Item Response Theory in a Multi-level Framework

Saralyn MillerMeg Oliphint

EDU 7309

Agenda

• Item Response Theory– Conceptual Overview– Show Models (Rasch, 2PL, 3PL)– Example– In Class Practice

• Differential Item Functioning– Conceptual Overview– Example– In Class Practice

Defining Terms• IRT– Item Response Theory - provides a framework for

evaluating how well assessments work, and how well individual items on assessments work

• DIF– Differential Item Functioning– people from

different groups with same ability function differently on certain items

• CTT – Classical Test Theory –– Observed Score + Error = True Score

IRT vs. CTT – Situating IRT

• IRT allows for greater reliability• IRT can be used in CAT• IRT allows for difficulty and ability to be on the

same scale• CTT is simple to compute• IRT can be analyzed using multi-level modeling

How does IRT work?

Defining Item Parameters

• ai – ability parameter – point on the ability scale (θ) that intersects with the probability of 0.5 - P(θ)

• bi – difficulty parameter – point on θ where the ICC has its maximum slope

• ci – guessing parameter – added to ability (based number of response choices) to account for possible guessing

IRT formula

3 PL Model (which includes 1PL and 2PL)

Rasch Model

Example: LSAT data• 5 Test items; 1000 examinees

library(ltm)head(LSAT) Item 1 Item 2 Item 3 Item 4 Item 51 0 0 0 0 02 0 0 0 0 03 0 0 0 0 04 0 0 0 0 15 0 0 0 0 16 0 0 0 0 1

Example: LSAT datadescript(LSAT) Descriptive statistics for the 'LSAT' data-set Sample: 5 items and 1000 sample units; 0 missing values Proportions for each level of response: 0 1 logitItem 1 0.076 0.924 2.4980Item 2 0.291 0.709 0.8905Item 3 0.447 0.553 0.2128Item 4 0.237 0.763 1.1692Item 5 0.130 0.870 1.9010  Frequencies of total scores: 0 1 2 3 4 5Freq 3 20 85 237 357 298

Example: LSAT dataPoint Biserial correlation with Total Score: Included ExcludedItem 1 0.3618 0.1128Item 2 0.5665 0.1531Item 3 0.6181 0.1727Item 4 0.5342 0.1444Item 5 0.4351 0.1215  Cronbach's alpha: valueAll Items 0.2950Excluding Item 1 0.2754Excluding Item 2 0.2376Excluding Item 3 0.2168Excluding Item 4 0.2459Excluding Item 5 0.2663  

Pairwise Associations: Item i Item j p.value1 1 5 0.5652 1 4 0.2083 3 5 0.1134 2 4 0.0595 1 2 0.0286 2 5 0.0097 1 3 0.0038 4 5 0.0029 3 4 7e-0410 2 3 4e-04

##Fitting the Rasch model## fitRasch1<-rasch(LSAT,constraint=cbind(length(LSAT)+1,1))summary(fitRasch1)Call:rasch(data = LSAT, constraint = cbind(length(LSAT) + 1, 1)) Model Summary: log.Lik AIC BIC -2473.054 4956.108 4980.646 Coefficients: value std.err z.valsDffclt.Item 1 -2.8720 0.1287 -22.3066Dffclt.Item 2 -1.0630 0.0821 -12.9458Dffclt.Item 3 -0.2576 0.0766 -3.3635Dffclt.Item 4 -1.3881 0.0865 -16.0478Dffclt.Item 5 -2.2188 0.1048 -21.1660Dscrmn 1.0000 NA NA Integration:method: Gauss-Hermitequadrature points: 21  Optimization:Convergence: 0 max(|grad|): 6.3e-05 quasi-Newton: BFGS

coef(fitRasch1,prob=TRUE,order=TRUE)

Dffclt Dscrmn P(x=1|z=0)Item 1 -2.8719712 1 0.9464434Item 5 -2.2187785 1 0.9019232Item 4 -1.3880588 1 0.8002822Item 2 -1.0630294 1 0.7432690Item 3 -0.2576109 1 0.5640489

patterns<-rbind("all.zeros"=rep(0,5),"mix1"=rep(0:1,length=5),"mix2"=rep(1:0,length=5),"all.ones"=rep(1,5))residuals(fitRasch1,resp.patterns=patterns,order=FALSE)

Item 1 Item 2 Item 3 Item 4 Item 5 Obs Exp Residall.zeros 0 0 0 0 0 3 5.016847 -0.9004457mix1 0 1 0 1 0 0 2.739417 -1.6551184mix2 1 0 1 0 1 28 23.314087 0.9704765all.ones 1 1 1 1 1 298 323.237052 -1.4037121

Item Characteristic Curveplot(fitRasch1,legend=TRUE,pch=rep(1:20,each=5),xlab="LSAT",col=rep(1:5,2),lwd=2,cex=1.2,sub=paste("Call:",deparse(fitRasch1$call)))

Item Information Curveplot(fitRasch1, type = "IIC", legend = TRUE, pch = rep(1:2, each = 5), xlab = "Attitude",col = rep(1:5, 2), lwd = 2, cex = 1.2, sub = paste("Call: ", deparse(fitRasch1$call)))

Test Information Curveinfo1<-plot(fitRasch1,type="IIC",items=0,lwd=2,xlab="LSAT")

Multi-level analysis of IRT

• Hierarchical generalized linear models (HGLM)– Framework used for the nesting structure of item responses.– We are going to focus on the intercept model where items are

dichotomous.• Items are nested in examinees.– Item Responses (1st level)– Examinees (2nd level)

• The HGLM model is a fully crossed design since all examinees answer all test items.

• We will use a type of Rasch modeling.

Fully Nested Design Fully Crossed Design

Person 1

Item 1

Item 2

Item 3

Item 4

Item 6

Item 5Person 2

Person 1

Item 1

Item 2

Item 3

Item 4

Item 6

Item 5Person 2

HGLM Rasch Model

• At level 1, all items are inserted into the model and usually the last item is used as the reference item (intercept).

• At level 2, we have fixed and random effects where examinee ability is random, but item difficulty is fixed.

Multi-level Formulas

At level 1 we are obtaining the log-odds of the probability that person j obtains a correct score (one) on item i:

At level 2 under this model, intercepts are random. This means we are allowing an examinee’s ability to be random. Slopes are not random. This means item difficulties are fixed.

Now we can substitute the formulas above, back into the equation for the probability that person j answers item I correctly.

Kyle’s data multi level

kyle<-read.table("mlm2.txt",header=T)

##All items must be factors to use nlme###kyle$person<-as.factor(kyle$person)kyle$resp<-as.factor(kyle$resp)kyle$i1<-as.factor(kyle$i1)kyle$i2<-as.factor(kyle$i2)kyle$i3<-as.factor(kyle$i3)kyle$i4<-as.factor(kyle$i4)kyle$i5<-as.factor(kyle$i5)kyle$i6<-as.factor(kyle$i6)kyle$i7<-as.factor(kyle$i7)kyle$i8<-as.factor(kyle$i8)kyle$i9<-as.factor(kyle$i9)kyle$i10<-as.factor(kyle$i10)

Kyle’s data multi levellibrary(nlme)glmm.fit.kyle<-glmmPQL(resp~i1+i2+i3+i4+i5+i6+i7+i8+i9,random=~1|person,family=binomial,data=kyle)summary(glmm.fit.kyle)

Linear mixed-effects model fit by maximum likelihood Data: kyle AIC BIC logLik NA NA NA

Random effects: Formula: ~1 | person (Intercept) ResidualStdDev: 1.134212 0.9028756

Variance function: Structure: fixed weights Formula: ~invwt Fixed effects: resp ~ i1 + i2 + i3 + i4 + i5 + i6 + i7 + i8 + i9 Value Std.Error DF t-value p-value(Intercept) -2.538158 0.7737393 171 -3.280379 0.0013i11 5.844407 1.2270676 171 4.762906 0.0000i21 4.554857 0.9629141 171 4.730283 0.0000i31 5.056955 1.0357069 171 4.882613 0.0000i41 3.551793 0.8852165 171 4.012344 0.0001i51 3.551793 0.8852165 171 4.012344 0.0001i61 2.548906 0.8606717 171 2.961531 0.0035i71 1.806033 0.8671231 171 2.082787 0.0388i81 2.307883 0.8604645 171 2.682136 0.0080i91 0.510344 0.9462893 171 0.539311 0.5904

Kyle’s data multi levellibrary(nlme)glmm.fit.kyle<-glmmPQL(resp~i1+i2+i3+i4+i5+i6+i7+i8+i9,random=~1|person,family=binomial,data=kyle)summary(glmm.fit.kyle)

###Rest of output###Correlation: (Intr) i11 i21 i31 i41 i51 i61 i71 i81 i11 -0.571 i21 -0.725 0.475 i31 -0.675 0.444 0.561 i41 -0.784 0.510 0.646 0.602 i51 -0.784 0.510 0.646 0.602 0.697 i61 -0.800 0.514 0.654 0.609 0.708 0.708 i71 -0.787 0.503 0.640 0.595 0.694 0.694 0.710 i81 -0.798 0.512 0.651 0.606 0.706 0.706 0.720 0.709 i91 -0.711 0.449 0.572 0.532 0.622 0.622 0.639 0.634 0.639

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -4.1891522 -0.5501785 0.1875289 0.4873082 4.6409578

Number of Observations: 200Number of Groups: 20

Calculate DifficultiesFixed effects: resp ~ i1 + i2 + i3 + i4 + i5 + i6 + i7 + i8 + i9

Value Std.Error (Intercept) -2.538158 0.7737393 i11 5.844407 1.2270676i21 4.554857 0.9629141i31 5.056955 1.0357069 i41 3.551793 0.8852165 i51 3.551793 0.8852165 i61 2.548906 0.8606717 i71 1.806033 0.8671231i81 2.307883 0.8604645 i91 0.510344 0.9462893

To calculate item difficulty, we must use the following:

I1 [-5.84-(-2.54)] = -3.3I2 -2.01I3 -2.52I4 -1.01I5 -1.01I6 -0.01I7 0.73I8 0.23I9 2.03I10 2.54

Kyle’s Data (single level)kyle<-read.table("mlm2.txt",header=T)library(psych)library(ltm)head(kyle) resp person id i1 i2 i3 i4 i5 i6 i7 i8 i9 i10 cons bcons denom1 1 1 1 1 0 0 0 0 0 0 0 0 0 1 1 12 1 1 2 0 1 0 0 0 0 0 0 0 0 1 1 13 1 1 3 0 0 1 0 0 0 0 0 0 0 1 1 14 1 1 4 0 0 0 1 0 0 0 0 0 0 1 1 15 1 1 5 0 0 0 0 1 0 0 0 0 0 1 1 16 1 1 6 0 0 0 0 0 1 0 0 0 0 1 1 1

Kyle’s Data (single level)##data is already stacked for multi-level analysis so data needs to be unstacked###

kyle$item<-rep(1:10, 20)kyle.new<-kyle[,c(1,2,17)]kyle1<-reshape(kyle.new, timevar="item", idvar="person", direction="wide")head(kyle1)

person resp.1 resp.2 resp.3 resp.4 resp.5 resp.6 resp.7 resp.8 resp.9 resp.101 1 1 1 1 1 1 1 1 1 1 011 2 1 1 1 1 1 1 1 1 0 121 3 1 1 1 1 1 1 1 1 0 031 4 1 1 1 1 1 1 0 1 1 041 5 1 1 1 1 1 0 1 1 0 051 6 1 1 1 1 1 1 0 1 0 0

##create new subset without “person” variable##kyle2<-subset(kyle1,select=c(resp.1,resp.2,resp.3,resp.4,resp.5,resp.6,resp.7,resp.8,resp.9,resp.10))

Kyle’s Data (single level)##constraints where disc=1###fitRasch1<-rasch(kyle2,constraint=cbind(length(kyle2)+1,1))summary(fitRasch1)Call:rasch(data = kyle2, constraint = cbind(length(kyle2) + 1, 1))

Model Summary: log.Lik AIC BIC -93.08932 206.1786 216.1360

Coefficients: value std.err z.valsDffclt.resp.1 -3.3917 1.0854 -3.1249Dffclt.resp.2 -2.0704 0.7076 -2.9260Dffclt.resp.3 -2.5864 0.8188 -3.1586Dffclt.resp.4 -1.0371 0.5806 -1.7862Dffclt.resp.5 -1.0371 0.5806 -1.7862Dffclt.resp.6 -0.0071 0.5445 -0.0129Dffclt.resp.7 0.7520 0.5652 1.3306Dffclt.resp.8 0.2395 0.5468 0.4380Dffclt.resp.9 2.0760 0.7124 2.9144Dffclt.resp.10 2.5993 0.8245 3.1527Dscrmn 1.0000 NA NA

Integration:method: Gauss-Hermitequadrature points: 21

Optimization:Convergence: 0 max(|grad|): 0.00025 quasi-Newton: BFGS

Kyle’s Data (single level)#items ordered by difficulty and probability of positive response by the average individual#

coef(fitRasch1,prob=TRUE,order=TRUE) Dffclt Dscrmn P(x=1|z=0)resp.1 -3.391711408 1 0.96744449resp.3 -2.586410763 1 0.92998186resp.2 -2.070364708 1 0.88798924resp.4 -1.037146071 1 0.73829896resp.5 -1.037111039 1 0.73829219resp.6 -0.007050493 1 0.50176262resp.8 0.239493993 1 0.44041105resp.7 0.751995163 1 0.32038672resp.9 2.076049140 1 0.11144661resp.10 2.599328664 1 0.06918164

Compare Difficulties (kyle data)Items Difficulty – IRT Difficulty – Multi-level IRT

Item 1 -3.39 -3.3

Item 2 -2.07 -2.01

Item 3 -2.59 -2.52

Item 4 -1.04 -1.01

Item 5 -1.04 -1.01

Item 6 -0.007 -0.01

Item 7 0.75 0.73

Item 8 0.24 0.23

Item 9 2.08 2.03

Item 10 2.60 2.54

Example in Class – Multi-level of LSAT data in ltm package

##need to reshape the data## LSAT1<-reshape(LSAT,varying=list(1:5),direction="long")LSAT1<-LSAT1[order(LSAT1$id),]colnames(LSAT1)<-c("item","score","id")

LSAT1$item1<-ifelse(LSAT1$item==1,1,0)LSAT1$item2<-ifelse(LSAT1$item==2,1,0)LSAT1$item3<-ifelse(LSAT1$item==3,1,0)LSAT1$item4<-ifelse(LSAT1$item==4,1,0)LSAT1$item5<-ifelse(LSAT1$item==5,1,0)

LSAT1[1:15,]

###MAKE VARIABLES FACTORS######RUN ANALYSIS######COMPUTE DIFFICULTIES###

1. MAKE VARIABLES FACTORS2. RUN ANALYSIS3. COMPUTE DIFFICULTIES

Item Difficulty – IRT Difficulty – Multi-level IRT

Item 1

Item 2

Item 3

Item 4

Item 5

Compare DifficultiesItem Difficulty – IRT Difficulty – Multi-level IRT

Item 1 -2.9 -2.6

Item 2 -1.06 -0.93

Item 3 -0.26 -0.21

Item 4 -1.39 -1.23

Item 5 -2.22 -1.99

Example in Class – Multi-level of LSAT data in ltm packageglmm.fit.LSAT<-glmmPQL(score~item1+item2+item3+item4,random=~1|id,family=binomial,data=LSAT1)summary(glmm.fit.LSAT)Linear mixed-effects model fit by maximum likelihood Data: LSAT1 AIC BIC logLik NA NA NA

Random effects: Formula: ~1 | id (Intercept) ResidualStdDev: 0.8172182 0.8986588

Variance function: Structure: fixed weights Formula: ~invwt Fixed effects: score ~ item1 + item2 + item3 + item4 Value Std.Error DF t-value p-value(Intercept) 1.997380 0.09013816 3996 22.159099 0item11 0.614355 0.13849689 3996 4.435876 0item21 -1.057513 0.10750549 3996 -9.836829 0item31 -1.776414 0.10453076 3996 -16.994170 0item41 -0.763684 0.11004239 3996 -6.939908 0 Correlation: (Intr) item11 item21 item31item11 -0.592 item21 -0.767 0.497 item31 -0.791 0.510 0.661 item41 -0.748 0.485 0.626 0.645

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -4.2531208 0.2231402 0.3957765 0.5798209 1.9214245

Number of Observations: 5000Number of Groups: 1000

Differential Item Functioning

• DIF is a way to detect biased (unfair) questions in a given test

• An item is said to have DIF if:• People in the same group• With the same ability• Answer the question differently

• Classic Example: Question about math that requires heavy reading load or Questions about calculating ERA

Differential Item Functioning

• Can be detected using logistic regression:– Looking for SS for an item to have DIF

• Can be detected in a multi-level modeling framework:– Looking at the interaction effect between the

grouping variable and that certain item– If the DIF estimate is larger than twice the

standard error, the item is biased• Keep in mind: DIF = bad!

DIF Example with LSAT Data> LSAT1$gender<-as.factor(rep(0:1, each=500))> head(LSAT1) item score id item1 item2 item3 item4 item5 gender1.1 1 0 1 1 0 0 0 0 01.2 2 0 1 0 1 0 0 0 01.3 3 0 1 0 0 1 0 0 01.4 4 0 1 0 0 0 1 0 01.5 5 0 1 0 0 0 0 1 02.1 1 0 2 1 0 0 0 0 0

Item 1glmm.dif.LSAT<-

glmmPQL(score~0+item1*gender+item2*gender+item3*gender+item4*gender,random=~1|id,family=binomial,data=LSAT1)

summary(glmm.dif.LSAT) Fixed effects: score ~ 0 + item1 * gender + item2 * gender + item3 * gender +

item4 * gender Value Std.Error DF t-value p-valueitem1 0.330415 0.144 3993 2.299060 0.0216gender0 1.534279 0.107 998 14.296862 0.0000gender1 2.954049 0.162 998 18.206689 0.0000item2 -0.665216 0.128 3993 -5.203006 0.0000item3 -0.940232 0.126 3993 -7.469119 0.0000item4 -0.695781 0.128 3993 -5.453253 0.0000item1:gender1 24.417536 19800.750 3993 0.001233 0.9990gender1:item2 -1.175118 0.219 3993 -5.360656 0.0000gender1:item3 -2.155689 0.216 3993 -9.990973 0.0000gender1:item4 -0.379341 0.226 3993 -1.675854 0.0938 Correlation: item1 gendr0 gendr1 item2 item3 item4 itm1:1 gnd1:2gender0 -0.604 gender1 0.000 0.000 item2 0.508 -0.684 0.000 item3 0.515 -0.697 0.000 0.583 item4 0.509 -0.686 0.000 0.574 0.584 item1:gender1 0.000 0.000 0.000 0.000 0.000 0.000 gender1:item2 -0.296 0.399 -0.676 -0.583 -0.340 -0.335 0.000 gender1:item3 -0.301 0.406 -0.695 -0.340 -0.583 -0.341 0.000 0.708gender1:item4 -0.287 0.387 -0.650 -0.324 -0.329 -0.564 0.000 0.669 gnd1:3gender0 gender1 item2 item3 item4 item1:gender1 gender1:item2 gender1:item3 gender1:item4 0.681

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -5.236435e+00 -3.620071e-11 4.086712e-01 7.028004e-01 1.950463e+00

Number of Observations: 5000Number of Groups: 1000

LOOK! This item does not have

DIF!

Item 2Fixed effects: score ~ 0 + item1 * gender + item2 * gender + item3 * gender +

item4 * gender Value Std.Error DF t-value p-valueitem1 0.330415 0.144 3993 2.299060 0.0216gender0 1.534279 0.107 998 14.296862 0.0000gender1 2.954049 0.162 998 18.206689 0.0000item2 -0.665216 0.128 3993 -5.203006 0.0000item3 -0.940232 0.126 3993 -7.469119 0.0000item4 -0.695781 0.128 3993 -5.453253 0.0000item1:gender1 24.417536 19800.750 3993 0.001233 0.9990gender1:item2 -1.175118 0.219 3993 -5.360656 0.0000gender1:item3 -2.155689 0.216 3993 -9.990973 0.0000gender1:item4 -0.379341 0.226 3993 -1.675854 0.0938 Correlation: item1 gendr0 gendr1 item2 item3 item4 itm1:1 gnd1:2gender0 -0.604 gender1 0.000 0.000 item2 0.508 -0.684 0.000 item3 0.515 -0.697 0.000 0.583 item4 0.509 -0.686 0.000 0.574 0.584 item1:gender1 0.000 0.000 0.000 0.000 0.000 0.000 gender1:item2 -0.296 0.399 -0.676 -0.583 -0.340 -0.335 0.000 gender1:item3 -0.301 0.406 -0.695 -0.340 -0.583 -0.341 0.000 0.708gender1:item4 -0.287 0.387 -0.650 -0.324 -0.329 -0.564 0.000 0.669 gnd1:3gender0 gender1 item2 item3 item4 item1:gender1 gender1:item2 gender1:item3 gender1:item4 0.681

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -5.236435e+00 -3.620071e-11 4.086712e-01 7.028004e-01 1.950463e+00

Number of Observations: 5000Number of Groups: 1000

Does this item have

DIF?

YES =(

Item 3Fixed effects: score ~ 0 + item1 * gender + item2 * gender + item3 * gender +

item4 * gender Value Std.Error DF t-value p-valueitem1 0.330415 0.144 3993 2.299060 0.0216gender0 1.534279 0.107 998 14.296862 0.0000gender1 2.954049 0.162 998 18.206689 0.0000item2 -0.665216 0.128 3993 -5.203006 0.0000item3 -0.940232 0.126 3993 -7.469119 0.0000item4 -0.695781 0.128 3993 -5.453253 0.0000item1:gender1 24.417536 19800.750 3993 0.001233 0.9990gender1:item2 -1.175118 0.219 3993 -5.360656 0.0000gender1:item3 -2.155689 0.216 3993 -9.990973 0.0000gender1:item4 -0.379341 0.226 3993 -1.675854 0.0938 Correlation: item1 gendr0 gendr1 item2 item3 item4 itm1:1 gnd1:2gender0 -0.604 gender1 0.000 0.000 item2 0.508 -0.684 0.000 item3 0.515 -0.697 0.000 0.583 item4 0.509 -0.686 0.000 0.574 0.584 item1:gender1 0.000 0.000 0.000 0.000 0.000 0.000 gender1:item2 -0.296 0.399 -0.676 -0.583 -0.340 -0.335 0.000 gender1:item3 -0.301 0.406 -0.695 -0.340 -0.583 -0.341 0.000 0.708gender1:item4 -0.287 0.387 -0.650 -0.324 -0.329 -0.564 0.000 0.669 gnd1:3gender0 gender1 item2 item3 item4 item1:gender1 gender1:item2 gender1:item3 gender1:item4 0.681

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -5.236435e+00 -3.620071e-11 4.086712e-01 7.028004e-01 1.950463e+00

Number of Observations: 5000Number of Groups: 1000

Does this item have

DIF?

YES =(

Item 4Fixed effects: score ~ 0 + item1 * gender + item2 * gender + item3 * gender +

item4 * gender Value Std.Error DF t-value p-valueitem1 0.330415 0.144 3993 2.299060 0.0216gender0 1.534279 0.107 998 14.296862 0.0000gender1 2.954049 0.162 998 18.206689 0.0000item2 -0.665216 0.128 3993 -5.203006 0.0000item3 -0.940232 0.126 3993 -7.469119 0.0000item4 -0.695781 0.128 3993 -5.453253 0.0000item1:gender1 24.417536 19800.750 3993 0.001233 0.9990gender1:item2 -1.175118 0.219 3993 -5.360656 0.0000gender1:item3 -2.155689 0.216 3993 -9.990973 0.0000gender1:item4 -0.379341 0.226 3993 -1.675854 0.0938 Correlation: item1 gendr0 gendr1 item2 item3 item4 itm1:1 gnd1:2gender0 -0.604 gender1 0.000 0.000 item2 0.508 -0.684 0.000 item3 0.515 -0.697 0.000 0.583 item4 0.509 -0.686 0.000 0.574 0.584 item1:gender1 0.000 0.000 0.000 0.000 0.000 0.000 gender1:item2 -0.296 0.399 -0.676 -0.583 -0.340 -0.335 0.000 gender1:item3 -0.301 0.406 -0.695 -0.340 -0.583 -0.341 0.000 0.708gender1:item4 -0.287 0.387 -0.650 -0.324 -0.329 -0.564 0.000 0.669 gnd1:3gender0 gender1 item2 item3 item4 item1:gender1 gender1:item2 gender1:item3 gender1:item4 0.681

Standardized Within-Group Residuals: Min Q1 Med Q3 Max -5.236435e+00 -3.620071e-11 4.086712e-01 7.028004e-01 1.950463e+00

Number of Observations: 5000Number of Groups: 1000

Does this item have

DIF?

NO =)

Thank you for a great two years!