TEST-OF-FIT STATISTICS

**Question:**

*What limitations apply to test-of-fit
statistics used in RUMM2030?*

**Explanation:**

- The
**Residual test-of-fit statistic:**is constructed as a standard normalised residual, but is*not perfectly normally distributed*: - a very
**positive**value implies poor discrimination; - a very
**negative**value implies too good a discrimination. - The
**Chi-square test-of-fit:**(and its probability) is constructed as an*approximate*chi square but is*not perfectly distributed*as the chi square.

**Overall:**
the tests-of-fit employed by RUMM2030 for a Rasch analysis should be used *relatively*,
and not strictly *absolutely* according to external criteria.

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**Question:**

*How do RUMM2030 fit indices relate to OUTFIT
and INTFIT statistics used in other programs?*

**Explanation:**

The outfit and infit statistics used in other programs are similar to the Residual statistic in RUMM2030:

These values are differently weighted statistics based on the residual between a person's response and the expected response according to the model given the person and item estimates.

- The Outfit in these programs is closer in value to that display in RUMM2030.
- All Residual statistics displayed in RUMM2030 have an expected mean of 0 and a standard deviation of 1 but, because they are approximations, the distributions are not strictly normal.
- All of the distributions of these Residuals:
- are affected by the relative locations of the persons and the items;
- the number of parameters estimated as well as
- the fit between the data and the model.
- In all of these Residual statistics:
- a very negative value implies overfit [Observations of means in successive class intervals steeper than the ICC curve] for some reason [perhaps violation of local independence], and
- a very large value implies underfit [Observations of means in successive class intervals flatter than the ICC curve] of some kind [perhaps a violation of unidimensionality].
- The chi square test of fit formalises the graphical display of the ICC curves.

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