| Log-linear vs. Logit-linear Rasch | ||
| Estimation | Log-linear Rasch (CMLE) | Logit-linear Rasch (JMLE) |
| Data matrix | Contingency table: one cell per response string and
demographic combination: 4 dichotomies + 2 genders: 2x2x2x2x2 = 32 cells (see TGK) 3 4-category items: 4x4x4 = 64 (Agresti) |
Response strings for all subjects. Persons coded with demographic variables. |
| Missing data | Must be imputed or subject omitted | Merely lessens precision |
| Basic element | Frequency of persons in cell: e.g., (TGK) F{X1010M} for response string "1010", Male |
Observation: Xni |
| Model | Log(F{X1010}) = 1*E1 + 0*E2 + 1*E3 + 0*E4 + š(1+0+1+0) (see TGK) |
log(Pni1/Pni0) = Bn + Ei |
| Interaction terms | Yes, but no longer Rasch model | Yes, post-hoc to explain residuals |
| Constraints | To eliminate terms, and establish local origin. | To establish local origin |
| Estimation bias | Negligible - equivalent to Conditional Maximum Likelihood (CMLE) Rasch | Up to 2, corrected by (L-1)/L |
| Global fit | Decisive as to acceptability of model. | Uninformative |
| Items | ||
| Maximum items | 13, i.e., 213 cells | >3,000 |
| Item calibrations | Yes, but relative to the anchored item | Yes, with mean calibration of zero or anchor item(s). |
| Item S.E. | Test-dependent, because relative to anchored item. Anchored item has S.E.=0 | As test-independent as possible. S.E.s reported for all items. |
| Item fit diagnosis | Unexpected cell frequencies, summarized by tests of local independence (see TGK) | Unexpected response patterns, summarized by sums of residuals |
| Persons | ||
| Maximum persons | Unlimited, because accumulated in cells | >20,000 |
| Person measures | Only obtained by secondary analysis | Yes, modeled |
| Person S.E. | Obtained by secondary analysis | Yes, modeled |
| Person fit diagnosis | Unexpected cell frequencies: Agresti: 8 strings of "322", but 2.9 expected |
Unexpected response patterns: in Agresti data: pattern "122". |
| Unexpected responses | No | Yes, by residual size |
| Best for | ||
| Item calibration | <=13 items with local S.E.s | >=5 items with general S.E.s |
| Person measurement | No | Yes |
| Misfit diagnosis | No | Yes |
| Software | Standard statistical: SAS, SPSS | Custom: BIGSTEPS, QUEST |
John Michael Linacre
Agresti: Agresti A (1993) Computing conditional maximum likelihood estimates for generalized Rasch models using simple log-linear models with diagonals parameters. Scandinavian Journal of Statistics 20(1) 63-71.
TGK: TenVergert E, Gillespie M, & Kingma J (1993) Testing the assumptions and interpreting the results of the Rasch model using log-linear procedures in SPSS. Behavior Research Methods, Instruments & Computers 25(3) 350-359.
Log-linear vs. Logit-linear Rasch. Linacre J.M. … Rasch Measurement Transactions, 1997, 11:3 p. 586.
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