Table 21 Probability curves

To produce these using other software, see GRFILE= or Graphs menu.

Table

21.2 Expected score ogive (means. model Item Characteristic Curve).

21.3 Cumulative category probability curves (medians, shows 50% cumulative probabilities).

1. Dichotomies: Category probabilities

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.

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

2. Polytomies: Individual category probabilities

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).

CATEGORY PROBABILITIES: MODES - Structure measures 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

3. Polytomies: Expected average ratings

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.

EXPECTED SCORE OGIVE: 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

4. Polytomies: Cumulative category probabilities

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.

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