﻿ Table 2 Polytomies: category and threshold statistics
 Table 2 Polytomies: Most probable, expected, cumulative, structure, average measures

(controlled by T2SELECT=, MRANGE=, CATREF=)

Table 2 for multiple-choice items.

Each plot answers a different question:

What category is most likely? The maximum probability (mode) plot.

What is the average or expected category value? The expected score (mean) plot.

What part of the variable corresponds to the category? The cumulative probability (median) plot.

The numeric information for these plots is in ISFILE=

Which Table should be used for a standard setting procedure?

Most standard setting is based on "average" or "frequency" considerations. For instance,

"If we observed 1000 candidates whose measures are known to be exactly at the pass-fail point, ...

..., we would expect their average score to be the pass-fail score." If this is how you think, then the Table you want is Table 2.2 (matches Table 12.5)

..., we would expect 50% to pass and 50% to fail the pass-fail score." If this is how you think, then the Table you want is Table 2.3 (matches 12.6)

..., we would expect more to be in the criterion pass-fail category of each item than any other category." If this is how you think, then the Table you want is Table 2.1 (no matching 12.)

"Our current sample is definitive, ...

..., we would expect the next sample to behave in exactly the same way this sample did." If this is how you think, then the Table you want is Table 2.5 (or Table 2.6, if the responses have been rescored.)

..., we would expect the next sample to behave the way this sample should have behaved, if this sample had conformed to the Rasch model." If this is how you think, then the Table you want is Table 2.7.

The left-side of this table lists the items in descending order of measure. Anchored items are indicated by an * between the sequence number and name. A particular category can be used as the reference for sorting the items by specifying the CATREF= variable.

Across the bottom is the logit (or user-rescaled) variable with the distribution of the person measures shown beneath it. An "M" marker represents the location of the mean person measure. "S" markers are placed one standard deviation away from the mean. "T" markers are placed two standard deviations away. An "M" inside a plot indicates the measure corresponding to missing data.

To produce all subtables of Table 2, request Table 2.0

Tables 2.1 & 2.11: The "Most Probable Response" Table, selected with CURVES=001, answers the question "which category is a person of a particular measure most likely to choose?" This is the most likely category with which the persons of logit (or user-rescaled) measure shown below would respond to the item shown on the left. The area to the extreme left is all "0"; the area to the extreme right is at the top category. Each category number is shown to the left of its modal area. If a category is not shown, it is never a most likely response. An item with an extreme, perfect (maximum possible) or zero (minimum possible), score is not strictly estimable, and is omitted here. Blank lines are used to help approximate the placement of the items on the latent variable.

This table presents in one picture the results of this analysis in a form suitable for inference. We can predict for people of any particular measure what responses they would probably make. "M" depicts an "average" person. The left "T" a low performer. The right "T" a high performer. Look straight up from those letters to read off the expected response profiles.

Table 2.1 to 2.7 reports with observed categories, i.e., those in the CODES= statement: (illustrated by an Observed Category).

Table 2.11 to 2.17 report with scored categories, i.e., after IVALUE=, RESCORE=, KEY1=, etc., but only if different from Table 2.1 to 2.7: (by Category Score).

Most Probable Response: Mode  (Between "0" and "1" is "0", etc.) (illustrated by an Observed Category)

-6       -4        -2         0         2         4         6

|---------+---------+---------+---------+---------+---------|  NUM   ACT

0                                     1       2             2    5  Find bottles and cans

|                                                           | deliberate space

0                                    1       2              2   23  Watch a rat

|                                                           |

0                                  1       2                2   20  Watch bugs

0                                  1       2                2    4  Watch grass change

0                                 1        2                2    8  Look in sidewalk cracks

|                                                           |

0         1        2                                        2   18  Go on picnic

|---------+---------+---------+---------+---------+---------|  NUM   ACT

-6       -4        -2         0         2         4         6

1

12 2 2117083563422342 111 1 1   1     1  KID

T      S      M      S     T

0    10 30 50 70 80 90               99  PERCENTILE

When there are disorders Andrich thresholds, soem categories are never most probable:

-4    -3     -2     -1      0      1      2      3      4

|------+------+------+------+------+------+------+------|  NUM   ITEM

0             2        4         6        8             8    1   disorder

Tables 2.2 & 2.12: In the "Expected Score" Table, the standard output (or selected with CURVES=010) answers the question "what is the average rating that we expect to observer for persons of a particular measure?" This rating information is expressed in terms of expected scores (with ":" at the half-point thresholds). Extreme scores are located at expected scores .25 score points away from the extremes. This plot also indicates the operational range of each item.

Expected Score: Mean  (Rasch-score-point threshold, ":" indicates Rasch-half-point threshold) (illustrated by an Observed Category)

-6       -4        -2         0         2         4         6

|---------+---------+---------+---------+---------+---------|  NUM   ACT

0                               0   :     1     :   2       2    5  FIND BOTTLES AND CANS

0                              0   :     1     :   2        2   23  WATCH A RAT

0                            0   :     1     :    2         2   20  WATCH BUGS

Operational range: ^--------------------^

....

0        0    :     1     :   2                             2   12  GO TO MUSEUM

0      0    :     1     :   2                               2   19  GO TO ZOO

0   0   :     1     :    2                                  2   18  GO ON PICNIC

|---------+---------+---------+---------+---------+---------|  NUM   ACT

-6       -4        -2         0         2         4         6

In this example of a 5 category rating scale, the operation range of the top item is indicated in red.

-5       -3        -1         1         3         5         7

|---------+---------+---------+---------+---------+---------|  NUM   ITEM

1                        1  :   2  : 3 :  4    :   5        5   13  M. STAIRS

|---------+---------+---------+---------+---------+---------|  NUM   ITEM

-5       -3        -1         1         3         5         7

Tables 2.3 & 2.13: The "Cumulative Probability" Table: Rasch-Thurstone thresholds, selected with CURVES=001, answers the question "whereabouts in the category ordering is a person of a particular measure located?" This information is expressed in terms of median cumulative probabilities (the point at which the probability of scoring in or above the category is .5).

50% Cumulative Probability: Median (equal-cumulative-probability Rasch-Thurstone thresholds) (illustrated by an Observed Category)

-6       -4        -2         0         2         4         6

|---------+---------+---------+---------+---------+---------|  NUM   ACT

0                                    1         2            2    5  Find bottles and cans

|                                                           |

0                                   1         2             2   23  Watch a rat

|                                                           |

0                                 1         2               2   20  Watch bugs

0                                 1         2               2    4  Watch grass change

0                                1         2                2    8  Look in sidewalk cracks

|                                                           |

0        1         2                                        2   18  Go on picnic

|---------+---------+---------+---------+---------+---------|  NUM   ACT

-6       -4        -2         0         2         4         6

Tables 2.4 & 2.14 show Rasch structure calibrations: Rasch-Andrich thresholds (step parameters, step measures, step difficulties, rating (or partial credit) scale calibrations). These are the relationships between adjacent categories, and correspond to the points where adjacent category probability curves cross, i.e., are equally probable of being observed according to a Rasch model.

Andrich Thresholds (Rasch model parameters: equal-adjacent-probability structure measures) (illustrated by an Observed Category)

-6       -4        -2         0         2         4         6

|---------+---------+---------+---------+---------+---------|  NUM   ACT

|                                     1       2             |    5  Find bottles and cans

|                                                           |

|                                    1       2              |   23  Watch a rat

|                                                           |

|                                  1       2                |   20  Watch bugs

|                                 1        2                |    4  Watch grass change

|                                 1        2                |    8  Look in sidewalk cracks

|                                                           |

|         1        2                                        |   18  Go on picnic

|---------+---------+---------+---------+---------+---------|  NUM   ACT

-6       -4        -2         0         2         4         6

Tables 2.5 & 2.15 plot the observed average person measures for each scored category. It reflects how this sample used these categories. The plotted values cannot fall outside the range of the sample.

Observed Average Measures for KID (scored) (illustrated by an Observed Category)

-2     -1       0       1       2       3       4       5

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

|                       000102                          |    5  FIND BOTTLES AND CANS

|                                                       |

|                       000102                          |   23  WATCH A RAT

|                                                       |

|                   00  01         02                   |   20  WATCH BUGS

|                 00    01               02             |    4  WATCH GRASS CHANGE

|                   0001                02              |    8  LOOK IN SIDEWALK CRACKS

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

-2     -1       0       1       2       3       4       5

1 2  11 2 114346533233332222 322 2 1 1 1  1  1      11  PUPILS

T        S         M         S         T

Tables 2.6 & Table 2.16 plot the observed average person measures for each observed category. It reflects how this sample used these categories. The plotted values cannot fall outside the range of the sample. "m" in the plot indicates the average measure of those for whom their observation is missing on this item. This Table is shown first from the Diagnosis pull-down menu.

Observed Average Measures for KID (unscored) (by Observed Category)

-2     -1       0       1       2       3       4       5

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

|                       000102                          |    5  FIND BOTTLES AND CANS

|                                                       |

|                       000102    m                     |   23  WATCH A RAT

|                                                       |

|                   00  01         02                   |   20  WATCH BUGS

|                 00    01               02             |    4  WATCH GRASS CHANGE

|                   0001                02              |    8  LOOK IN SIDEWALK CRACKS

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

-2     -1       0       1       2       3       4       5

Code for unidentified missing data: m

1 2  11 2 114346533233332222 322 2 1 1 1  1  1      11  PUPILS

T        S         M         S         T

Tables 2.7 & 2.17 plot the expected average person measures for each category score. It reflects how this sample were expected to use these categories. The plotted values cannot fall outside the range of the sample. This Table applies the empirical person distribution to Table 2.2.

Expected Average Measures for KID (scored) (illustrated by an Observed Category)

-2     -1       0       1       2       3       4       5

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

|                  00      01            02             |    5  FIND BOTTLES AND CANS

|                                                       |

|                  00     01            02              |   23  WATCH A RAT

|                                                       |

|                 00     01           02                |   20  WATCH BUGS

|                 00     01          02                 |    4  WATCH GRASS CHANGE

|                 00     01          02                 |    8  LOOK IN SIDEWALK CRACKS

|-------+-------+-------+-------+-------+-------+-------|  NUM   ACT

-2     -1       0       1       2       3       4       5

1 2  11 2 114346533233332222 322 2 1 1 1  1  1      11  PUPILS

T        S         M         S         T

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Rasch Publications
Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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