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Original Articles

Blood Pressure and Memory

Novel Approaches to Modeling Nonlinear Effects in Longitudinal Studies

Liu, Anran MSPH*; Sun, Zhaowen PhD; McDade, Eric M. DO; Hughes, Tiffany F. PhD, MPH§; Ganguli, Mary MD, MPH; Chang, Chung-Chou H. PhD*,¶

Author Information
Alzheimer Disease & Associated Disorders: October–December 2019 - Volume 33 - Issue 4 - p 291-298
doi: 10.1097/WAD.0000000000000346


There is increasing interest in vascular variables as potentially modifiable risk factors for cognitive impairment and dementia. This point is illustrated by a recent randomized clinical trial, the SPRINT-MIND study,1,2 which compared the effects of intensive versus standard blood pressure (BP) lowering strategies on the risk of mild cognitive impairment (MCI) and dementia. Although associations between BP and cognitive performance are well-established, their relationship is complex and likely dynamic across the late-life spectrum.3–5 As clinicians know, the effect of BP on cognition may be different at different ages, different BP levels, or with different comorbidities, that is, their relationships are not linear.6–9 We reported earlier that associations between changes in pulse pressure and cognitive decline depended on baseline age as well as baseline systolic blood pressure (SBP).10 Further, SBP and diastolic blood pressure (DBP) are highly correlated with each other11 and their individual relationships with a specific outcome will depend on each other. Thus, there is a need for statistical modeling techniques that can depict complex nonlinear associations.

In this paper, we report a technique involving smoothing splines to better explore the nonlinear relationships between BP and memory function. We also demonstrate an alternative data visualization procedure to illustrate SBP and DBP ranges associated with optimal memory performance in older adults.


Study Population

Our study cohort, the Monongahela-Youghiogheny Healthy Aging Team (MYHAT), was selected between 2006 and 2008 by age-stratified random sampling from the publicly available voter registration list of residents aged 65 years or older from targeted small-town communities in Southwestern Pennsylvania.

Details of sampling, recruitment, and cohort characteristics have been previously reported.12 Inclusion criteria were age 65 or older, not residing in a long-term care facility at study entry, no substantial sensory impairment, and decisional capacity. Initial screening was performed on 2036 participants, of whom 54 were excluded from the full evaluation based on substantial baseline cognitive impairment (scores <21) on the age-adjusted and education-adjusted Mini-Mental State Examination (MMSE).13,14 The full evaluation was conducted on the remaining 1982 participants who at study entry had a mean (SD) age of 77.6 (7.4) years; were 61.1% women and 94.8% of mixed European descent; and had a median educational level of high school graduate. All procedures were approved by the University of Pittsburgh Institutional Review Board and all participants provided written informed consent.

At baseline and at up to 9 follow-up visits, trained research staff performed evaluations including but not limited to: health history (self-report of receiving specific diagnoses from health care professional), lifestyle factors including current and past alcohol and tobacco use, currently taken prescription medications, a brief physical examination including BP measurement, and cognitive assessment described later.

BP Measurement

At each study visit, BP was measured ~90 minutes after the start of the assessment. Before the BP reading was taken, the participant remained in the seated position for at least 5 minutes with feet flat on the floor. After the appropriate cuff size was determined, BP was measured using an aneroid sphygmomanometer and stethoscope. A second reading was taken after having the participant stand for at least 3 minutes. If SBP was >175 mm Hg or DBP was >100 mm Hg in either position, a repeated measurement was performed later in the physical examination. The BP variables used here represent an average of all measurements taken at a given visit.

We defined 3 baseline BP categories based on BP measurements at study entry: normotensive (90≤SBP≤140 and 60≤DBP≤90), hypertensive (SBP>140 or DBP>90), or hypotensive [SBP<90 or (DBP <60 and SBP ≤140)]. Note that no participant had SBP<90 and DBP>90.

Memory Test Performance

We assessed participants’ memory performance at each visit using the following neuropsychological tests: WMS-R Logical Memory (immediate and delayed recall),15 WMS-R Visual Reproduction (immediate and delayed recall),15 and Fuld Object Memory Test with Semantic Interference.16 We transformed each test score into a standardized score by centering to its mean and dividing by its SD. We then calculated a composite memory score, for each participant, as the arithmetic mean of the standardized scores of all the memory tests at each visit. A standardized memory domain score of 0 represents the average of the study population, while score of 0.5 or −0.5 means that the individual’s memory test performance falls 0.5 SD above or below the mean, respectively, for the population.17 Memory scores were standardized to the cycle 1 (baseline) visit adjusting for age, sex, and education. For these analyses, we modeled repeated memory measures in each participant at up to 10 annual assessments. As would be expected in a longitudinal study of older adults randomly drawn from the population, we experienced steady attrition across cycles. By cycle 10, the cohort had reduced in size to 31.4% of the original cohort.


Sex, education (<high school, =high school, >high school), APOE*4, and baseline body mass index (BMI) level (<25, 25 to 29, ≥30) were included as time-fixed covariates. Time-varying covariates were age, smoking status (current vs. not current), alcohol intake (never, former, current: within the past year), antihypertensive drug(s) usage, physical activity, and self-reported history of a health care provider’s diagnosis of stroke or transient ischemic attack (TIA), diabetes, or high cholesterol.

Need for Alternative Statistical Approaches

In a linear mixed model (LMM) for repeated measurements, a unit change in the predictor is associated with a fixed difference in the outcome. However, for different values of the predictor, it is possible that the corresponding changes in the outcome may not be of the same magnitude or in the same direction. Thus, linear or quadratic terms of the covariate may not suffice.

The smoothing spline offers flexibility in adapting to unknown and/or nonuniform trends in the data. The method consists of piecewise polynomial functions with predefined smoothness criteria where the pieces meet, and the values of the spline terms directly describe the magnitude of the predictor(s)’ effects. The application of splines is not limited to the 1-dimensional case; they can also handle multidimensional-associated predictors (eg, SBP and DBP).

The thin-plate spline (TPS) is a 2-dimensional analog of smoothing spline that requires no a priori knowledge of the functional form of the data or of the relationship between predictors. Thus, it is ideal for examining the combined effect of 2 associated continuous predictors on a single outcome with their multidimensional appearance.18

In this study, we included (a) 1-dimensional cubic smoothing spline to handle nonuniform effects of age on memory, and (b) 2-dimensional TPS to examine the joint effects of SBP and DBP on memory.

Statistical Analyses

Descriptive Statistics

We examined demographic characteristics in the sample using descriptive measures (eg, mean, SD, frequency, and percentage). Because of the non-normal distributions of some continuous variables, we compared them among the 3 baseline BP groups (normotensive, hypotensive, and hypertensive) using analysis of variance or Kruskal-Wallis tests for continuous variables and χ2 or Fisher’s exact test for categorical variables.


We first fit LMMs with linear terms of age, SBP, and DBP, and the interaction between SBP and DBP, for all participants and in each baseline BP subgroup.

Generalized Additive Mixed Models (GAMMs)

We then fit GAMMs replacing linear terms with the aforementioned spline functions for age and SBP/DBP, allowing varying spline covariates among baseline BP subgroups.19,20 The F-statistic and corresponding P-value were used to perform model selection regarding whether a spline term adds significantly to the prediction of memory performance, while adjusting for all other covariates/terms.

Data Visualization

For each baseline BP group, we produced a heat map depicting the predicted memory domain score obtained from the GAMM from different combinations of SBP and DBP values. SBP and DBP constructed a surface and the predicted memory domain scores were plotted on the heat map with different colors representing different predicted memory score. The SBP and DBP points with the same color correspond to the same predicted memory domain score.

All analyses were performed in R 3.5.2 and all the GAMMs were fitted using the mgcv package. ( We used open source software Gnuplot to visualize the bivariate TPS of SBP and DBP against memory domain scores with 3-dimensional heat-map plots (


Among 1982 participants at baseline, we classified 1351 (68.16%) as normotensive, 527 (26.59%) as hypertensive, and 89 (4.49%) as hypotensive. The hypotensive participants were older than the normotensives, had lower baseline memory domain scores than the normotensives and hypertensives, and included a larger proportion of women and stroke/TIA history reporters, than the normotensives. Regarding proportions of antihypertensive drug users; while a 3-way comparison showed significant differences across the 3 groups, cell sizes were too small for pairwise comparisons; however, the largest proportion was in the hypotensives (Table 1).

Baseline Characteristics of MYHAT Participants by Blood Pressure Group

We had an average of 6 observations per participant over 10 annual assessment cycles.

The Pearson product moment correlation coefficient of SBP and DBP at baseline is 0.45 (P-value <0.001 for testing H0: correlation=0 vs. H1: correlation ≠0). For cycles 2 to 9, we also found significant correlations of similar magnitude of Pearson correlation coefficient between SBP and DBP (ranges from 0.41 to 0.50).

In the LMMs, the main effects and interaction of SBP and DBP were not significantly associated with the contemporaneously measured memory domain scores, either in the overall model or in the models stratified by baseline BP groups (results shown in Supplementary Table 1, Supplemental Digital Content 1,

In the GAMMs with spline terms, adjusting for covariates, when allowing the TPS of SBP and DBP to vary among 3 baseline BP subgroups, memory domain score was significantly associated with the TPS of SBP and DBP in the hypertensive group. There was a borderline significant association between the TPS of SBP and DBP and memory domain score (Table 2).

Generalized Additive Mixed Models

Being female, better educated, being free of stroke/TIA history, performing any physical activity, and having higher baseline BMI were associated with higher memory in both normotensives and hypertensives. Current consumption of alcohol, using any type of antihypertensive medication, and not being APOE*4 carrier, were associated with higher memory only in normotensives.

Although older age was associated with decreasing memory scores, the use of a 1-dimensional spline highlighted a nonlinear relationship where there was stable memory performance until approximately 75 years of age, followed by a steadier decline at older ages. This result demonstrates the value of examining the effect of age using 1-dimensional splines instead of merely fitting a linear line with negative slope (Table 2, Figs. 1A–C).

The solid line represents the relationship between predicted memory domain score and age. The dashed lines are the corresponding 95% confidence interval. Baseline normotensives (A); baseline hypertensives (B); baseline hypotensives (C). A, Memory domain score versus age (normotensive baseline). B, Memory domain score versus age (hypertensive baseline). C, Memory domain score versus age (hypotensive baseline).

Optimal BP for Memory Prediction

In the heat map derived from the GAMM results, the maximum and minimum values of the predicted memory domain score stand out as yellow and purple blocks, respectively, on the plot (Figs. 2A–C).

The 3-dimensional surface at the top of the figure represents the relationship among predicted memory domain score, systolic blood pressure (SBP) and diastolic blood pressure (DBP). The colors at the bottom are the corresponding heat maps. Yellow to purple corresponds to the highest to lowest predicted memory score. Baseline normotensives (A); baseline hypertensives (B); baseline hypotensives (C). A, Predicted memory domain score against SBP and DBP (normotensive baseline). B, Predicted memory domain score against SBP and DBP (hypertensive baseline). C, Predicted memory domain score against SBP and DBP (hypotensive baseline).

Among baseline normotensives, SBP around 140 mm Hg and DBP around 80 mm Hg were associated with the highest predicted memory domain scores. In contrast, the lowest predicted memory domain score was associated with SBP around 110 mm Hg and DBP around 60 mm Hg.

Among baseline hypertensive participants, BPs of 130/85 mm Hg were associated with the highest predicted memory domain score; the unique combination associated with the lowest predicted memory domain score was SBP circa 150 mm Hg with DBP of 65 mm Hg.

Among baseline hypotensives, no significant association between BP and memory score was found, probably due to their small sample size.

In normotensive and hypertensive groups, there was evidence that a DBP below 75 was associated with worse memory performance, whereas with SBP there was a wider range and a less uniform pattern across groups.


In a large population-based study of older adults followed over 10 years, we found that BP was significantly but nonlinearly associated with memory test performance; the relationship was only revealed when examined within subgroups that were normotensive or hypertensive at study entry.

Different SBP and DBP ranges, not necessarily in the conventional normotensive range, were associated with optimal memory scores in the different baseline BP subgroups. Low diastolic pressure was associated with poor memory in both hypertensive and normotensive subgroups. Notably, these associations escaped detection in a conventional LMM but were revealed when we used an alternative modeling approach designed to identify complex nonlinear relationships. Specifically, we identified the nonlinear impact of baseline BP measurements on the joint effects of 2 closely related predictors (SBP and DBP) on repeatedly measured memory in each of 3 BP subgroups. Further, we found a nonlinear association of memory with age, in that memory was relatively stable until age 75 and then declined steadily.

As regards other covariates: (a) among normotensives, being female, APOE*4 negative, better educated, free of stroke/TIA history, performing any physical activity, having higher baseline BMI, currently using alcohol any type of antihypertensive medication, were associated with higher memory. (b) Among hypertensives, being female, better educated, free of stroke/TIA history, performing any physical activity, and having higher baseline BMI were associated with higher memory. Our hypotensive group was likely too small (<5% of the cohort) to have sufficient power to allow the detection of significant associations between BP and memory. Nevertheless, it is notable that at study entry they were the oldest, had the lowest baseline memory scores, were the mostly likely to report history of stroke/TIA, and to be taking antihypertensive drugs.

Population studies have clearly established that cognitive impairment and decline in late life are associated with high BP in midlife (fourth to seventh decades) and low BP in late life (eighth and ninth decades).22 While the underlying mechanisms are not completely understood, there is general consensus that sustained hypertension in midlife causes arterial remodeling and stiffening, disrupting the neurovascular unit and its autoregulatory functions, reducing not only the brain’s perfusion but its ability to clear beta-amyloid.23 Thus, subsequent low BP in late life may be insufficient to perfuse the brain adequately through the now narrowed and stiffened arterioles. Further, the effect of hypertension on overall brain health is influenced by additional factors such as age, chronicity of hypertension, and antihypertensive medication use.22

BP monitoring and management are challenging in older adults who are the most likely to be affected by multiple comorbidities, longer durations of abnormal BP, and poorer adherence to prescribed antihypertensive regimens. Often, the goal of maintaining BP in the conventional “normal” range may be unachievable, and sometimes, possibly detrimental. Previous studies have drawn attention to the risks of excessively tight BP control with subsequent ischemic and/or hypoxic states leading to hypoperfusion and faster progression of cognitive decline, in older adults with MCI and dementia treated with antihypertensive drugs.24,25 Related to this, our findings of lower DBP being more consistently associated with worse memory function (except in the hypotensive baseline BP group) may support work suggesting that both cardiac and cerebral perfusion is more dependent on diastolic function.26–28

The recent SPRINT-MIND randomized clinical trial1,2 focused on management of SBP, comparing an intensive strategy of lowering SBP to 120 mm Hg with a standard strategy of lowering SBP to 140 mm Hg among adults with hypertension over 3 years. The study detected no significant difference in hazard of developing dementia, possibly due to insufficient power, but did find a significantly reduced hazard of MCI in those whose SBP was lowered to 120 mm Hg. Among MYHAT hypertensives, we found that an SBP around 130 was associated with higher predicted memory score. Our results are not directly comparable to the SPRINT-MIND trial in hypertensive patients because ours was a population-based observational study conducted over 10 years, and our outcome was memory test performance rather than discrete events of developing MCI or dementia. Further, MYHAT participants had a mean age of almost 78 years, compared with about 68 years in SPRINT-MIND; there is evidence that in the oldest-old a higher perfusion pressure (using hypertension as a surrogate) may provide protection against cognitive decline.29

However, we also examined DBP, and in our hypertensive subgroup, a “normal” or higher DBP appeared to be most important in maintaining better predicted memory scores. The longitudinal AGES-Reykjavik Study30 found that the combination of midlife hypertension and late-life low DBP were associated with smaller whole brain and gray matter volumes as well as lower memory scores. The majority of clinical trials, like SPRINT-MIND, have focused solely on SBP. The average DBP in that trial was 78 mm Hg, similar to our hypertensive group, but greater than both our normotensive and hypotensive groups. Potentially, future trials might evaluate the effect of loosening BP control to avoid diastolic hypotension in older adults.

By not focusing solely on people with hypertension, we were able to show that the optimal BP levels associated with the highest memory scores were not uniform across the cohort but rather depended on their BP at study baseline. However, all participants were at least 65 when they entered the study, so our findings do not reflect midlife BP. We demonstrate that the smallest range of changes in predicted memory score were noted in the normotensive group, which also had the widest range of BPs associated with persistently normal memory performance (denoted by a predicted standardized memory score at or above the population average). This finding supports the importance of maintaining BPs within “normal” levels throughout the lifespan and is consistent with BP goals endorsed by the Joint National Committee (eighth edition).31

Unlike our study, a previous population-based study in Indianapolis19 was able to find a single optimal BP range around 135/88 in that cohort as a whole, using a semiparametric mixed modelling approach. The participants in that study were also aged 65+ years but were all African American; the outcome variable in that study was a global cognitive composite (the Community Screening Instrument for Dementia) rather than a memory composite as in our study. These methodological differences might help explain the difference in our findings.

As regards our statistical approach, the main benefits of using GAMMs with smoothing splines is the ability to detect the complexities of nonlinear, yet correlated, measures with a specific outcome, and also the ability to incorporate model selection and inferences using the F-statistic and corresponding P-value, as in generalized linear models. When interpreting the smoothing splines, one option is to use the value of the splines term at specific values of the predictor (1-dimensional splines, eg, age). Alternatively, we can utilize visualization tools like heat maps using 2-dimensional splines to identify combinations of the predictors associated with the same value of the outcome (eg, SBP and DBP). Because GAMMs with smoothing splines are more parameter-enriched than mixed models, we caution that they require more data points to obtain convergence and maintain acceptable power. Another potential challenge when using splines is the risk of overfitting (ie, fitting the splines term in an excessively complex form). Although the splines terms are fitted in a data-driven scheme and software provides useful default setup and model check features, investigators should still carefully check the model fitting results, especially the choice of tuning parameter during the model building process, and perform model selection to ensure that covariates significantly associated with the outcome are included. It is worth noting that GAMMs is based on the assumption of missingness at random. Nonrandom dropout would have to be accounted for using more complicated modeling approaches, such as joint modeling of GAMM and survival model, which is beyond the scope of this paper.

A potential study limitation is that we defined our “hypertensive” subgroup using the conventional BP threshold of 140/90 mm Hg. While higher than in current guidelines, it was the standard during the period of our study, and does not preclude our using it to demonstrate the GAMM method. Further, we illustrated the method using only the example of memory. Other cognitive domains such as attention and executive functioning should also be examined in future studies.

In summary, our study adds to the evolving understanding of the association between BP and cognition, 2 complex, yet common contributors to morbidity in older adults. By employing a flexible modeling method to investigate longitudinal memory and BP measurements from a community cohort, we were able to identify a nuanced relationship between the 2. From a methodological perspective, we have demonstrated an approach that captures the underlying nonlinear, nonuniform effects of predictors and makes optimal use of longitudinal data for valid inference. From a clinical perspective, we have generated testable future hypotheses regarding the need to maintain BPs within the conventional “normal” range throughout adulthood, and to consider how diastolic BP and cognition are related in late life. Our approach is consistent with the overall goal of developing treatment strategies at an individual level that consider the specific person within the context of guidelines and evidence-based medicine.


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blood pressure; memory; smoothing splines

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