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 thestrength 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 Extraction Sums of Squared Loadings

Total% of

Variance 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 -- ModeledTotal 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 : 0.90Person separation: 3.05Strata : 4.4

Person Reliability : 0.92Person separation: 3.38Strata : 4.84

-4

-3

-2

-1

0

1

2

3

4

-4 -3 -2 -1 0 1 2 3 4

Measure (55)

Me

as

ure

(5

1)

r = 0.994

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