Lecture 14—Monday, February 6, 2006

What was covered?

Terminology Defined

Overview

Explanation of the G2 Test

Frequency n1 n2 n3 ... nm
Value 1 2 3 ... m

where .

Saturated Model

in which there are only m – 1 parameters to estimate. Usually though we keep the likelihood the way it was and we incorporate the constraint into the maximization algorithm explicitly using, for example, the method of Lagrange multipliers. The point I wish to make here though is that there are really only m – 1 parameters to estimate.

Experimental Model

Likelihood Ratio Test

This test has a large sample chi-squared distribution where the degrees of freedom are the difference in the number of estimated parameters in the two models. In the saturated model we estimated m – 1 parameters. In the experimental model we estimated 0 parameters. Thus for this example . (Note: in general the degrees of freedom would be m – 1 – p where p is the number of parameters needed to specify the experimental model.)

where in the last step I make the identifications: are the observed counts and are the expected counts under the experimental model.

Explanation of the Pearson Χ 2 Test

where the equality holds only for . The expression on the right is called a geometric series and is an example of a type of infinite series called a power series. This identity is easily derived using algebra or by applying Taylor's theorem from calculus.

for .

Setting these expressions equal to each other yields

where convergence occurs for –1 < x ≤ 1 . (We pick up one of the endpoints as a bonus.)

The usual way these infinite series expressions are used is to approximate functions by only keeping a finite number of terms. Suppose we use this infinite series to replace the corresponding logarithm term in the G2 expression above. After grouping terms I elect to drop everything after the quadratic term in the series. The steps are shown below.

This proves the claimed result that the Pearson Χ 2 test is an approximation to the G2 test.

Cited Reference

Course Home Page


Jack Weiss
Phone: (919) 962-5930
E-Mail: jack_weiss@unc.edu
Address: Curriculum in Ecology, Box 3275, University of North Carolina, Chapel Hill, 27516
Copyright © 2006
Last Revised--Feb 9, 2006
URL: http://www.unc.edu/courses/2006spring/ecol/145/001/docs/lectures/lecture14.htm