Global Statistics:
Active KID: 35, non-extreme: 34
Active TAP: 18, non-extreme: 14
Active datapoints: 630
Missing datapoints: none
Non-extreme datapoints: 476, ln(476) = 6.1654
Standardized residuals N(0,1): mean: -.009 P.SD: .827 count: 476.00
Fit Indicators:
Equal-item-discriminations test of Parallel ICCs/IRFs: 7.4177 with 13 d.f., probability = .9174
van den Wollenberg Q1 test of Parallel ICCs/IRFs: 64.5188 with 117 d.f., probability = 1.0000
Estimated Free Parameters = Lesser of (Non-extreme KID or Non-extreme TAP) - 1 + Sum(Thresholds - 1) = 17
Analysis Degrees of freedom (d.f.) by counting = 476 - 17 = 459
Log-Likelihood chi-squared: 221.6118 with 459 d.f., probability = 1.0000
Pearson Global chi-squared: 325.2167 with 459 d.f., probability = 1.0000
Akaike Information Criterion, AIC = (2 * parameters) + chi-squared = 255.6118
Schwarz Bayesian Information Criterion, BIC = (parameters * ln(non-extreme datapoints)) + chi-squared = 326.4239
Global Root-Mean-Square Residual: .2318 with expected value: .2344 count: 630.00
Capped Binomial Deviance: .0802 with expected value: .0858 count: 630.00 dichotomies
Global statistics:
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Statistics based on the currently-selected data, omitting permanently and temporarily deleted, deselected and dropped items and persons
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Active person, weighted, non-extreme
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Persons currently active in this analysis. Weighted count only if active persons are weighted. Non-extreme persons do not have extreme scores (0%, 100% scores)
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Active item, weighted, non-extreme
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Items currently active in this analysis. Weighted count only if active items are weighted.
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Active datapoints, weighted, non-extreme
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Observations/responses currently active in this analysis (includes extreme scores). Weighted count only if active items or persons are weighted. Non-extreme items do not have extreme scores (0%, 100% scores)
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Missing datapoints, % of Active+Missing datapoints
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Observations/responses not active in this analysis, usually because they do not have scored values. If any potentially active data are missing, then percent that are active
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Non-extreme datapoints, weighted, non-extreme
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Observations/responses currently active in this analysis (excludes extreme scores). Weighted count if any active persons or items are weighted.
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Standardized residuals N(0,1):
mean: P.SD: count:
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Standardized Residuals are modeled to have a unit normal distribution. Large departures from mean of 0.0 and standard deviation of 1.0 indicate that the data do not conform to the basic Rasch model specification that randomness in the data be normally distributed do that standardized residuals to be close to mean 0.0, P.SD 1.0. The count is of relevant observations.
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Fit Indicators:
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Equal-item-discriminations test of Parallel ICCs/IRFs: .... with .... d.f., probability = ....
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Parallel ICCs are tested using a fixed-effects chi-squared test of the item discriminations, DISCRIM=Yes in Table 14, etc. - The degrees of freedom are (number of active items - 1).
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van den Wollenberg Q1 test of Parallel ICCs/IRFs: .... with .... d.f., probability = ....
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The Q1 test (1982, 1995 - below) for parallel ICCs is modified to include polytomous ICCs and missing data by stratifying the person sample by ability (theta) instead of raw score.
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Estimated Free Parameters = Lesser of (Non-extreme persons or Non-extreme items) - 1 + Sum(Thresholds - 1)
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Count of free parameters. Anchoring is ignored. Item and Person weights are included
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Analysis Degrees of freedom (d.f.) by counting
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Weighted count of datapoints in non-extreme response strings less the count of free parameters.
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Log-Likelihood chi-squared: .... with ... d.f., probability = ..
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The chi-square value asymptotically = -2 * log-likelihood of the active datapoints. It is based on the currently-reported estimates which may depart noticeably from the "true" maximum likelihood estimates for these datapoints. The probability that these data fit the Rasch model globally. Despite good global fit, there can be considerable local misfit in Tables 6 and 10.
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Pearson Global chi-squared: ... with ... d.f., probability = ....
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This chi-squared is calculated using the squared standardized residuals, which approximate unit-normal deviates.
Cressie, N., & Read, T. R. (1989). Pearson's X2 and the log-likelihood ratio statistic G2: a comparative review. International Statistical Review, 57, 19-43.
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Akaike Information Criterion, AIC
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AIC = 2*k - 2*ln(L) where k = parameters, and -2*ln(L) is the log-likelihood chi-square.
Used for model comparison: higher values indicate worse adjusted fit
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Schwarz Bayesian Information Criterion, BIC
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BIC = ln(n)*k - 2*ln(L) where n is the number of Non-extreme datapoints. Used for model comparison: higher values indicate worse adjusted fit. ln() = natural logarithm of count of datapoints
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Global (Weighted) Root-Mean-Square Residual (RMSR) with expected value:
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This is √(∑(X-E)²) where the sum is across X, each of the observations, and E, the expectation of each observation according to the Rasch model. Weighting is applied to the data if IWEIGHT= or PWEIGHT= are specified. The expected value of the RMSR according to the Rasch model. RMSR values smaller than the expected value indicate better fit (or overfit) to the Rasch model.
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Capped (Weighted) Binomial Deviance (CBD) = ... with expected value ... for ... dichotomies
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This is the average of -[X*LOG10(E) + (1-X)*LOG10(1-E)] for all dichotomous observations where X=0,1 is the observation and E is its Rasch-model expectation. E is limited to the range 0.01 to 0.99. Weighting is applied to the data if IWEIGHT= or PWEIGHT= are specified.
Glickman, Mark E. "Parameter estimation in large dynamic paired comparison experiments." Journal of the Royal Statistical Society: Series C (Applied Statistics) 48.3 (1999): 377-394.
The expected value of the CBD according to the Rasch model. CBD values smaller than the expected value indicate better fit (or overfit) to the Rasch model.
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Example: Rating Scale Model (RSM) and Partial Credit Model (PCM) of the same dataset. When the models are nested (as they are with RSM and PCM), then we have:
RSM chi-square and RSM d.f.
PCM chi-square (which should be smaller) and PCM d.f. (which will be smaller)
Then the model choice could be based on: (RSM LL chi-square - PCM LL chi-square) with (RSM-PCM) d.f., However, global fit statistics obtained by analyzing your data with conditional Rasch estimation, CMLE, or log-linear models (e.g., in SPSS) will be more exact than those produced by Winsteps. For the more information about the choice between RSM and PCM, see www.rasch.org/rmt/rmt143k.htm.
if global fit statistics are the decisive evidence for choice of analytical model, then Rasch analysis may not be appropriate. In the statistical philosophy underlying Rasch measurement, the decisive evidence for choice of model is "which set of measures is more meaningful and useful" (a practical decision), not "which set of measures fit the model better" (a statistical decision).
However, local fit and misfit are more important that global fit for decision-making: "Evaluating Restrictive Models in Educational and Behavioral Research: Local Misfit Overrides Model Tenability" (2020) Tenko Raykov, Christine DiStefano. Educational and Psychological Measurement.
Glas C.A.W., Verhelst N.D. (1995) Testing the Rasch Model. In: Fischer G.H., Molenaar I.W. (eds) Rasch Models. Springer, New York, NY.
van den Wollenberg, A. L. (1982). Two new test statistics for the Rasch model. Psychometrika 47: 123–140.
Question: Is it possible in Winsteps to obtain the p-value corresponding to the chi-square test on the difference in model fit between the Rating Scale Model and Partial Credit Model?
Answer: Yes, do the two analyses. Output Table 44. Compare the two log-likelihood chi-squares. However, the big challenge is the d.f. of the difference between the chi-squares. You can estimate these as "number of PCM Andrich thresholds - number of RSM Andrich thresholds" which is probably different from the difference in the "d.f. by simulation" of the two analyses.
