Table 5 is the measurable data summary

Table 5 reports summary statistics 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 ordinally 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 - Resid^2

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

From Table 7:

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

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

v3 = (measure 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

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

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%

Identification	Meaning
Data log-likelihood chi-square	This is a estimate of the global fit of the data to the model.
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
Identification
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
Variance of residuals	square of S.D. (Population) of Resd.

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.