Score Tests in Semiparametric Models
Raymond J. Carroll
Department of Statistics
Faculties of Nutrition and Toxicology
Texas A&M University
http://stat.tamu.edu/~carroll
Papers available at my web site
Texas is surrounded on all sides by foreign countries: Mexico to the
south and the United States to the east, west and north
Palo Duro
Canyon, the
Grand
Canyon of
Texas
West Texas

East Texas 
Wichita Falls,
Wichita Falls,
that’s my
hometown
Guadalupe
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National
Park
College Station, home
of Texas A&M
University
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Big Bend
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Palo Duro Canyon of the Red River
Co-Authors
Arnab Maity
Co-Authors
Nilanjan Chatterjee
Co-Authors
Kyusang Yu
Enno Mammen
Outline
• Parametric Score Tests
• Straightforward extension to semiparametric
models
• Profile Score Testing
• Gene-Environment Interactions
• Repeated Measures
Parametric Models
• Parametric Score Tests
• Parameter of interest = b
 Nuisance parameter = q
 Interested in testing whether b = 0
 Log-Likelihood function = L (Y ; X ; Z; ¯ ; µ)
Parametric Models
• Score Tests are convenient when it is easy to
maximize the null loglikelihood
P
n
i = 1 L (Y i ; X i ; Z i ; 0; µ)
• But hard to maximize the entire loglikelihood
P
n
i = 1 L (Y i ; X i ; Z i ; ¯ ; µ)
Parametric Models
b ) be the MLE for a given value of b
• Let µ(¯
• Let subscripts denote derivatives
• Then the normalized score test statistic is just
S=
¡ 1=2 P n
n
i = 1L
b
f
Y
;
X
;
Z
;
0;
µ(0)g
i
i
i
¯
Parametric Models
• Let I
be the Fisher Information evaluated at
b = 0, and with sub-matrices such as I ¯ µ
• Then using likelihood properties, the score
statistic under the null hypothesis is
asymptotically equivalent to
·
¡ 1=2 P n
n
i = 1 L ¯ f Y i ; X i ; Z i ; 0; µg
¸
1
¡ I ¯ µ I ¡µµ
L µ f Y i ; X i ; Z i ; 0; µg
Parametric Models
• The asymptotic variance of the score statistic is
T = I ¯ ¯ ¡ I ¯ µ I ¡ 1 I µ¯
µµ
• Remember, all computed at the null b = 0
• Under the null, if b = 0 has dimension p, then
S > T ¡ 1 S ) Â2p
Parametric Models
• The key point about the score test is that all
computations are done at the null hypothesis
• Thus, if maximizing the loglikelihood at the null
is easy, the score test is easy to implement.
Semiparametric Models
• Now the loglikelihood has the form
L f Y i ; X i ; ¯ ; µ(Z i )g
• Here, µ(¢) is an unknown function. The
obvious score statistic is
¡ 1=2 P n
n
i = 1L
• Where
b i ; 0)g
f
Y
;
X
;
0;
µ(Z
i
i
¯
b i ; 0) is an estimate under the null
µ(Z
Semiparametric Models
• Estimating µ(¢)
in a loglikelihood like
L f Y i ; X i ; 0; µ(Z i )g
• This is standard
• Kernel methods used local likelihood
• Splines use penalized loglikelihood
Simple Local Likelihood
• Let K be a density function, and h a bandwidth
• Your target is the function at z
• The kernel weights for local likelihood are
 Zi -z 
K

h


• If K is the uniform density, only observations within h
of z get any weight
Simple Local Likelihood
Only observations
within h = 0.25 of
x = -1.0 get any
weight
Simple Local Likelihood
• Near z, the function should be nearly linear
• The idea then is to do a likelihood estimate
local to z via weighting, i.e., maximize
P
µ
n
i = 1K
Zi ¡ z
h
• Then announce
¶
L f Y i ; X i ; 0; ®0 + ®1 (Z i ¡ z)g
θ̂(z) = 0
Simple Local Likelihood
• It is well-known that the optimal bandwidth is
h / n¡
1=5
• The bandwidth can be estimated from data
using such things as cross-validation
Score Test Problem
• The score statistic is
S=
¡ 1=2 P n
n
i = 1L
b i ; 0)g
f
Y
;
X
;
0;
µ(Z
i
i
¯
¡ 1=5
• Unfortunately, when h / n
this
statistic is no longer asymptotically normally
distributed with mean zero
• The asymptotic test level = 1!
Score Test Problem
• The problem can be fixed up in an ad hoc
way by setting h / n ¡ 1=3
• This defeats the point of the score test, which
is to use standard methods, not ad hoc ones.
Profiling in Semiparametrics
• In profile methods, one does a series of steps
• For every b, estimate the function by using
local likelihood to maximize
µ
¶
Pn
Zi ¡ z
L f Y i ; X i ; ¯ ; ®0 + ®1 (Z i ¡ z)g
i = 1K
h
• Call it
b ¯)
µ(z;
Profiling in Semiparametrics
• Then maximize the semiparametric profile
loglikelihood
¡ 1=2 P n
b i ; ¯ )g
n
L
f
Y
;
X
;
¯
;
µ(Z
i
i
i= 1
• Often difficult to do the maximization, hence
the need to do score tests
Profiling in Semiparametrics
• The semiparametric profile loglikelihood has
many of the same features as profiling does
in parametric problems.
• The key feature is that it is a projection, so
that it is orthogonal to the score for µ(Z) ,
or to any function of Z alone.
Profiling in Semiparametrics
• The semiparametric profile score is
¡ 1=2 P n
n
i= 1
@
b i ; ¯ )g
L f Y i ; X i ; ¯ ; µ(Z
¯= 0
@¯
·
¡ 1=2 P n
b
¼n
i = 1 L ¯ f Y i ; X i ; 0; µ(Z i ; 0)g
@b
b
+ L µ f Y i ; X i ; 0; µ(Z i ; 0)g µ(Z i ; ¯ ) ¯ = 0
@¯
¸
Profiling in Semiparametrics
• The problem is to compute
@b
µ(Z i ; ¯ ) ¯ = 0
@¯
• Without doing profile likelihood!
Profiling in Semiparametrics
• The definition of local likelihood is that for
every b,
£
¤
0 = E L µ f Y ; X ; ¯ ; µ(Z; ¯ )gjZ = z
• Differentiate with respect to b.
Profiling in Semiparametrics
• Then
h
i
E L ¯ µ f Y ; X ; 0; µ(Z; 0)gjZ = z
@b
£
¤
µ(Z; 0) = ¡
@¯
E L µµ f Y ; X ; 0; µ(Z; 0)gjZ = z
• Algorithm: Estimate numerator and
denominator by nonparametric regression
• All done at the null model!
Results
• There are two things to estimate at the null
model
b
µ(Z;
0)
@b
b (Z; 0)
µ(Z; 0) = µ
¯
@¯
• Any method can be used without affecting
the asymptotic properties
• Not true without profiling
Results
• We have implemented the test in some cases
using the following methods:
•
•
•
•
Kernels
Splines from gam in Splus
Splines from R
Penalized regression splines
• All results are similar: this is as it should be:
because we have projected and profiled, the
method of fitting does not matter
Results
• The null distribution of the score test is
asymptotically the same as if the following
were known
µ(Z)
@
µ(Z; 0) = µ¯ (Z; 0)
@¯
Results
• This means its variance is the same as the
variance of
¡ 1=2 P n
n
i= 1
·
L ¯ f Y i ; X i ; 0; µ(Z i )g
¸
+ L µ f Y i ; X i ; 0; µ(Z i )gµ¯ (Z i ; 0)
• This is trivial to estimate
• If you use different methods, the asymptotic
variance may differ
Results
• With this substitution, the semiparametric
score test requires no undersmoothing
• Any method works
• How does one do undersmoothing for a
spline or an orthogonal series?
Results
• Finally, the method is a locally semiparametric
efficient test for the null hypothesis
• The power is: the method of nonparametric
regression that you use does not matter
Example
• Colorectal adenoma: a precursor of colorectal
cancer
• N-acetyltransferase 2 (NAT2): plays
important role in detoxification of certain
aromatic carcinogen present in cigarette smoke
• Case-control study of colorectal adenoma
• Association between colorectal adenoma and the
candidate gene NAT2 in relation to smoking
history.
Example
• Y = colorectal adenoma
• X = genetic information (below)
• Z = years since stopping smoking
More on the Genetics
• Subjects genotyped for six known functional
SNP’s related to NAT2 acetylation activity
• Genotype data were used to construct diplotype
information, i.e., The pair of haplotypes the
subjects carried along their pair of homologous
chromosomes
More on the Genetics
• We identifies the 14
most common
diplotypes
• We ran analyses on
the k most
common ones, for
k = 1,…,14
The Model
• The model is a version of what is done in
genetics, namely for arbitrary ° ,
© >
ª
>
pr (Y = 1jX ; Z) = H X ¯ + µ(Z i ) + ° X ¯ µ(Z i )
• The interest is in the genetic effects, so we
want to know whether b = 0
• However, we want more power if there are
interactions
The Model
• For the moment, pretend ° is fixed
© >
ª
>
pr (Y = 1jX ; Z) = H X ¯ + µ(Z i ) + ° X ¯ µ(Z i )
• This is an excellent example of why score
testing: the model is very difficult to fit
numerically
• With extensions to such things as longitudinal data
and additive models, it is nearly impossible to fit
The Model
• Note however that under the null, the model is
simple nonparametric logistic regression
pr (Y = 1jX ; Z) = H f µ(Z i )g
• Our methods only require fits under this simple
null model
The Method
• The parameter ° is not identified at the null
© >
ª
>
pr (Y = 1jX ; Z) = H X ¯ + µ(Z i ) + ° X ¯ µ(Z i )
• However, the derivative of the loglikelihood
evaluated at the null depends on °
• The, the score statistic depends on
S n (° )
°
The Method
• Our theory gives a linear expansion and an
easily calculated covariance matrix for each °
¡ 1=2 P n
n
i = 1 ª i (°
S n (° ) =
covf S n (° )g ! T (° )
) + op (n ¡
• The statistic S n (° ) as a process in °
converges weakly to a Gaussian process
1=2
)
The Method
• Following Chatterjee, et al. (AJHG, 2006), the
overall test statistic is taken as
-
n
= m ax a·
h
i
>
¡ 1
S
(°
)T
(° )S n (° )
°· c
n
• (a,c) are arbitrary, but we take it as (-3,3)
Critical Values
• Critical values are easy to obtain via simulation
• Let b=1,…,B, and let
N
ib
= N or mal(0; 1) Recall
¡ 1=2 P n
n
i = 1 ª i (°
S n (° ) =
) + op (n ¡ 1=2 )
• By the weak convergence, this has the same limit
distribution as (with estimates under the null)
S b n (° ) =
¡ 1=2 P n
b i (°
n
ª
i= 1
in the simulated world
)N
ib
Critical Values
• This means that the following have the same limit
distributions under the null N i b = N or mal(0; 1)
h
i
- n = m ax a· ° · c S >n (° )T ¡ 1 (° )S n (° )
h
i
- b n = m ax a· ° · c S >b n (° )T ¡ 1 (° )S b n (° )
• This means you just simulate - b n
times to get the null critical value
a lot of
Simulation
• We did a simulation under a more complex model
(theory easily extended)
© >
ª
>
>
pr (Y = 1jX ; Z) = H S ´ + X ¯ + µ(Z i ) + ° X ¯ µ(Z i )
• Here X = independent BVN, variances = 1, and
with means given as
¯ = c(1; 1) > ;
• c = 0 is the null
; c = 0; 0:01; :::; 0:15
Simulation
• In addition,
© >
ª
>
>
pr (Y = 1jX ; Z) = H S ´ + X ¯ + µ(Z i ) + ° X ¯ µ(Z i )
Z = U nifor m [¡ 2; 2]
µ(z) = sin(2z)
S = N or m al(0; 1); ´ = 1
¡ 3· ° · 3
• We varied the true values as
° t r ue = 0; 1; 2
Power Simulation
Simulation Summary
• The test maintains its Type I error
• Little loss of power compared to no interaction
when there is no interaction
• Great gain in power when there is interaction
• Results here were for kernels: almost numerically
identical for penalized regression splines
NAT2 Example
• Case-control study with 700 cases and 700
controls
• As stated before, there were 14 common
diplotypes
• Our X was the design matrix for the k most
common, k = 1,2,…,14
NAT2 Example
• Z was years since stopping smoking
• Co-factors S were age and gender
• The model is slightly more complex because of
the non-smokers (Z=0), but those details
hidden here
NAT2 Example Results
NAT2 Example Results
• Stronger evidence of genetic association seen
with the new model
• For example, with 12 diplotypes, our p-value
was 0.036, the usual method was 0.214
Extensions: Repeated Measures
• We have extended the results to repeated
measures models
• If there are J repeated measures, the
loglikelihood is
L f Y i 1 ; :::; Y i J ; X i 1 ; :::; X i J ; ¯ ; µ(Z i 1 ); :::µ(Z i J )g
• Note: one function, but evaluated multiple
times
Extensions: Repeated Measures
• If there are J repeated measures, the
loglikelihood is
L f Y i 1 ; :::; Y i J ; X i 1 ; :::; X i J ; ¯ ; µ(Z i 1 ); :::µ(Z i J )g
• There is no straightforward kernel method for
this
• Wang (2003, Biometrika) gave a solution in the
Gaussian case with no parameters
• Lin and Carroll (2006, JRSSB) gave the efficient
profile solution in the general case including
parameters
Extensions: Repeated Measures
• It is straightforward to write out a profiled
score at the null for this loglikelihood
L f Y i 1 ; :::; Y i J ; X i 1 ; :::; X i J ; ¯ ; µ(Z i 1 ); :::µ(Z i J )g
• The form is the same as in the non-repeated
measures case: a projection of the score for ¯
onto the score for µ(¢)
Extensions: Repeated Measures
@
µ(Z i ; ¯ ) ¯ = 0
@¯
• Here the estimation of
is not
trivial because it is the solution of a complex
integral equation
Extensions : Repeated Measures
• Using Wang (2003, Biometrika) method of
nonparametric regression using kernels, we
have figured out a way to estimate
@
µ(Z i ; ¯ ) ¯ = 0
@¯
• This solution is the heart of a new paper (Maity,
Carroll, Mammen and Chatterjee, JRSSB, 2009)
Extensions : Repeated Measures
• The result is a score based method: it is based
entirely on the null model and does not need to
fit the profile model
• It is a projection, so any estimation method can
be used, not just kernels
• There is an equally impressive extension to
testing genetic main effects in the possible
presence of interactions
Extensions : Nuisance Parameters
• Nuisance parameters are easily handled with a
small change of notation
Extensions: Additive Models
• We have developed a version of this for the
case of repeated measures with additive
models in the nonparametric part
Y ij =
X >ij
¯+
P
D
d= 1 µd (Z i j d )
(² i 1 ; :::; ² i J ) > = [0; § ]:
+ ² ij
Extensions: Additive Models
• The additive model method uses smooth
backfitting (see multiple papers by Park, Yu and
Mammen)
Summary
• Score testing is a powerful device in parametric
problems.
• It is generally computationally easy
• It is equivalent to projecting the score for ¯
onto the score for the nuisance parameters
Summary
• We have generalized score testing from
parametric problems to a variety of
semiparametric problems
• This involved a reformulation using the
semiparametric profile method
• It is equivalent to projecting the score for
onto the score for µ(¢)
¯
• The key was to compute this projection while
doing everything at the null model
Summary
• Our approach avoided artificialities such as ad
hoc undersmoothing
• It is semiparametric efficient
• Any smoothing method can be used, not just
kernels
• Multiple extensions were discussed