The protected survivor model: Using resistant successful cognitive aging to identify protection in the very old

Abstract

For some cardiovascular risk factors, association with risk for cognitive impairment observed in early old age is reduced, or paradoxically even reversed, as age of outcome increases. Successful cognitive aging is intact cognition in the oldest-old; we define resistant successful cognitive aging as successful cognitive aging despite high risk. The protected survivor model posits that a minority of the general population has a protective factor that mitigates the negative effect of a risk factor on successful cognitive aging for the unprotected majority. As age increases, differential failure rates increase the proportion of survivors with protection. Among the unprotected, the proportion with low risk increases, but among those with protection, high risk and low risk do not differ. Due to differential mortality, half the survivors are eventually protected – a majority among those with high risk, and a minority among those with low risk. According to the protective survivor model, an example of Simpson’s paradox, the association of the risk factor with survival does not change within an individual, but the association in the surviving population changes as its age increases.

We created quantitative illustrations of a simplified protected survivor model applied to successful cognitive aging to explain how the usual association of a risk factor with cognitive decline is reversed in the very old. In the illustrations, probability of subsequent survival was higher for survivors with high risk (mostly protected) than low risk (mostly not protected), an example of Simpson’s paradox. Resistance to disease despite the presence of risk factors is consistent with the presence of countervailing protection. Based on the protected survivor model, we hypothesize that studies seeking protective factors against cognitive decline will be more effective by limiting a successful cognitive aging sample to resistant successful cognitive aging – to contrast with a sample without successful cognitive aging.

Introduction

Although disease is usually investigated as an exception from non-diseased normality, a third, seldom-investigated status is resistance to disease. True resistance to disease – attributable to a protection – is difficult to distinguish from lucky absence of disease, but is more plausible in those who remain healthy despite high risk. For example, resistance to HIV infection despite high-risk behavior was used to identify subjects among whom the protective Δ32 mutation in the CCR5 gene [1] was discovered. This paper presents a model for resistance, the protected survivor model, and applies it to offer a hypothesis about finding protective factors against cognitive decline in the very old.

In addition to their implications for mortality, many risk factors for cardiovascular disease (CVRFs) are risk factors against intact cognition [2], mostly for cognitive outcomes in early old age (average age through 75). For later old age outcomes, such associations are few and there are even some reversals – CVRFs associated with better outcome. However, the associations of CVRFs with both mortality and cognitive outcomes are also stronger for studies with earlier ages of risk assessment [3,4], for which the age at outcome is also typically earlier. In the statistical analysis section, Table 1 presents longitudinal studies of cognitive risk in normal subjects, predicted by total cholesterol or C-reactive protein (CRP) – two examples of CVRFs.

Table 1.

Longitudinal studies associating higher total cholesterol and risk of bad cognitive outcomes.

Study	Nationality of sample	N in study	Mean age of risk factor measurement	Age of outcome assessment	Significant associations of CVRFs with bad cognition	Non-significant associations of CVRFs with bad cognition	Significant reversed associations of CVRFs with bad cognition
Cholesterol
Solomon et al., 2009 [12]a	USA	9844	42	69	Alzheimer’s disease (AD)
Kivipelto et al., 2001a,b [13,14]b	Finland	1449	50	71	Mild cognitive impairment (MCI), AD
Yaffe et al., 2002 [15]	USA	1037	67	71	Cognitive impairment
Toro et al., 2014 [16]	Germany	222	60	74	MCI, AD
Reynolds et al., 2010 [17]	Sweden	815	64	74 c		Cognitive decline
Whitmer et al., 2005 [18]a	USA	8845	42	76	Dementia
Solfrizzi et al., 2004 [19]	Italy	1445	73	76		MCI
Notkola et al., 1998 [20]b	Finland	444	47–66b	70–89d	AD
Lorius et al., 2015 [21]	USA	223	76	78		MCI
Mielke et al., 2010 [22]	Sweden	648	47b	79b		Dementia
Taniguchi et al., 2014 [23]	Japan	682	76	79		Cognitive decline
Yoshitake et al., 1995 [24]	Japan	826	74	81		AD
Li et al., 2005 [25]	USA	2112	72	81		Dementia, AD
Reitz et al., 2008 [26]	USA	854	76	81			MCI
Mielke et al., 2005 [27]	Sweden	382	70	81			Dementia
Tan et al., 2003 [28]	USA	1026	50c	83		AD
C Reactive Protein
Marioni et al., 2010 [29]	Scotland (AAA)	2091	62	67	Cognitive decline
Hoth et al., 2008 [30]	USA	78	71	72	Cognitive decline
Marioni et al., 2010 [29]	Scotland (EAS)	534	63	74		Cognitive decline
Yaffe et al., 2003 [31]	USA	2912	74	76	Cognitive decline
Wichmann et al., 2014 [32]	USA	1947	67	77		Cognitive decline
Ravaglia et al., 2007 [33]	Italy	804	74	78		AD
Eriksson et al., 2011 [34]	Sweden	543	74	78		Dementia, AD
Englehart et al., 2004 [35]	Netherlands	727	72	80		Dementia, AD
Schmidt et al., 2002 [36]	USA	1050	55	80	Dementia, AD
Lima et al., 2014 [37]	UK	266	77	81			Cognitive decline
Alley et al., 2008 [38]	USA	533	74	81		Cognitive decline
Sundelof et al., 2009 [39]	Sweden	1062	71	82		Dementia, AD
Laurin et al., 2009 [40]	USA	581	56	83		Cognitive decline
Sundelof et al., 2009 [39]	Sweden	749	78	83		Dementia, AD
Jenny et al., 2012 [41]	USA	833	76	85		Cognitive decline
Tan et al., 2007 [42]	USA	691	79	86		AD
van Himbergen et al., 2012 [43]	USA	840	73	86			Dementia, AD

^aDichotomized cholesterol ≥240 mg/dl

^bDichotomized cholesterol > 251 mg/dl

^cMedian age

^dOnly age range reported

What can explain a reversed association within a study at very high outcome ages? A possible explanation is that the causal effect on cognition of the CVRF could similarly reverse within an individual with increasing age. In an antithetical explanation, the effect of the CVRF within an individual does not reverse with age – or may even accelerate. We previously offered a qualitative explanation of paradoxical reversals of the usual association of bad outcome with high risk [5]. In very old probands who maintained intact cognition, we found those with higher CRP levels had better concurrent memory [6], and had lower rates of dementia in their relatives [5]. These subjects had successful cognitive aging (SCA), maintaining intact cognition in oldest-old ages – 85 and above. Many had resistant SCA (rSCA) – SCA despite high risk.

This paper defines the protected survivor model underlying the qualitative explanation for these results. The model posits that a minority of the general population has a protective factor that mitigates the negative effect of a risk factor for the unprotected majority. Applied to SCA, among those with risk factors, those who also possess protection are more likely to survive to very old age and remain cognitively intact (rSCA). This paper presents hypothetical quantitative illustrations that explain how the usual associations between presence of a risk factor and a bad outcome are reversed at very old age. The protected survivor model attributes the likelihood of maintaining SCA to the relative strengths of the protective and competing risk factors for mortality or impaired cognition. This implies that individuals with rSCA are more likely to have protective factors than others with SCA but only low risk.

Hypothesis

SCA would be promoted by the identification of protective factors in genetic or other studies of cognitive decline, but this is more difficult than identifying risk factors. Our hypothesis is that, to contrast with a sample of individuals without SCA, a sample with rSCA will be more useful for identifying protective factors than a sample with SCA and low risk.

Simplified illustrations of the protected survivor model

Risk and protective factors may be categorical or continuous; a particularly simple example has dichotomized protection, and dichotomized risk with equal probabilities that may represent a continuous variable split at the median. For simplicity, it is assumed that the effect of the protective factor begins only after a baseline age, at which the unprotected and protected subpopulations have the same probability of survival and distribution of the risk factor. Risk is highest for those who are unprotected with high risk, less for those who are unprotected with low risk, and even less for those who are protected. An additional simplification assumes that the protected factor not only mitigates but nullifies the risk factor, so that age-specific rates for a bad outcome (“failure”) are the same for those with either high or low risk who are protected.

Qualitatively, as age increases, differential failure rates – specifically for unprotected – make the survivors with low risk an increasing majority. The proportions of the protected increase among both high and low risk survivors, especially for high risk survivors. Eventually, about half the survivors are protected – a majority of the high risk, and a minority of the low risk. The quantitative examples compare overall survival rates – the complement of failure rates—for the high risk and low risk survivors.

Quantitative models for survival

For survival analysis, the hazard function is the conditional probability of occurrence of an event that has not yet occurred as a function of time – corresponding to an empirical incidence rate. The basic null hypothesis for survival analysis is a hazard function that is constant for all persons at all times. The unconditional probability of the event at time t has an exponential distribution, which decreases as t increases. In the simplest alternative hypothesis, subjects’ hazard functions are different constants. The subject’s average time for an event to occur increases as the value of the hazard function decreases. This is a “proportional hazards” model, since hazard functions are multiples of one another. More general proportional hazards models have some other shape for all the hazard functions, but they still are multiples of one another.

A different type of alternative hypothesis is a “time-dependent” model, with subjects’ hazard functions having different shapes [7]. The Weibull distribution is a generalization of the exponential distribution, with a second parameter, k, that controls the shape of the hazard function. If k = 1, the Weibull distribution is the exponential distribution. If k = 2, the hazard rate increases linearly with time. If k = 3, the hazard rate accelerates as a quadratic function; the probability of the event at time t looks similar to a normal distribution.

Statistical analysis

Specifying model parameters

In the context of SCA, “survival” may be defined as remaining alive with intact cognition, rather than simply not dying. In old age, both mortality and dementia have incidence that accelerates with age. Using United States mortality from age 60 through age 86 [8], with rates from 0.009 to 0.093, a quadratic model for accelerated mortality was a very good fit (R² = 0.992; F(2,24) = 1469, p < 0.0001). At age 85, 47% of those alive at age 60 were still alive.

In the absence of available data on the incidence of impaired cognition – including MCI – across a wide age range, we underestimated it by incidence of AD, the predominant dementia diagnosis in old age. The East Boston study [9] evaluated this for five-year intervals from 65–69 to 80–84 and 85+ that had annual rates from 0.006 to 0.084. A quadratic model for AD was a good fit (R² = 0.969; F(2,2)=31.5, p = 0.03) for the five data points. This quadratic curve had a minimum of 0.004 at age 70, so that estimates for younger ages were larger. For younger ages, instead of using this quadratic curve, the estimate at age 70 was applied. Using this adjusted quadratic model to estimate annual age-specific incidence rates, the estimated cumulative rate of survival without AD was 65% at age 85.

Beyond diagnosis of dementia, impaired cognition includes MCI, both amnestic (memory-related, often leading to AD) and non-amnestic (sometimes leading to other dementias). In a study comparing incidence rates, amnestic mild cognitive impairment was approximately two thirds that of dementia [10]. Longevity is greater for those without AD, so the probability of survival without AD is higher than if independence were assumed. By assuming independence of mortality and AD and combining them, that 31% of those alive at age 60 would still be alive without AD at age 85 was an underestimate. Nevertheless, taking account of all impairments of cognition, a rate between 15% and 20% is plausible for being alive with intact cognition at age 85 (no AD or MCI).

The primary illustration of reversal of association fitted Weibull hazard rates to represent risk accelerating with increasing age. We chose mean times for survival for the four groups and the proportion of protection in the population so that there were approximately equal numbers of unprotected and protected survivors at age 85. The total sample size was chosen to achieve statistical significance if the projected numbers were empirical data. The secondary illustration fitted an exponential model, so its constant hazard rates were not realistic for mortality and cognitive decline. Its parameters were chosen to approximate the survival model for the primary example well, without assuming accelerated hazard rates. A three-year follow-up interval was chosen for the exponential model, to accumulate about as many failures as a single follow-up year for the Weibull model.

Applying the Weibull model

Based on mortality and AD data, the Weibull distribution starting at age 60, with k = 3 for a quadratic hazard rate, was selected to model survival from age 60 to age 85, and also the outcome of subsequent survival to age 86. We selected the average time to failure as 13 years for unprotected high risk, 19 years for unprotected low risk, and 25 years for both high and low risk protected. Their respective rates of survival at age 85 were 0.6%, 20%, and 49%. Fig. 1 (top) shows the three Weibull survival curves for these four groups. These three curves show a minimal effect below age 65 even for unprotected high risk, which is consistent with the assumption that, at age 60, the presence or absence of protection is not associated with the probability of survival or the distribution of the risk factor.

Fig. 1.

Survival curves using Weibull (top) and exponential (bottom) distributions for hypothetical populations: high and low CVRF protected (open triangles), low CVRF non-protected (closed circles), high CVRF non-protected (closed squares).

With equal numbers of high and low risk, the overall unprotected survival rate was 10%, about one-fifth of the protected survival rate. Based on this ratio, assuming that the protected at age 60 were one-fifth as many as the unprotected—one-sixth of the cohort at age 60—achieved nearly equal numbers of unprotected and protected survivors at age 85. For a total sample of 10,000 at age 60, differential mortality reduced the unprotected to 51% of the survivors at age 85, with projected samples of 26 for high risk unprotected, 823 for low risk unprotected, and 409 for either high or low risk protected. Combining all four groups, the overall survival rate was 17%, consistent with the plausible range of 15% to 20% living to age 85 without cognitive impairment (AD or MCI).

According to the Weibull model, among survivors at age 85, the rates of survival one year later were 53% for high risk unprotected, 82% for low risk unprotected, and 91% for each of the protected groups, with respective samples of 14, 672, and 374. Combining protected and unprotected survivors at age 85, 89% of all 435 high risk cases survived to age 86, but only 85% of all the 1232 low risk cases. This discrepancy was surprising, since although the rates of survival to age 86 were equal for low and high risk among protected survivors, the rate was much higher for low risk than for high risk among unprotected survivors. Such an apparent contradiction of higher survival rates for high risk than low risk survivors is an example of Simpson’s paradox [11]. This paradox is based on an ecological fallacy – inference from overall characteristics to characteristics of subgroups or individuals. If this model was empirical data, Pearson’s chi square = 4.93, df = 1, p = .026; a significantly higher survival rate for high than for low risk.

A graphical explanation is provided by Fig. 2, which presents the frequency of survival at age 86 according to protection status, among high and low risk survivors at age 85. For high risk, the mean survival rate was close to that of protected survivors, who were the overwhelming majority of high risk survivors at age 85. For low risk, the mean survival rate was closer to that of unprotected survivors, who were the majority of low risk survivors at age 85. The surprising order of the means for high and low risk was due to high and low risk survivors having such different proportions of unprotected and protected at age 85.

Fig. 2.

Survival at age 86 for high (upper) and low (lower) risk protected (open bars) and unprotected (closed bars).

Applying the exponential model

To demonstrate that this illustration of reversal of rates in SCA did not depend on acceleration of the hazard rate in the Weibull model, it was approximated well by an example with a constant hazard rate –using the exponential distribution. This secondary example had the same sample of 10,000 at age 60, with the same one-sixth protection proportion. Average times to failure of 5 years for unprotected high risk, 15 years for unprotected low risk, and 35 years for protected were selected to have similar respective rates of successful survival at age 85 to the Weibull model: 0.7%, 19%, and 49%. Fig. 1 (bottom) shows the three exponential survival curves for these four groups. The projected samples were 28 for high risk unprotected, 787 for low risk unprotected, and 408 for either high or low risk protected. The respective rates of success after the three-year follow-up were 55%, 82%, and 92%, similar to the Weibull rates for one-year follow-up, with projected successful samples of 15, 644, and 374.

Simpson’s paradox was again illustrated, because when unprotected and protected successful survivors at age 85 were combined, there were successful outcomes at age 88 for 89% of the 436 high risk, but only 85% of the 1195 low risk – in contrast to the lower or equal rates of success for high risk in unprotected and protected separately. If these were empirical data, Pearson’s chi square = 4.74, df = 1, p = .029.

Longitudinal studies of cognitive risk

Studies of total cholesterol or of CRP risk factors were ordered by age at outcome – cognitive risk in normal subjects (Table 1). Each study had one or two cognitive evaluations (mild cognitive impairment [MCI], dementia, and/or Alzheimer’s disease [AD]). The direction of significance of the associations between CVRF and bad cognitive outcome are classified as significantly positive, non-significant, and the unconventional significantly negative. For each CVRF, Pearson correlation analyses associated the direction of significance (analyzed as +1, 0, −1, respectively) with age at outcome, and also age at risk factor assessment.

For cholesterol studies (n = 17), direction of significance was correlated with both at risk factor (r = −0.643, p = 0.005) and outcome (r = −0.726, p = 0.001) ages. The association of the risk factor and outcome ages for cholesterol studies approached significance (r = 0.455, p = 0.066); controlling for risk factor age, partial correlation of significance with outcome age was still significant (r = −0.635, p = 0.008).

For CRP studies (n = 18), the direction of significance was significantly correlated with outcome age (r = −0.597, p = 0.009), but the association for risk factor age was not significant (r = −0.399, p = 0.111). The ages were again correlated (r = 0.469, p = 0.0496); controlling for risk factor age, partial correlation of significance with outcome age was again significant (r = −0.510, p = 0.037).

For each CVRF, these negative associations between the direction of significance and outcome age constitute a reversal from the conventional association between a CVRF and a bad cognitive outcome as age increases. Since cognitive impairment – as well as general mortality –accelerates with increasing age, a reversal from the conventional association of a CVRF with bad cognitive outcome seems paradoxical. The significant partial correlations controlling for risk factor age indicates that these reversals are not attributable to associations of direction of outcome with risk factor age.

Discussion

The Weibull distribution model provides an illustration of survival to very old age, with “survival” referring to the combination of long life and intact cognition, representing SCA. The surprising higher subsequent survival rate for high risk than low risk individuals demonstrates that these are models for rSCA – intact cognition in long-lived individuals despite high risk. This model reflects a defining characteristic of the protected survivor model reflecting rSCA: the change in the observed association of the risk factor and the outcome as the age of outcome increases does not reflect a change over time in the underlying causal relationship within an individual. The change in observed association is an example of Simpson’s paradox, reflecting differences in attrition associated with the risk factor. (The secondary example using the exponential distribution also illustrates reversal and Simpson’s paradox, but is not intended to represent mortality and cognitive decline.)

In the Weibull model, among the 1667 SCA “survivors” at age 85, 49% were protected, but among the 8333 non-SCA “failures”, only 10% were protected. For a cross-sectional study at age 85 seeking protective factors, only the non-SCA who failed only by impaired cognition would be available, not those who failed by mortality. This would motivate comparison of SCA to non-SCA among those still alive at age 85. Furthermore, among SCA, the 409 with high risk (rSCA) were much more likely to be protected (94%) than the 409 with low risk (33%). In contrast, among non-SCA, there was little difference in the proportion protected between the 4565 with high risk (9%) and the 3768 with low risk (11%) – which plausibly would also be reflected among the sub-samples of those demented and still alive at age 85. This motivates the hypothesis that genetic or other studies seeking protective factors in very old subjects will be more effective by limiting SCA individuals to rSCA. The most appropriate non-SCA comparison group would be those with relatively similar high risk.

If a characteristic is identified in the rSCA sample – relative to the non-SCA high risk sample – the simple interpretation is that it represents a protective factor. Also finding this characteristic more in the rSCA sample than the low risk SCA sample – which has substantially less protection – would further support the interpretation as protection. Alternatively, if protection was not supported by this analysis, the identified characteristic might be some other specific risk factor for which by chance the high risk non-SCA sample had lower prevalence than the low risk non-SCA sample.

In the protected survivor model, a further distinction can be made between protection against a specific risk factor – used to define rSCA –versus global protection against a variety of risk factors. For a study of specific protection, it would appropriate to limit the contrasting sample to those with the risk factor. For global protection, even the low risk non-SCA could be included with the high risk non-SCA for comparison with rSCA, consistent with absence of global protection against whatever other risk factors they might have. Such a strategy would have the benefit of including the more numerous low-risk non-SCA, but have the drawback of potentially identifying irrelevant differences between high risk rSCA and low risk non-SCA. Whether for specific or global protection, the low risk SCA subjects would not be included in the primary comparison, since their rate of protection is intermediate between rSCA and non-SCA.

The key concept of the protected survivor model is that the population at very old age differs from the population at younger ages, due to reduced attrition in the minority with a protective factor. As the proportion of protection in the population increases with increasing age, the relationships of risk and outcome change. In the context of SCA – compared with those without SCA – protection is particularly likely in the subpopulation of rSCA. This generates the hypothesis that studies seeking protective factors will be strengthened by comparing rSCA –rather than simply SCA – with non-SCA subjects.

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.mehy.2017.10.022.