Stepwise Procedures In Discriminant Analysis

CHAPTER THREE

RESEARCH METHODOLOGY

STEPWISE METHODOLOGIES IN DISCRIMINANT ANALYSIS

Stepwise discriminant analysis is just like discriminant analysis. The only difference is the method of variables’ selection. It shares the same assumptions as well as tests with discriminant analysis. This is why deep knowledge on discriminant analysis is very necessary before one can apply the stepwise method.

Here, the independent variables are selected step by step (one at a time) on the basis of their discriminatory powers without the alteration of the model. Unlike in multiple regressions analysis, where independent variables are selected and the model altered as a result.

(1) The forward selection is based on Wilks’ ^ . At the first step we

calculate ^ (xi) for each individual variable and choose the one with

minimum         (xi) or maximum associated F.

With    m         =          degrees of freedom for error

n          =          degrees of freedom for hypothesis

P          =          Number of variables (dimension)

The partial ^ -Statistics and the partial F are distributed as ^ 1,n,m-P+1 and Fn, m - P+1, respectively. From the foregoing, in step one, the partial ^ (xi) and the accompanying partial F are found for each individual variable and the one with maximum partial F and minimum ^ (xi) is selected. x1 shows the variable selected at step 1 which is the variable with maximum F and minimum (xi).

At step two, we calculate ^ (xi/x1) for each of the P - 1 variables not entered at the first step, with x1 indicating the variable first entered. We then choose the next variable with maximum associated F or minimum ^ (xi/xi) or the variable which adds to the maximum separation to the one entered at step 1. The variable entered at step 2 is denoted by x2.

The above process is repeated for other available variables until the F falls below some predetermined threshold value. If the partial F is less than a second threshold value, Fout, the variable with the smallest partial F will be removed. There is also a very small possibility that the procedure will cycle continuously if Fout = F1n, though using a value of Fout slightly less than Fin, removes this possibility.

THE F-DISRTIBUTION

The F-Distribution is named after R.A. Fisher who originally developed it (Oyeka, 1996). This is also used in testing of hypothesis involving population means and variances. Let a random variable U be distributed as Chi-square with r1 degrees of freedom, and another random variable V, independent of U, be distributed as Chi-Square with r2 degrees of freedom. Then;

u/r

F = yt1 is said to have an F-distribution

/l2

with r1 and r2 degrees of freedom.

F-distribution is asymmetric and is generally, strongly skewed to the right. This is because the shape of F-distribution varies considerably with Changes in its degrees of freedom, r1 and r2 .

The cumulative probability distribution of F for the three commonly used probability values (^ = 0.05, 0.1 and 0.01), can be obtained from the Table of the F-Distribution. In the Table of the F-Distribution, the values in the main body of the table are Fi ri, r2, where â– => is the proportion of the area under the curve to the right hand tail of the F-Distribution with r1 = degrees of freedom of the numerator and r2 = degrees of freedom of the denominator. The F-ratio is obtained at the intersection of the column labeled r1 and the row labelled r2 in the Table of the F-Distribution.

F-Distribution is also related to Normal, x2 and t distributions based on the assumption that the underlying population is normally distributed.

The F-variable is a ratio of two independent Chi-square variables, each divided by its degrees of freedom as;

Sample size from population one (1)