﻿ Table 5 is the measurable data summary

# Table 5 Measurable data summary

Table 5 reports summary statistics about the data for the analysis.

+--------------------------------------------------+

| Cat  Score  Exp.  Resd StRes|                    |

|-----------------------------+--------------------|

| 4.80  4.80  4.80   .00  .00 | Mean (Count: 1152) |

| 1.63  1.63  1.03  1.27  .99 | S.D. (Population)  |

| 1.64  1.64  1.03  1.27  .99 | S.D. (Sample)      |

+--------------------------------------------------+

Column headings have the following meanings:

Cat  = Observed value of the category as entered in the data file.

Score = Value of category after it has been recounted cardinally commencing with "0" corresponding to the lowest observed category.

Exp. = Expected score based on current estimates

Resd = Residual, the score difference between Step and Exp.

StRes = The residual standardized by its standard error. StRes is expected to approximate a unit normal distribution.

Mean = average of the observations

Count = number of observations

S.D. (Population) = standard deviation treating this sample as the entire population

S.D. (Sample) = standard deviation treating this sample as a sample from the population. It is larger than S.D. (Population).

The raw-score error variance % is  100*(Resd S.D./Cat S.D.)²

When the parameters are successfully estimated, the mean Resd is 0.0. If not, then there are estimation problems - usually due to too few iterations, or anchoring.

When the data fit the Rasch model, the mean of the "StRes" (Standardized Residuals) is expected to be near 0.0, and the "S.D." (sample standard deviation) is expected to be near 1.0. These depend on the distribution of the residuals.

Explained variance by each facet can be approximated by using the element S.D.^2 (^2 means "squared").

From Table 5:

Explained variance = Score Population S.D.^2 - Resd^2

Explained variance % = Explained variance * 100 / Score Population S.D.^2

From Table 7:

|-------------------------------+--------------+---------------------+------+-------------+ +---------------------|

|   460.8    96.0     4.8   4.73|    .00   .08 | 1.00  -.1   .99  -.2|      |   .61       | | Mean (Cnt: 12)      |

|    29.5      .0      .3    .32|    .19   .00 |  .23  1.8   .22  1.7|      |   .05       | | S.D. (Population)   |

|    30.8      .0      .3    .33|    .20   .00 |  .24  1.9   .23  1.8|      |   .06       | | S.D. (Sample)       |

+------------------------------------------------------------------------------------------ ----------------------+

v1 = (measure Population S.D. facet 1)^2

v2 = (measure Population S.D. facet 2)^2

v3 = (measure Population S.D. facet 3)^2

vsum = v1 + v2 + v3 + .... (for all facets)

Compute Explained variance for each facet:

Explained variance % by facet 1 = (Explained variance %) * v1 /vsum

Explained variance % by facet 2 = (Explained variance %) * v2 /vsum

Explained variance % by facet 3 = (Explained variance %) * v3 /vsum

For a Rasch-based Generalizability Coefficient:

G = (Explained variance% by target facet) / 100

A more specific Generalizability Coefficient can be formulated by selecting appropriate variance terms from Table 5, Table 7, and Table 13.

Data log-likelihood chi-square = 3787.4307

Approximate degrees of freedom = 1093

Chi-square significance prob.  = .0000

Count   Mean   S.D.   Params

Responses after end-of-file        =         0   0.00   0.00        0 (only shown if not 0)

Responses only in extreme scores   =         0   0.00   0.00        0 (only shown if not 0)

Responses in two extreme scores    =         0   0.00   0.00        0 (only shown if not 0)

Responses with invalid elements    =         0   0.00   0.00        0 (only shown if not 0)

Responses invalid after recounting =         0   0.00   0.00        0 (only shown if not 0)

Responses used for estimation      =      1152   4.80   1.63       59

Responses in one extreme score     =         0   0.00   0.00        0 (only shown if not 0)

All Responses                      =      1152   4.80   1.63       59

 Identification Meaning Data log-likelihood chi-square This is a estimate of the global fit of the data to the model = -2 * log-likelihood of the empirical data. Likelihood = product of the probabilities of the observations. Approximate degrees of freedom The d.f. of the chi-square approximates the number of data points less the number of parameters estimated Chi-square significance prob. The probability of observing the chi-square value (or larger) when the data fit the model Response Type Responses not used for estimation: see Residual File Responses after end-of-file A Facets internal work-file has too many responses. Please report this to Winsteps.com and rerun this analysis. Responses only in extreme scores The category of the rating scale cannot be estimated. Responses in two extreme scores These cannot be estimated nor used for estimating element measures. Responses with invalid elements Elements for these observations are not defined. See Table 2 with Build option. Responses invalid after recounting A dichotomy or rating scale has less than two categories, so it cannot be estimated. See Table 8 for missing or one-category rating scales. Response Type Responses used for estimation: see Residual File Responses used for estimation This is the count of responses used in estimating non-extreme parameter values (element measures and rating scale structures). Responses in one extreme score These are only used for estimating the element with the extreme score All Responses Shown if there is more than one response type listed above

Count of measurable responses           =      1152

Raw-score variance of observations      =   2.67 100.00%

Variance explained by Rasch measures    =   1.06 39.57%

Variance of residuals                   =   1.61 60.43%

Variance explained by bias/interactions =   0.14 5.24%

Variance remaining in residuals         =   1.47 55.06%

 An approximate Analysis of Variance (ANOVA) of the data Identification Meaning Count of measurable responses All responses (including for extreme scores) with weighting (if any) Raw-score variance of observations Square of S.D. (Population) of Score Variance explained by Rasch measures Raw score variance - Variance of residuals. The size of the expected variance for 2-facet models is shown in www.rasch.org/rmt/rmt221j.htm Variance of residuals Square of S.D. (Population) of Resd. Variance explained by bias/interactions The variance explained by the bias/interactions specified with "B" in your Models= statements Variance remaining in residuals Variance of residuals - Variance of interactions

Nested models: Suppose we want to estimate the effect on fit of a facet.

Run twice:

First analysis: 3 facets

Models = ?,?,?, R

Second analysis: 2 facets:

Models = ?,?,X,R

We can obtain an estimate of the improvement of fit based on including the third facet:

Chi-square of improvement = Data log-likelihood chi-square (2 facets) - Data log-likelihood chi-square (3 facets) with d.f. (2 facets) - d.f. (3 facets).

If global fit statistics are the decisive evidence for choice of analytical model, then Facets is not suitable. In the statistical philosophy underlying Facets, the decisive evidence for choice of model is "which set of measures is more useful" (a practical decision), not "which set of measures fit the model better" (a statistical decision). The global fit statistics obtained by analyzing your data with log-linear models (e.g., in SPSS) will be more exact than those produced by Facets.

Help for Facets Rasch Measurement Software: www.winsteps.com Author: John Michael Linacre.

 Forum Rasch Measurement Forum to discuss any Rasch-related topic

Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez
Winsteps Tutorials Facets Tutorials Rasch Discussion Groups

Coming Rasch-related Events
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