Table 7 Reliability and Chi-squared Statistics |
This is for 32-bit Facets 3.87. Here is Help for 64-bit Facets 4
Table 7 also provides summary statistics by facet.
+------------------------------------------------------------------------------------------ ----------------------+
| Total Total Obsvd Fair-M| Model | Infit Outfit |Estim.| Correlation | | |
| Score Count Average Avrage|Measure S.E. | MnSq Zstd MnSq Zstd|Discrm| PtMea PtExp | | Nu Reader |
|-------------------------------+--------------+---------------------+------+-------------+ +---------------------|
.....
|-------------------------------+--------------+---------------------+------+-------------+ +---------------------|
| 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) |
+------------------------------------------------------------------------------------------ ----------------------+
Model, Populn: RMSE .08 Adj (True) S.D. .17 Separation 2.17 Strata 3.22 Reliability (not inter-rater) .82
Model, Sample: RMSE .08 Adj (True) S.D. .18 Separation 2.28 Strata 3.38 Reliability (not inter-rater) .84
Model, Fixed (all same) chi-squared: 66.3 d.f.: 11 significance (probability): .00
Model, Random (normal) chi-squared: 9.4 d.f.: 10 significance (probability): .49
Inter-Rater agreement opportunities: 384 Exact agreements: 108 = 28.1% Expected: 82.6 = 21.5%
or
With extremes, Model, Populn: RMSE 1.05 Adj (True) S.D. 1.98 Separation 1.88 Strata 2.84 Reliability .78
With extremes, Model, Sample: RMSE 1.05 Adj (True) S.D. 2.01 Separation 1.91 Strata 2.89 Reliability .79
Without extremes, Model, Populn: RMSE 1.02 Adj (True) S.D. 1.71 Separation 1.68 Strata 2.57 Reliability .74
Without extremes, Model, Sample: RMSE 1.02 Adj (True) S.D. 1.75 Separation 1.71 Strata 2.62 Reliability .75
With extremes, Model, Fixed (all same) chi-squared: 175.9 d.f.: 34 significance (probability): .00
With extremes, Model, Random (normal) chi-squared: 33.8 d.f.: 33 significance (probability): .43
In summary:
"model" = "the unexpectedness in this facet is considered to be the randomness predicted by the Rasch model"
"population" = the elements in this facet are the entire population of possible elements
"sample" = the elements in this facet are a sample from the entire population of possible elements
"fixed" = we are testing the hypothesis "all the elements of this facet have statistically the same measure"
"random" = we are testing the hypothesis "all the elements of this facet are a random sample from a normally-distributed population"
RMSE = root-mean-square-error: it is the statistical average of the standard errors of the measures. It reports the overall precision of the measurement of the elements in the facet. RMSE is heavily influenced by the count of observations of each element.
Mean = |
arithmetic average |
Count = |
number of elements reported |
S.D. (Populn) |
is the standard deviation when this sample comprises the entire population. If the element list includes every possible element for the facet: use the Population statistics, e.g., grade levels, genders (sexes), ... |
S.D. (Sample) |
is the standard deviation when this sample is a random sample from the population. If there are "more like this" elements in addition to the current elements: use the Sample statistics, e.g., candidates, items (usually), tasks, .... |
With extremes |
including elements with extreme (zero and perfect, minimum possible and maximum possible) scores |
Without extremes |
excluding elements with extreme (zero and perfect, minimum possible and maximum possible) scores |
Model |
Estimated as though all noise in the data is due to model-predicted stochasticity (i.e., the best-case situation for randomness in the data) |
Real |
Estimated as though all unpredicted noise is contradicting model expectations (i.e., the worst-case situation |
RMSE |
root mean square standard error (i.e., the average S.E. statistically) for all non-extreme measures. |
Adj (True) S.D. |
"true" sample standard deviation of the estimates after adjusting for measurement error |
Separation |
Adj "true" S.D. / RMSE, a measure of the spread of the estimates relative to their precision. The signal-to-noise ratio is the "true" variance/error variance = Separation². See also Separation. |
Strata |
(4*Separation + 1)/3, a measure of the spread of the estimates relative to their precisions, when extreme measures are assumed to represent extreme "true" abilities. See also Strata |
Reliability (not inter-rater) |
Spearman reliability: Rasch-measure-based equivalent to the KR-20 or Cronbach Alpha raw-score-based statistic, i.e., the ratio of "True variance" to "Observed variance" (Spearman 1904, 1911). This shows how different the measures are, which may or may not indicate how "good" the test is. High (near 1.0) person and item reliabilities are preferred. This reliability is somewhat the opposite of an interrater reliability, so low (near 0.0) judge and rater reliabilities are preferred. See also Reliability. |
Fixed (all same) chi-squared: |
A test of the "fixed effect" hypothesis: "Can this set of elements be regarded as sharing the same measure after allowing for measurement error?" The chi-squared value and degrees of freedom (d.f.) are shown. The significance is the probability that this "fixed" hypothesis is the case. Depending on the sub-Table, this tests the hypothesis: "Can these items be thought of as equally difficult?" The precise statistical formulation is: Or this tests the hypothesis: "Can these raters be thought of as equally lenient?" Is there a statistically significant rater effect? And so on .... |
Random (normal) chi-squared: |
A test of the "random effects" hypothesis: "Can this set of elements be regarded as a random sample from a normal distribution?" The significance is the probability that this "random" hypothesis is the case. This tests the hypothesis: "Can these persons (items, raters, etc.) be thought of as sampled at random from a normally distributed population?" The precise statistical formulation is: |
Rater agreement opportunities |
when Inter-rater= facet-number. |
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