Longitudinal Data Analysis and Survival Analysis

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Longitudinal Data Analysis and Survival Analysis

• Ming-Yu Fan, PhD
• June 25, 2008

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Outline

• Longitudinal data
• Methods for LDA
• Robust standard error
• Generalized Estimating Equation (GEE)
• Random effects model
• Survival data
• Methods for analyzing survival data

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Longitudinal Data

• Each individual has multiple observations
• The intervals between observations are
  approximately the same for each individual
• Even intervals are nice but not necessary
• Ex depression severity (SCL-20) evaluated at
  baseline, 3-, 6-, and 12-month

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Why are longitudinal data desirable?

• More information
• Can control for individual heterogeneity
• Can better assess causality than cross-sectional
  data

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Problem with longitudinal data

• Conventional statistical methods require
  independence between observations
• Longitudinal data are likely to violate this
  assumption
• Missing data due to attrition

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Notation

• yit, ith individual, tth observation
• i 1, 2, , n t 1, 2, , T
• yi1, yi2, yi3, ., yiT are very likely to be
  correlated

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Example

• Data from the IMPACT (PI Dr. Unützer) study
• 8 organizations, total N 1801
• Outcome SCL-20 measured at baseline and 3 months
• Comparison between the intervention and the usual
  care groups

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Example cont.

• General linear model
• ß -0.1323
• SE 0.0222
• t -5.96
• p-value lt0.0001
• Ignore the correlation
• coefficient (? 0.4)
• between SCL00 and
• SCL03
• Robust standard error
• ß -0.1323
• SE 0.0251
• z -5.26
• p-value lt0.0001

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Example 2

• Suppose we are only interested in two large study
  sites (N551)
• Outcome MCS12 (mental health component score
  from SF12) measured at baseline and 3 months
• Comparison between the intervention and the usual
  care groups

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Example 2 cont.

• General linear model
• ß 0.8088, SE 0.4117, t 1.96, p-value
  0.0497
• Ignore the correlation coefficient (? 0.23)
  between MCS1200 and MCS1203
• Robust standard error
• ß 0.8088, SE 0.4448, z 1.82, p-value
  0.0690
• The robust standard error is greater than the SE
  estimated without accounting for correlation
• Different methods lead to different conclusions

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Methods for LDA

• Robust standard error
• Generalized Estimating Equations (GEE)
• Random effects models (hierarchical models)

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Robust standard error

• Ordinary least squares covariance estimator
• Robust covariance estimator
• Residuals

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Robust standard error cont.

• Robust standard errors are usually larger than
  conventional standard errors, but its possible
  to see a smaller robust standard error
• Robust standard errors may be inaccurate if the
  sample sizes are small

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Generalized Estimating Equation (GEE)

• Estimate ß by solving the following equation
  (Wedderburn, 1974)

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Variance Structure

• Covariance matrix for yi1, yi2, yi3, ., yiT
• Corr(yik, yil) ?kl ? 0
• Corr(yik, yjl) 0 when i ? j (for both k l and
  k ? l )
• ?kl ?lk (symmetry)
• Ex T 4, need to estimate 6 correlation
  coefficients (?12, ?13, ?14, ?23, ?24, ?34)

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Possible Variance Structure

• Unstructured (UN)
• no constraints on ?s
• Exchangeable (EXCH)
• ?kl ? for all ks and ls
• 1st order autoregressive (AR)
• ?kl ?k-l
• Banded structure (MDEP(m))
• ?kl ?k-l when k-l lt m, otherwise ?kl 0
• Independent (IND) ? Robust standard errors
• ?kl 0 for all ks and ls

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Example of Variance Structure

• Outcome measured at 4 time points yi1, yi2, yi3,
  yi4
• In total 6 correlation coefficients (?12, ?13,
  ?14, ?23, ?24, ?34)
• Unstructured (UN) ? need to estimate all 6
  correlation coefficients
• Exchangeable (EXCH) ? need to estimate only 1
  correlation coefficient
• 1st order autoregressive (AR) ?12 ?, ?13 ?2,
  ?14 ?3 . ? need to estimate only 1 correlation
  coefficient
• (e.g. ?12 0.4, ?13 0.16, ?14 0.064..)
• Banded structure (MDEP(2)) ?12 ?23 ?34, ?13
  ?24, ?14 0 ? need to estimate 2 correlation
  coefficients (distance 1, 2)
• Independent (IND) ? no need to estimate the
  correlation coefficient

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GEE cont.

• GEE produces efficient estimates of the
  coefficients
• Can assume different variance structures. The
  results are usually robust to the choice
• GEE assumes the drop-outs are Missing Completely
  At Random

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Random effects models

• General linear model
• Random effects model
• Other names hierarchical models multilevel
  models mixed models
• This model is also called random intercepts
  model. It implies equal correlations and thus is
  equivalent to the exchangeable model in GEE
  estimation

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Random effects models cont.

• Can have more than 2 levels
• Allow for random coefficients / slopes
• The dependent variable can have missing data
  under a weaker assumption Missing At Random

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Compared with GEE

• Takes more computing time than GEE
• Computationally less stable than GEE
• More restrictive assumption on correlation
  structure (GEE can assume unstructured
  correlation)

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LDA Methods for Continuous Dependent Variables

• Robust standard errors
• Can be derived using SAS Proc GENMOD (variance
  structure IND)or Proc SURVEYREG procedures
• GEE
• Stata xtgee, xtreg
• SAS Proc GENMOD procedure
• Choices of variance structure
• Unstructured (UN)
• Exchangeable (EXCH)
• 1st order autoregressive (AR)
• Banded structure (MDEP(m))

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LDA Methods for Continuous Dependent Variables
cont.

• How to check variance structure?
• Assign UN (unstructured) first
• Use CORRW option to print out the estimated
  correlation matrix
• Random effects models
• Stata xtreg
• SAS Proc MIXED procedure

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LDA Methods for Categorical Dependent Variables

• Robust standard errors
• SAS Proc GENMOD procedure (with IND variance
  structure)
• Specify distribution family (e.g. Binomial for
  binary outcomes, Poisson for count data), default
  is Normal distribution
• Can also use Proc SURVEYLOGISTIC procedure for
  binary outcome
• GEE
• Stata xtlogit, xtpoisson, etc
• SAS Proc GENMOD procedure
• Specify distribution family for categorical
  dependent variables
• Assume variance structure to be UN, EXCH, AR, or
  MDEP(m)

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LDA Methods for Categorical Dependent Variables
cont.

• Random effects models
• Use SAS Proc NLMIXED (non-linear mixed models)
  procedure
• Can only estimate two-level models
• Syntax is quite complicated
• Computationally intensive and often unstable, and
  thus is not recommended (by Dr. Paul D. Alison)
• Can also use SAS Proc GLIMMIX procedure (need to
  download it from SAS web site (http//support.sas.
  com/rnd/app/da/glimmix.html)
• According to Dr. Alison
• Can handle more than two levels of data
• Much faster than NLMIXED
• Syntax is simpler
• Inaccurate for small numbers of time points (e.g.
  2-3 points per person)

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IMPACT Example

• Outcome SCL-20 measured at baseline, 3 months, 6
  months, 12 months, 18 months, and 24 months (time
  0, 3, 6, 12, 18, 24)
• Compare between intervention and usual care
  groups
• Models are adjusted for age, sex, education,
  ethnicity, number of chronic conditions, and time

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UN, AR, EXCH, or MDEP(5)?

• MDEP(5), perhaps?

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Survival Analysis

• Outcome failure failure time
• Unlike repeated measures, survival data have only
  1 outcome measure
• Methods for recurrent event are available
• Failure time is (often) the clock time between
  the time origin and failure
• Time origin should be precisely defined
• Ex the date of randomization in a randomized
  clinical trial
• Time origin doesnt have to be the same calendar
  time for all individuals
• Censoring individuals are not observed for the
  full time to failure

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BMT Example

• From Klein and Moeschberger (1997)
• Sample 137 patients who received bone marrow
  transplant
• At the time of transplant, each patient is
  classified into one of three risk categories
• ALL (Acute Lymphoblastic Leukemia)
• Low-risk AML (Acute Myeloid Leukemia)
• High-risk AML
• End point disease-free survival in days
• Time origin the date of transplant
• Failure death or relapse
• Censored no death or relapse by the end of the
  study

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Survival Function

• Failure time T
• Survival function S(t) P(T gt t)
• the probability that the failure time is greater
  than or equal to t
• Hazard function
• the chance that the failure occurs within the
  time interval t, t?t (let ?t be extremely
  small), given that the individual survives at t

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Estimating Survival Function

• Life-table
• Divide the period of observation into a series of
  time intervals (often of equal length)
• Compute the number of deaths and number of
  censored survival times
• Estimate the survival probability in each
  interval
• Take the product of the probabilities
• Kaplan-Meier estimate
• Similar to Life-Table method
• Each interval has only one death occurred at the
  start of the interval
• Cox proportional hazard model

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A subsample from the BMT example
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Life-Table

• D death C censored N number of
  individuals who are alive (at risk) at beginning
  of the interval
• N N (C/2) number of individuals who are at
  risk during the interval
• S(t) cumulative survival

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Kaplan-Meier Estimate

• The beginning of each interval is determined by
  death
• Each interval contains one death (or more if
  there are ties)
• N(t) includes individuals with censored data at t

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Cox Proportional Hazard Model

• h0(t) baseline hazard function
• The interpretation of b1

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BMT Example
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Reference

• Wedderburn, R.W.M. (1974). Quasi-likelihood
  functions, generalized linear models and the
  Gaussian method. Biometrika, 61, 439-47.
• Dr. Paul Alisons upcoming short courses
• http//www.statisticalhorizons.com/index.html
• Klein, J. P. and Moeschberger, M. L. (1997),
  Survival Analysis Techniques for Censored and
  Truncated Data, New York Springer-Verlag.