BOULDER WORKSHOP STATISTICS REVIEWED: LIKELIHOOD MODELS

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BOULDER WORKSHOPSTATISTICS REVIEWED LIKELIHOOD
MODELS

• Andrew C. Heath

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PRE-HISTORY (STATISTICS 101)

• Binomial distribution gives the probabilities
  of various possible numbers of successful
  outcomes in a fixed number of discrete trials,
  where all trials have the same probability of
  success.
• Probability that X is equal to a particular value
  x (x 0, 1, 2n) is given by
• Useful for genetics (e.g. transmission versus
  non-transmission of an allele)!

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LIKELIHOOD

• Focus on probability (likelihood) of the data,
  as a function of model parameters.
• E.g. Sham (1998) uses Neel Schull (1954) data
  on opalescent dentine in random sample of 112
  offspring of an affected parent, found 52
  affected, 60 normal. Compatible with hypothesis
  of rare autosomal dominant? Does the observed
  population (.464) differ from expected proportion
  of 0.5?
• Likelihood function for segregation ratio p is

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MAXIMUM LIKELIHOOD ESTIMATION

• Find the maximum value of the likelihood
  function in the range 0 ? p ?1. In more difficult
  problems, usual to maximize the log-likelihood,
  since computationally more convenient, this
  also maximizes the likelihood.
• In this simple case, maximum likelihood estimate
  (MLE) of p, r/n.
• For the opalescent dentine data, ignoring the
  constant term involving n, r which does not
  vary as a function of p, log-likelihood function
  is
• ln L(p) 52 ln(p) 60 ln(1p)

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Figure 2.1 Log-likelihood function of the
segregation ratio for the opalescent dentine data
(from Sham, 1998)
p
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LIKELIHOOD-RATIO STATISTIC

• Likelihood ratio statistic twice the difference
  between the log likelihood of the data at the MLE
  (i.e. ), L1, and the log likelihood of the
  data at the hypothesized value of 0.5, L0 2(ln
  L1 ln L0).
• For the segregation ratio example,
• 0.57 for the opalescent dentine example.
• Likelihood-ratio statistic in this case is
  distributed as chi-square on one degree of
  freedom, hence non-significant.

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MATRIX ALGEBRA BASICS

• A1 Inverse of A
• AT or A' Transpose of A
• A Determinant of A
• A ? B A postmultiplied by B
• r x c c x r conformable for multiplication
  since number of columns of A number of
  rows of B. Resulting matrix is r x r.
• Tr (A) Trace of matrix A

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HISTORY (MX INTRODUCTORY WORKSHOP)

• Maximum-likelihood estimation using linear
  covariance structure models, e.g. fitting models
  to twin data
• Let p be the number of observed variables, the
  expected covariance matrix be E, and the expected
  vector of means be ?, where E and ? are functions
  of q free parameters to be estimated from the
  data. Let x1, x2xn denote to observed variables.
  Assuming that the observed variables follow a
  multivariate normal distribution, the
  log-likelihood of the observed data is given by
• This is the formula used for maximum likelihood
  model-fitting to raw continuous data, assuming a
  multivariate normal distribution. (Often, ?2 ln
  L, is estimated). Requires that we provide
• (a) model for expected covariance matrix e.g.
  in terms of additive genetic, shared and
  non-shared environmental variance components
  that will vary as a function of relationship
• (b) model for expected means in simplest
  applications we might estimate a separate mean
  that might differ by gender, or possibly by twin
  pair zygosity.

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EXAMPLE

• MZ Pairs
• E
• ? m , m T
• DZ Pairs
• E
• ? m , mT
• Where m is estimate of population mean, VA, VC,
  VE and are additive genetic, shared environmental
  and non-shared environmental variances, all
  estimated jointly from the data.
• Compare e.g. ln L1 for VA VC VE model
• ln L0 for VC VE model
• 2(ln L1 ln L0) distributed as chi-square on one
  degree of freedom as before.

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HISTORY (MX INTRODUCTORY WORKSHOP) - II

• In practice most applications in the MX
  Introductory Workshop fitted models to summary
  covariance matrices.
• We can simplify (e.g. Sham, p. 238)
• (1)
• 
  
  
  (2)
• Where m is the vector of sample means of the p
  observed variables, S is the observed covariance
  matrix. For the simple applications in the
  Introductory Workshop, we had no predictions for
  the means structure, so we can saturate that
  component of the model (i.e. estimate a separate
  mean for every observed mean), equivalent to
  deleting the term (m??)T E-1 (m??) in (2). Thus
  the log-likelihood of the observed data becomes
• 
  
  
  (3)
• Under a saturated model, which equates every
  element of E to the corresponding element of S
  (i.e. a perfect fit model) we have for the
  log-likelihood

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HISTORY (MX INTRODUCTORY WORKSHOP) - II

• Thus the likelihood-radio test of the fitted
  model against the saturated model becomes
• (4)
• For multiple group problem, sum over groups.

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Analysis of Australian BMI data-young female MZ
twins pairs-MX DYI version

• Analysis of Australian BMI data-young female MZ
  twins pairs-MX DIY version
• !
• DA NG1 NI2 NO0
• begin matrices
• A LO 1 1 FR ! Additive genetic variance
• C LO 1 1 FR ! Shared environmental variance
• E LO 1 1 FR ! Non-shared environmental variance
• M FU 2 2 ! This will be observed MZ covariance
  matrix
• D FU 2 2 ! This will be observed DZ covariance
  matrix
• g fu 1 1 ! coefficient of 0.5 for DZ pairs
• n fu 1 1 ! sample size for MZ pairs (female
  in this illustration)
• k fu 1 1 ! sample size for DZ pairs (female
  in this illustration)
• p fu 1 1 ! order of matrices (i.e. number of
  variables 2 in this case)
• end matrices
• mat g 0.5
• mat p 2
• mat m
• 0.7247 0.5891
• 0.5891 0.7915

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Analysis of Australian BMI data-young female MZ
twins pairs-MX DYI version (ctd)

• BEGIN ALGEBRA
• tnk ! total sample size
• U(ACE AC _ AC ACE) ! Expected MZ
  covariance matrix
• V(ACE gAC _ gAC ACE) ! Expected DZ
  covariance matrix
• Hn(\ln (\det(U))-\ln (\det(M)) \tr((U
  M))-p) ! fit-function for MZ group
• Jk(\ln (\det(V))-\ln (\det(D)) \tr((V
  D))-p) ! fit-function for DZ group
• Fhj
• END ALGEBRA
• bo 0.01 1.0 e(1,1)
• bo 0.0 1.0 c(1,1) a(1,1)
• CO F
• option user df6
• end

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PRE-HISTORY (STATISTICS 101) - II

• LINEAR REGRESSION requires weaker assumptions
  than linear covariance structure models. Does not
  assume multivariate normal distribution, only
  homoscedastic residuals. Flexible for handling
  selective sampling schemes where we oversample
  extreme values of predictor variable(s).
• We can fit linear regression models by maximum
  likelihood e.g. using MX.

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HISTORY! MX INTRODUCTORY WORKSHOP - III

• Definition variables an option in MX when
  fitting to raw data, which allows us to model
  effects of some variables as fixed effects,
  modeling their contribution to expected means.
• Simple example controlling for linear or
  polynomial regression of a quantitative measure
  on age. Dont want to model covariance structure
  with age (which probably has rectangular
  distribution!)
• Definition variables variables whose values may
  vary from individual to individual, that can be
  read into matrices
• Important example genotypes at a given locus or
  set of loci.