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Principal Components Analysis

Principal Components Analysis with rotation

The first analysis is a 'pure' PCA using all variables, except for Region. Factor scores have been saved and all graph options selected.

Eigenanalysis of the Correlation Matrix

Eigenvalue

4.6505

2.1944

1.2284

1.1201

0.915

0.7011

Proportion

0.3880

0.1830

0.1020

0.0930

0.076

0.0580

Cumulative

0.3880

0.5700

0.6730

0.7660

0.842

0.9010


Eigenvalue

0.4477

0.308

0.210

0.1365

0.0514

0.0369

Proportion

0.0370

0.026

0.017

0.0110

0.0040

0.0030

Cumulative

0.9380

0.964

0.981

0.9930

0.9970

1.0000


Scree plot

Scree plot of eigen values.[D]

1
Effective dimensionality from the eigen values.

Which of the following statements is a reasonable description of the effective dimensionality of these data?

a) 3
b) 4
c) 5
d) 12
12 is incorrect, that is the original dimensionality. The other three answers can all be justified. The scree plot has an obvious break in slope at 3 components, 4 components have eigen values above 1 and the fifth eigen value is close to 1. In my opinion 4 or 5 would be the best options; the choice between these two depends on the ease of interpretation.
Check your answer

Variable

PC1

PC2

PC3

PC4

PC5

PC6

MaxAlt

0.407

0.053

-0.176

-0.351

-0.109

-0.013

MeanAlt

0.357

-0.110

0.023

-0.472

-0.151

0.031

SDAlt

0.384

0.186

-0.320

-0.091

0.042

-0.024

MeanSlope

0.357

0.196

-0.336

-0.040

0.267

-0.083

Mire

-0.235

-0.154

0.138

-0.563

-0.258

0.473

Heathland

0.326

-0.079

0.086

0.333

-0.035

0.302

WetHeath

-0.106

0.311

-0.115

0.026

-0.777

-0.446

Deer

0.326

0.070

0.225

0.248

-0.331

0.444

Sheep

-0.214

-0.366

-0.522

-0.152

0.053

0.038

Cattle

0.099

-0.617

-0.094

0.109

-0.124

-0.241

NPP

-0.196

0.096

-0.618

0.271

-0.208

0.458

Grazed

0.242

-0.507

0.008

0.213

-0.223

-0.100


Variable

PC7

PC8

PC9

PC10

PC11

PC12

MaxAlt

0.056

-0.105

-0.121

0.200

0.038

-0.771

MeanAlt

0.136

-0.460

-0.316

-0.225

-0.108

0.470

SDAlt

-0.069

0.334

0.120

0.632

-0.007

0.419

MeanSlope

-0.058

0.335

0.211

-0.694

0.008

-0.002

Mire

-0.219

0.485

0.063

-0.083

0.036

0.006

Heathland

-0.788

-0.156

-0.171

-0.027

-0.011

-0.021

WetHeath

-0.230

-0.004

0.091

-0.113

0.013

0.038

Deer

0.347

-0.117

0.563

-0.032

-0.134

-0.011

Sheep

-0.195

-0.420

0.523

0.038

0.177

0.021

Cattle

0.017

0.232

0.002

0.000

-0.676

-0.061

NPP

0.231

0.025

-0.419

-0.061

-0.098

0.001

Grazed

0.189

0.223

-0.137

-0.051

0.685

0.052


Loading plot (modified from Minitab default)

Loading plot for PC1 and PC2.[D]

2
Coefficients

Match the coefficient to the variable / component (PC1 to PC6 only).

a) 0.311
b) -0.777
c) 0.008
d) 0.407
e) -0.617
Look in the table (remember only PC1 to PC6) and find the intersection of the variable and PC.
Check your answer

3
Interpreting the coefficients

Which of the following statements can be justified?

a) It is difficult to interpret these coefficients since none of them are particularly large.
b) There is little evidence for a simple structure.
c) NNP and Mire make little contribution to components 1 and 2.
d) All of the topographic variables are associated with PC1
e) PC2 is mainly concerned with grazing by large herbivores.
f) A rotation of the components may improve the interpretation of the relationships between the original 12 variables.
All are correct. There are many coefficients in the range 0.25 - 0.5. There is some indication of simple structure, that could be clarified by a rotation.
Check your answer