Table 21 Category Probability curves and Expected score ogive

21.1 Dichotomies: Category probabilities

        DICHOTOMOUS CURVES                                            

P       -------------------------------------------------------------

R  1.0  0000000000000000                             1111111111111111 

O                       000000                 111111                 

B   .8                        000           111                       

A                                00       11                          

B   .6                             00   11                            

I                                    ***                              

L   .4                             11   00                            

I                                11       00                          

T   .2                        111           000                       

Y                       111111                 000000                 

    .0  1111111111111111                             0000000000000000 

        -------------------------------------------------------------

       -6        -4        -2         0         2         4         6 

                            PERSON [MINUS]  ITEM  MEASURE          

The probability of each response is shown across the measurement continuum. The measure to be used for determining the probability of any particular response is the difference between the measure of the person and the calibration of the item. For dichotomies, only one curve is shown plotting the probability of scoring a "1" (correct), and also of scoring a "0" (incorrect) for any measure relative to item measure. For 'S' and 'F' models these curves are approximations.

21.1 Polytomies: Individual category probabilities

        CATEGORY PROBABILITIES: MODES - Andrich Thresholds at intersections

P       -------------------------------------------------------------

R  1.0  00000000                                             22222222 

O               0000000                               2222222         

B   .8                 000                         222                

A                         000                   222                   

B   .6                       00               22                      

I   .5                         00*111111111*22                        

L   .4                        111 00     22 111                       

I                          111      00 22      111                    

T   .2                 1111         22*00         1111                

Y               1111111        22222     00000        1111111         

    .0  ********222222222222222               000000000000000******** 

        -------------------------------------------------------------

       -5    -4    -3    -2    -1     0     1     2     3     4     5 

                            PERSON [MINUS]  ITEM  MEASURE             

See Rating scale conceptualization.

When there are more than two categories, the probability of each category is shown. The points of intersection of adjacent categories are the Rasch-Andrich thresholds (structure calibrations).

21.2 Polytomies: Expected average ratings

        EXPECTED SCORE ICC/IRF: MEANS

E       -------------------------------------------------------------

X    2                                                   222222222222 

P                                                 2222222             

E  1.5                                        2222                    

C                                          111                        

T                                       111  |                        

E    1                               111     |                        

D                                 111 *      |                        

    .5                         111    *      |                        

S                          0000|      *      |                        

C                   0000000    |      *      |                        

O    0  000000000000           |      *      |                        

R       -------------------------------------------------------------

E      -5    -4    -3    -2    -1     0     1     2     3     4     5 

                            PERSON [MINUS]  ITEM  MEASURE             

The Expected score ogive is also called the Model Item Characteristic Curve (ICC) and Item Response Function (IRF). For response structures with three or more categories, two further graphs can be drawn. The second graph depicts the expected score ogive. The vertical "*" characters correspond to integer expected scores, and the "|" characters correspond to half-point expected scores, the Rasch-half-point thresholds. The intervals between the Rasch-half-point thresholds can be thought of as the intervals corresponding to the observed categories. For the purposes of inference, measures in the zone on the x-axis between '|' and '|' correspond, on average, to the rating given on the 'y' axis, '1'. Similarly ratings on the y-axis can be thought of as corresponding to measures in the matching zone on the x-axis. The degree to which the data support this is given by the COHERENCE statistics in Table 3.2. Empirical item characteristic curves are shown in Table 29 and from the Graphs menu.

21.3 Polytomies: Cumulative category probabilities

        MEDIANS - Cumulative probabilities      

P       -------------------------------------------------------------

R  1.0  ********222222222222222                                       

O       0       1111111        22222                                  

B   .8  0              111          222                               

A       0                 111          22                             

B   .6  0                    11          22                           

I   .5  0----------------------111---------222----------------------- 

L   .4  0                       | 11        | 22                      

I       0                       |   11      |   222                   

T   .2  0                       |     111   |      222                

Y       0                       |        11111        2222222         

    .0  0                       |           | 111111111111111******** 

        -------------------------------------------------------------

       -5    -4    -3    -2    -1     0     1     2     3     4     5 

                            PERSON [MINUS]  ITEM  MEASURE

The third graph is of the zone curves which indicate the probability of an item score at or below the stated category for any particular difference between person measure and item calibration. The area to the left of the "0" ogive corresponds to "0". The right-most area corresponds to the highest category. The P=0.5 intercepts are the median cumulative probabilities. "|" indicate the Rasch-Thurstone thresholds.