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 ID: 712342
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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
9Slide10
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
18
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
19
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
20
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
21Slide22
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.
22
Age
FMP
Censoring
W=0
W=0
W=0
W=1
48.0
48.07
48.16
48.28
52.3Slide23
23
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
26
Results: Individual Level Parameters
Histogram of Slide27
27
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
28
Population Level Parameters
Posterior mean and 95% predictive intervals for mean population level parameters :Slide29
29
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
30
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
31
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
32Slide33
33
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
34
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.
36Slide37
Appendix: Gibbs Sampling
37
are the corresponding part of prior multivariate normal mean and covariance matrix conditional on other parametersSlide38
Gibbs Sampling - Continue
38
Slide39
Gibbs Sampling - Continued
39Slide40
Gibbs Sampling - Continued
40Slide41
Appendix – Survival Model of FMPs
Assume that last observed ages of all subjects are from piecewise exponential distribution
Use prior:
The posterior distribution is
41Slide42
Appendix – Predict FMPs
The cumulative hazard and survival function:
42
Conditional and unconditional distribution of FMP occurrence by time Slide43
43
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
44Slide45
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
46Slide47
Changepoints
47Slide48
Principle Component Analysis of Pattern Measures
48Slide49
Sensitivity Analysis
49