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UNIDIMENSIONALITY – MULTIDIMENSIONALITY (An example) Panayiotis Panayides

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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

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PCA of raw scores ComponentInitial EigenvaluesExtraction Sums of Squared Loadings Total % of VarianceCumulative %Total % of VarianceCumulative % 1 11,91921,671 11,91921,671 2 3,0025,458 27,1293,0025,45827,129 3 2,1043,82530,955 2,1043,82530,955 4 1,9163,48334,4381,9163,48334,438 5 1,8013,27537,7131,8013,27537,713 6 1,5642,84440,5571,5642,84440,557 7 1,4742,67943,2361,4742,67943,236 8 1,3872,52145,7571,3872,52145,757 9 1,3092,38048,1381,3092,38048,138 10 1,2612,29250,4301,2612,29250,430 11 1,2192,21652,6461,2192,21652,646 12 1,1802,14654,7931,1802,14654,793 13 1,1362,06656,8581,1362,06656,858 14 1,0831,97058,8281,0831,97058,828 15 1,0531,91460,7421,0531,91460,742 16 1,0041,82662,5671,0041,82662,567

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x + x = ………..x.x = ……..

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PCA of standardised residuals (Linacre, 1998)

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Test – 51 itemsTest – 55 items Person Reliability :0.90 Person separation:3.05 Strata :4.4 Person Reliability :0.92 Person separation:3.38 Strata :4.84

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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

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