**Statistical Computing '99 - Schloß Reisensburg**

## A Semiparametric Approach to Analysis of Covariance

### Michael G. Schimek

Karl-Franzens-Universität Graz

Institut für Medizinische Informatik, Statistik und Dokumentation

Engelgasse 13, A-8010 Graz, Austria

Michael.Schimek@kfunigraz.ac.at

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

for
,
where
are known *k*-dimensional
vectors forming a design matrix ,
is an unknown parameter vector, *f*is an unknown smooth spline function, and the errors
are independent,
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

where
,
,
and
.
The equation of (linear) analysis of covariance is

and has the same structure as that of the partially linear model.
is a
design matrix and
is a
matrix of
covariate measurements.
and
are the corresponding unknown parameter vectors. Futher
it is assumed that
and
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.

REFERENCES

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*,
**38**, 597-611.

Schimek, M. G. (1999). Estimation and Inference in Partially Linear Models with
Smoothing Splines. *JSPI*, to appear.

31. Statistical Computing '99