﻿ Polytomous mean-square fit statistics

# 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

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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