1 date name, department statistical analysis of longitudinal data ziad taib biostatistics, az april...
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Name, department
1 Date
Statistical Analysis of Longitudinal Data
Ziad Taib
Biostatistics, AZ
April 2011
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Outline of lecture 1
1. An introduction
2. Two examples
3. Principles of Inference
4. Modelling continuous longitudinal data
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Part 1: An introduction
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Why longitudinal data?
Very useful for their own sake. With longitudinal data, we have the possibility of
understanding what mixed models are about in a relatively simple but yet rich enough context.
___________________________________
A good reference is the book ”Designing experiments and analyzing data” by Maxwel l& Delaney (2004)
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Longitudinal Data
Repeated measures are obtained when a response is measured repeatedly on a set of units• Units:
• Subjects, patients, participants, . . .
• indivduals, plants, . . .
• Clusters: nests, families, towns, . .
• . . .
• Special case: Longitudinal data
Obs! Possible to handle several levels
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A motivating example
Consider a randomized clinical trial with two treatment groups and repeated measurements at baseline, 3 and 6 months later. As it turned out some of the data was missing. Moreover patients did not always comply with time requirements. Our first reaction is to try to compensate for the missing values by some kind of imputation, or to use list-wise deletion.
Both ”methods” having their shortcomings, wouldn't it be nice to be able to use something else? There is in fact an alternative method: using the idea of mixed models.
With mixed models,1. we can use all our data having the attitude that ”what is missing is
missing”. 2. we can even account for the dependencies resulting from measurements
made on the same individuals at different times. 3. we don’t need to be consistent about time.
A
B
Baseline 3 months 6 months
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Mixed effects models
Ordinary fixed effects linear model usually assume:
1) independence with the same variance.2) normally distributed errors.3) constant parameters
If we modify assumptions 1) and 3), then the problem becomes more complicated and in general we need a large number of parameters only to describe the covariance structure of the observations. Mixed effects models deal with this type of problems.
In general, this type of models allows us to tackle such problems as: clustered data, repeated measures, hierarchical data.
constant. ),,0( is , 2 INXY
nnn x
x
Y
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Various forms of models and relation between them
LM: Assumptions:
1. independence,
2. normality,
3. constant parameters
GLM: assumption 2) Exponential family
LMM: Assumptions 1) and 3) are modified
GLMM: Assumption 2) Exponential family and assumptions 1) and 3) are modified
Repeated measures: Assumptions 1) and 3) are modified
Longitudinal dataMaximum likelihood
Classical statistics (Observations are random, parameters are unknown constants)
Bayesian statistics
LM - Linear model
GLM - Generalised linear model
LMM - Linear mixed model
GLMM - Generalised linear mixed model
Non-linear models
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Part 2: Two examples
Rat data Prostate data
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Example 1: Rat Data (Verbecke et al)
Research question How does craniofacial growth in the wistar rat depend on testosteron production?
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Simplifie
d
(univariate) re
sponse
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•Randomized experiment in which 50 male Wistar rats are randomized to:
Control (15 rats) Low dose of Decapeptyl (18 rats) High dose of Decapeptyl (17 rats)
Treatment starts at the age of 45 days. Measurements taken every 10 days, from day 50
on. The responses are distances (pixels) between two
well defined points on x-ray pictures of the skull of each rat. Here, we consider only one response, reflecting the height of the skull.
Prevents the production of testesterone
45
Days
60 7050 80
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Individual profiles:
1. Connected profiles better that scatter plots2. Growth is expected but is it linear3. Of interest change over time (i.e. Relationship between response and age)
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Complication: Many dropouts due to anaesthesia imply less power but
no bias.
Without dropouts easier problem because of balance.
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Remarks:
Much variability between rats Much less variability within rats Fixed number of measurements scheduled per
subject, but not all measurements available due to dropout, for known reason.
Measurements taken at fixed time points
Research question: How does craniofacial growth in the wistar
rat depend on testosteron production ?
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Example 2: The BLSA Prostate Data
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Example 2: The BLSA Prostate Data (Pearson et al., Statistics in Medicine,1994). Prostate disease is one of the most common and
most costly medical problems in the world. Important to look for biomarkers which can detect the disease at an early stage.
Prostate-Specific Antigen is an enzyme produced by both normal and cancerous prostate cells. It is believed that PSA level is related to the volume of prostate tissue.
Problem: Patients with Benign Prostatic Hyperplasia also have an increased PSA level
Overlap in PSA distribution for cancer and BPH cases seriously complicates the detection of prostate cancer.
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Research question: Can longitudinal PSA profiles be used to detect prostate cancer in an early stage ?
A retrospective case-control study based on frozen serum samples:
16 control patients 20 BPH cases 14 local cancer cases 4 metastatic cancer cases
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Individual profiles:
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Remarks:
Much variability between subjects Little variability within subjects Highly unbalanced data
Research question: Can longitudinal PSA profiles be used to
detect prostate cancer in an early stage ?
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Part 3: Principles of Inference
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Fisher´s likelihood Inference for observable y and fixed parameter q Data Generation : Given a stochastic model
, Generate data, y, from
Parameter Estimation : Given the data y, make inference about q by using the likelihood
Connection between two processes :
)(yf
)/( yL
)()/( yfyL
)(yf
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(Classical) Likelihood Principle
Birnbaum (1962) All the evidence or information about the parameters in the data is in the likelihood.
Conditionality principle& Sufficiency principle
Likelihood principle
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Bayesian Inference for observable y and unobservable n Data Generation : Generate data according to
1. n, from
2. For n fixed generate y from
Combine into Parameter Estimation : Given the data y, make
inference about n by using The connection between two processes:
)(f
)/()()/()( yfyfyff
)/()( yff
)/( yf
)/( yf
prior
posterior
Compare with )/( yL
)/()(),()/()()(
),()/( yffyfyfyf
yf
yfyf
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Extended likelihood inference: (Lee and Nelder) for observable y, fixed parameter q and unobservable n
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Parameter estimation )()/( yfyL
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Extended Likelihood Principle
Björnstad (1996) All information in the data about the unobservables and the parameters is in the “likelihood”.
Conditionality principle& Sufficiency principle
Likelihood principle
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Prediction: predict the number of seizures during the next week
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Bayesian Predictive Inference
Given n, the observations y are assumed to be independent. How do we predict the next value, Y, of the observable? In a Bayesian setting we may determine the posterior and define the predictive density of Y given y as:)/( yxfY
)/( yf
Obs!
Jefreys’ Priors
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Bayesian inference (Pearson, 1920)
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Nelder and Lee (1996)
?
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Part 4: A Model for Longitudinal Data
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Introduction
In practice: often unbalanced data due to (i) unequal number of measurements per subject (ii) measurements not taken at fixed time points.
Therefore, ordinary multivariate regression techniques are often not applicable.
Often, subject-specific longitudinal profiles can be well approximated by linear regression functions. This leads to a 2-stage model formulation:
Stage 1: A linear (e.g. regression) model for each subject separately
Stage 2: Explain variability in the subject-specific (regression) coefficients using known covariates
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A 2-stage Model Formulation: Stage 1 Response Yij for ith subject, measured at time tij, i = 1, . . . , N,
j = 1, . . . , ni Response vector Yi for ith subject:
Zi is a (ni x q) matrix of known covariates and
bi is a (ni x q) matrix of parameters
Note that the above model describes the observed variability within subjects
iiiiiiii
iniii
InNZY
YYYYi
2
21
often ),,0(~ ,
)',...,,(
Possibly after some convenient transformation
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Stage 2
Between-subject variability can now be studied from relating the parameters bi to known covariates
Ki is a (q x p) matrix of known covariates and
b is a (p-dimensional vector of unknown regression
parameters Finally
iii bK
),0(~ ii Nb
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The General Linear Mixed-effectsModel The 2-stages of the 2-stage approach can now be
combined into one model:
Average evolution Subject specific
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Convenient using multivariate normal.Very difficult with other distributions
The general mixed effects models can be summarized by:
Terminology:• Fixed effects: b• Random effects: bi
• Variance components: elements in D and Si
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Remarks
1. It is occasionally unclear if we should treat an effect as a fixed or a mixed effect. For example in clinical trials with treatment and clinic as “factors” should we consider clinics as random?
2. Considering the general form of a mixed effects model
notice that the fixed effects are involved only in mean values (just like in ordinary linear models) while random effects modify the covariance matrix of the observations.
iiiii bZXY
?
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Example: The Rat Data
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Transformation of the time scale to linearize the profiles:
Note that t = 0 corresponds to the start of the treatment (moment of randomization)
• Stage 1 model:
]10
)45(1ln[
ij
ijij
AgetAge
iijijiiij njtY ,1,... ,21
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Stage 1
i
ii
2
1
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Stage 2 model:
In the second stage, the subject-specific intercepts and time effects are related to the treatment of the rats
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The hierarchical versus the marginal Model
The general mixed model is given by It can be written as
It is therefore also called a hierarchical model
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f(yi I bi)f(bi)
f(yi)
Marginally we have that is distributed as
Hence
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Example: The Rat Data
Linear model where eachrat has its own interceptand its own slope
Can be negative or positivereflecting individual deviationfrom average
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Notice that the model assumes that thevariance function is quadratic over time.
Comments:• Linear average evolution in each group• Equal average intercepts• Different average slopes
Moreover, taking
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),cov()(
),cov(
),cov(1
,
),cov(1
,1
),cov(1
)cov(,1
),1,,1(
))(),((
112221122111
11222112212111
112
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tdtddtd
tdd
ddt
tt
ttCov
ttCov
YY
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The prostate data
iijijiijii
ij
ij
njtt
PSA
Y
,1,... ,
)1ln(2
321
A model for the prostate cancer Stage 1
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The prostate data
Age could not be matched
jiiiii
jiiiii
jiiiii
i
i
i
bMLBCAge
bMLBCAge
bMLBCAge
31514131211
2109876
154321
3
2
1
A model for the prostate cancer Stage 2
Ci, Bi, Li, Mi are indicators of the classes: control, BPH, local or
metastatic cancer. Agei is the subject’s age at diagnosis. The parameters in the first row are the average intercepts for the different classes.
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The prostate data
This gives the following model
eij
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Stochastic components in general linear mixed model
Average evolution
Subject 2
Subject 1
Time
Res
pons
e
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References
Aerts, M., Geys, H., Molenberghs, G., and Ryan, L.M.(2002). Topics in Modelling of Clustered Data. London: Chapman and Hall.
• Brown, H. and Prescott, R. (1999). Applied Mixed Models in Medicine. New-York: John Wiley & Sons.
• Crowder, M.J. and Hand, D.J. (1990). Analysis of Repeated Measures. London: Chapman and Hall.
• Davidian, M. and Giltinan, D.M. (1995). Nonlinear Models For Repeated Measurement Data. London: Chapman and Hall.
Davis, C.S. (2002). Statistical Methods for the Analysis of Repeated Measurements. New York: Springer-Verlag.
Diggle, P.J., Heagerty, P.J., Liang, K.Y. and Zeger, S.L. (2002). Analysis of Longitudinal Data. (2nd edition). Oxford: Oxford University Press.
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