iclass.05

CLASSROOM DAY

Session Slot: Monday, 2:00 - 2:55

AudioVisual Request: None

Session Organizer: Scott, David W.

Session Title: Classroom Day

1. Quotient Spaces in Statistical Models

McCullagh, Peter,   University of Chicago

Address: Department of Statistics University of Chicago 5734 University Ave Chicago, IL 60637

Phone: 773-702-8340

Fax: 773-702-9810

Email: pmcc@roxbee.uchicago.edu

Abstract: A statistical variable, or variate for short, is a function, usually real-valued, on the statistical units $\Omega$.Each variate is thus an element in the vector space ${\cal V} = {\cal R}^\Omega$.Many statistical models, including all linear models and generalized linear models, are specified in terms of subspaces, usually in the form $\mu\in{\cal X}$, $\log\mu\in{\cal X}$, or $g(\mu) \in{\cal X}$, where ${\cal X}$ is the span of the model matrix X. Quotient spaces are rarely considered explicitly in statistical work, but it will be argued in this talk that this oversight is a mistake. Quotient spaces do arise naturally in various settings of which the following are typical examples.

In the case of the linear model $E(Y)\in{\cal X}$,the residual $Y+{\cal X}$ lies in the quotient space ${\cal V}/{\cal X}$.An elementary calculation shows that $Y+{\cal X}$ is normally distributed in the quotient space, whatever this may mean, and the likelihood function based on $Y+{\cal X}$ is the REML likelihood.

In testing a composite null hypothesis H₀ against an alternative $H_A\supset H_0$,if both both hypotheses correspond to vector subspaces, the standard test statistics are based on the quotient space projection on to H_A/H₀.

Interaction between two factors A, B, is the quotient-space projection Y+(A+B) of the response Y on to the space ${\cal V}/(A+B)$,i.e. Y modulo additivity. Regardless of whether the design is balanced, the interaction sum of squares is $\Vert {Y+(A+B)} \Vert^2$, or more generally $\Vert {P Y + (A+B)} \Vert^2$, where P is the orthogonal projection on to A.B.

In the case of multinomial response models, the probability vector $(\pi_1,\ldots,\pi_k)$ is a point in the probability simplex in ${\cal R}^k$.The usual link transformation $\eta_j = \log\pi_j$ has inverse

$$\pi_r = \exp(\eta_r)\big/\sum\exp( \eta_j)$$

so that $\eta_j$ and $\eta_j+c$ are equivalent points. In other words, $\eta$ is a point in ${\cal R}^k/1$, and the logistic link is 1-1 from the interior of the simplex onto ${\cal R}^k/1$.

Bayes's theorem is quotient-space vector addition.

For incomplete data, if the relation between the incomplete response and the complete response is linear, the incomplete response is most naturally viewed as a point in a certain quotient space. Although more restrictive than the EM algorithm, this formulation leads to a simple theory of estimation using estimating functions.

The lecture will be in the form of an expository review session, beginning with the definition of a quotient space, continuing with inner products norms and Lebesgue measure on quotient spaces, and deriving the normal distribution on ${\cal V}/{\cal X}$.Details of this and other examples listed above can be found at http://galton.uchicago.edu/~pmcc/quotient.ps

KeyWords: interaction, analysis of variance, multinomial response model, REML

AudioVisual Request: None