Implementation of Quasi-Least Squares using xtgee in Stata

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Implementation of Quasi-Least Squares using xtgee
in Stata

• Justine Shults
• Assistant Professor of Biostatistics
• Department of Biostatistics
• University of Pennsylvania School of Medicine

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Outline

• Motivational Study
• -Traditional Approach (GEE)
• Describe quasi-least squares (QLS)
• -Advantages
• Implementation of QLS using xtgee
• Example
• Conclusion
• Broad Overview of Work in Progress (Papers to be
  submitted to Biometrics and Stata journal)

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Motivational Study-Traditional approach (GEE)

• Study of Interstitial Cystitis (IC) in women
• IC is a disease of urinary tract
• Subjects recorded number of 24 hour voids

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Timeline for a subject with all measurements

• 0 months 3 months 6 months 9 months 12
  months
• xxx xxx xxx xxx
  __ xxx
• Bout 1 Bout 2 Bout 3 Bout 4
  Bout 5
• Time between bouts gtgtgtgtgt time within bouts

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• Goal of analysis Relate covariates with
  expected number of 24 hour voids.
• Should adjust for potential intra-subject
  correlation.
• Traditional approach generalized estimating
  equation approach (GEE) of Liang and Zeger
  (1986).

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• Limitation of GEE Can be difficult to implement
  complex correlation structures.
• Banded Toeplitz structure (Mazurick and Landis,
  Journal of Urology, 2000)
• 1 r1 r1 r2 r2 r2 r3 r3 r3 r4 r4 r4 r5 r5 r5
• r1 1 r1 r2 r2 r2 r3 r3 r3 r4 r4 r4 r5 r5 r5
• r1 r1 1 r2 r2 r2 r3 r3 r3 r4 r4 r4 r5 r5 r5
• r2 r2 r2 1 r1 r1 r2 r2 r2 r3 r3 r3 r4 r4 r4
• r2 r2 r2 r1 1 r1 r2 r2 r2 r3 r3 r3 r4 r4 r4
• r2 r2 r2 r1 r1 1 r2 r2 r2 r3 r3 r3 r4 r4 r4
• r3 r3 r3 r2 r2 r2 1 r1 r1 r2 r2 r2 r3 r3 r3
• r3 r3 r3 r2 r2 r2 r1 1 r1 r2 r2 r2 r3 r3 r3
• r3 r3 r3 r2 r2 r2 r1 r1 1 r2 r2 r2 r3 r3 r3
• r4 r4 r4 r3 r3 r3 r2 r2 r2 1 r1 r1 r2 r2 r2
• r4 r4 r4 r3 r3 r3 r2 r2 r2 r1 1 r1 r2 r2 r2
• r4 r4 r4 r3 r3 r3 r2 r2 r2 r1 r1 1 r2 r2 r2
• r5 r5 r5 r4 r4 r4 r3 r3 r3 r2 r2 r2 1 r1 r1
• r5 r5 r5 r4 r4 r4 r3 r3 r3 r2 r2 r2 r1 1 r1
• r5 r5 r5 r4 r4 r4 r3 r3 r3 r2 r2 r2 r1 r1 1

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• Allows for decline in correlation with time, but
  not as severe as for Markov structure
• Mazurick and Landis (2000) could only implement
  in ad-hoc approach with GEE
• Will directly implement using Quasi-Least Squares
  (QLS).
• Developed in Chaganty (JSPI, 1997), Shults and
  Chaganty (Biometrics,1998), Chaganty and Shults
  (JSPI, 1999)

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Advantages of QLS

• In framework of GEE- extends generalized linear
  models for correlated data
• Guarantees feasible estimates for some
  structures, i.e. positive definite matrix
• Allows for easier implementation of some complex
  correlation structures
• e.g. Structure appropriate for data with
  multiple sources of correlation. Shults, Whitt,
  Kumanyika (Statistics in Medicine, in press)
• Banded Toeplitz structure

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• Two stage procedure
• Based on GEE (Liang and Zeger, 1986).
• Stage one Alternates till convergence
• Estimate ? via GEE estimating equation.
• Estimate p by minimizing an objective function
  (residuals). Solve stage one estimating equation.
• Stage two
• Solve stage two equation to update estimate of p
  .
• Obtain final estimate of ? by again solving GEE.

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Implementation of QLS using xtgee

• Basic Idea and Outline for program
• important xtgee allows for implementation of a
  correlation structure that is fixed and known

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• Outline for program
• Let p (r1,r2,r3,r4,r5).
• To implement stage one of QLS
• Let p 0 so that working structure Identity.
• Estimate ? by solving GEE estimating equation via
  xtgee, with identity correlation structure.
• Repeat till convergence
• Obtain Pearson residuals.
• Use algorithm based on method of bisection to
  solve stage one estimating equation for p.
• Construct correlation matrix WORK based on
  updated estimate of p.
• Update estimate of ? via xtgee, with correlation
  structure WORK treated as fixed and known.

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• Stage two
• Use algorithm based on method of bisection to
  solve stage two estimating equation for p .
• Construct correlation matrix FINAL based on
  updated estimate of p.
• Obtain final estimate of ? via xtgee with fixed
  and known structure FINAL.

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• Nice features of Stata
• Matrix commands are helpful in setting up working
  structure.
• The xtgee procedure allows for implementation of
  a structure that is fixed and known.
• Programming, e.g. bisection method, is
  straightforward.

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Syntax of command

• Under development- programs for stats paper done,
  but not tied together
• Almost identical to xtgee
• Simplest syntax
• xtqls depvar varlist, family(family) link(link)
  corr(correlation) i(varname) t(varname)

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• Description
  In Example
• Depvar dependent variable
  24hourvoid
• varlist covariates
  age volumebladder
• family(family)
  Poisson
• link(link)
  log
• corr(correlation)
  Toepband(5,3)
• i(varname)
  subjectid
• t(varname)
  time

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What is different about syntax?

• Will use xtqls instead of xtgee
• The list of possible correlation structures will
  be expanded.
• Previous list
• Independent
• Exchangeable
• Autoregressive
• Stationary
• Non-stationary
• Unstructured
• User-Specified

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• Will add Toepband(b,v) Toeplitz banded
  structure for b bouts with v visits per bout.
• In future will add additional structures.

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• Grant Number 1R01CA096885-01A2
• PI Name SHULTS, JUSTINE
• PI Email jshults_at_cceb.upenn.edu
• Project Title Longitudinal Analysis for
  Diverse Populations
• Aim 5 of Abstract .. (5) To implement the
  methods for analysis (Aim 1) and planning (Aim 2)
  in Stata programs, for use by other
  statisticians. Further, to widely disseminate the
  programs, and their documentation, on a web site
  developed for this project.
• 
  

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Example

• Estimated correlation matrix
• r1 0.8478
• r2 0.7165
• r3 0.6831
• r4 0.6653
• r5 0.6654
• This does suggest some decline, but not as severe
  as that indicated by a Markov structure.
• Note This structure includes equicorrelated and
  Markov Banded as special cases.

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Conclusion

• QLS is based on GEE, but allows for easier
  implementation of some complex structures.
• Nice features of Stata allow for relatively
  straightforward programming of QLS.
• Am building xtqls procedure based on xtgee.
• e-mail jshults_at_cceb.upenn.edu