DOCTORAT HEC

Exercices Analyse factorielle

(Extrait de A.C. Rencher: Methods of multivariate analysis, Wiley, 1995)

1. Use the ramus bone data of Table 1

Table 1 : Ramus bone length at four age for 20 boys

a)Extract loadings by the principal component method and do a varimax rotation. Use two factors.

b)Do all variables have a complexity of 1? (Complexity of a variable = number of moderate or high loadings.) Carry out an oblique rotation to improve the loadings.

c)What is the angle between the oblique axes? Would a single factor (m=1) be more appropriate here?

2. In a classical experiment carried out from 1918 to 1934, apple trees of different rootstocks were compared. The data for eight trees from each of six rootstocks are given in Table 2. The variables are

Y1 = trunk girth a 4 years (mm  100)

Y2 = extension growth at 4 years (m)

Y3 = trunk girth at 15 years (mm  100)

Y4 = weight of tree above ground at 15 years (lb  1000)

Table 2 : Rootstock data

a)Carry out a factor analysis of the rootstock data of table 2. Combine the groups into a single sample.

b)Estimate the loadings for two factors by the principal component method and do a varimax rotation.

c)Did the rotation improve the loadings?

3. Baten, Tack and Baeder compared judges’ scores on fish prepared by three methods. Twelve fish were cooked by each method, and several judges tasted fish samples and rated each on four variables: y1 = aroma, y2 = flavour, y3 = texture, and y4 = moisture. The data are in Table 3. Each entry is an average score for the judges on that fish.

Table 3 : Judges’scores on fish prepared by three methods

Use the fish data of table 3. Combine the groups into a single sample.

a)Obtain loadings on two factors by the principal component methods and do a varimax rotation.

b)Notice the similarity of loadings for y1 and y2. Is there any indication in the correlation matrix as to why this is so.

c)Compute factor scores.

d)Construct error bars for both factors comparing the three methods.

e)Carry out a MANOVA on the factor scores comparing the three groups.

4. Twenty engineer apprentices and 20 pilots were given six tests. The variables were

Y1 = intelligence

Y2 = form relation

Y3 = dynamometer

Y4 = dotting

Y5 = sensory motor coordination

Y6 = perseveration

The data are given in Table 4.

Table 4 : Comparison of six tests on engineer apprentices and pilots

Use the engineer data of Table 4. Combine the groups into a single sample.

a)Using a scree plot; the number of eigenvalues greater than 1, and the percentages, is there a clear choice of m?

b)Extract three factors by the principal component method and carry out a varimax rotation.

c)Extract three factors by the principal factor method and carry out a varimax rotation.

d)Compare the results of (b) and (c).

5. Four measurements were made on two species of flea beetles (Haltica oleracea and Haltica carduorum). The variables were

Y1 =distance of transverse groove from posterior border of prothorax (m)

Y2 = length of elytra (in 0.01 mm)

Y3 = length of second antennal joint (m)

Y4 = length of third antennal joint (m)

The data are given in Table 5.

Carry out a factor analysis of the flea data in Table 5. Combine the groups into a single sample.

a)From an examination of the eigenvalues greater than 1, the scree plot and the percentages, is there a clear choice of m.

b)Extract two factors by the principal component method and carry out a varimax rotation.

c)Is the rotation an improvement? Try an oblique rotation.

Table 5 : Four measurements on two species of flea beetles

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