Li & Olejnik (1997) investigate six person fit statistics and conclude that "Practitioners need no longer be confused by the large number of possible personfit indexes available to detect nonfitting examinees. The l_{z} index will provide as reliable and accurate identification of unusual responding as other person fit statistics" (p. 229). Oh joy! Oh rapture! Our fit detection problems are finally over!  But wait, Klauer (1995) cautions "a given person [index] implies a bias for detecting the [index]specific deviations and a bias against detecting other kinds of deviations" (p. 109). Since Li & Olejnik consider only dichotomous tests, let us contain our joy while we evaluate the utility of l_{z} for polytomies.
l_{z} (Hulin et al., 1983) is a likelihoodbased statistic, computed in several steps:
Step 1. Compute the likelihood, _{n}, of the observed response string for person n of estimated ability B_{n},
where P_{nix} is the Raschmodel probability that person n on item i would respond in the observed category x of a 0,M rating scale. This has loglikelihood,
Step 2. Compute the expected loglikelihood,
Step 3. Compute the loglikelihood variance,
Step 4. Compute the l_{z} index,
The l_{z} index is standardized, so that a value of 0.0 is intended to reflect a perfectly typical response string. Values greater than 2.0 could indicate unexpectedly good fit (overfit). Values below 2.0 could indicate unexpectedly poor fit (noise).
Applying this to Smith's (1996, RMT 10:3 p.5167) simulated polytomous data produces the results in the Table. In Section A, l_{z} is skewed. Response strings simulated to fit the model have values around .75, rather than the proclaimed 0.0.
In Section B, overfitting (Guttmanlike) response strings reach only as far as 1.9, and so are not large enough to be flagged. Response patterns corresponding to misuse of the ratings scale (central tendency, extremism, erratic behavior), often detected by INFIT and OUTFIT, are not detected by l_{z}.
In Section C, large negative values of l_{z} do usefully pinpoint grossly aberrant response patterns. Even so, there was only one instance in which l_{z} flagged misbehavior which INFIT and OUTFIT missed. This was for the "one category" response set (Block V, line 1) which the l_{z} value of 2.06 flags as unexpected. However, the RPM (pointmeasure correlation, similar to pointbiserial) value of 0.0 provides a powerful and immediate diagnosis of this response string. In general, l_{z} does not flag borderline response strings.
Alas, l_{z} is markedly less useful than INFIT and OUTFIT meansquares. Our joy must be deferred.
John M. Linacre
Later note: Adjustment to l_{z} is suggested in "Assessment of Person Fit for MixedFormat Tests", Sandip Sinharay, Journal Of Educational And Behavioral Statistics July 13, 2015
Snijders, T. (2001). Asymptotic null distribution of personfit statistics with estimated person parameter. Psychometrika, 66, 331342 (for dichotomous items)
Sinharay, S. (2016). Asymptotically correct standardization of personfit statistics beyond dichotomous items. Psychometrika, 81, 9921013 (for polytomous items)
Hulin, C.L., Drasgow F., Parsons C. (1983) Item Response Theory: Applications to Psychological Measurement. Homewood Il: Dow & Jones Irwin.
Klauer K.C. (1995) The assessment of person fit. p. 97110. In G.H. Fischer & I.W. Molenaar (Eds.) Rasch Models: Foundations, Recent Developments and Applications. New York: Springer Verlag.
Li M.F, Olejnik S. (1997) The power of Rasch personfit statistics in detecting unusual response patterns. Applied Psychological Measurement 21:3, p. 215231
"It is recommended that with real tests, identification of significantly nonfitting examinees be based on empirical distributions of l_{z} generated from Monte Carlo simulations using item parameters estimated from real data."
l_{z} PersonFit Index to Identify Misfit Students With Achievement Test Data. Dong Gi Seo and David J. Weiss. Educational and Psychological Measurement 2013;73 9941016. epm.sagepub.com
Investigation of the l_{z} Person Fit Statistic  
Response String Easy..........Hard 
INFIT MnSq 
OUTFIT MnSq 
RPM Corr. 
l_{z} Index  Diagnosis 
Diagnostic use:  1.0 typical >1.3 noisy <0.7 overfit 
1.0 typical >1.3 noisy <0.7 overfit 
<0.0 reversed 0.0 useless 
0.0 typical <2.0 noisy >2.0 overfit 

A. Data fits model  l_{z} biased positive, OUTFIT and INFIT at expectation  
I. modelfitting: 33333132210000001011 31332332321220000000 33333331122300000000 33333331110010200001 
.98 .98 1.06 1.03 
.99 1.04 .97 1.00 
.78 .81 .87 .81 
.49 .75 1.12 .72 
l_{z} bias of 0.7 misleadingly diagnoses data as too deterministic 
B. Poor fit, merits attention  l_{z} flags 1 of 12, OUTFIT 9 of 12  
II. overfitting (muted): 33222222221111111100 33333222221111100000 33333333221100000000 32222222221111111110 32323232121212101010 
.18 .31 .80 .21 .52 
.22 .35 .77 .26 .54 
.92 .97 .93 .89 .82 
1.95 2.67 1.95 1.39 1.27 
most expected most likely high discrimination low discrimination tight progression 
III. limited categories: 33333333332222222222 22222222221111111111 33333322222222211111 
.24 .24 .16 
.24 .34 .20 
.87 .87 .93 
1.27 .67 1.95 
high (low) categories 2 central categories only 3 categories 
IV. informativenoisy: 32222222201111111130 33233332212333000000 33333333330000000000 33133330232300101000 
.94 1.25 1.37 1.49 
1.22 1.09 1.20 1.40 
.55 .77 .87 .72 
.85 .21 .67 .56 
noisy outliers erratic transitions extreme categories noisy progression 
C. Obvious gross misfit, requires attention  l_{z} and RPM flag 10 of 10  
V. noninformative: 22222222222222222222 12121212121212121212 03202002101113311002 01230123012301230123 03030303030303030303 
.85 1.50 2.99 3.62 5.14 
1.21 1.96 3.59 4.61 6.07 
.00 .09 .01 .19 .09 
2.06 3.73 6.73 9.34 12.54 
one category central flipflop random responses rotate categories extreme flipflop 
VI. contradictory: 11111122233222111111 22222222223333333333 11111111112222222222 00111111112222222233 00000000003333333333 
1.75 2.11 2.56 4.00 8.30 
2.02 4.13 3.20 5.58 9.79 
.00 .87 .87 .92 .87 
4.13 6.64 7.33 11.82 23.35 
folded pattern high reversal central reversal Guttman reversal extreme reversal 
This is the BIGSTEPS control file for the data above: &inst TITLE='COMPUTING STATISTICS' NI=20 ITEM1=1 ; include response strings in person name name1=1 namlen=30 CODES=0123 ptbis=no ; compute pointmeasure correlation INUMB=YES ; no item labels TFILE=* 6 ; Table 6  persons in fit order 18 ; table 18  persons in entry order * IAFILE=* ; item anchor values  uniform 1 1.9 2 1.7 3 1.5 4 1.3 5 1.1 6 0.9 7 0.7 8 0.5 9 0.3 10 0.1 11 0.1 12 0.3 13 0.5 14 0.7 15 0.9 16 1.1 17 1.3 18 1.5 19 1.7 20 1.9 * SAFILE=* ; step anchor values 0 0 1 1 2 0 3 1 * &end 33333132210000001011 modelled 31332332321220000000 modelled 33333331122300000000 modelled 33333331110010200001 modelled 33222222221111111100 most expected 33333222221111100000 most likely 33333333221100000000 high discrimination 32222222221111111110 low discrimination 32323232121212101010 tight progression 33333333332222222222 high (low) categories 22222222221111111111 2 central categories 33333322222222211111 only 3 categories 32222222201111111130 noisy outliers 33233332212333000000 erratic transitions 33333333330000000000 extreme categories 33133330232300101000 noisy progression 22222222222222222222 one category 12121212121212121212 central flipflop 03202002101113311002 random responses 01230123012301230123 rotate categories 03030303030303030303 extreme flipflop 11111122233222111111 folded pattern 22222222223333333333 high reversal 11111111112222222222 central reversal 00111111112222222233 Guttman reversal 00000000003333333333 extreme reversal
An AllPurpose Person Fit Statistic? Linacre J.M. … Rasch Measurement Transactions, 1997, 11:3 p. 5823.
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