MMU - Clustering and Classification

Golden Eagle Core Areas: Stepwise Analysis

The default stepwise method (minimize Wilk's lambda) is used.

**Variables Entered/Removed(a,b,c,d)**
	Entered	Wilks' Lambda
		Statistic	df1	df2	df3	Exact F
Step		Statistic	df1	df2	df3	Statistic	df1	df2	p
1	WET	0.313	1	2	37.000	40.687	2	37	0.000
2	LT200	0.108	2	2	37.000	36.810	4	72	0.000
3	STEEP	0.087	3	2	37.000	27.937	6	70	0.000
At each step, the variable that minimizes the overall Wilks' Lambda is entered.
a Maximum number of steps is 16.
b Minimum partial F to enter is 3.84.
c Maximum partial F to remove is 2.71.
d F level, tolerance, or VIN insufficient for further computation.

**Variables in the Analysis**
Step		Tolerance	F to Remove	Wilks' Lambda
1	WET	1.000	40.687
2	WET	.836	48.013	0.396
2	LT200	.836	34.167	0.313
3	WET	.815	15.506	0.164
	LT200	.587	30.987	0.240
	STEEP	.609	4.249	0.108

Only three variables are entered: Wet, LT200 & Steep, giving 2 significant discriminant functions. Although the ideal number of predictors was 2 (cases to predictor ratio), 3 is much better than 8.

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Summary of Canonical Discriminant Functions

**Eigenvalues**
Function	Eigenvalue	% of Variance	Cumulative %	Canonical Correlation
1	3.816(a)	73.3	73.3	0.890
2	1.393(a)	26.7	100.0	0.763
a First 2 canonical discriminant functions were used in the analysis.

**Wilks' Lambda**
Test of Function(s)	Wilks' Lambda	Chi-square	df	Sig.
1 through 2	0.087	87.998	6	0.000
2	0.418	31.409	2	0.000

**Standardized Canonical Discriminant Function Coefficients**
	Function
	1	2
WET	.0836	-0.199
STEEP	.0621	-0.159
LT200	.0955	0.794

Using the structure matrix to interpret the differences suggests that:

function 1 is mainly associated with large areas of wet heath and small areas of bog (negative correlation)
function 2 is mainly associated with large areas of land below 200m and small areas of steep land, land between 400 & 600m and wet heath.

Indeed these are effectively the same axes obtained using all 8 predictors.

**Structure Matrix**
	Function
	1	2
WET	.674(*)	-.577
BOG(a)	-.481(*)	-.269
POST(a)	-.072(*)	-.037
LT200	.236	.972(*)
STEEP	.339	-.717(*)
L4_600(a)	.049	-.568(*)
CALL(a)	-.250	.290(*)
PRE(a)	.046	.088(*)
Pooled within-groups correlations between discriminating variables and standardized canonical discriminant functions Variables ordered by absolute size of correlation within function.
* Largest absolute correlation between each variable and any discriminant function
a This variable not used in the analysis.

Since the axes are similar it is perhaps not too surprising that the separation is similar to the previous analysis.

**Functions at Group Centroids**
	Function
REGION	1	2
1.00	-3.503	-1.263
2.00	1.836	-.838
3.00	-.285	1.309
Unstandardized canonical discriminant functions evaluated at group means

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Classification Statistics

Territorial Map

Map of function 2 (y) against function 1 (x) scores. The map is split
into 3 zones indicating combinations of function scores that will result in
the allocation of a particular group label.

Symbols used in territorial map
Symbol Group Label
------ ----- --------------------
1         1
2         2
3         3
* Indicates a group centroid

You may be suprised that the classification accuracy is better using fewer predictors! This is not as surprising as it sounds, it can happen when the extra predictors do not discriminate but introduce noise that 'clouds' the distinction between the groups.

**Classification Results(b,c)**
			Predicted Group Membership			Total
		REGION	1	2	3	Total
Original	Count	1	7	0	0	7
		2	0	14	2	16
		3	0	0	17	17
	%	1	100.0	.0	.0	100
		2	.0	87.5	12.5	100
		3	.0	.0	100.0	100
Cross-validated(a)	Count	1	7	0	0	7
		2	0	14	2	16
		3	0	0	17	17
	%	1	100.0	.0	.0	100
		2	.0	87.5	12.5	100
		3	.0	.0	100.0	100
a Cross validation is done only for those cases in the analysis. In cross validation, each case is classified by the functions derived from all cases other than that case.
b 95.0% of original grouped cases correctly classified.
c 95.0% of cross-validated grouped cases correctly classified.

Although this stepwise analysis has produced some promising results it can be criticised because it involved the rather unthinking application of a stepwise selection procedure. In the next example judgement is used to select the predictors.

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Golden Eagle Core Areas: Stepwise Analysis

Summary of Canonical Discriminant Functions

Classification Statistics