Joint Modeling of Longitudinal Item Response Data and Survival - - PowerPoint PPT Presentation

▶
joint modeling of longitudinal item response data and
SMART_READER_LITE
LIVE PREVIEW

Joint Modeling of Longitudinal Item Response Data and Survival - - PowerPoint PPT Presentation

Aug 11, 2023 371 likes •1.2k views

Joint Modeling of Longitudinal Item Response Data and Survival Jean-Paul Fox University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede, Netherlands J.-P. Fox Bayesian Item

slide-1
SLIDE 1

Joint Modeling of Longitudinal Item Response Data and Survival

Jean-Paul Fox

University of Twente Department of Research Methodology, Measurement and Data Analysis Faculty of Behavioural Sciences Enschede, Netherlands

J.-P. Fox Bayesian Item Response Modeling

slide-2
SLIDE 2

Overview

Introduction Cross-classified Response Data

J.-P. Fox Bayesian Item Response Modeling

slide-3
SLIDE 3

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE)

J.-P. Fox Bayesian Item Response Modeling

slide-4
SLIDE 4

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes

J.-P. Fox Bayesian Item Response Modeling

slide-5
SLIDE 5

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion

J.-P. Fox Bayesian Item Response Modeling

slide-6
SLIDE 6

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion

J.-P. Fox Bayesian Item Response Modeling

slide-7
SLIDE 7

Responses to Test Items

Collection of responses on tests, i = 1, . . . , N persons who answer k = 1, . . . , K items, resulting in N × K 0/1 responses Y : Y =      1 1 . . . Y1K 1 1 . . . Y2K . . . ... . . . YN1 1 . . . YNK     

J.-P. Fox Bayesian Item Response Modeling

slide-8
SLIDE 8

Responses to Test Items

Collection of responses on tests, i = 1, . . . , N persons who answer k = 1, . . . , K items, resulting in N × K 0/1 responses Y : Y =      1 1 . . . Y1K 1 1 . . . Y2K . . . ... . . . YN1 1 . . . YNK     

  • Develop a model to say something about the structure of

this data set

J.-P. Fox Bayesian Item Response Modeling

slide-9
SLIDE 9

Responses to Test Items

Collection of responses on tests, i = 1, . . . , N persons who answer k = 1, . . . , K items, resulting in N × K 0/1 responses Y : Y =      1 1 . . . Y1K 1 1 . . . Y2K . . . ... . . . YN1 1 . . . YNK     

  • Develop a model to say something about the structure of

this data set

  • Structure: person and item effects.

J.-P. Fox Bayesian Item Response Modeling

slide-10
SLIDE 10

Stage 1: Modeling Success Probabilities

P (Yik = 1 | θi, ξk) = F (akθi − bk) θi ∼ N

  • µθ, σ2

θ

  • J.-P. Fox

Bayesian Item Response Modeling

slide-11
SLIDE 11

Stage 1: Modeling Success Probabilities

P (Yik = 1 | θi, ξk) = F (akθi − bk) θi ∼ N

  • µθ, σ2

θ

  • Response observations k are nested within persons, random

person effects (latent variable)

J.-P. Fox Bayesian Item Response Modeling

slide-12
SLIDE 12

Stage 1: Modeling Success Probabilities

P (Yik = 1 | θi, ξk) = F (akθi − bk) θi ∼ N

  • µθ, σ2

θ

  • Response observations k are nested within persons, random

person effects (latent variable)

  • Response observations k are nested within items,

fixed/random item effects.

J.-P. Fox Bayesian Item Response Modeling

slide-13
SLIDE 13

Two-Parameter Item Response Model

Individual i

qi yik

Item k

ak bk mq sq

J.-P. Fox Bayesian Item Response Modeling

slide-14
SLIDE 14

Likelihood-Model

Collection of N × K responses, N persons and K items

J.-P. Fox Bayesian Item Response Modeling

slide-15
SLIDE 15

Likelihood-Model

Collection of N × K responses, N persons and K items P (Yik = 1 | θi, ak, bk) =

  • exp(d(akθi−bk))

1+exp(d(akθi−bk))

Logistic Model Φ(akθi − bk) Probit Model

J.-P. Fox Bayesian Item Response Modeling

slide-16
SLIDE 16

Likelihood-Model

Collection of N × K responses, N persons and K items P (Yik = 1 | θi, ak, bk) =

  • exp(d(akθi−bk))

1+exp(d(akθi−bk))

Logistic Model Φ(akθi − bk) Probit Model p (y | θ, a, b) =

  • i
  • k

F (ηik)yik (1 − F (ηik))1−yik

  • where ηik = akθi − bk

J.-P. Fox Bayesian Item Response Modeling

slide-17
SLIDE 17

Population Model for Item Parameters

Stage 2: Prior for Item Parameters (ak, bk)t ∼ N (µξ, Σξ) IAk(ak), where the set Ak = {ak ∈ R, ak > 0}

J.-P. Fox Bayesian Item Response Modeling

slide-18
SLIDE 18

Population Model for Item Parameters

Stage 2: Prior for Item Parameters (ak, bk)t ∼ N (µξ, Σξ) IAk(ak), where the set Ak = {ak ∈ R, ak > 0} Stage 3: Hyper prior Σξ ∼ IW(ν, Σ0) µξ | Σξ ∼ N(µ0, Σξ/K0).

J.-P. Fox Bayesian Item Response Modeling

slide-19
SLIDE 19

Population Model for Person Parameter

Stage 2: Prior for Person Parameters θi ∼ N(µθ, σ2

θ).

Respondents are sampled independently and identically distributed.

J.-P. Fox Bayesian Item Response Modeling

slide-20
SLIDE 20

Population Model for Person Parameter

Stage 2: Prior for Person Parameters θi ∼ N(µθ, σ2

θ).

Respondents are sampled independently and identically distributed. Stage 3: Hyper prior σ2

θ ∼ IG(g1, g2)

µθ | σ2

θ ∼ N(µ0, σ2 θ/n0).

J.-P. Fox Bayesian Item Response Modeling

slide-21
SLIDE 21

Longitudinal Item Response Data

  • Discrete response data Yijk : (subject i, measurement
  • ccasion j, item k)

J.-P. Fox Bayesian Item Response Modeling

slide-22
SLIDE 22

Longitudinal Item Response Data

  • Discrete response data Yijk : (subject i, measurement
  • ccasion j, item k)
  • Several measurement occasions j = 1, . . . , ni, several points

in time.

J.-P. Fox Bayesian Item Response Modeling

slide-23
SLIDE 23

Longitudinal Item Response Data

  • Discrete response data Yijk : (subject i, measurement
  • ccasion j, item k)
  • Several measurement occasions j = 1, . . . , ni, several points

in time.

  • Latent Growth Modeling

J.-P. Fox Bayesian Item Response Modeling

slide-24
SLIDE 24

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

J.-P. Fox Bayesian Item Response Modeling

slide-25
SLIDE 25

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

  • 1. Subjects not measured on the same time points across time

(include all data)

J.-P. Fox Bayesian Item Response Modeling

slide-26
SLIDE 26

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

  • 1. Subjects not measured on the same time points across time

(include all data)

  • 2. Number of observations per subject may vary

J.-P. Fox Bayesian Item Response Modeling

slide-27
SLIDE 27

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

  • 1. Subjects not measured on the same time points across time

(include all data)

  • 2. Number of observations per subject may vary
  • 3. Follow-up times not uniform across subjects (time a

continuous variable, individualized schedule)

J.-P. Fox Bayesian Item Response Modeling

slide-28
SLIDE 28

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

  • 1. Subjects not measured on the same time points across time

(include all data)

  • 2. Number of observations per subject may vary
  • 3. Follow-up times not uniform across subjects (time a

continuous variable, individualized schedule)

  • 4. Handle time-invariant and time-varying covariates

J.-P. Fox Bayesian Item Response Modeling

slide-29
SLIDE 29

Latent Growth Modeling

  • Model Latent Developmental Trajectories:

θij = β0i + β1iTimeij + eij β0i = γ00 + u0i β1i = γ10 + u1i

  • 1. Subjects not measured on the same time points across time

(include all data)

  • 2. Number of observations per subject may vary
  • 3. Follow-up times not uniform across subjects (time a

continuous variable, individualized schedule)

  • 4. Handle time-invariant and time-varying covariates
  • 5. Estimate subject-specific change across time (average

change)

J.-P. Fox Bayesian Item Response Modeling

slide-30
SLIDE 30

Latent Growth Modeling

  • Specify curvilinear individual change, e.g., polynomial

individual change of any order

J.-P. Fox Bayesian Item Response Modeling

slide-31
SLIDE 31

Latent Growth Modeling

  • Specify curvilinear individual change, e.g., polynomial

individual change of any order

  • Model the covariance structure of the level-1 measurement

errors explicitly.

J.-P. Fox Bayesian Item Response Modeling

slide-32
SLIDE 32

Latent Growth Modeling

  • Specify curvilinear individual change, e.g., polynomial

individual change of any order

  • Model the covariance structure of the level-1 measurement

errors explicitly.

  • Model change in several domains simultaneously.

J.-P. Fox Bayesian Item Response Modeling

slide-33
SLIDE 33

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion

J.-P. Fox Bayesian Item Response Modeling

slide-34
SLIDE 34

Mini-Mental State Examination (MMSE)

  • Data: 4016 measurements of 668 subjects (4-16

measurement occasions)

J.-P. Fox Bayesian Item Response Modeling

slide-35
SLIDE 35

Mini-Mental State Examination (MMSE)

  • Data: 4016 measurements of 668 subjects (4-16

measurement occasions)

  • 26 MMSE (binary) items;

J.-P. Fox Bayesian Item Response Modeling

slide-36
SLIDE 36

Mini-Mental State Examination (MMSE)

  • Data: 4016 measurements of 668 subjects (4-16

measurement occasions)

  • 26 MMSE (binary) items;
  • What day of the week is it? (orientation)

J.-P. Fox Bayesian Item Response Modeling

slide-37
SLIDE 37

Mini-Mental State Examination (MMSE)

  • Data: 4016 measurements of 668 subjects (4-16

measurement occasions)

  • 26 MMSE (binary) items;
  • What day of the week is it? (orientation)
  • pencil - What is this? (language)

J.-P. Fox Bayesian Item Response Modeling

slide-38
SLIDE 38

Mini-Mental State Examination (MMSE)

  • Data: 4016 measurements of 668 subjects (4-16

measurement occasions)

  • 26 MMSE (binary) items;
  • What day of the week is it? (orientation)
  • pencil - What is this? (language)
  • subtract 7 from 100 (attention/concentration)

J.-P. Fox Bayesian Item Response Modeling

slide-39
SLIDE 39

Demographics

Table: Demographic information of the study participants.

Participants (N = 668) Gender Male 329 Female 339 Age start mean 50-59 55 41 60-69 195 149 70-79 323 315 80-89 149 215 90-100 9 14 Average sum score 24 − 26 302 22 − 23 66 20 − 21 47 18 − 19 61 15 − 17 66 < 14 126

J.-P. Fox Bayesian Item Response Modeling

slide-40
SLIDE 40

2 4 6 Follow-up Time (years) 5 10 15 20 25 MMSE Sum Scores

J.-P. Fox Bayesian Item Response Modeling

slide-41
SLIDE 41

Mixture IRT Modeling

Modeling of asymmetrical data: Define latent groups g1 and g2 p (θij | Ω) =

2

  • g=1

πigp (θij | Ωg)

J.-P. Fox Bayesian Item Response Modeling

slide-42
SLIDE 42

Mixture IRT Modeling

Modeling of asymmetrical data: Define latent groups g1 and g2 p (θij | Ω) =

2

  • g=1

πigp (θij | Ωg) P (Gi = 1 | yi, θi) = πi1 ni

j=1 p (yij | θij) p (θij | Ω1)

  • g=1,2 πig

ni

j=1 p (yij | θij) p (θij | Ωg)

J.-P. Fox Bayesian Item Response Modeling

slide-43
SLIDE 43

Likelihood-model M1

Measurement Part M1 P(Yijk = 1 | θij, ak, bk) = F (akθij − bk)

J.-P. Fox Bayesian Item Response Modeling

slide-44
SLIDE 44

Likelihood-model M1

Measurement Part M1 P(Yijk = 1 | θij, ak, bk) = F (akθij − bk) Latent Growth Part M1 p

  • θij | γ00, τ 2, σ2

= πi1φ

  • µij,1, σ2

+ πi2φ

  • µij,2, σ2

µij,1 = γ00,1 + ui0,1 µij,2 = γ00,2 + ui0,2

  • γ(2) < γ(1)

J.-P. Fox Bayesian Item Response Modeling

slide-45
SLIDE 45

Table: MMSE: Parameter Estimates of Model M1.

Mixture MLIRT M1 Decline Stable Mean SD Mean SD Fixed Effects γ00 Intercept

  • .998

.037 .689 .030 Random Effects Within-individual σ2

θ Residual variance

.133 .003 .133 .003 Between-individual τ 2

00 Intercept

.211 .015 .211 .015

J.-P. Fox Bayesian Item Response Modeling

slide-46
SLIDE 46

Likelihood-model M2

Measurement Part M2 P(Yijk = 1 | θij, ak, bk) = F (akθij − bk)

J.-P. Fox Bayesian Item Response Modeling

slide-47
SLIDE 47

Likelihood-model M2

Measurement Part M2 P(Yijk = 1 | θij, ak, bk) = F (akθij − bk) Latent Growth Part M2 p

  • θij | γ, T , σ2

= πi1φ

  • µij,1, σ2

+ πi2φ

  • µij,2, σ2

µij,g = βi0,g + βi1,gTimeij βi0,g = γ00,g + ui0,g βi1,g = γ10,g + ui1,g, and ui,g ∼ N (0, Tg) with Tg a diagonal matrix with elements τ 2

00,g and τ 2 11,g for g = 1, 2.

J.-P. Fox Bayesian Item Response Modeling

slide-48
SLIDE 48

Table: MMSE: Parameter estimates of Model M2

Mixture MLIRT M2 Decline Stable Mean SD Mean SD Fixed Effects γ00 Intercept −.332 .037 .913 .012 Time Variables γ10 Follow-up time −.274 .013 −.007 .004 Random Effects Within-individual σ2

θ Residual variance

.043 .001 .043 .001 Between-individual τ 2

00 Intercept

.471 .038 .016 .003 τ 2

11 Follow-up time

.047 .004 .002 .000

J.-P. Fox Bayesian Item Response Modeling

slide-49
SLIDE 49

Estimated random effects of cognitive impairment

  • 2
  • 1

1 2 Initial Score

  • 1.0
  • 0.8
  • 0.6
  • 0.4
  • 0.2

0.0 0.2 Individual Trend Patients Controls Not Classified J.-P. Fox Bayesian Item Response Modeling

slide-50
SLIDE 50

Survival Time Data

  • Time to certain (non-repeatable) events (e.g., death,

response, failure time)

J.-P. Fox Bayesian Item Response Modeling

slide-51
SLIDE 51

Survival Time Data

  • Time to certain (non-repeatable) events (e.g., death,

response, failure time)

  • Persons were followed to death or (right-)censored in a

study survival time vi = ti ti ≤ ci observed (uncensored) ci ti > ci not observed (censored)

J.-P. Fox Bayesian Item Response Modeling

slide-52
SLIDE 52

Distribution of Survival Times

  • Survivor function: probability survives longer than t

S(t) = P(T > t) = 1 − F(t)

  • Probability density function

f(t) ≥ 0, t ≥ 0

  • Hazard function: conditional failure rate

h(t) = f(t) 1 − F(t) = f(t) S(t)

J.-P. Fox Bayesian Item Response Modeling

slide-53
SLIDE 53

Right-Censored Observations

Joint probability of observing data v: f(v | η) =

r

  • i=1

f(ti, η)

n

  • i=r+1

S(ci, η)

J.-P. Fox Bayesian Item Response Modeling

slide-54
SLIDE 54

Problems in Survival Modeling

  • Censoring: data missingness, subject does not undergo the

event

J.-P. Fox Bayesian Item Response Modeling

slide-55
SLIDE 55

Problems in Survival Modeling

  • Censoring: data missingness, subject does not undergo the

event

  • Unobserved between-individual variation in the probability

to experience the event

J.-P. Fox Bayesian Item Response Modeling

slide-56
SLIDE 56

Problems in Survival Modeling

  • Censoring: data missingness, subject does not undergo the

event

  • Unobserved between-individual variation in the probability

to experience the event

  • Presence of time-varying covariates (e.g., prognostic

factors)

J.-P. Fox Bayesian Item Response Modeling

slide-57
SLIDE 57

Identify Prognostic Factors

  • Prognosis, course, outcome of a disease

J.-P. Fox Bayesian Item Response Modeling

slide-58
SLIDE 58

Identify Prognostic Factors

  • Prognosis, course, outcome of a disease
  • Model probability of surviving given (possible) prognostic

factors (risk factors, individual characteristics)

J.-P. Fox Bayesian Item Response Modeling

slide-59
SLIDE 59

Identify Prognostic Factors

  • Prognosis, course, outcome of a disease
  • Model probability of surviving given (possible) prognostic

factors (risk factors, individual characteristics)

  • Popular model: Cox Proportional Hazards Model:

h(t | x1) h(t | x2) = constant h(t | x) = h0(t)g(x) = h0(t) exp(ηtx)

J.-P. Fox Bayesian Item Response Modeling

slide-60
SLIDE 60

Identify Prognostic Factors

  • Prognosis, course, outcome of a disease
  • Model probability of surviving given (possible) prognostic

factors (risk factors, individual characteristics)

  • Popular model: Cox Proportional Hazards Model:

h(t | x1) h(t | x2) = constant h(t | x) = h0(t)g(x) = h0(t) exp(ηtx)

  • Violate proportional hazards assumption when using

time-varying covariates

J.-P. Fox Bayesian Item Response Modeling

slide-61
SLIDE 61

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion

J.-P. Fox Bayesian Item Response Modeling

slide-62
SLIDE 62

Joint Modeling Approach

  • Patients and controls with different (latent,

time-continuous) backgrounds may have different survival prognosis

J.-P. Fox Bayesian Item Response Modeling

slide-63
SLIDE 63

Joint Modeling Approach

  • Patients and controls with different (latent,

time-continuous) backgrounds may have different survival prognosis

  • Longitudinal factor/covariates are measured infrequently

and with measurement error

J.-P. Fox Bayesian Item Response Modeling

slide-64
SLIDE 64

Joint Modeling Approach

  • Patients and controls with different (latent,

time-continuous) backgrounds may have different survival prognosis

  • Longitudinal factor/covariates are measured infrequently

and with measurement error

  • Subjects enter the study at different time-points, measured

at different times, different number of measurements

J.-P. Fox Bayesian Item Response Modeling

slide-65
SLIDE 65

Joint Modeling: Two-stage procedure

  • Survival information not used in modeling the covariate

process.

J.-P. Fox Bayesian Item Response Modeling

slide-66
SLIDE 66

Joint Modeling: Two-stage procedure

  • Survival information not used in modeling the covariate

process.

  • New growth curves are estimated at each new event

(interpretability)

J.-P. Fox Bayesian Item Response Modeling

slide-67
SLIDE 67

Joint Modeling: Two-stage procedure

  • Survival information not used in modeling the covariate

process.

  • New growth curves are estimated at each new event

(interpretability)

  • Handle measurement error in time-dependent (latent)

covariate(s) survival model

J.-P. Fox Bayesian Item Response Modeling

slide-68
SLIDE 68

Joint Modeling

  • Joint distribution (survival data, response data):

p (t, y | x) =

  • p (t, y | η, x) p (η | x) dη

= p (t | η, x) p (y | η) p (η | Ω, x) p(Ω)dηdΩ = p (t | η, Ω, x) p (y | η) p (η | Ω, x) p(Ω)dηdΩ

J.-P. Fox Bayesian Item Response Modeling

slide-69
SLIDE 69

Time Density Function

  • Define vi = min(ti, ci),

Di = 1 Event observed Censored observation

J.-P. Fox Bayesian Item Response Modeling

slide-70
SLIDE 70

Time Density Function

  • Define vi = min(ti, ci),

Di = 1 Event observed Censored observation

  • Density Function:

f (vi, di | η, Ω) = h (vi | η, Ω)di S (vi | η, Ω) = h (vi | η, Ω)di exp

  • −

vi h (u | η(u), Ω) du

  • J.-P. Fox

Bayesian Item Response Modeling

slide-71
SLIDE 71

Time Density Function

  • Define vi = min(ti, ci),

Di = 1 Event observed Censored observation

  • Density Function:

f (vi, di | η, Ω) = h (vi | η, Ω)di S (vi | η, Ω) = h (vi | η, Ω)di exp

  • −

vi h (u | η(u), Ω) du

  • Define subject-specific time-intervals, til − ti(l+1);

fil

  • di, ti(l+1), til | ηl
  • =

S

  • ti(l+1) | ηl

1−di f

  • ti(l+1) | ηl

di S (tl | ηl)

J.-P. Fox Bayesian Item Response Modeling

slide-72
SLIDE 72

Time Density Function

  • For subject i

fi (di, ti | η) =

l=Li−1

  • l=0

fil

  • di, til, ti(l+1) | ηl
  • =

l=Li−1

  • l=0

S

  • ti(l+1) | ηl

1−di f

  • ti(l+1) | ηl

di S (tl | ηl)

J.-P. Fox Bayesian Item Response Modeling

slide-73
SLIDE 73

Overview

Introduction Cross-classified Response Data Longitudinal Response Data Mini-Mental State Examination (MMSE) Survival Analysis Joint Modeling of Latent Developmental Trajectories and Survival Joint (Response, Survival) Analysis Outcomes Discussion

J.-P. Fox Bayesian Item Response Modeling

slide-74
SLIDE 74

Estimated Parameters Mixture Model

Decline Stable EAP SD EAP SD Fixed Effects γ00 Intercept

  • .709

.031 .776 .036 γ01 Time Slope

  • .112

.012

  • .009

.006 Variance Components τ 2 Between Individual .244 .124 .134 .118 σ2 Residual .219 .029 .219 .029 Mixture Proportion π .432 .020 .568 .020

J.-P. Fox Bayesian Item Response Modeling

slide-75
SLIDE 75

Results: Comparing Models

Model Density Covariates Groups DIC M1 Exponential 1 2 2146.0 M2 Weibull 1 2 1872.3 M3 Lognormal 1 2 1905.4 M4 Exponential 1, θ 2 2041.7 M5 Weibull 1, θ 2 1836.0 M6 Lognormal 1, θ 2 1816.1 M7 Weibull 1, θ, Male, Age 2 1775.0 M8 Lognormal 1, θ, Male, Age 2 1768.1 M9 Weibull 1, θ, Male, Age 1 1856.3 M10 Lognormal 1, θ, Male, Age 1 1858.3

J.-P. Fox Bayesian Item Response Modeling

slide-76
SLIDE 76

Stratified Lognormal Survival Model M10

Decline (g=2) Stable (g=1) EAP SD EAP SD Fixed Effects β0,g Intercept 1.986 .072 2.561 .081 β1,g Male

  • .270

.064

  • .282

.076 β2,g Age (standardized)

  • .179

.080

  • .212

.090 Λg Cognitive Function .369 .041 .254 .046 Variance Components σ2

S Residual

.362 .031 .362 .031

J.-P. Fox Bayesian Item Response Modeling

slide-77
SLIDE 77

Time (years) P(Survival)

2 4 6 8 10

0.0 0.2 0.4 0.6 0.8 1.0 Stable Decline

J.-P. Fox Bayesian Item Response Modeling

slide-78
SLIDE 78

Cognitive Function P(Survival)

  • 0.6
  • 0.4
  • 0.2

0.0 0.6 0.7 0.8 0.9 1.0 1 year 2 years 3 years 4 years 5 years Stable Fifty-Fifty Decline J.-P. Fox Bayesian Item Response Modeling

slide-79
SLIDE 79

Sum score Latent variable estimate 5 10 15 20 25 30

  • 2
  • 1

1

MMSE: Cognitive Function

Time in years P(survival) 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0

Sum score = 15

Time in years P(survival) 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0

Sum score = 20

Time in years P(survival) 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0 1 2 3 4 5 0.4 0.6 0.8 1.0

Sum score = 25 J.-P. Fox Bayesian Item Response Modeling

slide-80
SLIDE 80

Discussion

  • Mental health (individual trajectory of cognitive

impairment) serves as a time-varying covariate

J.-P. Fox Bayesian Item Response Modeling

slide-81
SLIDE 81

Discussion

  • Mental health (individual trajectory of cognitive

impairment) serves as a time-varying covariate

  • Making inferences at the level of individuals (patients) and

their disease trajectories

J.-P. Fox Bayesian Item Response Modeling

slide-82
SLIDE 82

Discussion

  • Mental health (individual trajectory of cognitive

impairment) serves as a time-varying covariate

  • Making inferences at the level of individuals (patients) and

their disease trajectories

  • Combine cognitive test outcomes with other indicators to

serve as a diagnostic tool

J.-P. Fox Bayesian Item Response Modeling