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Explanatory Secondary Dimension Modeling of Latent Different Item
FunctioningPaul De Boeck, Sun-Joo Cho, and
Mark Wilson
DIF
• Conditional on person latent trait, different manifest groups have different probabilities to answer correctly/endorse an item. (manifest DIF)
• DIF as the consequence of neglecting the “nuisance dimensions”. (secondary dimensions)
This study
• Latent DIF: DIF between latent classes, i.e., a non-DIF vs a DIF latent class
group membership not observed or unobservableThe observed membership not reliable or not valid
• Mixed dimensionality: the secondary dimension in the DIF latent class, but not in the non-DIF latent class
Multidimensional model for DIF (MMD) (Roussos & Stout, 1996)
One dimensionality approach: Multiple-indicator multiple-cause (MIMIC) approach
• MMD: a secondary dimension in both manifest groups
• MIMIC: neither of the classes has a secondary dimension but a direct effect of the group variable on the primary dimension (exceptional: Glockner-Rist & Hoijtink, 2003)
• Mixed dimensionality: a secondary dimension but only in the DIF latent class (but may have some problems)
Measurement Invariance Notions
• Scalar variance: difference in item difficulties (intercept in factor model) and implies uniform DIF
• Metric variance: difference in item discriminations (loadings in factor model) and implies nonuniform DIF
• Relate the idea of mixed dimensionality to measurement invariance: With secondary dimension, scalar invariance (equivalence of difficulties) are regain (M2a vs M1a).
Outline of the Study
• Aim: to show the implication and application of a mixed dimensionality model into the latent DIF
• (1) Models specification• (2) Model Estimation and Evaluation• (3) Model application (Speededness &
Arithmetic operation)
Models
• Non-DIF One-Dimensional Mixture Models (1d-non-DIF; Model 0)
• αi and i are the same in both groups;• , (model identification)
• 2I + 3 para (I difficulties, I discriminations, a mean and variance of DIF latent class, and a mixing probability
ijgi βθlogit (Mixture 2PL model)
12 DIFnon 0 DIFnon
• One dimensional difficulty DIF mixture model (1d-dif; Model 1a)
• i are different between groups• Lack of scalar invariance, uniform DIF• 2I+D+3 para (D items showing latent DIF)
ifjfi βθlogit
• One dimensional difficulty and discrimination DIF mixture model (1d-dif; Model 1b)
• αi and i are different between groups• Lack of scalar and metric invariance, uniform and nonuniform DIF • 2(I+D)+3 para (D items showing latent DIF)
ifjfif βθlogit
• Mixture of one-dimensional and two-dimensional model(1&2d-DIF, Model 2a)
• For DIF latent class, (f1,f2)’ ~ BVN(,).
( )• 2I+D+4 free paras• One discrimination for a DIF items in dimension 2
of DIF latent class is fixed to 1.00. (over constrain)
ijfijfi βθθlogit 2211
122 f
• Features of Model 2a: • (1) no difference in difficulty between two latent
traits• (2) metric invariance of primary dimension & scalar
invariance• Implication of Model 2a:• (1) Respondents from the DIF latent class differ with
respect to how much DIF they show (pf2);
• All DIF is assumed to rely on the secondary dimension
• Mixture of one-dimensional and two-dimensional model with discrimination DIF (1&2d-dif, Model 2b)
• 2(I+D)+3 free parameters• Scalar invariance• Cov f12 is fixed to zero (over constrained)
ijfijfif βθθlogit 2211
• Two-dimensional mixture models(2d-DIF and 2d-DIF/ Models 3a and 3b): introduce the second dimension to the non-DIF latent class
• Model 3a: Measurement-invariance• Model 3b: scalar invariance, configural
invariance
Aim of Applications
• Preliminary question: whether any DIF occurs at the level of latent classes (M0 vs M1a & 1b)
• Whether a secondary dimension in the DIF latent class can explain the DIF? (M1a vs M2a; or M1b vs M2b)
• Is the DIF of a kind that the discrimination on the primary dimension is affected? (M1a vs M1b or M2a vs M2b)
• Is the secondary dimension limited to the DIF latent class or does it also apply to the non-DIF latent class? (M2a vs M3a, or M2b vs M3b)
Model Estimation & Evaluation
• LatentGOLD 4.5 Syntax Cluster module (Vermunt & Magidson, 2007)
• Two major problems in mixture model: (1) Label switch; (2) multiple local maxima
• Model specification in LatentGOLD
• LRT (for nested models), AIC, BIC
Results of Speededness Data
First 23 items were assumed no speededness while last 8 items has (item location).
Speeded vs no speeded latent class
• M1a and 1b fit much better than M0, suggesting that DIF exist at the level of latent class;
• First horizontal comparison (M2a> M1a; M2b>M1b) suggests the mixed dimensionality approach is favored.
• Vertical comparison (M1b>M1a;M2b>M2a) suggests discrimination DIF is related to the primary dimension. Thus, M2b is chose.
• Second comparison (M3b is not significantly better than M2b) suggests secondary dimension not need to extend to the non-DIF latent class.
• =0.311, speeded class is a minority class;• Nonzero discriminations on the second dimension
for the DIF items in the speeded class; the discrimination of items on the primary dimension differ between two latent classes.
• For DIF class, mean=(-0.634,-0.987)’,var-cov=(0.787, 0; 0, 1), suggesting that the speeded class has a lower ability and being speeded at the end of the test seems not to help.
• Validation check for the DIF items
• Confirmatory and exploratory two-dimensional models without latent classes were not supported by AIC and BIC.
Results of Arithmetic Operations Data
• Class 1 (Poorer on the division items) vs class 2 (proficient)
• DIF exist between two latent classes;• Secondary dimension helps;• The primary dimension discrimination DIF is
not strongly supported. Hence, M2a is favored.
• No needed to extend the second dimension to the non-DIF latent class.
• For DIF class, mean=(-1.150,-2.945)’,var-cov=(1.034, 0.353*sqrt(1.034); 0.353*sqrt(1.034), 1).
• =0.292• AIC supported exploratory two-dimensional
model without latent class but BIC did not.
Conclusions
• Mixed dimensionality approach are supported by the generally better goodness of fit.
• It was sufficient to include the secondary dimension in only one of the latent class.
• Mixed dimensionality approach has some merits (for latent DIF, explaining DIF…)
• Mixed dimensionality approach should primarily be used for confirmatory and explanatory purposes, but not for exploratory or detection purpose. (problematic)
Discussions & Future Studies
• The mixed dimensionality approach have some problems (not usual; local dependence).
• To define item properties which are not available as the clue of the suspected DIF before the secondary dimension approach can be implemented.
• Applied mixed dimensionality approach to manifest groups, to study cognitive development, mutiple strategies…
• Applied the mixture approach within the cognitive development theoretical framework
• Whether individual differences in DIF is proper?