Response to J.A. Nelder: What is the mixed models controversy?.

Abstract

We begin by apologizing for the lamentable omission of Professor Nelder’s interesting andcontroversial papers both in Lencinaet al.(2005) and in Lencina & Singer (2006). Although he focus of our papers is much narrower than that of his articles, we feel that any manuscript touching on the topic of parametrization and definition of hypotheses in linear mixed models should necessarily refer to his numerous contributions to this field.Essentially, we compare two stochastic models (one with constrained parameters, denotedCPand the other with unconstrained parameters, denoted UP by Voss (1999)) that are extensively employed to analyze data from balanced experiments with a fixed and a random factor. The twomodels generate different test statistics presumably directed at the same hypothesis, namely thatof the non-existence of the random factor main effect in the presence of the interaction. Differingopinions over which statistic should be used is what we call the ‘mixed-model controversy’following Voss (1999). Our objective is to provide an answer to this very specific question. Toretain this focus, we do not discuss the general issue of model building or add to a debate overthe reasonableness of the hypothesis under investigation. We explicitly state this in the abstractof Lencinaet al.(2005).