﻿ Bayesian prior distributions

# Bayesian prior distributions

Winsteps does not support Bayesian prior distributions directly, but the effect of a Bayesian prior on the final estimates can be obtained  in other ways.

Example 1: You have predicted the item difficulties based on your construct theory. You want to use these predicted difficulties as Bayesian Priors on the estimation of the empirical Item Difficulties

1.  Constructing some dummy person records. These persons would respond to your items in the way predicted by your desired Bayesian prior.

2. Analyze the dummy person records by themselves. Add and alter dummy person records until the estimated item difficulties match your Bayesian predictions.

3. Analyze the dummy records with your actual sample. PWEIGHT= the dummy records until the influence of the dummy records on the final estimates is the same as your desired hypothetical Bayesian prior.

Illustration: There are 6 items. Our theory says: items 1-2 are easy, items 3-4 are moderate, items 5-6 are hard. Dummy data records could be these. Notice that they all have the same raw score. Fit to the Rasch model is irrelevant for dummy data.

101010

110100

110100

011001

Example 2: You want to impose a Bayesian prior on the person-ability distribution.

There are usually two noticeable effects of a Bayesian prior:

(i) extreme person scores participate in the estimation of item difficulties in the same way as non-extreme scores.

(ii) the person-ability estimates are more central with a smoother distribution.

These two effects can be obtained by adding item records to the dataset.

1. Analyze the dataset without the Bayesian prior. Verify that all is correct.

2. Define two more dichotomous items in the control file. These are dummy items. Use ISGROUPS= to place the dummy items in a separate item-group.

3. Append to each person's response string two responses. "01" for the first half of the person sample. "10" for the second half.

Illustration: the original control file has 100 dichotomous items and 500 persons:

CODES=01

NI=100

the Bayesian control file has 102 dichotomous items

CODES=01

NI=102  ; two more items

ISGROUPS=*

1-100 A  ; all the original dichotomous items in one group

101-102 D ; the dummy items in their own group

*

EDFILE=*     ; using EDFILE=, we don't change the actual data

1-250 101 0

1-250 102 1

251-500 101 1

251-500 102 0

*

IWEIGHT=*

101 1

102 1

*

4. Analyze the enlarged dataset. Verify that (i) all previously extreme scores are not shown as extreme, Table 17, and (ii) the person-ability S.D. has reduced. We expect that the fit statistics for all the items, except for the two dummy items, are tending towards overfit (lower mean-squares).

5. Adjust the weighting of the dummy items using IWEIGHT= until the influence of the dummy records on the final estimates is the same as your desired hypothetical Bayesian prior.

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