Table 5 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

 

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

 

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%

 

 

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.