1 Differential Item Functioning in Mplus Summer School Week 2

Differential Item Functioning Differential item functioning (DIF) occurs when people from different groups (e.g gender or ethnicity) with the same underlying latent trait score have a different probability of responding to an item in a particular way. Group differences in item responses (or on latent variables) do not reflect DIF per se (e.g females score higher than males on a particular item or scale). DIF is only present if people from different groups with the same underlying ability (or trait level) have a different probability of response. Reise, Widaman, Pugh 1993; Psych Bulletin, Vol 114, Embretson,S.E., Reise,S.P. (2000). Item Response Theory for Psychologists. Definition from Laura Gibbons: ‘when a demographic characteristic interferes with relationship expected between ability level and responses to an item’

DIF – Measurement Non-Invariance If the probablity of item response is the same (among different sub-groups with the same underlying ability) measurement invariance is assumed If the probablity of response is different (among different sub-groups with the same underlying ability) than measurement non-invariance is assumed.

Types of DIF Uniform: DIF occurs uniformly at all levels along the latent trait Non-Uniform : DIF does not occur equally at all points on the latent trait (e.g. gender differences in response) may only be evident at high or low levels of the construct Crane et al (2004) describe uniform DIF to be analogous to ‘confounding’ in epidemiology and non-uniform DIF with ‘effect modification’ – i.e. interaction between trait level, group assignment and item responses

Example of Item with Uniform DIF From: Jones R (2006), Medical Care Volume 44, Number 11 Suppl 3, (Figure 2)

Example of Item with Non-Uniform DIF From: Mellenbergh, G. (1989)

7 Definition of DIF (Mellenbergh, 1989) Item Group

8 Definition of DIF (Mellenbergh, 1989) P(u = 1| G, θ) = P(u = 1| θ) An item is unbiased if... i.e. the probability of an item response only depends on the values x of the variable X Item Group Trait

9 Definition of DIF (Mellenbergh, 1989) P(u = 1| G, θ) ≠ P(u = 1| θ) An item is biased if... i.e. the probability of an item response depends on the combination of values x of the variable X and values g of the variable G Item Group Trait

Differential Item Functioning Important first step in the evaluation of test bias For construct validity items of a scale ideally should have little or no DIF Items should function in the same way across subgroups of respondents who have the same underlying ability (or level on the latent trait) Presence of DIF may compromise comparison across subgroups – give misleading results Confound interpretation of observed variables Camilli and Shephard, 1994

Methods to identify DIF Parametric – Mantel-Haenszel (MH) (Holland & Thayer, 1988) Non-parametric methods – Logistic regression (Zumbo, 1999) – Ordinal logistic regression (Crane et al, 2004) – MIMIC models (Muthen, 2004) – Multiple group models – IRT based methods (Thissen, 1991) Good review by Teresi (2006) Medical Care Vol 44

What to do if DIF present Remove items? 1) Ok if you have a large item pool and the item can be replaced with a item measuring similar threshold / discrimination parameters 2) But dropping items might adversely affect the content validity of the instrument. 3) May end up with an instrument that is not comparable to other research using that instrument Look for causes of DIF What do all the DIF items have in common e.g. – Are they all negatively or positively worded – Are they all at end of study – Readability etc How do they differ from the invariant items?

How to adjust for DIF Adjust for DIF in the model – in Mplus can do this by adding direct effect between the covariate and the item Crane et al (2004, 2006) a) items without DIF have item parameters estimated from whole sample – (anchors) b) items with DIF have parameters estimated separately in different subgroups

Two Examples of Identifying DIF Mplus : MIMIC Model (Multiple Indicators, Multiple Causes) Uniform DIF Stata: DIFd program (Crane et al, 2004) Non-Uniform and Uniform DIF

Mplus Example - MIMIC Model: BCS70 Externalising (Conduct) Scale 03 Teenager often destroys belongings 04 Teenager frequently fights with others 10 Teenager sometimes takes others' things 14 Teenager is often disobedient 18 Teenager often tells lies 19 Teenager bullies others Mother’s rating of teenager on Rutter Scale age 16 Ordinal 3 category scale (0=does not apply, 1=applies somewhat, 2=certainly applies)

CFA Model for BCS70 Externalising Observed items F1 Conduct problems RUT03 RUT04 RUT10 RUT14 RUT18 Latent Variable ε ε ε ε ε SEX

Mimic Model Stages of identifying potential DIF 1. Run CFA model without covariates 2. Include MIMIC model (add covariate but no direct effects) 3. Add paths from covariate to indicator constrained to 0 - i.e.assuming there is no direct effect (Y1 on 4. Check modification indices 5. Add direct path from covariate to indicator for indicator with highest modification indices - rerun model 6. Repeat steps 4 & 5 until there are no further significant modification indices, evaluate model fit and significance of the direct effects

Stage 1-3: Mplus CFA USEVARIABLES are rut03 rut04 rut10 rut14 rut18 sex; CATEGORICAL are rut03 rut04 rut10 rut14 rut18; Missing are all ( ); ANALYSIS: ESTIMATOR IS wlsmv; ITERATIONS = 1000; CONVERGENCE = ; MODEL: CONDUCT by rut03 rut04 rut10 rut14 rut18; ! (define latent variable) OUTPUT: SAMPSTAT STANDARDIZED RES MOD(10) ; CONDUCT on sex; ! (MIMIC model - add regression latent var on SEX rut03- rut18 on !(assume no direct effect of sex on item)

CFA Mimic Model Observed items Conduct problems SEX RUT03 RUT04 RUT10 RUT14 RUT18 CovariateLatent Variable O n

Check MOD indices M.I. E.P.C. Std E.P.C. StdYX E.P.C. ON Statements RUT03 ON SEX RUT04 ON SEX Include item with largest MI as a direct effect in model RUT03 on SEX; RUT04- RUT18 ON Recheck mod indices and repeat if necessary

Stage 4: Mplus MIMIC DIF USEVARIABLES are rut03 rut04 rut10 rut14 rut18 sex; CATEGORICAL are rut03 rut04 rut10 rut14 rut18; Missing are all ( ); ANALYSIS: ESTIMATOR IS wlsmv; ITERATIONS = 1000; CONVERGENCE = ; MODEL: CONDUCT by rut03 rut04 rut10 rut14 rut18; ! (define latent variable) RUT04 ON sex ; ! MI in run 4b so may not be required ?? OUTPUT: SAMPSTAT STANDARDIZED RES MOD(10) ; CONDUCT on sex; ! (MIMIC model - add regression latent var on SEX ) rut04- rut18 on !(assume no direct effect of sex on item) rut03 on sex; !(adds direct effect of sex on item 03)

CFA Mimic Model (DIF) Observed EXT items Conduct problems SEX RUT03 RUT04 RUT10 RUT14 RUT18 O n CovariateLatent Variable

CFA Mimic model fit 1. CFA2&3 MIMIC4a. MIMIC (1 direct effect) 4b. MIMIC (2 direct effects) Chi Square37.9 (df=3)131.8 (df=7)51.9 (df=6)38.1 (df=5) CFI TLI RMSEA WRMR

Mplus results Model (2) Initial Mimic Model (no direct effects) Estimate S.E. Est./S.E. P-Value Std CONDUCT ON SEX Model 4(b) Add 2 direct effects Estimate S.E. Est./S.E. P-Value Std CONDUCT ON SEX RUT03 ON SEX RUT04 ON SEX

Mplus results Model (2) Initial Mimic Model (no direct effects) Estimate S.E. Est./S.E. P-Value Std CONDUCT ON SEX Model 4(b) Add 2 direct effects Estimate S.E. Est./S.E. P-Value Std CONDUCT ON SEX RUT03 ON SEX RUT04 ON SEX Is this practically meaningful?

In a Graded Response Model... Analysis: TYPE = general missing h1 ; estimator=mlr ; algorithm=integration ; MODEL: RUT16EX BY rut03 rut04 rut10 rut14 rut18; ! rut19; Rut16ex on sex; ! rut03- rut18 on Output: residual modindices(1.00) sampstat standardized tech1 tech5 cinterval;

In a Graded Response Model... Analysis: TYPE = general missing h1 ; estimator=mlr ; algorithm=integration ; MODEL: RUT16EX BY rut03 rut04 rut10 rut14 rut18; ! rut19; Rut16ex on sex; ! rut03- rut18 on Output: residual modindices(1.00) sampstat standardized tech1 tech5 cinterval;

In a Graded Response Model... Analysis: TYPE = general missing h1 ; estimator=mlr ; algorithm=integration ; MODEL: RUT16EX BY rut03 rut04 rut10 rut14 rut18; ! rut19; Rut16ex on sex; rut03 ON sex; rut04- rut18 on Output: residual modindices(1.00) sampstat standardized tech1 tech5 cinterval; Odds ratios

Exercise Work through MIMIC modelling stages using multivariate probit regression model implemented by WLSMV (equivalent to normal ogive IRT model for polyomous items) 2....using the Graded Response Model implemented by full-information maximum likelihood (MLR) Note: references are included at the end of the next (DifDetect) presentation