WMLE = Warm's Weighted Maximum Likelihood Estimation (WLE) = No

WMLE=Yes reports Warm's Weighted Maximum Likelihood estimates in the measure tables. Also called Warm's Weighted Likelihood Estimation (WLE).

 

For persons, Thomas Warm's exact method is used. Warm's bias corrected estimate, WLE is when (raw score - J / ( 2 * I ) ) = Expected score. Where I is the test information for the WLE measure, and J is a function of the density.

 

For items, Frederick Lord's MLE Bias Correction is used. This approximates WLE. Lord's bias corrected estimate = MLE + (J / ( 2 * I^2 ) )

 

See Shudong Wang and Tianyou Wang.(1999) Precision of Warm's Weighted Likelihood Estimation of Ability for a Polytomous Model in CAT.

 

Example: With WMLE=Yes, using the dataset in Table 4.1 on page 59 of Fischer, G.H. and Molenaar, I.W. (1995) Rasch Models. Foundations, Recent Developments and Applications. Springer, New York.

 

Title= "Fischer & Molenaar Table 4.1"

item1= 1

ni= 5

name1= 1

namelength= 30

IAFILE=*

1 -1.5 ; item values from Fischer Table 1

2 -0.75

3 0

4 0.75

5 1.5

*

CODES=01

WMLE=YES

LCONV=.0001

RCONV=.01

CONVERGE= Both

TFILE=*

18.1  ; persons in measure order

14.1  ; items in measure order

20.1  ; score table

*

&END

END LABELS

00000 MLE=        WMLE= -2.95

00001 MLE=  -1.70 WMLE= -1.45

00011 MLE=   -.51 WMLE=  -.46

00111 MLE=    .51 WMLE=   .46

01111 MLE=   1.70 WMLE=  1.45

11111 MLE=        WMLE=  2.95

 

Output:

 

Persons: Warm's WLE estimates:

TABLE 18.1 Fischer & Molenaar Table 4.1

         PERSON STATISTICS:  ENTRY ORDER

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

|ENTRY   TOTAL  TOTAL    JMLE  |  WARM |                              |

|NUMBER  SCORE  COUNT  MEASURE |  WMLE | PERSON: Fischer's results    |

|------------------------------+-------+------------------------------|

|     1      0      5   -3.20  |  -2.95| 00000 MLE=        WMLE= -2.95|

|     2      1      5   -1.70  |  -1.45| 00001 MLE=  -1.70 WMLE= -1.45|

|     3      2      5    -.51  |   -.46| 00011 MLE=   -.51 WMLE=  -.46|

|     4      3      5     .51  |    .46| 00111 MLE=    .51 WMLE=   .46|

|     5      4      5    1.70  |   1.45| 01111 MLE=   1.70 WMLE=  1.45|

|     6      5      5    3.20  |   2.95| 11111 MLE=        WMLE=  2.95|

|------------------------------+-------+------------------------------|

| MEAN     2.5    5.0     .00  |    .00|                              |

| P.SD     1.7     .0    2.12  |   1.92|                              |

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

 

Items: Lord's MLE Bias Correction approximates Warm's WLE estimates:

TABLE 14.1 Fischer & Molenaar Table 4.1

         ITEM STATISTICS:  ENTRY ORDER

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

|ENTRY   TOTAL  TOTAL    JMLE  |  WARM | <- This is Lord's Approximation

|NUMBER  SCORE  COUNT  MEASURE |  WMLE |

|------------------------------+-------+

|     1      1      6   -1.50A |  -1.24|

|     2      2      6    -.75A |   -.65|

|     3      3      6     .00A |    .00|

|     4      4      6     .75A |    .65|

|     5      5      6    1.50A |   1.24|

|------------------------------+-------+

| MEAN     3.0    6.0     .00  |    .00|

| P.SD     1.4     .0    1.06  |    .88|

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