Table 3.1 Summaries of persons and items

(controlled by REALSE=, UMEAN=, USCALE=, ISUBTOTAL=, PSUBTOTAL=)

This table summarizes the person, item and structure information. Extreme scores (zero, minimum possible and perfect, maximum possible scores) have no exact measure under Rasch model conditions, so they are dropped from the main summary statistics. Using a Bayesian technique, however, reasonable measures are reported for each extreme score, see EXTRSC=. Totals including extreme scores are also reported, but are necessarily less inferentially secure than those totals only for non-extreme scores.

Table 3.1: Gives summaries for all persons and items.

Table 27.3: Gives subtotal summaries for items, controlled by ISUBTOT=

Table 28.3: Gives subtotal summaries for persons, controlled by PSUBTOT=

    SUMMARY OF     34 MEASURED (NON-EXTREME) KIDS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN       6.9      14.0        -.20    1.03      1.03    -.3    .73    -.3 |

| S.D.       2.1        .0        2.07     .11      1.01    1.2   1.45     .5 |

| MAX.      11.0      14.0        3.89    1.15      4.43    2.3   6.86    1.3 |

| MIN.       2.0      14.0       -4.48     .82       .17   -1.6    .08    -.8 |

|-----------------------------------------------------------------------------|

| REAL RMSE   1.23  ADJ.SD    1.66  SEPARATION  1.35  KID    RELIABILITY  .65 |

|MODEL RMSE   1.03  ADJ.SD    1.79  SEPARATION  1.73  KID    RELIABILITY  .75 |

| S.E. OF KID MEAN = .36                                                      |

+-----------------------------------------------------------------------------+

 MINIMUM EXTREME SCORE: 1 KIDS

 MINIMUM EXTREME SCORE: 46 PUPILS

     LACKING RESPONSES: 8 PUPILS

    SUMMARY OF     35 MEASURED(EXTREME AND NON-EXTREME)KIDS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN       6.7      14.0        -.38    1.05                                |

| S.D.       2.4        .0        2.31     .18                                |

| MAX.      11.0      14.0        3.89    1.88                                |

| MIN.        .0      14.0       -6.79     .82                                |

|-----------------------------------------------------------------------------|

| REAL RMSE   1.25  ADJ.SD    1.95  SEPARATION  1.56  KID    RELIABILITY  .71 |

|MODEL RMSE   1.07  ADJ.SD    2.05  SEPARATION  1.92  KID    RELIABILITY  .79 |

| S.E. OF KID MEAN = .40                                                      |

+-----------------------------------------------------------------------------+

KID RAW SCORE-TO-MEASURE CORRELATION = 1.00

CRONBACH ALPHA (KR-20) KID RAW SCORE RELIABILITY = .73

UMEAN=.000 USCALE=1.000

476 DATA POINTS. LOG-LIKELIHOOD CHI-SQUARE: 221.61 with 429 d.f. p=1.0000

RAW SCORE-TO-MEASURE CORRELATION is the Pearson correlation between raw scores and measures, including extreme scores. When data are complete, this correlation is expected to be near 1.0 for persons and near -1.0 for items.

KID RAW SCORE-TO-MEASURE CORRELATION is the correlation between the marginal scores (person raw scores and item scores) and the corresponding measures. The item correlation is expected to be negative because higher measure implies lower probability of success and so lower item scores.

CRONBACH ALPHA (KR-20) KID RAW SCORE RELIABILITY is the conventional "test" reliability index. It reports an approximate test reliability based on the raw scores of this sample. It is only reported for complete data. See more at Reliability.

UMEAN=.000 USCALE=1.000 are the current settings of UMEAN= and USCALE=.

476 DATA POINTS is the number of observations that are used for standard estimation, and so are not missing and not in extreme scores.

LOG-LIKELIHOOD CHI-SQUARE: 221.61 is the approximate value of the global fit statistic. The chi-square value is approximate. It is based on the current reported estimates which may depart noticeably from the "true" maximum likelihood estimates for these data. The degrees of freedom are the number of datapoints used in the free estimation (i.e., excluding missing data, data in extreme scores, etc.) less the number of free parameters. For an unanchored analysis, free parameters = non-extreme items + non-extreme persons - 1 + (categories in estimated rating-scale structures - 2 * rating-scale structures). It is typical in Rasch analysis that the probability of the chi-square is 0.0.

A log-likelihood ratio test for pair of models (e.g., rating-scale and partial-credit), where one nests within the other, would be the difference between the chi-square values from the two analyses, with d.f. given by the difference between the d.f.

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 LL chi-square and RSM d.f.

PCM LL chi-square (which should be smaller) and PCM d.f. (which will be smaller)

Then the model choice is based on:

(RSM LL chi-square - PCM LL chi-square)  with (RSM-PCM) d.f.

(RSM-PCM) d.f. is the number of extra categories in the PCM model over the RSM model. The number of categories is reported in the heading of most Winsteps tables, e.g., a PCM analysis might have 50 categories, and an RSM analysis might have 5 categories. So the d.f. differ by 50-5 = 45.