Likelihood Ratio Testing under Nonidentifiability: Theory and Biomedical Applications

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Likelihood Ratio Testing under
Non-identifiability Theory and Biomedical
Applications

• Kung-Yee Liang and Chongzhi Di
• Department Biostatistics
• Johns Hopkins University
• July 9-10, 2009
• National Taiwan University

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Outline

• Challenges associated with likelihood inference
• Nuisance parameters absent under null hypothesis
• Some biomedical examples
• Statistical implications
• Class I alternative representation of LR test
  statistic
• Implications
• Class II
• Asymptotic null distribution of LR test statistic
• Some alternatives
• A genetic linkage example
• Discussion

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Likelihood Inference

• Likelihood inference has been successful in a
  variety of scientific fields
• LOD score method for genetic linkage
• BRCA1 for breast cancer
• Hall et al. (1990) Science
• Poisson regression for environmental health
• Fine air particle (PM10) for increased mortality
  in total cause and in cardiovascular and
  respiratory causes
• Samet et al. (2000) NEJM
• ML image reconstruction estimate for nuclear
  medicine
• Diagnoses for myocardial infarction and cancers

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Challenges for Likelihood Inference

• In the absence of sufficient substantive
  knowledge, likelihood function maybe difficult to
  fully specify
• Genetic linkage for complex traits
• Genome-wide association with thousands of SNPs
• Gene expression data for tumor cells
• There is computational issue as well for
  high-dimensional observations
• High throughput data

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Challenges for Likelihood Inference (cont)

• Impacts of nuisance parameters
• Inconsistency of MLE with many nuisance
  parameters (Neyman-Scott problem)
• Different scientific conclusions with different
  nuisance parameter values
• Ill-behaved likelihood function
• Asymptotic approximation not ready

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Challenges for Likelihood Inference (cont)

• There are situations where some of the
  regularity conditions may be violated
• Boundary issue (variance components, genetic
  linkage, etc.)
• Self Liang (1987) JASA
• Discrete parameter space
• Lindsay Roeder (1986) JASA
• Singular information matrix (admixture model)
• Rotnitzky et al (2000) Bernoulli

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Nuisance Parameters Absent under Null

• Ex. I.1 (Polar coordinate for bivariate normal)
• Under H0 d 0, ? is absent
• Davies (1977) Biometrika
• Andrews and Ploberger (1994) Econometrica

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Examples (cont)

• Ex. I.2 (Sterotype model for ordinal categorical
  response)
• For Y 2, .., C,
• log Pr(Y j)/Pr(Y 1) aj ßjtx, j 2,..,C
• aj fj ßtx
• 0 f1 f2 fC 1
• Under H0 ß 0, fj s are absent
• Anderson (1984) JRSSB

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Examples (cont)

• Ex. I.3 (Variance component models)
• In certain situations, the covariance matrix of
  continuous and multivariate observations could be
  expressed as
• dM(?) ?1M1 ?qMq
• A hypothesis of interest is H0 d 0 (? is
  absent)
• Ritz and Skovgaard (2005) Biometrika

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Examples (cont)

• Ex. I.4 (Gene-gene interactions)
• Consider the following logistic regression model
• logit Pr(Y 1S1, S2) a Sk dkS1k Sj ?kS2j
• ? Sk Sj dk ?j S1k S2j
• To test genetic association between gene one (S1)
  by taking into account potential interaction with
  gene two (S2), the hypothesis of interest is
• H0 d1 dK 0 (? is absent)
• Chatterjee et al. (2005) American Journal of
  Human Genetics

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Examples (cont)

• Ex. II.1 (Admixture models)
• f(y d, ?) d p(y ?) (1 d) p(y ?0)
• d proportion of linked families
• ? recombination fraction (?0 0.5)
• Smith (1963) Annals of Human Genetics
• The null hypothesis of no genetic linkage can be
  cast as
• H0 d 0 (? is absent) or H0 ? ?0 (d is
  absent)

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Examples (cont)

• Ex. II.2 (Change point)
• logit Pr(Y 1x) ß0 ßx d(x ?)
• (x ?) x ? if x ? gt 0 and 0 if otherwise
• Alcohol consumption protective for MI when
  consuming less than ?, but harmful when exceeding
  the threshold
• Pastor et al. (1998) American Journal of
  Epidemiology
• Hypothesis of no threshold existing can be cast
  as
• H0 d 0 (? is absent) or H0 ? 8 (d is
  absent)

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Examples (cont)

• Ex. II.3 (Non-linear alternative)
• logit Pr(Y 1x) ß0 ßx dh(x ?)
• e.g., h(x ?) exp(x?) 1
• The effect of alcohol consumption on risk,
  through log odds, of MI is non-linear if ? ? 0
• The hypothesis of linearity relationship with a
  specific non-linear alternative can be cast as
• H0 d 0 (? is absent) or H0 ? 0 (d is
  absent)
• Gallant (1977) JASA

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Characteristics of Examples

• Majority of examples can be characterized as
• f(y, x dhy,x(?, ß), ß)
• Class I Class II
• H0 d 0 H0 d 0 or ? ?0 ( hy,x(?0, ß) 0
  )

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Figure expected log likelihood function for
three cases
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Implications

• Under H0, conventional asymptotic results may not
  be applied
• Likelihood ratio test statistic distribution not
  ?2
• Difficult to assess evidence against H0 (e.g.
  p-value)
• Maximum likelihood estimators not normally
  distributed
• Impact parameter space near H0
• Q. How to deal with this phenomenon?

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Class I Asymptotic

• For H0 d 0,
• 1. LRT 2logL( ) logL(0, )
• sup? 2logL( , ?) logL(0, )
• sup? LRT(?)
• 2. LRT(?) S(?)t I-1(?)S(?) op(1) W2(?)
  op(1),
• where S(?) ?logL(d, ?)/?dd0, I(?) varS(?)
  
• W(?) I-1/2(?) S(?)
• and W(?) is a Gaussian process in ? with mean 0,
  variance 1 and autocorrelation ?(?1, ?2)
  covW(?1), W(?2)

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Class I Asymptotic (cont)

• Results were derived previously by Davies (1977,
  Biometrika)
• No analytical form available in general
• Approximation, simulation or resampling methods
• Kim and Siegmund (1989) Biometrika
• Zhu and Zhang (2006) JRSSB
• Q. Can simplification be taken place for
• Asymptotic null distribution?
• Approximating the p-value?

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Class I Principal Component Representation

• Principal component decomposition
• K could be finite or 8
• ?1, ,?K are independent r.v.s
• ?k N(0, ?k), k 1, .., K
• ?(?, ?) Sk ?k ?k(?)2 1

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Class I Principal Component Representation
(cont)

• W2(?) Sk ?k ?k(?)2
• Sk ?k2/?k Sk ?k ?k(?)2 Sk ?k2/?k
• Consequently, one has
• sup? W2(?) sup? Sk ?k ?k(?)2 Sk ?k2/?k
  
• The asymptotic distribution of LRT under Ho is
  bounded by

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Class I Simplification

• Simplify to if K lt 8 and for almost every
• (?1, , ?K), there exists ? such that
• ?1?1(?)/?1 .. ?K?K(?)/?K
• Ex. I.1. (Polar coordinate for bivariate normal)
• For any , there exists ? e 0,
  p) such that
• LRT instead of
• H0 d 0 ? H0 µ1 µ2

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Class I Simplification (cont)

• Simplify to if S(?) h(?) g(Y)
• ?(?1, ?2) 1
• Ex. Modified admixture models
• ? p(y d) (1 ?) p(y d0), ? e a, 1 with a
  gt 0 fixed
• H0 d d0 (? is absent) and the score function
  for d at d0 is
• S(?) ? ?logp(y d0)/?d
• Known as restricted LRT for testing H0 d d0
• Has been used in genetic linkage studies
• Lamdeni and Pons (1993) Biometrics
• Shoukri and Lathrop (1993) Biometrics

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Class I Approximation for P-values

• When simplification fails
• Step 1 Calculate W(?) and ?(?1, ?2)
• Step 2 Estimate eigenvalues ?1, , ?K and
  eigenfunctions ?1, , ?K, where K is chosen so
  that first K components explain more than 95
  variation
• Step 3 Choose a set of dense grid ?1, , ?M
  and for i 1, , N, repeat the following steps
• Simulate ?ik N(0, ?k) for k 1, , K
• Calculate Wi(?m) Sk ?ik ?k(?m)2 for each m
• Find the maximum of Wi(?1), , Wi(?M), Ri say
• R1, , RN approximates the null distribution of
  LRT

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Class II Some New Results

• Consider the class of family
• f(y, x dhy,x(?, ß), ß),
• where hy,x(?0, ß) 0 for all y and x
• H0 d 0 or ? ?0
• Tasks 1. Derive asymptotic distribution of LRT
  under H0
• 2. Present alternative approaches
• Illustrate through Ex. II.1 (Admixture models)
• d Binom (m, ?) (1 d) Binom (m, ?0)
• d e 0, 1 and ? e 0, 0.5, hy,x(?, ß) p(y
  ?) p(y ?0)
• For simplicity, assuming ß is absent

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Class II LRT Representation

• Under H0, f(y, x d, ?) f(y, x 0, ) f(y,
  x , ?0 ),
• LRT supd,? 2logL(d, ?) logL(0, )
  supd,? LRT(d, ?)
• max sup1,4 LRT(a), sup2,4 LRT(b), sup3
  LRT(a, b),
• here for fixed a, b gt 0 and ?0 0.5
• Region 1 d e a, 1, ? e 0.5 b, 0.5
• Region 2 d e 0, a, ? e 0, 0.5 b
• Region 3 d e 0, a, ? e 0.5 b, 0.5
• Region 4 d e a, 1, ? e 0, 0.5 b

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Class II Four Sub-Regions of Parameter Spaces
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Class II Regions 1 4

• With d e a, 1, this reduces to Class I, and
• sup1,4 LRT(a) supd op(1)
• W1(d) I1-½(d)S1(d)
• S1(d) ?logL(d, ?)/??? ?0, I1(d)
  var(S1(d))
• For the admixture models,
• S1(d) d ?log p(y ?0)/??, which is
  proportional to d
• is independent of d

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Class II Regions 2 4

• With ? e 0, ?0 b, this reduces to Class I,
  and
• sup2,4 LRT(b) sup? op(1)
• W2(?) I2-½(?)S2(?)
• S2(?) ?logL(d, ?)/?dd 0, I2(?)
  var(S2(?))
• For the admixture models,
• S2(?) p(y ?) p(y ?0)/p(y ?0)
• W2(?) ? ?log p(y ?0)/?? W1 as ? ? ?0 (or b ?
  0)

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Class II Region 3

• With d e 0, a, ? e 0.5 b, 0.5, expand at
  0, 0.5
• sup3 LRT(a, b) supd,? op(1)
• W3 I3-½S3
• S3 ?2logp(y 0, 0.5)/?d??, I3 var(S3)
• For the admixture models,
• S3 ?log p(y ?0)/?? and W3 W1

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Class II Asymptotic Distribution of LRT

• Combining three regions and let a, b ? 0,
• LRT max sup1,4 LRT(a), sup2,4 LRT(b), sup3
  LRT(a,b)
• max , sup W2(?)2,
• ? sup W2(?)2 ,
• where
• The asymptotic null distribution of LRT is
  supremum of squared Gaussian process w.r.t. ?
• Simplification (null distribution and
  approximation to p-values) can be adopted from
  Class I

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Class II Alternatives

• Question Can one find alternatives test
  statistics with conventional asymptotic null
  distributions?
• Restricted LRT limit range for d to a, 1 with
  a gt 0 (Region 1 and 4)
• TR(a) sup1,4 LRT(a)
• TR(a) decreases in a
• How to choose a?
• Smaller the a, the better
• Chi-square approximation maybe in doubt

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Class II Alternatives (cont)

• 2. Smooth version (penalized LRT)
• Instead of excluding (d, ?) values in Regions 2
  3, they are penalized toward d 0 by
  considering penalized log-likelihood
• PL(d, ? c) log L(d, ?) c g(d),
• where g(d) 0 is a smooth penalty with
  , maximized at d0 and c gt 0
  controlling the magnitude of penalty
• Bayesian interpretation
• g(d) could be prior on d

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Class II Penalized likelihood

• Define the penalized LR test statistic for H0 ?
  ?0
• PLRT(c) 2 sup PL(d, ? c) PL(d0, ?0 c)
• Under H0 PLRT(c) ? as n ? 8
• Proof is more demanding
• Similar concerns to restricted LRT
• Decreasing in c
• Chi-square approximation maybe invalid with small
  c
• Lose power if d is small (linked families is
  small in proportion)
• Different approach for mixture/admixture model
  was provided by
• by Chen et al. (2001, 2004) and Fu et al. (2006)
  

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Application Genetic Linkage for Schizophrenia

• Conducted by Ann Pulver at Hopkins
• 486 individuals from 54 multiplex families
• Interested in marker D22S942 in chromosome 22
• Schizophrenia is relatively high in prevalence
  with strong evidence of genetic heterogeneity
• To take into account of this phenomenon, consider
• d Binom (m, ?) (1 d) Binom (m, ?0),
• where d e 0, 1, ? e 0, 0.5 with ?0 0.5

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Genetic Linkage Study of Schizophrenia (cont)

• With this admixture model considered,
• LRT 6.86, p-value 0.007
• PLRT with
• PLRT(3.0) 5.36, p-value 0.010
• PLRT(0.5) 5.49, p-value 0.009
• PLRT(0.01) 6.84, p-value 0.004
• Which p-value do we trust better?

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Figure asymptotic vs empirical distribution of
the LRT for genetic linkage example
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Discussion

• Issue considered, namely, nuisance parameters
  absent under null, is common in practice
• Examples can be classified into Class I and II
• Class I H0 can only be specified through d ( 0)
• Class II H0 can specified either in d ( 0) or ?
  ( ?0)
• For Class I, asymptotic distribution of LRT is
  well known, and through principal component
  representation
• Deriving sufficient conditions for simple null
  distribution
• Proposing a means to approximate p-values

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Discussion (cont)

• For Class II, less well developed
• Deriving asymptotic null distribution of LRT
• Through this derivation, we observe
• Connection with Class I
• Connection with RLRT and PLRT
• Proof on asymptotic of PLRT non-trivial
• Shedding light on why penalty applied to d not to
  ??
• Pointing out some peculiar features and
  shortcoming of these two approaches
• A genetic linkage example on schizophrenia was
  presented for illustration

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Discussion (cont)

• Some future work
• Constructing confidence intervals/regions
• Generalizing to partial/conditional likelihood
• Cox PH model with change point (nuisance
  function)
• Extending to estimating function approach in the
  absence of likelihood function to work with
• Linkage study of IBD sharing for affected
  sibpairs
• E(S(t)) 1 (1 - 2?t,?)2 E(S(?)) 1 (1 -
  2?t,?)2 d
• ?t,? (1 exp(0.02t ?)/2

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Figure log likelihood and penalized log
likelihood function for genetic linkage example