Polytomous mean-square fit statistics

For a general introduction, see Diagnosing Misfit, also Dichotomous mean-square fit statistics.

 

Response String

Easy............Hard

INFIT
Mean-square

OUTFIT
Mean-square

Point-measure correlation

Diagnosis


 

 

 

 

I. modeled:

 

 

 

 

33333132210000001011

0.98

0.99

0.78

Stochastically

31332332321220000000

0.98

1.04

0.81

monotonic in form

33333331122300000000

1.06

0.97

0.87

strictly monotonic

33333331110010200001

1.03

1

0.81

in meaning

 

 

 

 

 

II. overfitting (muted):

 

 

 

 

33222222221111111100

0.18

0.22

0.92

Guttman pattern

33333222221111100000

0.31

0.35

0.97

high discrimination

32222222221111111110

0.21

0.26

0.89

low discrimination

32323232121212101010

0.52

0.54

0.82

tight progression

 

 

 

 

 

III. limited categories:

 

 

 

 

33333333332222222222

0.24

0.24

0.87

high (low) categories

22222222221111111111

0.24

0.34

0.87

central categories

33333322222222211111

0.16

0.2

0.93

only 3 categories

 

 

 

 

 

IV. informative-noisy:

 

 

 

 

32222222201111111130

0.94

1.22

0.55

noisy outliers

33233332212333000000

1.25

1.09

0.77

erratic transitions

33133330232300101000

1.49

1.4

0.72

noisy progression

33333333330000000000

1.37

1.2

0.87

extreme categories

 

 

 

 

 

V. non-informative:

 

 

 

 

22222222222222222222

0.85

1.21

0

one category

12121212121212121212

1.5

1.96

-0.09

central flip-flop

01230123012301230123

3.62

4.61

-0.19

rotate categories

03030303030303030303

5.14

6.07

-0.09

extreme flip-flop

03202002101113311002

2.99

3.59

-0.01

random responses

 

 

 

 

 

VI. contradictory:

 

 

 

 

11111122233222111111

1.75

2.02

0

folded pattern *

11111111112222222222

2.56

3.2

-0.87

central reversal

22222222223333333333

2.11

4.13

-0.87

high reversal

00111111112222222233

4

5.58

-0.92

Guttman reversal

00000000003333333333

8.3

9.79

-0.87

extreme reversal

from Smith R.M. (1996) Rasch Measurement Transactions 10:3 p. 516

 

The z-score standardized statistics report, as unit normal deviates, how likely it is to observe the reported mean-square values, when the data fit the model. The term z-score is used of a t-test result when either the t-test value has effectively infinite degrees of freedom (i.e., approximates a unit normal value) or the Student's t-statistic value has been adjusted to a unit normal value.

 

* "folded data" can often be rescued by imposing a theory of "not reached" and "already passed" on to the observations. For instance, in archaeological analysis, the absence of bronze implements can mean a "stone age" or an "iron age" society. A useful recoding would be "1" = "stone age", "2" = "early bronze", "3" = "bronze", "2=>4" = "late bronze", "1=>5" = "iron age". This can be done iteratively to obtain the most self-consistent set of 4's and 5's. (Folding is discussed in Clive Coombes' "A Theory of Data".)


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