Multivariate Statistics
Thomas Asendorf, Steffen Unkel
Study sheet 9
Summer term 2017
Exercise 1:
(a) Show that the covariance matrix


1.0 0.63 0.45


Σ =  0.63 1.0 0.35 
0.45 0.35 1.0
for p = 3 standardized variables z1 , z2 , and z3 can be generated by the exploratory
factor analysis (EFA) model
z1 = 0.9ξ1 + δ1
z2 = 0.7ξ1 + δ2
z3 = 0.5ξ1 + δ3 ,
where Var(ξ1 ) = 1, Cov(δ, ξ1 ) = 0, and


0.19
0
0


Cov(δ) =  0
0.51
0  .
0
0
0.75
(b) Calculate communalities λ2i1 (i = 1, 2, 3) and interpret these quantities.
(c) Obtain the correlations between the observed variables and the common factor. Which
variable might carry the greatest weight in “naming” the common factor? Justify your
answer!
Exercise 2:
Verify that the method of Anderson and Rubin leads to a matrix of common factor scores,
say F ∈ Rn×k , in which the columns are uncorrelated (see slides 17 and 18 of the set of slides
“Factor Analysis”).
Exercise 3:
Verify that for the EFA model, maximum likelihood factor analysis is scale invariant.
Date: 30 June 2017
Page 1
Exercise 4:
In the following Figure, the effect of a rotation of axes on the coordinates of a point is
illustrated.
(a) Based on the Figure above, deduce formally the following statement: If we know the
coordinates (x1 , x2 ) of a point P referred to a pair of orthogonal axes, and we wish to
derive its new coordinates (y1 , y2 ) when the axes are rotated (anticlockwise) through
an angle α, we can use
y1 = x1 cos α + x2 sin α ,
y2 = −x1 sin α + x2 cos α .
(b) Put the coefficients of x1 and x2 in the two equations above in a matrix A and state
its properties.
(c) The reverse operation to an anticlockwise rotation of axes through an angle α is a
clockwise rotation through an angle α. Given the point (y1 , y2 ), suppose you would like
to rotate the axes clockwise to find the coordinates (x1 , x2 ) relative to its new axes.
Which matrix would you use for rotation?
Date: 30 June 2017
Page 2
Exercise 5:
Calsyn and Kenny (1977)1 recorded the values of the following six variables for 556 white
eighth-grade students:
SCA: self-concept ability;
PPE: perceived parental evaluation;
PTE: perceived teacher evaluation;
PFE: perceived friend’s evaluation;
EA: educational aspiration;
CP: college plans.
Calsyn and Kenny (1977) postulated that two underlying latent variables, namely “ability”
and “aspiration”, generated the relationships between the observed variables. The first four
of the manifest variables were assumed to be indicators of ability and the last two indicators
of aspiration; the latent variables are assumed to be correlated. The observed correlations
between the six manifest variables are given in Table 1.
Tabelle 1: Correlation matrix of ability and aspiration data (values above the main diagonal
are omitted).
SCA
PPE
PTE
PFE
EA
CP
SCA
1.00
0.73
0.70
0.58
0.46
0.56
PPE
PTE
PFE
EA
CP
1.00
0.68
0.61
0.43
0.52
1.00
0.57
0.40
0.48
1.00
0.37
0.41
1.00
0.72
1.00
(a) Write down the regression-like equations that specify the postulated confirmatory factor
analysis (CFA) model.
(b) Read the data given in Table 1 into R.
(c) Use the function sem() in the R package sem to fit the postulated CFA model to
the data. The first argument in sem() is a model specified in the so-called reticular
action model (RAM) notation. You may use specifyModel() along with the text file
ability_model.txt, which is available on the course website, to create such a model
object. Once you fitted the CFA model to the data, give an interpretation of the fitting
results. Does the CFA model fit the data well?
(d) Draw a path diagram of the fitted CFA model with the estimated parameters
superimposed, using the function semPaths() in the R package semPlot.
1
Calsyn, R. J. and Kenny, D. A. (1977): Self-concept of ability and perceived evaluation of others. Cause
or effect of academic achievement, Journal of Educational Psychology, Vol. 69, pp. 136-145.
Date: 30 June 2017
Page 3