Table 7 Reliability and Chi-square Statistics

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-square: 66.3  d.f.: 11  significance (probability): .00

Model,  Random (normal) chi-square: 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-square: 175.9  d.f.: 34  significance (probability): .00

With extremes, Model,  Random (normal) chi-square: 33.8  d.f.: 33  significance (probability): .43

 

 

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-square:

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-square 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:
wi = 1/SE²i for i=1,L, where L is the number of items, and Di is the difficulty/easiness of item i.
chi-square = S(wi.D²i) - S( wi.Di)²/ Swi  with d.f. = L-1

Or this tests the hypothesis: "Can these raters be thought of as equally lenient?" Is there a statistically significant rater effect?
The precise statistical formulation is:
wj = 1/SE²j for j=1,J, where J is the number of raters, and Cj is the leniency/severity of rater j.
chi-square = S(wj.C²j) - S( wj.Cj)²/ Swj  with d.f. = J-1

And so on ....

Random (normal) chi-square:

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:
var(D) = S(Di-Dmean)²/(L-1) - ( SSE²i)/L
wi = 1/(var(D)+SE²i)
chi-square = S(wi.D²i) - ( Swi.Di)²/ Swi  with d.f. = L-2

Rater agreement opportunities

see Table 7 Agreement statistics


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