Statistical Computing '99 - Schloß Reisensburg
A Semiparametric Approach to Analysis of Covariance
Michael G. Schimek
Institut für Medizinische Informatik, Statistik und Dokumentation
Engelgasse 13, A-8010 Graz, Austria
We discuss a partially linear (i.e. semiparametric) regression model with a predictor function
consisting of a categorical parametric component and a metrical nonparametric component.
We assume that responses
are obtained at non-stochastic values
of a covariable t. Let us now consider
are known k-dimensional
vectors forming a design matrix ,
is an unknown parameter vector, fis an unknown smooth spline function, and the errors
zero mean random variables with a common variance .
When the design matrix
characterizes k treatment conditions this setting can be interpreted as analysis
of covariance. Our semiparametric model in matrix notation takes the form
The equation of (linear) analysis of covariance is
and has the same structure as that of the partially linear model.
design matrix and
are the corresponding unknown parameter vectors. Futher
it is assumed that
are of full rank and uncorrelated. These assumptions
can be relaxed in the semiparametric approach apart from the advantage that the relation
between y and t can be arbitrary apart from certain smoothness requirements.
Schimek (1999) has proposed a general cubic spline-based algorithm
for partially linear models which also allows for testing of the treatment conditions.
The smoothing parameter choice is crucial as might be expected. We recommend an unbiased risk criterion introduced in Eubank et al. (1998).
The connection to so-called nonparametric analysis of covariance (Quade, 1982) is pointed out.
Finally a real data example is given.
Eubank, R. L., Kambour, E. L., Kim, J. T., Klipple, K., Reese, C. S.
and Schimek, M.(1998). Estimation in Partially Linear Models. CSDA, 29, 27-34.
Quade, D. (1982). Nonparametric Analysis of Covariance by Matching. Biometrics,
Schimek, M. G. (1999). Estimation and Inference in Partially Linear Models with
Smoothing Splines. JSPI, to appear.
31. Statistical Computing '99