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This paper explains score tests in semiparametric models and discusses their application in gene-environment interactions and repeated measures parametric models.
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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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAAA
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 Mountains National Park College Station, home of Texas A&M University I-45 Big Bend National Park I-35
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 =
Parametric Models • Score Tests are convenient when it is easy to maximize the null loglikelihood • But hard to maximize the entire loglikelihood
Parametric Models • Let be the MLE for a given value of b • Let subscripts denote derivatives • Then the normalized score test statistic is just
Parametric Models • Let be the Fisher Information evaluated at b = 0, and with sub-matrices such as • Then using likelihood properties, the score statistic under the null hypothesis is asymptotically equivalent to
Parametric Models • The asymptotic variance of the score statistic is • Remember, all computed at the null b = 0 • Under the null, if b = 0 has dimension p, then
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 • Here, is an unknown function. The obvious score statistic is • Where is an estimate under the null
Semiparametric Models • Estimating in a loglikelihood like • 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 • 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 • Then announce
Simple Local Likelihood • It is well-known that the optimal bandwidth is • The bandwidth can be estimated from data using such things as cross-validation
Score Test Problem • The score statistic is • Unfortunately, when 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 • 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 • Call it
Profiling in Semiparametrics • Then maximize the semiparametric profile loglikelihood • 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 , or to any function of Z alone.
Profiling in Semiparametrics • The semiparametric profile score is
Profiling in Semiparametrics • The problem is to compute • Without doing profile likelihood!
Profiling in Semiparametrics • The definition of local likelihood is that for every b, • Differentiate with respect to b.
Profiling in Semiparametrics • Then • 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 • 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
Results • This means its variance is the same as the variance of • 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 NAT2acetylation 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 , • 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 • 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 • Our methods only require fits under this simple null model
The Method • The parameter is not identified at the null • However, the derivative of the loglikelihood evaluated at the null depends on • The, the score statistic depends on
The Method • Our theory gives a linear expansion and an easily calculated covariance matrix for each • The statistic as a process in converges weakly to a Gaussian process
The Method • Following Chatterjee, et al. (AJHG, 2006), the overall test statistic is taken as • (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 Recall • By the weak convergence, this has the same limit distribution as (with estimates under the null) in the simulated world
Critical Values • This means that the following have the same limit distributions under the null • This means you just simulate a lot of times to get the null critical value
Simulation • We did a simulation under a more complex model (theory easily extended) • Here X = independent BVN, variances = 1, and with means given as • c = 0 is the null
Simulation • In addition, • We varied the true values as
