Residuals Diagnostic in Functional Errors in Variables Models

Abdolrahman Rasekh

Department of , Ahwaz University, Ahwaz, Iran [email protected]

1. Introduction

In the ordinary linear models, response variables are the only variables that will be observed with error and so we have residuals due only to this kind of error. However, in the measurement errors models there are errors in response and predictors variables and consequently residuals in both directions. Fuller (1987, p. 25), defines of the measurement errors models. Instead of defining residuals in horizontal and vertical directions he suggests a function of both residuals as a residual of the model. He also proposes using diagnostic techniques analogous to those for the ordinary linear models. Miller (1990) provides partial theoretical justification for the use of these techniques. Both Fuller and Miller consider only the models with no replicated observations. In this paper we consider residuals in functional measurement errors models with emphasis on the replicated cases. We show that in this case we can not use Fuller’s definition of the residuals except in the special cases which we shall derive. The functional measurement errors model with replicated is defined by

=+ = Yyeil i il in1, ..., =+ = (1) Xxuij i ij jr1, ..., i = β ′ = yxii ls1,. ..., i

β In this model is the matrix of coefficients, xi and yi are the vectors of unobservable fixed values with k and 1 dimensions, which can be observed through X ij and Yil , respectively.

Furthermore, eil and uij are random errors of the individual observations which have normal σ Σ distribution with zero and matrices ee and uu , respectively. We allow the ≠ model to have srii for at least one of the ith and we refer to this model as the unequally replicated functional measurement errors model. In this case there is no natural pairing among Xij and Yil . = Thus we assume that cov(euil , mj ) 0 for all i, j, l, and m. If we consider the of the observations at each level of the model (1), then we will have a different model among the means of observations. Since we have replicated observations we will assume that consistent estimates of the β σ Σ β$ σ$ Σ$ , ee and uu exist and are given by , ee and uu .

2. Residuals in the unequally replicated case

== = $ =−σν$ −1Σ$ $ Suppose that in model (1) srii1, in1, ..., , then xXiiνν ui ν will be an νβ$ =−$′ σ =+σ β ′Σ β ΣΣ=−β ′ estimate of the xi , with iiYX i, νν ee uu and uuuν . Fuller (1987, p. ν$ $ =−$ $ =−$ 25) defines i as the residual of the model. If we define uXxiii and eYyiii as the $ = σν$ −1Σ$ $ $ =+βσ$′ $ −1Σ$ ν$ residuals in each direction then we have uiuiνν ν and eiui(1 νν ν ) . It is obvious that ν$ $ $ residuals in both directions are functions of i and so we no longer need to look at ui and ei separately for diagnostic procedures in this case. In the replicated case we have two different kinds of errors and hence residuals. One is due to the individuals and the other one is due to the means of observations at each level. Carroll & = Speigelman (1992) refer to the errors and the residuals in the special case of model (1) with si 1 = ν =−β ′ νβ$ =−$′ for in1, ..., as ijYX i ij and ijYX i ij , respectively. An extension of this definition > for the cases with si 1 is to match Yi with each individual X ij observation, that is define νβ=−′ νβ$ =−$′ ijYX i ij and ijYX i ij . However, there is some weakness in this definition. Firstly, this definition of the errors leads to a set of correlated errors. Secondly, there is no relation between ν$ the residuals in both directions and ij . Following the argument proposed by Fuller (1987, chapter 3) the maximum likelihood −1 $ $ −1 estimate of the x will be xX$ =−σν$ Σ $ in which νβ$ =−YX′ , ΣΣ=−r (),β ′ i iiννii ui ii ν ii i uiuuiiν −− σσββ=+sr11′Σ . Thus, the residuals of the mean observations in both directions will be ννii ieei uu

$ −1 $ $ $ $ $ −1 $ (2) uXx$$=−=σνννΣ ν and eYy$$=− =+(1 βσν′Σ ννν) in= 1, ..., . iiiii ui ii iii uii ii i

Furthermore, the residuals of the individual observations are

$$=−=−+σν$ −1 Σ$ $ in= 1, ..., uXxXXij ij i ij iνν u ν i (3) ii ii jr= 1, ..., $ $ $ −1 $ i eYyYY$$=−=−++(1 βσν′Σ ννν) ililiili uii ii i = ls1,. ..., i

$$ ν$ From (2) it is obvious that ueii and are not only functions of the i , but also depend on the $ σ$ and Σ . On the other hand from (3) we find that u$ and e$ will depend both on residuals of ννiiu ii ν ij il the mean observations and how far the individual observation is from its mean at that level. Furthermore, there is no relation between these residuals and those suggested by Carroll & Speigelman (1992). Therefore in the unequally replicated case we suggest using residuals of both individual and mean observations in each direction as the residuals of the model rather than using ν$ ν$ ij and i . = = $ $ If scrii for in1, ..., , where c is a constant, then the residuals ui and ei in unequally ν$ replicated case will be functions of the i and so are the same as those for the unreplicated case.

REFERENCES

Carroll, R. J. & Spiegelman, C. H. (1992). Diagnostics for nonlinearity and in errors-in-variables regression. Technometrics 34, 186-196. Fuller, W. A. (1987). Measurement error models. Wiley, New York. Miller, S. M. (1990). Analysis of residuals from measurement error models. In statistical analysis of measurement error models and applications. Brown, P. and Fuller, W. A. (Eds). Proc. of the AMS- IMS-SIAM joint summer research conference held June 10-16, 1989.