Bayesian Change Point Models for Analysis of Menstrual Diary Data at the Approach of Menopause - PowerPoint Presentation

Slide1

Bayesian Change Point Models for Analysis of Menstrual Diary Data at the Approach of Menopause

Michael

R.

Elliott

2

, Xiaobi (Shelby) Huang

1

,

Sioban

Harlow

3

1.

Genzyme

, a Sanofi

Company

2

.

Department of Biostatistics, University of

Michigan

3.

Department

of

Epidemiology,

University of Michigan

Modern Model Methods 2016Slide2

2Slide3

Introduction

Goal for women’s menstrual studies: identify associations between women’s menstrual characters and women’s health

Why?

Menstrual cycles are the most easily observed markers of ovarian functions.

Alterations in bleeding are a significant source of

gyne-cologic

morbidity, especially in late reproductive life. Menopausal transition is a period of critical change in women's biology and health status.

3Slide4

How do we define and identify ONSET of the transition?

1981: Metcalf and

Livesey

. Pituitary–ovarian function in normal women during the menopausal transition.

1994:

Brambilla

et al. Defining the Perimenopause for Application in Epidemiologic Investigations. 2000: Mitchell et al. Three stages of the menopausal transition from the Seattle Midlife Women's Health Study: Toward a more precise definition.

2001:

Soules

et al. Executive summary: Stages of reproductive aging workshop (STRAW). 2002: Taffe and Dennerstein. Menstrual patterns leading to the final menstrual period. 2007: The ReSTAGE Collaboration. Recommendations from a multi-study evaluation of proposed criteria for Staging Reproductive Aging.

4Slide5

Previous Approaches

Visual browsing of menstruation patterns.

Summary statistics of sliding windows over age.

Linear mixed model.

Major

proble

m: lack of precision - Traditional longitudinal models tend to underutilize information from subject-level in clinical and epidemiological research settings, at least in part because of the lack of methods for such analyses.

5Slide6

GoalsInitial goal: compare menstrual pattern changes between two generations of women

Subsequent goals:

Model how menstrual cycle length and variability change when women approach menopause.

Develop method to impute various types of

missingness

.

Find potential biomarkers for women’s menopausal transition.Define subgroups of menstruation patterns.6Slide7

Outline

TREMIN Trust Data

Bayesian

Changepoint

Model

Missing Data Imputation

Menstruation Patterns7Slide8

8

TREMIN Trust Data

TREMIN:

Ongoing 70 year longitudinal menstrual calendar study

Initiated by Dr. Alan

Treloar

of University of Minnesota in 1934Cohort I: 1936-1939, 2350 U. Minnesota undergraduates

Cohort II: 1961-1964, 1367 U. Minnesota undergraduates

One of the only two data sets worldwide for individual women’s menstrual diary data across their reproductive life span.Slide9

Data in analysis

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10

Missing Data

Missing due to hormone use

Hysterectomy or bilateral

oophorectomy

surgeries

Non-reporting or withdrawal from the studyNon- menstrual intervals are not treated as missing:

Pregnancy intervals

First two cycles after a birth

First cycle after a spontaneous abortionSlide11

11

Four Typical Women in TREMIN Cohort

-

Blue line

: cycle

lengths (on log scale).

-Black dot (●): Observed FMP.

-

Red dot (●):

Truncated by surgery.-

Green bars (||)

:

Pregnancy interval.

-

Red bars (||)

: Missing

due to hormone use.

-Black bars (||)/circle (

○

):

Intermittent

missingness

due to

nonreporting

.Slide12

Outline

TREMIN Trust Data

Bayesian

Changepoint

Model

Missing Data Imputation

Menstruation PatternsSubgroups of Menstruation Patterns12Slide13

Patterns of Menstruation Cycle Lengths

13

Regular cycling

Premenopausal irregularity

(plot form

Lisabeth

et al. 2004) Slide14

Thoughts of Modeling

Common pattern: how

menstrual cycle length changes over

age

Variability has the same pattern

Despite the overall pattern, individual women have their unique change points, intercepts and slopesSlide15

15

Bayesian Changepoint Model for Mean and Variance

Subject Level:

Population Level:

Some notations:

-

i

th

subject’s

tth cycle length. - age of ith

subject’s

t

th

menstruation cycle.

- covariates of

i

th

subject.Slide16

InferenceJoint posterior distribution:

16Slide17

Outline

TREMIN Trust Data

Bayesian

Changepoint

Model

Missing Data Imputation

Menstruation PatternsSubgroups of Menstruation Patterns17Slide18

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Imputation of Missing Data - Complexities

Large amount of

missingness

Various reasons of missing: hormone, surgery, loss of follow up

Cycle lengths and ages should match

When to stop if FMP was not observed? How to impute FMP?Slide19

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Imputation Procedure

Step 1: Obtain initial parameters from complete data analysis:

subjects with complete cases, assign

subjects with missing data, draw

Step 2: Impute the missing data using :

Imputation draws are from the model prediction:

Update ages and cycle lengths together:Slide20

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Age

42.0 42.35

End

Start

(L)

Original data

Imputation

Adjusting

Adjusted imputed age

42.0 42.07 42.16 42.28 42.35

0.07 0.09 0.12 0.08

Adjusted imputed cycle length (year) of one set

Imputed age

Imputed cycle length (year)

42.0 42.07 42.16 42.28 42.40

0.07 0.09 0.12 0.12

(L’)

Imputation: How

to fill the missing

gaps

Cut the last segment length to fit the gap length

Find 50 sets of imputations and perform importance samplingSlide21

Imputation: Final Menstruation Periods

If FMPs are not observed: impute and update the data until imputed FMP or when , whichever happens first.

Model the age at FMP as a piecewise exponential distribution with hazard , for

Knots are set at one year or 0.5 year gaps between age 40 and 60, assuming the risk of having FMP before age 40 is zero.

Find the probability of FMP occurring between time interval

, given the event has not occurred before

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Imputation: Gaps till FMPs Every time after a segment is imputed, draw a

bernoulli

variable

to judge whether it is the final menstruation period.

If any imputed cycle is longer than 365 days or an imputed age is larger than 60, stop imputing and treat the corresponding age as FMP.

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Age

FMP

Censoring

W=0

W=0

W=0

W=1

48.0

48.07

48.16

48.28

52.3Slide23

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Imputation Procedure

Step 3: Update parameters using Gibbs steps based on the imputed data set we obtained in step 2.

Step 4: Using the updated parameters in 3 to impute another imputed data set using method stated in step 2.

Step 5: Repeat step 3 and 4 for many times until we obtain converged MCMC chains.Slide24

Posterior Model CheckConvergence:

Two MCMC chains with different starting values;10,000 iterations each after “burn-in”.

Gelman

and Rubin statistic: 99.2% individual level parameters and all population level parameters achieved convergence.

Model fit:

Posterior predictive Chi-square test for cycle lengths.

Compare observed FMP with replicated FMPs.24Slide25

Outline

TREMIN Trust Data

Bayesian

Changepoint

Model

Missing Data Imputation

Menstruation PatternsSubgroups of Menstruation Patterns25Slide26

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Results: Individual Level Parameters

Histogram of Slide27

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Individual Level Parameters

Posterior mean and associated 95% posterior predictive interval of the cycle length mean and the upper and lower 2.5 percentiles for the cycle distribution:Slide28

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Population Level Parameters

Posterior mean and 95% predictive intervals for mean population level parameters :Slide29

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Menstruation Pattern Characteristics

Mean cycle length declines slightly until

changepoint

, then increases rapidly.

Cycle lengths are stable on average until change-point, then variability explodes.

Variability begins increasing well in advance (3 years) of longer cycle lengths.Slide30

Population Level Parameters

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Mean intercept

Mean

slope before change-point

Mean

slope after change-

point

Mean

Change-point

Log-

Var

intercept

Log-

Var

slope before change-point

Log-

Var

slope after change-

Point

Var

Change-point

Mean intercept

1

-

0.13

-

0.01

0.29

0.17

-

0.14

0.17

0.27

Mean slope before

changepoint

1

-0.02

0.00

-

0.07

0.03

-

0.00

0.01

Mean slope after

changepoint

1

0.25

0.08

-0.08

0.33

0.24

Changepoint

for mean

1

0.15

-0.25

0.43

0.79

Log-Variance intercept

1

-

0.69

0.44

0.02

Log-Variance slope before

changepoint

1

-

0.74

0.09

Log-Variance slope after

changepoint

1

0.34

Variance

changepoint

1

Posterior mean for correlations:Slide31

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Correlations Among Characteristics

Later change points for variance are highly associated with later change points

for mean.

Later change points for both mean and variance are also correlated with longer

and more

variable segment lengths, and more rapid increases in mean and variance after the change point; consequently mean and variance slopes after change points are positively correlated.Greater mean length at age 35 is associated with greater declines in

variability before

the variance change point and greater increases in variability after

.Larger segment variability is associated with longer mean segment length.Larger segment variability is highly associated with more rapid declines in variability before but larger increases in variability after the variance change point: thus change in variability before and after the variance change point is negatively correlated.Slide32

Menstruation Patterns and Menopause

Accelerated failure time model with

gaussian

link:

Age of FMP ~ pattern parameters

Women with late menopause have:

Later changepointsSmaller variance of cycle lengths at age of 35 Less rapid decrease in variance of cycle lengths before changepointsLess rapid increase in mean and variance of cycle lengths after changepoints Less abrupt changes of variance slopes before and after changepoints

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Publication and Related

Work

Publication of this work:

"

Modeling Menstrual Cycle Length and Variability at the Approach of Menopause Using Bayesian

Changepoint Model," X. Huang, S. D. Harlow, M. R. Elliott, 2014, Journal of the Royal Statistical Society C: Applied Statistics, 63(3): 445-466

Comparing

changepoints to previously defined transition markers.Publication: "Distinguishing 6 Population Subgroups by Timing and Characteristics of the Menopausal Transition," X. Huang, S. D. Harlow, M. R. Elliott, 2012. American Journal of Epidemiology, 175(1): 74-83Include data from cohort II and study the difference of women’s menstruation patterns between cohort I and cohort II.Slide34

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Acknowledgement

Grant R01HD055524 from the National Institute of Child Health and Development.

Data from TREMIN Trust.Slide35

35

Thank You!Slide36

Additional Literatures

1987:

Davidian

and

Caroll

. Variance function estimation.

2000: Harlow et al. Analysis of menstrual diary data across the reproductive life span: Application of the bipartite model approach and the importance of within-woman variance. 2001: Thum and Bhattacharya. Detecting a change in school performance: a Bayesian analysis for a multilevel joint point problem.

2003: Hall et al.

Bayesian and profile likelihood

changepoint methods for modeling cognitive function over time.2004: Lisabeth et al. A new statistical approach demonstrated menstrual patterns during the menopausal transition did not vary by age at menopause2007: Crainiceanu et al. Spatially adaptive Bayesian penalized splines with heteroscedastic

errors.

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Appendix: Gibbs Sampling

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are the corresponding part of prior multivariate normal mean and covariance matrix conditional on other parametersSlide38

Gibbs Sampling - Continue

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Gibbs Sampling - Continued

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Gibbs Sampling - Continued

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Appendix – Survival Model of FMPs

Assume that last observed ages of all subjects are from piecewise exponential distribution

Use prior:

The posterior distribution is

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Appendix – Predict FMPs

The cumulative hazard and survival function:

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Conditional and unconditional distribution of FMP occurrence by time Slide43

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Posterior Predictive Model Check -Cycle Length

Posterior predictive Chi-square test:

Created histogram of p-values of Chi-square tests for all subjects, each test based on 200 replications.Slide44

Observed and Predicted FMPs

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Observed and Imputed FMPs

-

observed FMP

45

-

imputed FMP and 95% predictive interval

x -

age at censoringSlide46

Posterior Model Check – FMPsReplicate imputations for FMPs for subjects with observed FMPs

Compare each observed FMPs with corresponding 200 draws of predicted FMPs

Histogram of proportion of mean(

FMP

rep

) > Observed FMP

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Changepoints

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Principle Component Analysis of Pattern Measures

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Sensitivity Analysis

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