simulation scenario were obtained through hand calibration (plugging in values and testing resulting dementia incidence rates) so that dementia incidence rates for men matched those reported in the ACT study. The parameters used for the model in each simulation scenario are presented in eTable 2. Average cognitive trajectories for men and women and samples of individual trajectories in each simulation scenario are presented in eFigure 2. We determined the rate of "random shock" dementia from incidence rates for men in the youngest age bands of the ACT study [65,70). We set the rate of "random shock" dementia in our simulations to 7/1000 person-years for every age band, based on the dementia incidence rate reported for men in the youngest age band, [65,70).
There was no interval censoring in this simulation study. For those diagnosed with dementia due to cognitive decline in a specified time interval, we solved for the time within that interval that the individual's cognitive trajectory fell below the age-constant dementia cut point. For an individual who developed dementia based on a "random shock" event in a specified time interval, their time to dementia was determined by first drawing a random variable uniformly from the interval (0, 5) (corresponding to the time between "study visits"). This random draw was then added to the years from baseline visit at the start of the time interval.
Dementia incidence rates reported in the ACT study were used as a guide rather than a strict calibration criterion because of the likely chance variation in ACT results across age bands (reflected by wide confidence intervals). To ensure the validity of our simulation results, we verified that each simulation scenario was as wellcalibrated to the ACT study data as the other simulation scenarios. The consistency of our calibration is shows in eFigure 3, which depicts the dementia incidence rates for men in each of our simulation scenarios compared to the rates reported for men in the ACT study. The average dementia ̂ in all our simulations compared to those reported in the ACT study are presented in eTable 3.

eFigure 3. Dementia Calibration
Average dementia incidence rates for men (used as the reference for calibration) in each simulation scenario* compared to those reported in the ACT study.