UNIDIMENSIONALITY – MULTIDIMENSIONALITY
(An example)
Panayiotis Panayides
Rules of thumb for the existence of a second dimension
1. In the unexplained variance a secondary dimension must have the
strength of at least 3 items.
Eigenvalue < 3 (in a reasonable length test) then the test is probably
unidimensional. (Linacre, 2005)
2. The first factor must explain a significant % of the unexplained variance
(more than 20%)
3. A significant % of the total variance in the data
(Linacre, 2005, eigenvalue 2.7, N = 14, 0.2% of total variance)
Example: Maths (27) and Language (28) diagnostic tests
PCA of raw scores
Component
Initial Eigenvalues
Total
% of
Variance
Extraction Sums of Squared Loadings
Cumulative %
Total
% of
Variance
Cumulative %
1
11,919
21,671
21,671
11,919
21,671
21,671
2
3,002
5,458
27,129
3,002
5,458
27,129
3
2,104
3,825
30,955
2,104
3,825
30,955
4
1,916
3,483
34,438
1,916
3,483
34,438
5
1,801
3,275
37,713
1,801
3,275
37,713
6
1,564
2,844
40,557
1,564
2,844
40,557
7
1,474
2,679
43,236
1,474
2,679
43,236
8
1,387
2,521
45,757
1,387
2,521
45,757
9
1,309
2,380
48,138
1,309
2,380
48,138
10
1,261
2,292
50,430
1,261
2,292
50,430
11
1,219
2,216
52,646
1,219
2,216
52,646
12
1,180
2,146
54,793
1,180
2,146
54,793
13
1,136
2,066
56,858
1,136
2,066
56,858
14
1,083
1,970
58,828
1,083
1,970
58,828
15
1,053
1,914
60,742
1,053
1,914
60,742
16
1,004
1,826
62,567
1,004
1,826
62,567
x + x = ………..
x.x = ……..
PCA of standardised residuals (Linacre, 1998)
Table of STANDARDIZED RESIDUAL variance (in Eigenvalue units)
-- Empirical -Modeled
Total raw variance in observations
=
109.8 100.0%
100.0%
Raw variance explained by measures
=
54.8 49.9%
51.0%
Raw variance explained by persons =
33.5 30.5%
31.2%
Raw Variance explained by items
=
21.3 19.4%
19.8%
Raw unexplained variance (total)
=
55.0 50.1% 100.0%
49.0%
Unexplned variance in 1st contrast =
4.1
3.7%
7.5%
Test – 51 items
Test – 55 items
Person Reliability : 0.90
Person separation: 3.05
Strata
: 4.4
Person Reliability : 0.92
Person separation: 3.38
Strata
: 4.84
4
3
2
Measure (51)
1
0
-4
-3
-2
-1
0
1
2
3
-1
r = 0.994
-2
-3
-4
Measure (55)
4
Given
• the small % of unexplained variance (7.5%) explained by the first contrast
• the small % of Total variance (3.7%) explained by the first contrast
• the variance explained by the second dimension is about 14 times smaller
than the variance explained by the dimension measured by the test
• the closeness of the points to a straight line passing through the origin
• the extremely high correlation (r = 0.994) between the person measures
• the fact that the two tests were on different subjects but were very easy
Unidimensional