# BFF: Bayesian, Fiducial, and Frequentist Analysis of Cognitive Engagement among Cognitively Impaired Older Adults

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## Abstract

**:**

## 1. Introduction

#### 1.1. Physical and Mental Health

#### 1.2. Stress

#### 1.3. Frequentist Paradigm

_{p}are available, they are not commonly used within the cognitive aging literature. We extend previous studies that examined normative aging processes within the frequentist paradigm to cognitively impaired older adults across three statistical paradigms to rigorously test principles from selective engagement theory [1].

#### 1.4. Bayesian Paradigm

#### 1.5. Fiducial Paradigm

#### 1.6. Present Study

## 2. Materials and Methods

#### 2.1. Participants

#### 2.2. Measures

#### Cognitive Engagement and Cardiovascular Monitoring

#### 2.3. Procedure

## 3. Analysis

#### 3.1. Data Preparation

_{00}= 27.88, SE = 28.92, p = 0.168), so the analyses treat the tertiles (Level 1) as nested within persons (Level 2).

#### 3.2. Modeling

Level 1 (within-person, t): |

SBP-R_{ti} = ß_{0ti} + ß_{1ti}(Tertile) + r_{ti} |

Level 2 (between-person, i): |

ß_{0i} = γ_{10} + γ_{01}(SBP baseline) + γ_{02}(Cognition) + γ_{03}(Physical health challenges) + u_{0i} |

ß_{1i} = γ_{10} + γ_{11}(Cognition) + γ_{12}(Physical Health Challenges) |

_{0}) and slopes (ß

_{1}) become the outcomes at the person level (Level 2). The within-person time effect was indexed by tertiles within tests (γ

_{10}). Main effects of SBP at baseline (γ

_{01}), cognition (γ

_{02}), and physical health challenges (γ

_{03}) were included as predictors of the intercept. The Cognition × Tertile (γ

_{11}) cross-level interaction tests for cognition differences in changes in SBP-R. The Physical Health Challenges × Tertile (γ

_{12}) cross-level interaction tests for physical health challenge differences in SBP-R. The random intercept (u

_{0i}) was allowed to vary across persons and was assumed to follow a multivariate normal distribution with a mean of zero and covariance matrix ∑. The residual error r

_{ti}is independent of the random intercept and follows a normal distribution with a mean of zero and variance σ

^{2}. Notice that technically speaking only the γs, σ

^{2}, and ∑ are the parameters of this model.

_{a}is a $n\times n$ matrix with (i,j) entry equal to 1 if the observations i and j are on the same subject and 0 otherwise, I

_{n}is the n × n identity matrix, and U is a vector of independent standard Gaussian random variables. For this model, the Jacobian matrix simplifies to a matrix obtained by concatenation of columns

## 4. Results

^{2}= 131.22, SE = 8.62) of the variance is within-person and 49% (τ

_{00}= 125.97, SE = 35.79) of the variance is between-person, indicating sufficient variability at each level for subsequent analyses.

#### 4.1. Frequentist Results

_{10}). People with higher cognition (γ

_{02}) as well as those with more physical health challenges (γ

_{03}) had lower SBP-R. Although changes in SBP-R did not differ by physical health challenges (γ

_{12}), changes in SBP-R did depend on cognition (γ

_{11}) (see Figure 1). SBP-R was stable for those with higher cognition (b = −0.11, SE = 0.14, p = 0.413), but SBP-R significantly decreased for those with lower cognition (b = −0.75, SE = 0.34, p = 0.027). There was a significant average decrease in SBP-R over time (γ

_{10}) in Model 2, as in Model 1. People with higher cognition (γ

_{02}) as well as those with more mental health challenges (γ

_{03}) had lower SBP-R. As in Model 1, we also found evidence of a Cognition × Tertile interaction (γ

_{11}) in Model 2 (Figure 1 also applies here). There was no evidence of mental health challenge differences in changes in SBP-R (γ

_{12}).

_{02}) had lower SBP-R, and those differences predicted changes in SBP-R (γ

_{11}), as in the previous models. We found evidence of cortisol differences in SBP-R changes (γ

_{12}, see Figure 2), such that those with low cortisol experienced a significant decrease in SBP-R (b = −0.54, SE = 0.17, p = 0.001), whereas those with high cortisol did not experience significant changes (b = 0.14, SE = 0.08, p = 0.081).

_{10}) and those with higher cognition had lower SBP-R (γ

_{02}). Although cognition (γ

_{11}) and physical health challenges (γ

_{12}) did not predict changes in SBP-R, changes in physical health challenges from Wave 1 to Wave 2 did modify the association between cognition and SBP-R (γ

_{05}, see Figure 3). Changes in physical health challenges were associated with a bigger difference in SBP-R for those who were lower in cognition compared to those higher in cognition. Increases in physical health challenges from Wave 1 to Wave 2 for those low in cognition corresponded with the lowest SBP-R.

_{02}), low mental health challenges at Wave 1 (γ

_{03}), and high mental health challenges at Wave 2 (γ

_{04}) had lower SBP-R. We also found evidence of an interaction where changes in mental health challenges from Wave 1 to Wave 2 did modify the association between cognition and SBP-R (γ

_{05}, see Figure 4). Changes in mental health challenges were associated with a bigger difference in SBP-R for those who were higher in cognition compared to those lower in cognition. Increases in mental health challenges from Wave 1 to Wave 2 for those higher in cognition corresponded with the lowest SBP-R.

_{02}). We also found evidence of two 2-way interactions (Tertile × Cortisol, γ

_{12}; Cognition × Cortisol, γ

_{05}) which were qualified by a significant 3-way interaction (Tertile × Cortisol × Cognition, γ

_{13}; see Figure 5). The steepest decrease in SBP-R was for those with low cognition who experienced a decrease in cortisol from Wave 1 to Wave 2.

#### 4.2. Bayesian Results

^{53}). However, the model that included the above effects plus the addition of the cross-level interaction between cognition and tertile (K = 7.75 × 10

^{53}see Figure 1) was only slightly less likely than the best fitting model (the ratio of the Bayes factor comparing the best fitting model with the model that includes the interaction is 0.89). Similar results were found for the models regarding cognition and mental health challenges. The best fitting model included SBP at baseline and main effects of cognition and mental health challenges (K = 1.64 × 10

^{62}), but the model that included the cross-level interaction between cognition and tertile (K = 5.53 × 10

^{61}) was ⅓ as likely as the best fitting model. With respect to the question addressing cortisol differences in changes in SBP-R, the best fitting model only included SBP at baseline and a main effect of cognition (K = 3.97 × 10

^{55}). The model that included the best fitting effects plus tertile, cortisol, Cognition × Tertile, and Cortisol × Tertile (K = 8.86 × 10

^{54}), was somewhat unlikely (0.22), indicating that the evidence is not strong enough to claim a null effect, but that the Cortisol × Tertile interaction (Figure 2) should be interpreted with some caution.

^{49}, see Figure 3). The next best fitting model included the terms above as well as physical health challenges at Wave 1 (K = 1.33 × 10

^{49}) and was 0.56 as likely as the best fitting model. With respect to mental health challenges, the best fitting model contained SBP at baseline, cognition, mental health challenges at Wave 1, mental health challenges at Wave 2, and the interaction between cognition and mental health challenges at Wave 2 (K = 1.32 × 10

^{27}, see Figure 4). The final model examined cognition and changes in cortisol differences in SBP-R over time. The best fitting model included SBP at baseline, cognition, cortisol at Wave 1, cortisol at Wave 2, and the interaction between cognition and cortisol at Wave 2 (K = 1.80 × 10

^{34}). The model that contained the 3-way interaction (Figure 5) was 0.01 as likely as the best fitting model, so we interpret this interaction with some caution.

#### 4.3. Fiducial Results

_{10}) in Model 4, the pattern of results for the fixed effects replicated all of the results from the frequentist models. However, the fiducial models provide more information (i.e., confidence intervals for all parameters, including random effects) and have the benefit of the fiducial distribution which does not rely on a prior (as in Bayesian models) but instead derives the information from the data.

## 5. Discussion

#### 5.1. Findings That Converge across the Paradigms

#### 5.2. Findings that Diverge across the Paradigms

#### 5.3. Limitations and Future Directions

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Additional Participant Information

#### Appendix A.2. Additional Measures Information

#### Appendix A.2.1. Physical Health Challenges

#### Appendix A.2.2. Mental Health Challenges

#### Appendix A.2.3. Cortisol

#### Appendix A.2.4. Cognitive Ability Assessment

#### Appendix A.3. Additional Procedure Information

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**Figure 3.**Changes in Physical Health Challenges Modifying the Relationship between Cognition and SBP-R.

**Figure 4.**Changes in Mental Health Challenges Modifying the Relationship between Cognition and SBP-R.

**Figure 5.**Changes in Cortisol Modifying Cognition Differences in Changes in SBP-R (Cortisol × Cognition × Tertile).

Variables | M | SD |
---|---|---|

SBP-R | 4.41 | 11.56 |

SBP at baseline | 130.92 | 21.31 |

Cognition Factor | −0.03 | 0.90 |

Recall (HCAP) | 3.90 | 1.88 |

Serial Subtraction (HCAP) | 3.56 | 1.80 |

CSI-D | 7.80 | 1.38 |

Letter-Number Sequencing | 8.42 | 3.04 |

Digit-Symbol Substitution | 46.30 | 15.71 |

Plus-Minus Task | 69.64 | 74.22 |

Stroop Task | 36.30 | 24.01 |

Short Blessed | 8.58 | 4.45 |

Mental Health Challenges | 0.01 | 0.68 |

Geriatric Depression Scale | 0.89 | 1.44 |

SF-36 Mental | 48.36 | 11.54 |

Physical Health Challenges | −0.04 | 0.67 |

Number of chronic conditions | 3.58 | 2.18 |

SF-36 Physical | 46.51 | 6.47 |

Cortisol (log) | 2.35 | 1.47 |

**Table 2.**Unstandardized Estimates, Standard Errors, and 95% Confidence Intervals Predicting Systolic blood pressure responsivity (SBP-R) using Frequentist and Generalized Fiducial Inference (GFI) Multilevel Models.

Variable | Parameter | Frequentist Estimates | GFI Estimates | 95% GFI CI |
---|---|---|---|---|

Model 1 | ||||

Intercept | γ_{00} | 5.82 (3.65) | 5.83 (3.87) | (−1.92, 13.30) |

Tertile | γ_{10} | −0.33 * (0.15) | −0.33 (0.15) | (−0.64, −0.03) |

SBP at baseline | γ_{01} | 0.22 ***(0.06) | 0.22 (0.07) | (0.09, 0.35) |

Cognition | γ_{02} | −11.82 (2.57) | −11.98 (3.16) | (−18.28, −6.11) |

Physical health challenges | γ_{03} | −6.59 * (2.57) | −6.53 (2.60) | (−11.65, −1.48) |

Tertile × Cognition | γ_{11} | 0.32 * (0.16) | 0.32 (0.16) | (0.01, 0.64) |

Tertile × Physical Health | γ_{12} | 0.13 (0.22) | 0.13 (0.21) | (−0.29, 0.56) |

Between-person variance | τ_{00} | 318.9 (17.86) | 356.37 | (160.57, 720.52) |

Within-person variance | σ^{2} | 121.5 (11.02) | 122.02 | (105.97, 140.45) |

Model 2 | ||||

Intercept | γ_{00} | 5.34 (5.42) | 5.31 (5.71) | (−5.89, 16.70) |

Tertile | γ_{10} | −0.31 * (0.14) | −0.31 (0.14) | (−0.58, −0.04) |

SBP at baseline | γ_{01} | 0.22 *** (0.06) | 0.22 (0.06) | (0.10, 0.33) |

Cognition | γ_{02} | −23.33 *** (2.68) | −23.27 (3.17) | (−29.33, −16.99) |

Mental health challenges | γ_{03} | −23.57 *** (3.13) | −23.58 (3.35) | (−30.08, −16.96) |

Tertile × Cognition | γ_{11} | 0.36 * (0.15) | 0.36 (0.15) | (0.07, 0.66) |

Tertile × Mental Health | γ_{12} | 0.15 (0.20) | 0.15 (0.20) | (−0.25, 0.53) |

Between-person variance | τ_{00} | 742.2 (27.24) | 808.12 | (384.35, 1542.38) |

Within-person variance | σ^{2} | 103.1 (10.15) | 103.76 | (90.12, 119.77) |

Model 3 | ||||

Intercept | γ_{00} | 4.74 (3.38) | 4.76 (3.54) | (−2.19, 11.89) |

Tertile | γ_{10} | −0.23 (0.15) | −0.23 (0.15) | (−0.53, 0.06) |

SBP at baseline | γ_{01} | 0.21 *** (0.05) | 0.22 (0.06) | (0.11, 0.33) |

Cognition | γ_{02} | −10.12 ***(2.32) | −10.20 (3.16) | (−15.62, −4.99) |

Cortisol | γ_{03} | −1.08 (1.16) | −1.07 (1.20) | (−3.40, 1.25) |

Tertile × Cognition | γ_{11} | 0.38 * (0.16) | 0.38 (0.16) | (0.06, 0.69) |

Tertile × Cortisol | γ_{12} | 0.22 * (0.10) | 0.22 (0.10) | (0.01, 0.42) |

Between-person variance | τ_{00} | 260.2 (16.13) | 289.03 | (135.09, 576.38) |

Within-person variance | σ^{2} | 121.5 (11.02) | 122.02 | (106.42, 139.90) |

Model 4 | ||||

Intercept | γ_{00} | −11.56 (17.21) | −12.14 (20.26) | (−52.43, 27.09) |

Tertile | γ_{10} | −0.37 * (0.19) | −0.37 (0.19) | (−0.75, 0.004) |

SBP at baseline | γ_{01} | 0.74 *** (0.06) | 0.75 (0.06) | (0.62, 0.87) |

Cognition | γ_{02} | −15.99 *** (3.01) | −16.02 (3.04) | (−22.02, −0.10) |

Physical challenges Wave 1 | γ_{03} | 13.00 (35.49) | 13.39 (42.37) | (−67.28, 96.14) |

Physical challenges Wave 2 | γ_{04} | −28.45 (29.14) | −29.09 (34.88) | (−96.27, 36.09) |

Tertile × Cognition | γ_{11} | 0.04 (0.24) | 0.04 (0.24) | (−0.43, 0.51) |

Tertile × Physical Wave 2 | γ_{12} | 0.02 (0.24) | 0.02 (0.24) | (−0.44, 0.49) |

Cognition × Phys. Wave 2 | γ_{05} | 51.37 *** (4.38) | 51.39 (4.46) | (42.60, 60.15) |

Tertile × Cog. × Phys. w2 | γ_{13} | 0.40 (0.30) | 0.40 (0.30) | (−0.19, 0.99) |

Between-person variance | τ_{00} | 2603.9 (51.03) | 3482.92 | (1127.96, 9974.09) |

Within-person variance | σ^{2} | 100.3 (10.02) | 101.26 | (84.28, 121.27) |

Model 5 | ||||

Intercept | γ_{00} | −1.68 (6.14) | −1.74 (7.13) | (−16.20, 12.70) |

Tertile | γ_{10} | −0.31 (0.25) | −0.31 (0.25) | (−0.81, 0.50) |

SBP at baseline | γ_{01} | 0.35 *** (0.07) | 0.35 (0.08) | (0.20, 0.87) |

Cognition | γ_{02} | −14.02 ** (4.26) | −14.08 (4.53) | (−23.19, −5.21) |

Mental challenges Wave 1 | γ_{03} | 48.08 * (19.32) | 47.43 (22.18) | (4.2887, 91.98) |

Mental challenges Wave 2 | γ_{04} | −35.44 * (14.73) | −34.94 (16.78) | (−69.057, −2.80) |

Tertile × Cognition | γ_{11} | 0.14 (0.31) | 0.14 (0.31) | (−0.46, 0.74) |

Tertile × Mental Wave 2 | γ_{12} | −0.04 (0.30) | −0.05 (0.30) | (−0.64, 0.55) |

Cognition × Ment.Wave 2 | γ_{05} | −7.55 * (3.65) | −7.55 (3.70) | (−14.63, −0.22) |

Tertile × Cog. × Ment. w2 | γ_{13} | 0.21 (0.34) | 0.21 (0.34) | (−0.47, 0.89) |

Between-person variance | τ_{00} | 371.2 (19.27) | 501.73 | (146.46, 1464.99) |

Within-person variance | σ^{2} | 176.5 (13.28) | 178.25 | (148.90, 212.98) |

Model 6 | ||||

Intercept | γ_{00} | −5.79 (9.29) | −5.86 (10.35) | (−26.25, 14.53) |

Tertile | γ_{10} | 0.22 (0.25) | 0.23 (0.25) | (−0.26, 0.72) |

SBP at baseline | γ_{01} | 0.43 *** (0.06) | 0.44 (0.07) | (0.31, 0.57) |

Cognition | γ_{02} | −11.31 *** (3.10) | −11.29 (3.18) | (−17.66, −5.11) |

Cortisol Wave 1 | γ_{03} | −8.59 (8.64) | −8.43 (9.62) | (−28.48, 10.68) |

Coritsol Wave 2 | γ_{04} | −6.09 (10.87) | −6.34 (12.07) | (−30.29, 17.62) |

Tertile × Cognition | γ_{11} | −0.02 (0.25) | −0.02 (0.25) | (−0.51, 0.48) |

Tertile × Cortisol Wave 2 | γ_{12} | 0.79 * (0.34) | 0.80 (0.34) | (0.13, 1.46) |

Cognition × Cort. Wave 2 | γ_{05} | 24.10 *** (3.95) | 24.32 (4.10) | (16.33, 32.35) |

Tertile × Cog. × Cort. w2 | γ_{13} | −0.83 * (0.38) | −0.84 (0.38) | (−1.58, −0.09) |

Between-person variance | τ_{00} | 928.7 (30.47) | 1146.28 | (429.03, 2817.93) |

Within-person variance | σ^{2} | 132.5 (11.51) | 133.46 | (113.88, 156.87) |

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## Share and Cite

**MDPI and ACS Style**

Neupert, S.D.; Growney, C.M.; Zhu, X.; Sorensen, J.K.; Smith, E.L.; Hannig, J.
BFF: Bayesian, Fiducial, and Frequentist Analysis of Cognitive Engagement among Cognitively Impaired Older Adults. *Entropy* **2021**, *23*, 428.
https://doi.org/10.3390/e23040428

**AMA Style**

Neupert SD, Growney CM, Zhu X, Sorensen JK, Smith EL, Hannig J.
BFF: Bayesian, Fiducial, and Frequentist Analysis of Cognitive Engagement among Cognitively Impaired Older Adults. *Entropy*. 2021; 23(4):428.
https://doi.org/10.3390/e23040428

**Chicago/Turabian Style**

Neupert, Shevaun D., Claire M. Growney, Xianghe Zhu, Julia K. Sorensen, Emily L. Smith, and Jan Hannig.
2021. "BFF: Bayesian, Fiducial, and Frequentist Analysis of Cognitive Engagement among Cognitively Impaired Older Adults" *Entropy* 23, no. 4: 428.
https://doi.org/10.3390/e23040428