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

24.1 24.99 Tables of persons with highly correlated residuals.

24.2 Plot of loadings on first contrast in residuals vs. person measures.

24.3 Persons in contrast loading order.

24.4 Items exhibiting contrast.

24.5 Persons in measure order.

24.6 Persons in entry order.

24.7 etc. Subsequent contrasts.

24.99 Tables of persons with highly correlated residuals.

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 over-optimistic 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)
	-- Empirical --			Modeled
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 Rasch-model-expected 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%
(More smaller contrasts)	Eigenvalue units	Percentage of total variance	Percentage of unexplained variance

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.

Empirical: variance components for the observed data

Model: 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 raw-score variance in the observations

Raw variance explained by measures: raw-score variance in the observations explained by the Rasch item difficulties, person abilities and rating scale structures

Raw variance explained by persons: raw-score variance in the observations explained by the Rasch person abilities (and apportioned rating scale structures)

Raw variance explained by items: raw-score variance in the observations explained by the Rasch item difficulties (and apportioned rating scale structures)

Raw unexplained variance (total): raw-score variance in the observations not explained by the Rasch measures

Unexplned variance in 1st, 2nd, ... contrast: size of the first, second, ... contrast (component) in the principal component 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 off-dimensional 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 non-Rasch explanation for the "unexplained". This is what the "contrasts" are reporting.

How Variance Decomposition is done ...

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 "Empirical" columns:

3.	"Total raw variance in observations =" the sum-of-squares of the observations around their central values.

4.	"Raw unexplained variance (total)=" is the sum-of-squares 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 unexpained 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 inter-person correlation matrix, with as many columns as there are non-extreme 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 Rasch-predicted 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 inter-person correlations are statistically zero. The PCA analysis would report each person as its own component. Simulation studies indicate that even Rasch-conforming 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 multi-dimensionality in the persons. In the "Model" columns:

17.	"Raw variance explained by measures=" is the sum-of-squares of the Rasch-predicted 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 Rasch-model 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 Rasch-model "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 x-axis	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)

-- Empirical --

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, 71.7% of the variance in the data is explained by the measures. If the data fit the model perfectly, 71.1% 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, 48.2%, is twice the variance explained by the person abilities, 23.6%. This usually reflects the person and item measure standard deviations in Table 3.1.

The unexplained variance in the data is 28.3%. This includes the Rasch-predicted 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 8.9, corresponding to a strength of almost 9 persons, and considerably larger than the largest eigenvalue expected by chance. It corresponds to 7.4% of the variance in the data, 15% of the variance explained by the person measures, 48.2%. For comparison, when the positive-loading persons are split from the negative-loading persons, the correlation between the item raw scores on each half of the sample is r=0.84.