Table 24.0 Variance components scree plot for persons 
Table 24.0 shows a variance decomposition of the observations for the persons. This is not produced for PRCOMP=O.
Table 24.0 Variance components scree plot for persons
Table 24.1, 24.11 Principal components plots of person loadings
Table 24.2, 24.12 Person Principal components analysis/contrast of residuals
Table 24.3, 24.13 Person contrast by items
Table 24.4, 24.14 Person contrast loadings sorted by measure
Table 24.5, 24.15 Person contrast loadings sorted by entry number
Table 24.99 Largest residual correlations for persons
Table of STANDARDIZED RESIDUAL variance in Eigenvalue units = KID information units
Eigenvalue Observed Expected
Total raw variance in observations = 150.8183 100.0% 100.0%
Raw variance explained by measures = 76.8183 50.9% 50.7%
Raw variance explained by persons = 30.5375 20.2% 20.2%
Raw Variance explained by items = 46.2809 30.7% 30.6%
Raw unexplained variance (total) = 74.0000 49.1% 100.0% 49.3%
Unexplned variance in 1st contrast = 18.2333 12.1% 24.6%
Unexplned variance in 2nd contrast = 6.9841 4.6% 9.4%
Unexplned variance in 3rd contrast = 6.1491 4.1% 8.3%
Unexplned variance in 4th contrast = 4.9831 3.3% 6.7%
Unexplned variance in 5th contrast = 4.8045 3.2% 6.5%
STANDARDIZED RESIDUAL VARIANCE SCREE PLOT
VARIANCE COMPONENT SCREE PLOT
++++++++++++
100%+ T +
 
V 63%+ +
A  M 
R 40%+ U +
I  
A 25%+ I +
N  P 
C 16%+ +
E  
10%+ 1 +
L  
O 6%+ +
G  
 4%+ 2 3 +
S  4 5 
C 3%+ +
A  
L 2%+ +
E  
D 1%+ +
 
0.5%+ +
++++++++++++
TV MV PV IV UV U1 U2 U3 U4 U5
VARIANCE COMPONENTS
If your Table says "Total variance in observations", instead of "Total raw variance in observations", then please update to the current version of Winsteps, or produce this Table with PRCOMP=R.
Extreme items and persons (minimum possible and maximum possible raw scores) are omitted from this computation because their correlations are 0.
Simulation studies, and the empirical results of Winsteps users, indicated that the previous computation of "variance explained" was overoptimistic in explaining variance. So a more conservative algorithm was implemented. Technically, the previous computation of "variance explained" used standardized residuals (by default). These are generally considered to have better statistical properties than the raw residuals. But the raw residuals (PRCOMP=R) were found to provide more realistic explanations of variance, so the current Winsteps computation uses raw residuals for "variance explained" in the top half of the variance table.
The "Unexplained variance" is controlled by PRCOMP=, which defaults to standardized residuals (PRCOMP=S). Set PRCOMP=R to express the entire table in terms of raw residuals.
Table of STANDARDIZED RESIDUAL variance in Eigenvalue units = ITEM information units 


Eigenvalue 
Observed percentage of total variance 
Observed percentage of unexplained variance 
Expected percentage of total variance 
Total raw variance in observations = 
50.9 
100.0% 
← Expected values if these data fit the Rasch model perfectly →
← If these match reasonably, then the measures explain the expected amount of variance in the data → 
100.0% 
Raw variance explained by measures = 
25.9 
50.9% 
46.5% 

Raw variance explained by persons = 
10.3 
20.2% 
18.5% 

Raw Variance explained by items = 
15.6 
30.7% 
28.0% 

Raw unexplained variance (total) = 
25.0 = count of items (or persons) 
49.1% 
100.0% 
53.5% 
Unexplned variance in 1st contrast = 
4.6 
9.1% 
18.5% 
← Use simulations to estimate the Raschmodelexpected values SIFILE= 
Unexplned variance in 2nd contrast = 
2.9 
5.8% 
11.8% 

Unexplned variance in 3rd contrast = 
2.3 
4.5% 
9.2% 

Unexplned variance in 4th contrast = 
1.7 
3.4% 
6.9% 

Unexplned variance in 5th contrast = 
1.6 
3.2% 
6.5% 
Table of STANDARDIZED RESIDUAL variance: the standardized residuals form the basis of the "unexplained variance" computation, set by PRCOMP=
in Eigenvalue units: variance components are rescaled so that the total unexplained variance has its expected summed eigenvalue.
ITEM information units: the eigenvalue units are rescaled to match the number of items, so these values are equivalent to "strength in item units".
Observed: variance components for the observed data
Expected: variance components expected for these data if they exactly fit the Rasch model, i.e., the variance that would be explained if the data accorded with the Rasch definition of unidimensionality.
Total raw variance in observations: total rawscore variance in the observations
Raw variance explained by measures: rawscore variance in the observations explained by the Rasch item difficulties, person abilities and polytomous scale structures.
Raw variance explained by persons: rawscore variance in the observations explained by the Rasch person abilities (and apportioned polytomous scale structures)
Raw variance explained by items: rawscore variance in the observations explained by the Rasch item difficulties (and apportioned polytomous scale structures)
Raw unexplained variance (total): rawscore variance in the observations not explained by the Rasch measures
Unexplned variance in 1st, 2nd, ... contrast: variance that is not explained by the Rasch measures is decomposed into Principal Component Analysis, PCA, components = Contrasts. The size of the first, second, ... contrast (component) in the PCA decomposition of standardized residuals (or as set by PRCOMP=), i.e., variance that is not explained by the Rasch measures, but that is explained by the contrast.
The important lines in this Table are "contrasts". If the first contrast is much larger than the size of an Eigenvalue expected by chance, usually less than 2  www.rasch.org/rmt/rmt191h.htm  please inspect your Table 23.3 to see the contrasting content of the items which is producing this large offdimensional component in your data, or Table 24.3 to see the contrasting persons. The threat to Rasch measurement is not the ratio of unexplained (by the model) to explained (by the model), or the amount of explained or unexplained. The threat is that there is another nonRasch explanation for the "unexplained". This is what the "contrasts" are reporting.
How Variance Decomposition is done ...
1.A central person ability and a central item difficulty are estimated. When the central ability is substituted for the estimated person abilities for each observation, the expected total score on the instrument across all persons equals the observed total score. Similarly, when the central ability is substitute for the estimated item difficulties for each observation, the expected total score on the instrument across all items equals the observed total score.
2.For each observation, a central value is predicted from the central person ability and the central item difficulty and the estimated rating scale (if any).
In the "Observed" columns:
3."Total raw variance in observations =" the sumofsquares of the observations around their central values.
4."Raw unexplained variance (total)=" is the sumofsquares of the difference between the observations and their Rasch predictions, the raw residuals.
5."Raw variance explained by measures=" is the difference between the "Total raw variance" and the "Raw unexplained variance".
6."Raw variance explained by persons=" is the fraction of the "Raw variance explained by measures=" attributable to the person measure variance (and apportioned rating scale structures).
7."Raw variance explained by items=" is the fraction of the "Raw variance explained by measures=" attributable to the item measure variance (and apportioned rating scale structures).
8.The reported variance explained by the items and the persons is normalized to equal the variance explained by all the measures. This apportions the variance explained by the rating scale structures.
9.The observation residuals, as transformed by PRCOMP=, are summarized as an interperson correlation matrix, with as many columns as there are nonextreme persons. This correlation matrix is subjected to Principle Components Analysis, PCA.
10.In PCA, each diagonal element (correlation of the person with itself) is set at 1.0. Thus the eigenvalue of each person is 1.0, and the total of the eigenvalues of the matrix is the number of persons. This is the sum of the variance modeled to exist in the correlation matrix, i.e., the total of the unexplained variance in the observations.
11.For convenience the size of the "Raw unexplained variance (total)" is rescaled to equal the total of the eigenvalues. This permits direct comparison of all the variance terms.
12.The correlation matrix reflects the Raschpredicted randomness in the data and also any departures in the data from Rasch criteria, such as those due to multidimensionality in the persons.
13.PCA reports components. If the data accord with the Rasch model, then each person is locally independent and the interperson correlations are statistically zero. The PCA analysis would report each person as its own component. Simulation studies indicate that even Raschconforming data produce eigenvalues with values up to 2.0, i.e., with the strength of two persons.
14.Multidimensionality affects the pattern of the residuals. The residual pattern should be random, so the "contrast" eigenvalue pattern should approximately match the eigenvalue pattern from simulated data. When there is multidimensionality the residuals align along the dimensions, causing the early contrast eigenvalues to be higher than those from random (simulated) data. So multidimensionality inflates the early PCA contrasts above the values expected from random data, and correspondingly must lower the later ones, because the eigenvalue total is fixed.
15."Unexplned variance in 1st contrast =" reports the size of the first PCA component. This is termed a "contrast" because the substantive differences between persons that load positively and negatively on the first component are crucial. It may reflect a systematic second dimension in the persons.
16."Unexplned variance in 2nd contrast =". Consecutively smaller contrasts are reported (up to 5 contrasts). These may also contain systematic multidimensionality in the persons.
In the "Expected" columns:
17."Raw variance explained by measures=" is the sumofsquares of the Raschpredicted observations (based on the item difficulties, person abilities, and rating scale structures) around their central values.
18."Raw variance explained by persons=" is the fraction of the "Raw variance explained by measures=" attributable to the person measure variance (and apportioned rating scale structures).
19."Raw variance explained by items=" is the fraction of the "Raw variance explained by measures=" attributable to the item measure variance (and apportioned rating scale structures).
20.The reported variance explained by the items and the persons is normalized to equal the variance explained by all the measures. This apportions the variance explained by the rating scale structures.
21."Raw unexplained variance (total)=" is the summed Raschmodel variances of the observations around their expectations, the unexplained residual variance predicted by the Rasch model.
22."Total raw variance in observations =" is the sum of the Raschmodel "Raw variance explained by measures=" and the "Raw unexplained variance (total)="
23.The "Model" and the "Empirical" values for the "Total raw variance in observations =" are both rescaled to be 100%.
24.Use the SIFILE= option in order to simulate data. From these data predicted model values for the contrast sizes can be obtained.
STANDARDIZED RESIDUAL VARIANCE SCREE PLOT
VARIANCE COMPONENT SCREE PLOT
++++++++++++
100%+ T +
 
V 63%+ +
A  M 
R 40%+ U +
I  
A 25%+ I +
N  P 
C 16%+ +
E  
10%+ +
L  1 
O 6%+ +
G  2 
 4%+ 3 +
S  4 5 
C 3%+ +
A  
L 2%+ +
E  
D 1%+ +
 
0.5%+ +
++++++++++++
TV MV PV IV UV U1 U2 U3 U4 U5
VARIANCE COMPONENTS
Scree plot of the variance component percentage sizes, logarithmically scaled:
On plot 
On xaxis 
Meaning 
T 
TV 
total variance in the observations, always 100% 
M 
MV 
variance explained by the Rasch measures 
P 
PV 
variance explained by the person abilities 
I 
IV 
variance explained by the item difficulties 
U 
UV 
unexplained variance 
1 
U1 
first contrast (component) in the residuals 
2 
U2 
second contrast (component) in the residuals, etc. 
For the observations (PRCOMP=Obs), a standard Principal Components Analysis (without rotation, and with orthogonal axes) is performed based on the scored observations.
Table of OBSERVATION variance (in Eigenvalue units)
 Observed 
Raw unexplained variance (total) = 13.0 100.0%
Unexplned variance in 1st contrast = 9.9 76.1%
Unexplned variance in 2nd contrast = .9 7.2%
Unexplned variance in 3rd contrast = .6 4.3%
Unexplned variance in 4th contrast = .3 2.6%
Unexplned variance in 5th contrast = .3 2.0%
Here "contrast" means "component" or "factor".
Example:
In this example, 50.9% of the variance in the data is explained by the measures. If the data fit the model perfectly, 50.7% would be explained. These percentages are close, indicating that the estimation of Rasch measures has been successful.
The variance in the data explained by the item difficulties, 30.7%, is larger than the variance explained by the person abilities, 20.2%. For dichotomies, this usually reflects the person and item measure standard deviations in Table 3.1.
The unexplained variance in the data is 49.1%. This includes the Raschpredicted randomness and any departures in the data from Rasch criteria, such as those due to multidimensionality in the items.
The first contrast has an eigenvalue of 18.2, corresponding to a strength of over 18 persons, and considerably larger than the largest eigenvalue expected by chance. It corresponds to 12.1% of the variance in the data.
Help for Winsteps Rasch Measurement Software: www.winsteps.com. Author: John Michael Linacre
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