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WINMIRA 2001

1. threshold 1 threshold 2 thresholc item Parameters in Class 1 with size 0 47969 Fe 1 000 0 9962 1 000 WINMIRA 2001 c 2000 2001 by Matthias von Davier Table of Contents WINMIRA user manual 1 The WINMIRA THANK YOU PAGE sssss0ossssssossssssnsnsnsnnssssnnnsnssnnnnne 3 Preface ua rain sobeudelovaudsescasseoseacesesosoesdososessents 5 How tose WINMIRA 3 000 ia 7 How to do a Latent Class Analysis eeeeeeeeseseseeeeeseesssnnnnnennnnennnnnnnn 8 How to analyse data with the Rasch Model 10 How todo a Mixed Rasch M del uueseseen 12 How to analyse data with the Hybrid Model o 14 Open data ee seien 17 Open SPSS Data Files a een ie 17 Hnipore ASC Da su ses es te 18 Recode data with WINM IRA nannte 20 How to edit variables with WINMIRAN seseennnenn 2 SEIS Uy AS 2 ee ee 22 Choose Number Of Classes ea Dr a 25 WINMIRA 2001 Table of Contents Choosing Output Ophons a ae eu ei 26 Appendins variables t hedatare aa a 28 Edi Tlenames een en ANES 29 Edit deraul yalues an sa a a E EA 31 Testing the fit of a model with the Bootstrap eeenneee 33 SEI RUN senken 35 Parameter constramts nse nenn 37 Probability consu ann aaa ee ae 38 Logistic parameter constant 40 Class size CONSE ALINES ee KES T EE 41 The rationale of Mixture Distribution Models ccsccsssssoesse
2. ipecification Job Definition Stat Graphs Window Help Latent class analysis Rasch model Mixed Rasch model Eee This will open the model selection dialog for the Latent Class Analysis see below Rating Scale Model Equidistance Model Dispersion Model Ordinal Partial Credit Model W Class specific parameters Help Cancel If you use WINMIRA for the first time please do not change anything WINMIRA 2001 10 How to analyse data with the Rasch Model Clicking OK with the default settings will choose the most general LCA model which is suitable both for dichotomous and polytomous data The remaining options impose restrictions on the model parameters of the LCA More information on restricted models can also be found in the references e Change Output Options if necessary e choose Bootstrap Fit Statistics if your data are sparse e Start Run e Display Graphical Output How to analyse data with the Rasch Model e Open Ascii or SPSS Data Files e Recode Data if necessary e Select Variables e Do not choose the number of latent classes not necessary for the ordinary Rasch model this model assumes that the same parameters hold for the whole population Choose the Rasch model from the Job Definition gt Select Model submenu WINMIRA 2001 How to analyse data with the Rasch Model 11 as shown in the following screenshot F simula9x MIW Data Specification VAR2 v4 0 0 0
3. Expected Score Frequencies and Personparameters Raw Expected MLE SE MLE WLE SE WLE score freq estimate estimate WINMIRA 2001 90 0 28 92 KK KK 1 61 99 A Aa 2 68 12 2 288 3 85 94 1 897 4 54 58 1 614 5 62 72 1 388 6 37 86 1 198 7 34 08 1 030 8 21 82 0 878 9 34 41 0 737 10 311 0 605 11 38 97 0 478 12 28 80 s0397 13 32 78 0 238 14 28 91 SOs EZA 15 32 26 0 005 16 34 43 0 111 17 30 11 0 228 18 23 74 0 347 19 33 34 0 469 20 27 10 0 596 2 1 33 59 1 0 729 22 23 65 0 870 23 31 81 1 024 24 19 25 1 193 25 20 26 1 5387 26 8 06 1 616 WINMIRA 2001 0 Example Output file KKKKKKKK EI 1 33 984 2i 530 694 2 072 5 09 NEF DD 499 1 512 453 1 5312 422 14139 399 0 983 382 0 841 369 0 708 0 359 0 582 0 352 0 462 0 347 0 344 0 343 0 230 0 341 O TL 0 340 0 005 341 0 107 343 0 220 347 0 333 352 0 452 360 0 573 370 0 699 383 0 833 401 0 976 424 1 133 457 1 308 504 1 512 0 0 4 0 795 621 8l Example Output file 91 27 0 82 1 904 0 576 1 761 0 538 28 2 87 2 305 0 701 2 087 0 629 29 2 00 2 9851 ih 049930 2 582 04807 30 0572 AAKAKAKKK KKKKKKKK 3 656 1 397 The table above shows the expected rawscore frequencies in class 1 the person parameter
4. Job Definition Start Graphs Window Help N of Classes S e vi odel b Parameter constraints Dutput Options Bootstrap GoF Filenames Edit Defaults 0 0 0 0 1 0 0 0 00 0 0 je This will open the model selection dialog for the Mixed Rasch Model see below MIW calion Job Definition Stat Graphe Window Help Choose a Mixed Rasch model C Rating Scale Model Eguidistance Model Dispersion Model Ordinal Partial Cedi Model Bi Peer If you use WINMIRA for the first time please do not change anything Clicking OK with the default settings will choose the most common Mixed Rasch Model which is suitable both for dichotomous and polytomous data WINMIRA 2001 14 How to analyse data with the Hybrid Model The remaining options impose restrictions on the model parameters of the Mixed Rasch Model More information on restricted Rasch models can also be found in the references e Change Output Options if necessary e choose Bootstrap Fit Statistics if your data are sparse e Start Run e Display Graphical Output How to analyse data with the Hybrid Model e Open Ascii or SPSS Data Files e Recode Data if necessary e Select Variables e Choose the number of latent classes Choose the Hybrid model from the Job Definition gt Select Model submenu as shown in the following screenshot WINMIRA 2001 How to analyse data with the Hybrid Model Iw ation Job Definition Start Graphs Window
5. hardcopies of the graphical output is also required Display Graphical Output By choosing any of the function keys F5 F6 or F7 or the corresponding entries in the graphs menu plots of class specific model parameters are displayed The plots are resizeable and can be printed and or saved to a file see below WINMIRA 2001 Display Graphical Output 83 Category probabilities or the function key F5 displays a histogramm of the class specific response probabilities for all items f WINMIRA 32 professional File Edit Search Data a in ails at Graphs Window Help f amp re Probability Plot ws P JE ALD 2 Wr maks m category m category mw category 2 m category 3 Category Probabilities in Class 3 with size 0 23146 Item parameters or F6 displays class specific threshold parameters The spin button on the left side of the graphic control panel can be used to select the latent class to be displayed WINMIRA 2001 84 Display Graphical Output Class e Bl gt 2 W lines M marks Select class to be displayed a tresno select class to be displayed threshold 3 item Parameters in Class 2 with size 0 28165 0 5 Threshold o 0 5 Each graph can be printed or saved directly from within the graphs control panel The person parameter graph shows the absolute raw score frequencies for each class and if the class was assumed to be Rasch homogenous a simultanous pers
6. 676 item fit assessed by the Q index itemlabel item item item item item item item item 1641 1969 Q index zq 0 7426 0 6670 0 2980 0 4262 O 7969 0 3487 1 4552 0153 390 p X gt 2q 0 7711 0 25238 0 6171 0 33497 0 7754 0 36367 0 07281 0 6311 0 472 0 345 0 3 19 0 061 0 094 0 119 0 265 0 536 0 619 5 ua AY ee Ol Bef Se wath Oe 99 WINMIRA 2001 100 Example Output file 3temd9 0 1499 0 47769 0578138 sur 9 itemLO 0 1625 5020382 021223 1 See Onn 7 p0 057 2 P gt 0 95 p lt 0 01 p gt 0 99 According to the Q index there is no item with a significant deviation from the expected characteristic as predicted by the Rating Scale model in this latent class Similarily to the results in class 1 there is no indication to assume that the model does not fit the item responses in this class Nevertheless a decision regarding model fit should only be based on the goodness of fit statistics for the whole model which are given at the end of the output file The third latent class is a LCA type class i e it is assumed there are no systematic differences between the members of this class In our example dataset about 20 percent of the observed patterns can be fitted by this class Final Estimates in CLASS 3 of 3 with size 0 22534 LCA Latent Class Analysis class s
7. D B Maximum Likelihood from Incomplete Data via the EM Algorithm Journal of the Royal Statistical Society Series B 39 1 38 1977 Drasgow F Levine M Williams E Appropriateness measurement with polychotomous item response models and standardized residuals British J of Math and Stat Psychology 1985 38 67 86 Efron B The Jackknife the Bootstrap and other Resampling Plans SIAM Society for Industrial and Applied Mathematics 1982 Efron B amp Tibshirani R J An Introduction to the Bootstrap Monographs on Statistics and Applied Probability 57 New York Chapman amp Hall 1993 Everitt B S amp Hand D J Finite Mixture Distributions London Chapman amp Hall 1981 Giegler H amp Rost J 1990 Ordinale manifeste Variablen Nominale latente WINMIRA 2001 References 111 Variablen Latent Class Analyse f r ordinale Variablen In Faulbaum F Haux R amp J ckel K H Hrsg SoftStat 89 Fortschritte der Statistik Software 2 Stuttgart Gustav Fischer Hoijtink H amp Boomsma A On Person Parameter Estimation in the Dichotomous Rasch Model Chapter 4 in Fischer G H amp Molenaar I eds Rasch Models Foundations Recent Developments and Applications New York Springer 1995 Langeheine R amp Rost J Latent Trait and Latent Class Models New York Plenum 1988 Lazarsfeld P F The Logical and Mathematical Foundations of Latent Structure Analys
8. The saturated likelihood is the theoretical maximum of the likelihood function that can be reached This maximum can only be met by the saturated model by assuming one parameter for each observed response pattern The saturated likelihood is used in the likelihood ratio Goodness of HFit test number of different patterns 1834 number of possible patterns 1048576 The ratio of observed to possible patterns indicates that many of the possible pattern have zero frequencies i e they haven t been observed This implies that traditional Goodness of Fit statistics see below at the end of this example output can not be used for testing a model for this dataset Number of iterations needed 117 WINMIRA 2001 Example Output file 89 117 iterations were needed to reach the default accuracy criterion and terminating estimation fitted model Hybrid model in 3 latent classes A discrete mixture of different models for each class was fitted in this example The model of class one and two is a polytomous Rasch model the Rating Scale model in this example In the third class local independence according to the ordinary Latent Class model is assumed Final Estimates in CLASS 1 of 3 with size 0 49311 MIRA Mixed Rasch Model according to the rating scale model The class size indicates that about 50 percent of the population can be fitted by a polytomous Rasch model which was assumed to hold in this class
9. and additionally models with more than one IRT class can be specified v Davier 1994 This type of Hybrid model integrates a finite number of mixed Rasch models and a finite number of Latent Class models so that each latent class of this new model family can have it s own structure WINMIRA 2001 80 WINMIRA 2001 The Hybrid Model Using the output of WINMIRA The output files generated by WINMIRA are organized as follows An ASCII text file contains tables with summaries of the sufficient statistics the final parameter estimates and Goodness of Fit statistics In addition graphs of item and person parameters can be produced and saved as bitmaps or metafiles Finally person specific information can be appended to the data file printing output files e graphical output e append variables to the datafile e example output file a little outdated Printing the output Please print output files by choosing the File gt Print Output menu entry WINMIRA 2001 82 Display Graphical Output E WINMIRA 32 pro DUTP ef File Edit Search Data S Open gt Close Save gt ro 4 7 PrinterSetup 11 by IP Ol ff 241 Alternatively you can use the printer speed button in the panel WINMIRA 32 pro File Edit Search Jael Xi Winmira 32 Printing the ouput will produce a number of pages with all tables contained in the output file Please use the print option in the graph windows if
10. 0 0 0 0 0 0 0 0 0 Parameter constraints Qutput Options Mixed Rasch model Bootstrap GoF Hybrid model Filenames Edit Defaults This will open the model selection dialog for the Rasch Model and the Mixed Rasch Model see below Cation Job Definition Stat Graphs Window Help oe eli 4 Choose a Rasch model oo 0 0 0 0 0 0 n Rating Scale Model C Equidistance Model Dispersion Model sevessnsnserevenseeseseeesenssenssasesesssanessvsnuessugesenessn y inhenanensbeavasnatgaseenssnansnssnsuansssnerventeesuanensneanens If you use WINMIRA for the first time please do not change anything Clicking OK with the default settings will choose the most common Rasch model which is suitable both for dichotomous and polytomous data The WINMIRA 2001 12 How to do a Mixed Rasch Model remaining options impose restrictions on the model parameters of the model More information on restricted Rasch models can also be found in the references e Change Output Options if necessary e choose Bootstrap Fit Statistics if your data are sparse e Start Run e Display Graphical Output How to do a Mixed Rasch Model e Open Ascii or SPSS Data Files e Recode Data if necessary e Select Variables e Choose the number of latent classes Choose the Mixed Rasch Model from the Job Definition gt Select Model submenu as shown in the following screenshot WINMIRA 2001 How to do a Mixed Rasch Model 13
11. 0 0 10 nn nn nn nn nn This will open the Output Options Dialog which allows to specify a number of tables to be generated by WINMIRA and additional information WINMIRA 2001 Choosing Output Options 27 to be written in additional files or to be appended to the dataset ou Opto IV category probabilities i gt IV item threshold parameters I item discrimination index I standard errors of item parameters Resssssansascousuenesocnosustenssenesessensesenensnenessvsenpenseansbeseasensansvaseyenees standard detailed T pattern frequencies file IV person parameter estimates IV item fit Q Index gt oro o o a o o T compact bootstrap table e Cee le As a default both the categrory probabilities and the item parameter estimates for each latent class are written in tabular form to the output file For Rasch Models and Mixed Rasch Models and also for the Rasch Classes in Hybrid models person parameter estimates and the item fit measure Q Index are printed out in addition to the other two default tables In this example add person parameters etc to datafile was chosen in addition to the default output options This will gererate additional variables for each person case in the dataset These additional WINMIRA 2001 28 Appending variables to the data variables contain information about the most probable class the personparameter estimate an
12. 668 0 358 0 368 0 793 0 366 0 380 0 925 0 377 0 397 1 066 0 392 0 419 1 219 0 411 0 450 1 390 0 437 0 495 1 587 0 473 0 564 1 824 0 527 0 686 2 133 0 614 Mean 0 974 2 595 0 782 ARRAS 560 5 asad 0 026 Stdev 0 890 97 WINMIRA 2001 98 Example Output file Reliability 0 794 Raw score Mean 14 476 Stdev 6 398 expected category frequencies and item scores Item Item s relative category label Score Stdev frequencies I 0 1 42 3 item01 1 575 1 05 0 137 0 297 0 242 item02 1 78 1 03 0 128 0 284 0 267 item03 1 69 1 04 0 142 0 323 0 240 item04 1 67 1 02 0 134 0 339 0 250 item05 1 49 1207 0 202 0 352 0 205 item06 L238 1 01 0 216 0 371 0 231 item07 1 36 1 02 0 225 0 371 0 224 item08 1 26 1 00 0 259 0 369 0 228 item09 1 07 0 97 0 326 0 387 0 175 item10 L302 0 94 0 342 0 385 0 183 Sum 14 47 threshold parameters rating scale model item threshold parameters label 1 2 3 location item0l 1 447 0 019 0 122 0 435 WINMIRA 2001 0 295 0 277 0 181 0 180 0 090 item02 1 484 0 018 item03 1 356 0 110 item04 123831 0 135 item05 1 073 03393 item06 0 918 0 548 item07 0 893 0 573 item08 0 747 0 719 item09 0 476 0 990 item10 0 392 1 074 mean threshold distances 1 466 0 103 Example Output file 0 0 0 0 085 213 238 496 651
13. Class Analysis may be applied whenever a latent i e non observable typology or classification is to be identified for a set of persons or objects which are characterized by several categorical variables Examples are e 400 persons responses to a 9 item questionnaire aimed at assessing their attitudes towards ways of environmental protection Each item is rated on a 5 point scale Two groups of persons are assumed WINMIRA 2001 What can LCA be used for 57 which attribute responsibility for environmental protection to external or to internal factors respectively e 1000 individuals rated their proximity to four political parties on a 5 point scale The question is whether the four parties can be located on a latent continuum or whether latent types of persons with different patterns of proximity have to be assumed e All patients of a psychiatric clinic were rated according to a list of clinical symptoms The aim is to analyze whether the classical psychiatric categories of mental disease can be reproduced by identifying latent types of persons with corresponding symptom patterns on the basis of these data These examples share e a relatively large number of persons observations e a relatively small number of manifest variables which can be responses in a test or questionnaire expert ratings standardized behavior observations or all kinds of observable variables like hair color sex or social status e that these var
14. Classes Select Model b Parameter constraints VARS Dutput Options 0 Bootstrap GoF Filenames Edit Defaults oo oOo oO This menu provides options to specify filenames and the destination path for the output files i e for the final estimates the pattern frequencies file The Output file can be overwritten by selecting the respective option in this submenu Otherwise the old Output file will be kept and the name WINMIRA 2001 30 Edit filenames of the new output file will be modified in the following way WINMIRA 32 pro ia File Edt Seach Dats Specification Job Definition Stat Graphs Window Help aas xaa alele eli Enter filenames Output file D MvD data daten Simula9 015 Class membership file If lt filename OUT gt is already existing the new output will be renamed to lt filename OU1 gt If lt filename OU1 gt exists then the current will be in lt filename OU2 gt and so on The pattern frequency file s name is not modified automatically as usually only one pattern frequency the one according to the finally selected model is required Please modify the pattern frequency file name manually if more than one pattern frequency file is required for additional analyses The class membership filename is now obsolete as all person related statistics can be appended directly to the datafile in the WINMIRA 32 pro version of the software WINMIRA 2001 Edit default values 3
15. Typically this happens when the accuracy criterion was not met 1 e the difference of the log likelihood between two subsequent iterations was greater than the criterion In these cases the maximum number of iteration should be increased The random start value is used in some of the random number generators for the initial response pattern splitting in the EM algorithm Please change this value whenever you think the results obtained by WINMIRA could be a local maximum mainly due to small sample size The number of start values is used for the initial search for the relatively WINMIRA 2001 Testing the fit of amodel with the Bootstrap 33 best starting point in the parameter space Increase this number if you suspect that WINMIRA 32 might get stuck in local maxima The sort output by class size checkbox allows to choose between an unsorted output file where classes are assigned by chance or by the within class restrictions like in Hybrid models or if models with unique parameter constraints imply the order of the classes and an output sorted by class sizes whenever all mixture components carry the same structure like in the ordinary LCA or the ordinary mixed Rasch model The step width for minimization is used in some of the estimation procedures please do not change this value because it is the result of a lot of numerical fine tuning experiments Testing the fit of a model with the Bootstrap Please ch
16. categories in the sense that the thresholds are ordered i e have decreasing easiness parameters fgix Whether the latter is the case can be seen from the results of a data analysis Models 1 and 5 result from models 4 and 8 by equating the threshold distances over the variables 5 feix wis Axg and Ix Ars 0 1 feix uis Ax and x Ax 0 Models 2 and 6 however are not so easily obtained since simple equating over categories would make the index x disappear from the model equation An appropriate coefficient is required generating the individual threshold location by means of the mean location wis and a distance parameter dig This coefficient is m 1 2x in order to avoid non integer WINMIRA 2001 The threshold approach in the LCA cont 53 coefficients only half the distance is parameterized so that the models are 2 feix uis m 1 2x di and for class specific distances 6 feix uis m 1 2x dig Models 3 and 7 finally are a combination of these last models and will not be derived in detail here 3 feix pig Ax m 1 2x 8i with Lisi 0 7 feix uis Axg m 1 2x dis with Lidiz 0 The normalizing condition for the dig parameters is necessary because the basic threshold distances for all variables are already defined through the Axs parameters The dis s parameterize the deviation of all distances of a particular variable from the mean distance The lat
17. estimate and the standard error of estimation of the individual parameter for all rawscores in class one In this table both maximum likelihood MLE and bayes weighted likelihood estimates WLE are shown The WLE should usually be preferred as they are less biased and give reasonable estimates even for the to extreme score groups In the case of mixture distribution models the class specific expected frequencies can not be compared to the observed frequencies as only the overall observed frequencies are known Nevertheless it can be seen from the expected frequencies for example in which class most of the higher scoring persons belong WLE estimates Mean 0 726 Stdev 1 203 Reliability 0 813 Raw score Mean 10 515 Stdev 7 672 This is the mean and the standard deviation of the WLE person parameter and the raw score WINMIRA 2001 92 Example Output file expected category frequencies and item scores Item Item s relative category label Score Stdev frequencies Pe De ee a Ul item01 0 54 0 84 0 646 0 216 0 095 0 043 item02 0 65 0 92 0 602 0 205 0 135 0 058 item03 0 75 0 99 0 561 0 213 0 140 0 087 item04 0 82 1 02 0 525 0 233 0 140 0 102 item05 0 93 1 07 0 478 0 238 OL LST 0 127 item06 1 13 14 12 0 404 0 229 0 197 0 170 item07 1 24 715 0 367 0 224 0 210 0 199 item08 1 533 1 18 0 348 0 212 0 205 0 235 item09 1 47 Tak 7 0 290 0 222 0 216 0
18. of the ability parameters Q varies between 0 and 1 where 0 indicates perfect Guttman pattern fit and 1 indicates perfect misfit anti Guttman pattern or deviance from the model A value Q 0 5 indicates independence of the trait and the item 1 e random response behavior Rost amp von Davier 1994 presented another standardization of the item Q index with zero mean and unit variance and which can be assumed to be asymptotically normal In WINMIRA both the Q index as well as its asymptotically normal standardization are listed in a table in the output file if the respective output option is activated WINMIRA 2001 78 WINMIRA 2001 The Q Index The Hybrid Model The Hybrid Model Yamamoto 1989 assumes that the data can be described by a mixture of an IRT model like the Rasch model and the Latent Class Analysis This implies that each observed response pattern either stems from a latent subpopulation where the IRT model holds or that the response pattern can be fitted by one of the latent classes in the LCA part of the model Hybrid mixtures of IRT models and Latent Class models can be written as P X xv I Sg 1 mg PIRT X xl Ovg Ce 1 ne PLCAX xlc where Sg 1 ng YCc 1 me 1 i e the sum of all class sizes is one The first mixture sum stands for the IRT mixture components and the second sum stands for the Latent Class type mixture components In WINMIRA Hybrid models can be estimated for polytomous data
19. the subsequent output for class 2 and continued for latent class 3 as that class is assumed to follow a different model namely the traditional latent class model Expected Score Frequencies and Personparameters Raw Expected MLE SE MLE WLE SE WLE score freq estimate estimate a pr a ee 0 2 08 EERE RES ROPER 4 101 1 474 1 0 01 3 321 1 029 2 940 0 871 2 2 87 2 570 0 746 2 368 0 688 3 0 01 2 110 0 623 1 972 0 592 4 12 25 1 769 0 549 1 667 0 530 5 18 72 1 495 0 500 1 415 0 486 6 25 30 1 264 0 463 1 198 0 453 WINMIRA 2001 21229529 L 5063 8 37 70 0 884 9 24 28 0 720 10 32 38 0 570 11 27 25 0 429 12 44 25 0 295 13 21 60 0 167 14 17 67 0 043 15 24 73 0 078 16 21 58 0 198 17 14 06 0 318 18 25 60 0 438 19 15 99 0 561 20 30 30 0 688 21 29371 0 820 22 42 05 0 960 23 T5351 1 171 24 19 22 1 277 25 9 10 1 465 26 13 68 1 687 27 6 08 1 964 28 03 11 2 347 29 0 01 3 001 30 0 28 Kk KK WLE estimates Example Output file 0 435 1 008 0 428 0 413 0 838 0 408 0 395 0 682 0 392 0 381 0 538 0 379 0 370 0 403 0 368 0 361 0 275 0 360 04399 0 152 0 354 03 90 0 033 0 350 0 347 0 084 0 347 0 346 0 199 0 346 0 346 0 314 0 346 0 349 0 429 0 348 0 353 0 547 0 352 0 359 0
20. u the estimate of u in iteration step t The number of iterations is restricted to 10 within each M step for the Latent Class models and 1 for the Mixed Rasch models The parameter estimates of the previous M step serve as start values for succeeding M steps WINMIRA 2001 The LCA for ordinal variables WINMIRA 32 is capable of estimating the parameters of 8 different latent class models for manifest variables with ordered categories Four of these models assume class specific and four models assume class independent distances between response categories The models in each of the two groups result from the assumption of e ordered categories only e equidistant categories for each variable equal distances between categories for all variables e scaled distances between categories but different dispersions for each variable WINMIRA 32 provides e parameter estimates for all models e response probabilities e various goodness of fit statistics Likelihood Ratio Cressie Read Pearson X and Freeman Tukey e capabilities for performing bootstrap or monte carlo tests for these statistics WINMIRA 2001 48 The threshold approach in the ordinal LCA e information criteria AIC BIC CAIC e output of most likely class membership for each person The threshold approach in the ordinal LCA The 8 LCA models covered by LACORD LAtent Class analysis for ORDinal variables which are included in WINMIRA 32 can b
21. 001 66 The Equidistance Model The Equidistance Model Oixe Mig m 1 2x Sig In the Equidistance Model formerly called Dispersion Model all threshold distances are assumed to be constant within each item as indicated by the missing threshold index x for the dispersion parameters 6 ig The figure below shows an example where the threshold distance is 2 0 units for the first and 3 5 units for the second item WINMIRA 2001 The Equidistance Model 6 WINMIRA 2001 68 The Dispersion Model The Dispersion Model Oixe Hig Axg m 1 2x 8ig In the Dispersion Model formerly called Successive Interval Model there are both equidistance parameters dig and threshold parameters Axs The A parameters define basic distances for the thresholds which can be increased or decreased by the 6 parameters negative values 6 lt 0 decrease the distances positive values gt 0 increase the distances In this example the second threshold distance is larger than the first one within both items WINMIRA 2001 The Ordinal Partial Credit Model 69 more The Ordinal Partial Credit Model aixe without restrictions In the Ordinal or Partial Credit Model there is one parameter o ixg for each threshold of each item without any restriction except the inevitable normalizing conditions WINMIRA 2001 70 Person Parameters Person Parameters The previous subsection presented the model equation of the mix
22. 1 If WINMIRA 32 refuses to start the analysis it is good advice to check whether the destination path is existing especially if data and definition files for running WINMIRA 32 are moved from one computer to another Edit default values Please select the menu entry Job Definition gt Edit Defaults as shown in the screenshot below 8 iecification Job Definition Start Graphs Windo N of Classes Select Model gt Parameter constraints VARS Output Options Bootstrap GoF Filenames Edit Defaults co JE O O0 0 This will open the edit defaults dialog box as shown below Please be careful when modifying any of the values in this dialog box as some very important basic parameters of WINMIRA 32 can be modified here Please examine your output files carefully if you have changed any of these values WINMIRA 2001 32 Edit default values WINMIRA 32 pro File Edit Search Dats Specification Job Definiion Stat Grapl Edit Defaults E Max N of iterations 300 at Accuracy criterion fo 0005 oO Help 4 2 OEIS 0 0 Step Width indicator 650 3 z 0 0 4 Random start value 4321 0 1 5 0 0 6 N of start values 5 o o T Sort output by class size 0 D RB 4 n 4 e The maximum number of iterations limits the process of parameter estimation by default to a maximum of 500 iterations Please examine the output file to find out whether this number was reached
23. 189206 466 1072625 593 13213 9483 13 14870 321 24462 530 19184 418 183339 043 1064587 444 13102 2003 14 14868 631 24076 854 18416 446 180624 497 1193808 517 12879 2104 15 14817 184 24093 470 18552 571 178027 561 1050349 133 12828 3274 16 14850 098 24197 824 18695 452 184627 623 1177492 758 12930 2590 17 14857 015 24029 929 18345 826 173529 495 1062309 443 12879 3370 18 14865 694 24052 424 18373 459 182297 386 1273164 429 12868 3238 19 14846 662 23975 028 18256 733 179065 513 1209296 487 12802 8427 20 14832 243 23909 898 18155 309 166976 243 1020880 850 12800 8061 2 0 668 1 018 0 593 0 4293 P X gt Z 0 252 0 154 0 724 0 6661 Mean 19119 452 184623 133 1088481 165 13061 0666 Stdev 553 577 6255 476 65621 729 166 4833 p values emp PDF 0 350 0 100 0 850 0 6000 For very sparse data tables and small or moderate sample sizes the bootstrap procedure should be used only for the Cressie Read and the Pearson Chi Square statistics v Davier 1996 None of the four statistics in the table above rejects the model in this example But nevertheless both the FT and the Likelihood Ratio statistics can not be recommended as a very large sample size seems to be necessary to make the bootstrap reliable for these statistics WINMIRA 2001 108 Example Output file WINMIRA 2001 References Andrich D A rating formulation for ordered res
24. 5272 item10 1 64 6 0 22 39 0 198 0 244 0 319 Sum 10 50 The expected category frequencies show the descriptive characteristics of the items in each class Item 1 and item 2 for example are very difficult in this class because about 80 to 90 percent of the individuals choose only the lowest two categories 0 and 1 These relative frequencies are overall or mean values because in Rasch Model Classes these relative frequencies depend on the distribution of the individual parameter in the class In the ordinal Rasch model there is a strictly positive relationship between WINMIRA 2001 Example Output file 93 the probability of choice of the higher of two adjacent categories and the individual parameter Subjects with a high individual parameter have higher probabilities for the upper categories than in the table above Accordingly a subject with a low parameter will have higher probabilities for the lower categories threshold parameters rating scale model item threshold parameters label 1 2 3 location item01 0 468 0 861 1 419 0 916 item02 0 229 0 622 1 180 0 677 item03 0 035 0 428 0 985 0 483 item04 0 086 0 307 0 865 0 362 item05 0 280 0 113 0 670 0 167 item06 0 609 0 216 0 341 0 162 item07 SOA1 9 0 386 OTL 0 332 item08 O 9177 0 524 0 034 0 469 item09 1 134 0 741 0 184 0 687 item10 1 405 1 012 0 455 0 957 mean threshold dis
25. 707 The table above shows some descriptives of this assignment procedure The expected class size is printed in the second column the mean of the assignment probability maximum posterior probability is listed in the third column The table is completed by a list of mean posterior probabilities for all classes given that the current class row number has maximum posterior probability Goodness of fit statistics WINMIRA 2001 104 estimated saturated model model Log Likelihood Number of parameters geom mean likelihood Information Criteria ndex Power emp Likelihood ratio Cressie Read Pearson Chisquare Freeman Tukey Chi 2 Degrees of freedom To evaluate the fit of a specified model the goodness of fit table has to be examined carefully In the case of many items with more than 2 response categories there are a lot of possible response patterns most of which are not observed Data with many zero frequencies are referred to as sparse WINMIRA 2001 19489 21 Example Output file 24568 35 14823 74 112 1048575 0 29293527 0 47672437 ndex 49360 70 2126797 49 ndex 49988 06 8000287 94 50100 06 9048862 94 Divergence GoF statistics value chi square p value p 1 0000 190988 37 p 1 0000 1049539 31 p 0 2287 12989 60 p 1 0000 1048463 Example Output file 105 data In case of sparse data the trad
26. Help N of Classes Select Model Latent class analysis Parameter constraints Rasch model Qutput Options __ Mixed Rasch model _ Bootstrap GoF v Hybrid model Filenames Ta Edit Defaults 00 0 0 0 0 0 0 00 10 0 0 00 0 0 This will open the model specification dialog for the Hybrid Model as depicted below sation Job Brefinition Start Graphe Window Help ta 2 i Y Fr Choose Hybrid Model 1 0 Rasch model Latent Class model 0 1 0 0 Class 1 Class 3 1 0 J 0 gt gt 1 0 0 1 0 amp 1 1 0 0 1 0 1 n Restrictions Restrictions 15 Please click on the Class n lines for choosing one of the available Rasch or Latent Class Submodels A doubleclick on the line Class 2 as shown above will open the following dialog and lets you choose between any of the available models like it can be done for the Rasch Model or the Mixed WINMIRA 2001 16 How to analyse data with the Hybrid Model Rasch Model MIW Galion Job Definition Stat Graphe Window Help 4 Choose a Mixed Rasch model Rating Scale Model Equidistance Model C Dispersion Model Ordinal Partial Credit Model IV Smooth score distribution In contrast to the Mixed Rasch Model or the Latent Class Analysis you can specify a different model within each of the classes so that there are no similarity restrictions between the classes despite the fact that all models assume a logistic distribu
27. Latent Class Analysis the Mixed Rasch Model and the Hybrid Model WINMIRA 2001 2 WINMIRA user manual WINMIRA 2001 The WINMIRA THANK YOU PAGE I have maintained this software for almost 10 years now and I would like to thank the following people for their support not only during the revision of the current version of the software and the manual J rgen Rost amp Kentaro Yamamoto and Rolf Langeheine Ivo Molenaar Thorsten Meiser Knud Sievers Olaf K ller Claus Carstensen and all other colleagues and users of the software that came up will valuable hints and recommendations Many thanks also to my family Alina amp Thomas von Davier for their patience and support Thanks also to the component writers of TRegisterApp TxyGraph HugeArray and to Borland Inprise and SPSS WINMIRA 2001 4 The WINMIRA THANK YOU PAGE WINMIRA 2001 Preface The development of WINMIRA was aimed at producing an easy to use software tool for categorical data analysis with the a variety of models including the Latent Class Analysis and the Mixed Rasch Model Even though the user interface is more or less self explaining there will still remain difficulties in using this software Some of the models which can be estimated with WINMIRA are comparably complex and therefore quite a few selections have to be made As this software is thought to be a scientific tool for data analysis I tried not to restrict the use of the software
28. ach in the LCA cont 51 The threshold approach in the LCA cont If the class specific response probabilities equation 11 are substituted for the category probabilities in the general latent class model Pvix y Gg 1 Peix where ns is the relative size of class g the model structure underlying all 8 latent class models of WINMIRA 32 is obtained They only differ in their specification of fgix which is outlined in the following If there should be no restriction at all fgix itself may be considered a model parameter and nothing more than a reparameterization of the polytomous LC model is obtained In the ordinal model of the Latent Class module of WINMIRA 32 however another reparameterization is used in order to make the results comparable with the other models 8 fgix wig Aixg and Lx Aixg 0 In this specification Lig can be interpreted as the mean location of all thresholds of variable i in class g and because of the normalizing condition the Aixg parameterize the deviation of threshold x from that WINMIRA 2001 52 The threshold approach in the LCA cont mean Model 4 follows from equating the threshold distances not their mean location over classes 4 feix uis Aix and Xx Aix 0 Both models 8 and 4 assume an order of the categories insofar as for model derivation it must be known which categories are adjacent see the first formula in this chapter Neither of the models require ordered
29. also possible Please make sure that class sizes have to add up to 1 000 Please click through the classes by means of the Class up down arrow on the top left of the window panel to see all constraints Constraints can be made within and across classes WINMIRA 2001 The rationale of Mixture Distribution Models Mixture distribution models MDM relax the assumption that the observed data were drawn from a homogeneous population It is rather assumed that the sample is drawn from an unknown mixture of distributions Everitt amp Hand 1981 which are reffered to as latent classes in this context Mixture distribution models are more flexible as compared to classical statistical modeling where we usually apply a statistical model to a set of data and assume that the model is valid for the data and that all model parameters e g factor loadings path coefficients item parameters are the same for all individuals of the population Discrete MDM in contrast are based on the idea that different sets of model parameters are valid for different subpopulations In the case of latent subpopulations their number is not known but must be identified when the model is applied These subpopulations are solely defined by their property of being homogeneous in the sense that a particular model holds for this latent class In particular latent classes are not defined by manifest variables like gender age or socio economic status where the partition is don
30. ass specific response probabilities carrying the same letter CaSE SenSitiVE will be set to the same numerical value in each iteration of the estimation algorithm The example shows that all item locations are equal all cells show an a in class 3 In WINMIRA 2001 Class size constraints 41 addition the threshold distances are also carrying constraints Items VAR9 to VAR13 are constraints analog to the Rating scale model whereas items VAR4 to VAR8 carry constraints like in the equidistance model This shows that constraining parameters can be more flexible than using the models hard wired in WINMIRA Nevertheless be aware that using constraints can also mean that one specifies a model that cannot be estimated or that at least will slow down or disturb the convergence of the algorithm Please click through the classes by means of the Class up down arrow on the top left of the window panel to see all constraints Constraints can be made within and across classes Class size constraints The types of parameter constraints probability constraints and logistic item and threshold parameter constraints cannot be mixed Nevertheless both types allow constraining the clas sizes to constants like it is shown in the following picture WINMIRA 2001 42 Class size constraints Class E Size p 200 Level probabilities thresho In this example the size of class 3 was fixed at 20 or 0 200 Equality constraints are
31. be computed by means of these relative category frequencies by simple multiplication threshold parameters ordinal partial credit model item threshold parameters label 1 2 3 location ee iteml 0 663 0 2073 0 126 0 239 item2 0 151 0 180 0 151 0 060 item3 0 038 0 207 0 177 0 003 item4 0 294 0 181 0 044 0 052 item 0 011 0 096 0 177 0 023 item6 0 320 0 034 0 257 0 009 item 0 133 0 219 0 064 0 007 item8 0 187 0 167 0 187 0 069 item9 0 244 0 110 0 089 0 015 item10 0 236 0 207 0 058 0 010 The threshold parameters in class 3 are listed in the table above As in classes 1 and 2 before the last column is an overall difficulty parameter It can be seen that as compared to the class 1 these parameters do not vary a WINMIRA 2001 Example Output file 103 lot This holds also for the threshold parameters as the category frequencies are more or less equally distributed for all items The class specific output ends here The following part contains some general information on whether and where the class membership information has been saved and overall goodness of fit measures statistics of expected class membership exp mean class size prob 1 2 3 ee rg men en Te re 1 0 521 0 865 0 865 0 051 0 084 2 0 289 0 770 0 066 0 770 0 164 3 0 190 0 707 0 126 0 167 0
32. cial feature of this parameterization is the relative invariance of shape WINMIRA 2001 74 Category Characteristic Curves of the score distribution w r t how many score frequencies have to be fitted In the figure above the same parameters were used to smooth 14 and 88 raw score frequencies in each of the four diagrams Category Characteristic Curves The socalled category characteristic curve CCC can be used to visualize the relationship between item parameters and response probabilities The x axis represents the latent dimension and the y axis shows the response probability for each category x 0 m The intersections of the response probability curves are given by the item parameters in the de cumulated notation i e the threshold parameters In the example below these thresholds take the values 1 1 and 4 5 respectively WINMIRA 2001 The Q Index 75 Given a person with parameter 6 the response probability for this person can be directly taken from the CCC as both the item threshold parameters and person parameters are located on the same latent dimension proceed The Q Index The item Q index Rost and von Davier 1994 is an item fit index which makes use of the statistical properties of Rasch models i e parameter separability and conditional inference The item Q includes no assumption about the scale level of the response variable Instead it is based on the WINMIRA 2001 76 The Q Inde
33. d person fit measures Appending variables to the data When append person parameters etc to datafile was chosen in the Output Options menu a number of variables will be added to the datafile that was used for estimating the model The picture below shows the realisations of these variables for the cases 760 to 764 in our example data E WINMIRA 32 pro SPSS lel Es lg Eje Edit Search Data Specification Job Definition Start Graphs Window Hep x aael XE aes elf reo s 0 5755 03592 1 0664 1 2425 09446 1 0000 0 5755 03592 04565 04181 09342 1 0000 0 9258 O3773 0 7235 0 9811 0 5165 2 0000 0 9805 0 3957 09188 0 9620 0 9761 1 0000 1 2355 1 7002 0 7220 3 0000 M O wo Ww oO wn N The variable PERSPAR contains the trait estimate STDERR contains the standard error of estimation for the sufficient statistic MAXPT contains the maximum of the posterior probabilities of being member of one of the latent classes given the observed response vector of the respective person MAXCLASS is the latent class carrying the maximum WINMIRA 2001 Edit filenames 29 probability given the response vector OLDFIT is a heavily skewed person fit index NEWFIT contains an almost normally distributed Person Fit Index Edit filenames Please select the menu entry Job Definition gt Edit Defaults as shown in the screenshot below _ ification Job Definition Start Graphs Window Hli N of
34. e by a manifest observable moderator variable and parameter estimation can be performed for each manifest group Hence the aim of MDM is twofold to unmix the data into homogeneous subpopulations and to estimate the parameters for each subpopulation separately The general structure of WINMIRA 2001 44 The rationale of Mixture Distribution Models discrete MDM is P X x VGe 1 me P X xl Oe 1 where x x1 xK is a vector valued observation on k variables items and q g is the group class specific vector valued parameter of the conditional distribution in class g It is assumed that the overall probability of an observation x is a weighted sum of conditional probabilities within these subpopulations The weights 7 s are the mixing proportions which are often referred to as class sizes and represent the relative sizes of the subpopulations The family of discrete MDM can be divided into at least two groups of models Firstly MDM which assume the same type of model in all subpopulations but with different sets of model parameters these models are commonly referred to as MDM Secondly discrete MDM which can be defined with a different type of model in each subpopulation WINMIRA 32 can handle both types of MDM namely any possible combination of the class specific models available in the program can be specified and analyzed with WINMIRA 32 as long as there are no identification problems The section on What can LCA be u
35. e derived from a very simple assumption regarding the ratio of response probabilities of each two adjacent categories pvix pvi amp D for x 1 m where pvix is the probability of person v for scoring in category x on item 1 As in log linear models it is assumed that the logarithm of this ratio is a linear function of some parameters depending on the variable i the category x and the latent class g to which a person v belongs Without further specification of that linear function fgix the assumption may be written as Inc pvix J pvi D feix for x in 1 m 10 WINMIRA 2001 The threshold approach in the ordinal LCA 49 This simple approach leads to a very handy model structure because it can be derived without further assumptions that the category or response probabilities in a latent class g are paix exp L s Ofeis 0exp L s Ofgis 11 Although this is a very straightforward model derivation the parameter interpretation remains somewhat diffuse All that is known about the linear components of fsix which have not been specified yet is that the response probability of a category x in relation to its preceding category increases monotonously with fsix A very convincing interpretation of the model parameters is found with the transformation of model assumption equation 10 above into its equivalent form prix prix pvio D exp feis L exp faix 12 The ra
36. e right hand side of the dialog In order to start recoding click on the OK button The data will be then recoded into the same variables A good idea is to rename the variables that have been recoded This can be done with the edit variable dialog How to edit variables with WINMIRA by doubleclicking on the first row in the data file the fixed row where the variable labels are the variable menu is activated Delete Variable Edit Variable Please choose Delete Variable if you need to remove variables from the dataset or click on Edit Variable in order to open the following dialog WINMIRA 2001 22 In the example dialog above the variables name is VARS and the label is mein kleiner gr ner Kaktus my small green succulent plant You may change the variable s name label and output format i e it s printed width and decimals Click on the menu entry Data specification gt Select Variables as seen below Select variables WINMIRA 32 pro le Ey y mm o TE Name VAR5 G numenc ter Width 2 Decimals 0 vara z z vars Label mein kleiner gr ner Kaktusl cone l oo oo 2 Select variables Tie Social Data ee Job Be WINMIRA 2001 Select variables 23 Indicate which items of the current dataset are to be included in the analysis Please do not choose all items think of which items represent the construct you want to measure and select t
37. ed Rasch model in the conditional notation Conditional maximum likelihood WINMIRA 2001 Person Parameters 71 estimation CML is used in WINMIRA in order to avoid estimating biased person and item parameters simultaneously This ensures that consistent item parameters are obtained which can be used to obtain person parameter estimates Estimation of person parameters is carried out by an UML procedure after the CML estimation of the item parameters is completed Maximum likelihood estimates MLE as well as weighted likelihood estimates WLE see Warm 1989 can be computed with WINMIRA Warm s WLE estimates have as compared to the MLE estimates two main advantages First their bias is smaller Warm 1989 Hoijtink amp Boomsma 1995 and second they produce reasonable estimates even for the two extreme response patterns 1 e for the patterns with zero and maximum score The estimation of person parameters is optional and can be chosen with the corresponding option in the Output Options menu of WINMIRA If compute person parameters was chosen a table with MLE and WLE estimates along with the corresponding standard errors will be included in the output file for each Rasch model class The person parameter estimates are also printed in a separate file if append person parameters to datafile was selected in the options menu This file can be used in subsequent analyses WINMIRA 2001 72 Latent Score Distributions Late
38. es 43 Parameter estimation m MD Mi une A 45 The LCA for ordinal variables o00 000000s00sonsssssosssssnnnsssnnnnnsnsnnnnnnnnee 47 The threshold approach in the ordinal LCA see 48 The threshold approach in the LCA cont enennee 51 WINMIRA 2001 Table of Contents What does the LCA part of the program 55 What Cam LCA be used Tor ann a eg 56 The Mixed Rasch Model nusse0nisns sein 61 The dichotomous Mixed Rasch model uu une 62 The polytomous Mixed Rasch model 63 he Rating Scale Modelirne EAE EE 65 The Equidistance Model RR 66 The Dispersion Models un 68 The Ordinal Partial Credit Model eo 69 Person Parameter ansehen 70 Latent Score Distiibutons sat 72 Category Characteristic Civesaie een 74 CFO Tin Ci seele mark 75 The Hybrid Model uu0unssiussnsseninamuish 79 Using the output of WINMIRA ssssssssssssssssnsssssssssnunssnssnssssssnsnnsnssnnunnse 81 WINMIRA 2001 Table of Contents Primes SUNS OUP WG naar estere 81 Display Graphical Output men lee 82 Example Oup tfle une aklen 85 References nase 109 WINMIRA 2001 WINMIRA user manual by Matthias von Davier e mail winmira von davier de WINMIRA is a software for analyses with a variety of discrete mixture distribution models for dichotomous and polytomous categorical data This software can be used for the Rasch Model the
39. he items accordingly in Start Grap sch D pgp x Select items lx A SLALA maLa le After clicking the ok button the selected items will be analysed in order to obtain the number of categories and the missing values for each item and in addition whether all categories have at least been chosen once If the category codes do not start from 0 zero as it is required by the algorithms of WINMIRA the items will be autorecoded i e the minimum code say 1 one instead of 0 zero will be subtracted from all observed values Below the view selected items dialog is depicted This dialog shows the selected variables together with their number of categories as extracted from the data by the testing algorithm together WINMIRA 2001 24 Select variables with the minimum and maximum code If the minimum code is larger than 0 zero autorecode will be enabled in the dialog automatically E WINMIRA 32 pro Fite Edt Search Data ia Job Definition Stat Graj el ele ele ee ee ooooooooo6o WO WO www www Case 1 2 z 4 6 E A 4 i Cast T autorecode 1 2 3 data 1 Please click the OK button in order to confirm the selection This will enable WINMIRA 32 s other menus that have been disabled before Now the model selection and other more detailed specifications of the model to be estimated are made available as the data specification is completed by accept
40. iables are discrete i e they have a limited number of categories and each individual has one and only one value or category on each variable and WINMIRA 2001 58 What can LCA be used for e that a latent classification or typology of the individuals is aimed at which is latent because no observable or manifest variable can produce this classification It is probabilistic because probabilities of membership in the latent classes instead of deterministic assignments are obtained for each individual The least restrictive model in the program is the unrestricted latent class model which can be applied to unordered categories nominal scale variables or be used to check if categories are ordered If the categories of all variables are defined in the same way and hence all variables have the same number of categories eight different models can be computed and checked for their fit to the data see below The program s use lies in the identification of a latent classification for the individuals So far it is only a general description of latent class analysis and its results One property of the Latent Class module of WINMIRA 32 is its applicability to ordered categories However it is not known whether the distances between the categories are the same between all categories whether the distances have the same size for all variables nor whether they are the same for all groups of persons A systematic combination of these
41. ile and in the graphical display of the item parameters are fgix parameters as introduced in the formulae above What does the LCA part of the program Parameter estimation and related computations can only be made for a fixed number of latent classes since the number of classes in LCA is not a model parameter but an a priori model assumption Starting values for parameter estimation are generated by a random number generator The program can estimate the parameters for the 8 different latent class models for ordinal variables by using an extended EM algorithm The extension consists of a short Newton algorithm within each M step for maximizing the likelihood function Some models cannot be applied if either the number of categories is too small less than 4 or if the manifest variables have different numbers of WINMIRA 2001 56 What can LCA be used for categories In the latter case only models 4 and 8 are applied assuming ordered categories without further restrictions In case of only two categories for all variables the first model is sufficient to reproduce the parameters of the unrestricted dichotomous LCA because two categories have only one threshold and no threshold distance In case of 3 categories models 1 2 5 and 6 can be estimated because 3 categories have 2 thresholds and one distance Model 6 is equivalent to the unrestricted LCA in this Case What can LCA be used for The Latent
42. in order to make it even more easy to use This lack of control puts full responsibility in the hand of the user It is especially important to be sure about which model has been specified and estimated as the lack of input restrictions makes it even more important to examine the results of the analyses very carefully For example please consider a model with to many classes and only a few items The algorithm may still run though the final solution is not identified and there may be some classes with a class size near to zero In addition some parameters may diverge to minus or plus infinity so that some categories in the respective latent class have expected frequencies close to zero Perhaps most of these cases could be prevented by the software but that would mean to have a program with many warnings in the output and nasty message boxes during runtime WINMIRA 2001 6 Preface Therefore it can not be guaranteed that the software fulfills all requirements of the users see the license agreement and responsibility has to be taken by the users to interpret and examine the output carefully Support in using the software can be given of course e g by supplying references or helping to interpret some pieces of output Problems with the software and ideas for improving it should be reported directly to my e mail address winmira von davier de Of course not everything can be taken into account but in the past I ve been able to i
43. ing the selection in this dialog WINMIRA 2001 Choose Number of Classes 25 Choose Number of Classes Please choose the Job Definition gt N of Classes menu entry like shown in the figure below Select Model gt Parameter constraints Output Options Bootstrap GoF Filenames Edit Defaults This will open the Number of Classes Dialog which allows to specify a upper and lower bounds for the number of classes to be computed during the computation with WINMIRA hon Job Definition Stait Graphe Window Help jelel elf Nunber of Classes az From tn of Classes p 4 To N of Classes B Help Cancel IE nn nn nn nn nn nr WINMIRA 2001 26 Choosing Output Options In this example only one model with three latent classes will be computed Increasing the number in the To N of Classes field to 5 will make WINMIRA 32 to compute three models with three four and five latent classes Decreasing the From N of Classes field to 2 will make WINMIRA to compute both the two and the three class solution for the specified model Choosing Output Options Please choose the Job Definition gt Output Options menu entry like shown in the figure below Specification Job Definition Start Graphs Windor tf tal i N of Classes Select Model r Parameter constraints gt Iva Qutput Options Bootstrap GoF Filenames 00 Edit Defaults 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
44. is In S A Stoufller et al Measurement and Prediction Princeton Princeton University Press 1950 Lazarsfeld P F amp Henry N W Latent Structure Analysis New York Houghton Mifflin 1968 Masters G A Rasch model for partial credit scoring Psychometrika 1982 47 147 174 McLachlan G J amp Basford K E Mixture Models New York Marcel Dekker 1988 Meehl P E Factors and Taxa Traits and Types Differences of Degree and Differences in Kind Journal of Personality 60 1 117 174 1992 WINMIRA 2001 112 References Mooney C Z amp Duval R D Bootstrapping A Nonparametric Approach to Statistical Inference Sage Series Quantitative Applications in the Social Sciences Vol 95 1993 Rasch G Probabilitic Models for some Intelligence and Attainment Tests Copenhagen Denmarks Paedagogiske Institut 1960 Read N A C amp Cressie T R C Goodness of Fit Statistics for Discrete Multivariate Data Springer Series in Statistics New York Springer 1988 Rost J Rating Scale Analysis with Latent Class Models Psychometrika 53 327 348 1988 Rost J Measuring attitudes with a threshold model drawing on a traditional skaling concept Applied Psych Measurement 1988 12 397 409 Rost J Rasch models in latent classes An integration of two approaches to item analysis Applied Psychol Measurement 1990 14 271 282 Rost J A logistic mixture distribution model fo
45. it Statistics which is implemented in WINMIRA 32 see section 2 9 In the example above 20 bootstrap samples have been simulated At the bottom of the table the empirical p values for these 20 samples are listed It can be seen that between 2 and 17 bootstrap samples showed a higher Goodness of Fit value than the real data Therefore the assumption that the data were generated by the specified HYBRID model is not falsified Parametric Bootstrap estimates for Goodness of Fit No Satlik LogLik LR CressieRead Pearson X 2 FT 1 14836 781 24541 457 19409 352 190450 757 1069253 060 12984 9860 2 14883 365 24563 595 19360 461 188222 734 1073851 379 13063 5891 3 14898 684 24566 254 19335 140 183807 288 1048087 972 13191 3068 4 14940 153 24682 554 19484 801 187564 883 1069896 114 13257 5533 5 14927 206 24750 710 19647 008 188408 323 1059873 202 13285 5702 6 14891 243 24556 653 19330 819 185447 350 1051618 126 13143 8871 7 14888 905 24668 821 19559 833 189667 564 1060352 931 13162 2576 8 14869 646 24694 736 19650 179 191346 730 1054225 383 13145 5455 9 14864 434 24710 431 19691 993 189267 848 1041317 125 13196 0404 10 14885 997 24752 450 19732 907 191276 684 1066221 749 13251 3002 WINMIRA 2001 Example Output file 107 11 14909 753 24742 399 19665 291 189308 678 1050411 606 13234 0416 12 14903 741 24674 261 19541 040
46. itional goodness of fit significance tests in the table Likelihood ratio up to Neyman Chisquare cannot be used compare v Davier 1997 WARNING Number of cells is larger than number of different patterns obs patterns cells 0 001749038696289060 number of ro cells 1046742 WARNING Number of cells is larger than number of subjects subjects cells 0 001908302307128910 The data might be very sparse please do not use the chi square p value approximation for the Power Divergenc Goodness of Fit Statistics Consider to use the parametric bootstrap procedure instead In addition several start values should be used s defaults menu in order to examine the occurance of local likelihood maxima If the data table is sparse some researchers rely on socalled information criteria see above to compare different models Information criteria IC are based on the log likelihood and the number of estimated model parameters The number of parameters is included as a penalty term so that more parsimonious models are preferred These IC s for instance the BIC are compared for different models which were estimated for the same data and the model with the smallest IC is chosen WINMIRA 2001 106 Example Output file A better way than evaluating the models by means of information criteria is to use the parametric bootstrap procedure for the evaluation of Goodness of F
47. mprove the software mainly because there have been users of previous versions sending very helpful comments WINMIRA 2001 How to use WINMIRA E WINMIRA 32 pro Ele Est Search Dats Specification Job Definition Stat Graphe Window Help il S Xe 1 N sll et 2 7 D 5215 4 3626 E 5412 Si 0 5937 0 2696 04565 0 15412 0 1579 A207 W threshold 1 threshold 2 thresholc item Parameters in Class 1 with size 0 47969 E OUTPUT This HTML coursework provides cookbook examples of how to use the software WINMIRA 32 It will not tell much about the mathematical background of the models implemented in WINMIRA Please refer to the references part of this hypertext in order to find out more about the math These pages are intended to provide a quick reference for enabling novice users to start using the software without a human tutor Contents WINMIRA 2001 8 How to do a Latent Class Analysis How to analyse data with e the Latent Class Analysis e the Rasch Model e the Mixed Rasch Model e the Hybrid Model WINMIRA related WWW Links e Winmira Homepage e ProGAMMA e ASC How to do a Latent Class Analysis e Open Ascii or SPSS Data Files e Recode Data if necessary e Select Variables e Choose the number of latent classes WINMIRA 2001 How to do a Latent Class Analysis Choose the Latent class analysis from the Job Definition gt Select Model submenu as shown in the following screenshot
48. nt Score Distributions The latent score distributions determine the score probabilities in each Rasch type latent class These score probabilities are necessary in the conditional maximum likelihood estimation for conditioning out the person parameters in order to obtain unbiased item parameter estimates The latent score distributions can be fully parameterized i e one parameter for each score in each latent class or estimated by assuming a two parameter model According to this restricted model the score distribution within each class g is parameterized by a location parameter T s and a dispersion parameter 6 g assuming that the following relationship holds 1 olan E rk Se een r F k r Mi gl aE e EEN 5 9 where m 1 is the number of response categories k is the number of items r mk Ar mr r mk normalizing coefficient The parameters of this distribution have the mathematical property of a WINMIRA 2001 Latent Score Distributions 73 location parameter T and a dispersion parameter The following figure shows the shape of the score distribution as a function of 4 different sets of parameters It can be seen that the model is capable of approximating very different shapes like symmetric unimodal extremely dislocated and u shaped distributions Figure la Figure 1b Tau 0 Delta 3 Tau 6 Delta 6 Figure 1c Figure Id Tau 9 Delta 2 Tau 0 Delta 3 A spe
49. on parameter plot for the Maximum likelihood and the Warm person parameter WINMIRA 2001 Example Output file 85 Person Parameter Plot Class fh y S gt z M Freg M WLE V MLE Save Graph as Metafie frequency Person Parameters in Class 1 with size 0 49311 s 80 3 p 70 2 a 60 gt 0 a gt r z p 40 at a y 30 4 a 3 20 4 10 0 0 12345 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Rawscore Example Output file A Hybrid model combining ordinal Rasch models and Latent Class models Below an output file as produced by WINMIRA 32 is commented in order to give a guideline for reading the results of an analysis To learn more about discrete mixture distribution models a book like Lehrbuch der Testtheorie by J Rost is highly recommended To learn more about using the program WINMIRA 32 please refer to the on line manual or the user manual WINMIRA 2001 86 Example Output file WINMIRA 32 beta v0 97 c 1998 1999 by Matthias von Davier IPN institute for science education Olshausenstrasse 62 24098 Kiel Germany email vdavier ipn uni kiel de or rost ipn uni kiel de date of analysis 24 11 98 time 12 05 01 Filenames data simula9x dat output simula9x out member simula9x mem patterns simula9x pat number of persons 2001 number of items 10 numbe
50. oose the Job Definition gt Bootstrap GoF menu entry like shown in the figure below WINMIRA 2001 34 Testing the fit of amodel with the Bootstrap u Z scification Job Definition Start Graphs Window N of Classes Select Model gt Parameter constraints Output Options Bootstrap GoF Filenames Edit Defaults This will open the Bootstrap Dialog which allows to specify the number of simulations and other paramters in order to perform the parametric bootstrap test for Chisquared Goodness of Fit statistics Il Bootstrapping options ak Bootstrap ML options Accuracy criterion 0 0007500 Max N of iterations 75 Person parameter options Use WLE estimates C Use MLE estimates e ee The bootstrap will be performed for four different Chisquared Goodness of Fit statistics namely the Pearson X the Cressie Read statistic the Likelihood Ratio and the Freeman Tukey statistic PLEASE WINMIRA 2001 Start Run 35 NOTE that only the Pearson X and the Cressie Read statistic work well when the data are extremely sparse In this example 40 bootstrap samples are generated The number of iterations starting from the parameter estimates of the real data are only 75 in this example as the original data s estimates are extremely good starting values for the simulated dataset which was generated using exactly these parameters The data are generated based on CML estimates Condi
51. pecific thresholds according to the ordinal partial credit model The partial credit model reserves one parameter for each threshold 1 e in WINMIRA 2001 Example Output file 101 contrast to the two Rasch model classes before no restrictions are imposed on the threshold parameters expected category frequencies and item scores Item Item s relative category label Score Stdev frequencies I I0 1 2 3 item01 72 1 05 0 147 0 286 0 266 0 301 item02 1 41 1 15 0 293 0 252 0 210 0 245 item03 152 1 10 0 242 0 233 0 286 0 240 item04 14253 1 09 0 216 0 290 0 242 02253 item05 1 49 1 10 0 246 0 249 0 274 0 230 item06 1 50 L205 0 211 0 291 0 281 UAE item07 1 46 1 11 0 248 0 283 0 227 0 242 item08 125 6 1 12 0 224 OPER AR 0 229 0 276 item09 1250 1 08 0 224 0 286 0 256 0 234 item10 1 46 1 08 0 234 0 297 0 241 0 228 Sum 2 22 2316 The expected category frequencies are sufficient statistics of the item parameters in the latent classes These conditional frequencies have to be estimated in each E step of the EM algorithm in WINMIRA 32 for both Latent Class models and Rasch models WINMIRA 2001 102 Example Output file These category frequencies hold for all subjects in the case of a LCA class because subjects do not differ systematically in the latent classes of LCA Because of the local independence assumption in the LCA the probability of any pattern can
52. ponse categories Psychometrika 1978 43 561 573 Andrich D Application of a psychometric rating model to ordered categories which are scored with successive integers Applied Psychological Measurement 1978 2 4 581 594 Andrich D An extension of the Rasch model for ratings providing both location and dispersion parameters Psychometrika 1982 47 105 113 Bozdogan H Model Selection and Akaike s Information Criterion AIC The General Theory and its Analytical Extensions Psychometrika 1987 52 3 345 370 Cressie T R C amp Read N A C Multinomial Goodness of Fit Statistics Journal of the Royal Statistical Society Series B 46 440 464 1984 Davier von M Neue Probabilistische Testmodelle und ihre Anwendungen Poster pr sentiert auf dem 39 Kongre der Deutschen Gesellschaft f r Psychologie 1994 Davier von M Methoden zur Pr fung probabilistischer Testmodelle IPN Schriftenreihe Band 157 1997 WINMIRA 2001 110 References Davier von M amp Rost J Self Monitoring A Class Variable In Rost J amp Langeheine R Eds Applications of Latent Trait and Latent Class Models Proceedings of the IPN Symposium in Sankelmark 1994 Davier von M amp Rost J Polytomous Mixed Rasch Models Chapter 20 in Fischer G amp Molenaar I Eds Rasch Models Foundations Recent Developments and Applications New York Springer 1995 Dempster A P Laird N M amp Rubin
53. port ASCII Data 19 sone De lal allen simuladx 113 simula9x 114 simula9x 115 fai simula9x 116 a simuladx 117 simula9x 118 simula9x 119 simula9x 120 a simula9x 121 a Simula9x BAK Simula9s dat Simula9x FIx a Simula9s MIW a simula9x 010 simula9x 011 a simula9x 012 a simula9x 013 a simula9x 014 4 i j Dateiname Simula dat Dateityp All files 7 Clicking theOK button will open another window which asks for information needed to import ASCII data He Edt Search DataSpecification Job Definition Stat Graphe Window EE Pred format Ie Per en Please select the separator character in most cases this will be the space or the tab character and click OK in order to import ASCII data WINMIRA 2001 20 Recode data with WINMIRA Recode data with WINMIRA please choose the Data Specification gt Recodings menu entry as depicted in the following figure e Edit Search Data Specification Job Definition Start X Siba idk B rl a SPSS 0 io 0 0 0 This will open the Recode dialog which contains a list of all variables in the dataset URE Zur Fr aa E AE 24 i Old and new integer values In this example the variables VAR4 VAR6 VAR8 and VAR9 are chosen WINMIRA 2001 How to edit variables with WINMIRA 21 to be recoded Please enter all values to be recoded together with the new codes in the grid on th
54. quality constraints are one letter entries in the constraints dialog All cells or better the corresponding class specific response probabilities carrying the same letter CaSE SenSitiVE will be set to the same numerical value in each iteration of the estimation algorithm Nevertheless certain regularity conditions are imposed afterwards If you try to constrain two items of extremely different difficulty to carry the same reponse probabilities the algorithm will nevertheless assume that the overall response probabilities are like given by the dataset and will adapt the equality constraints to match the overall probabilities Please click through the classes by means of the Class up down arrow on the top left of the window panel to see all constraints Constraints can be made within and across classes WINMIRA 2001 40 Logistic parameter constraints Logistic parameter constraints WINMIRA 32 pro provides the means to impose both parameter fixations and equality constraints on item difficulties as well as on threshold distances when estimating polytomous models The picture below shows the constraint dialog when using it for these logistic constraints WINMIRA 32 pro Parameter fixations are entered as numerical values both negative and positive in contrast to probability constraints in the constraints dialog Equality constraints are one letter entries in the constraints dialog All cells or better the corresponding cl
55. r of categories 4 number of classes 3 max number of iterations 350 accuracy criterion 0 0010 WINMIRA 2001 Example Output file 87 random start value 4321 The output file starts with a summary of the data specification and a description of the selected model In this example three latent classes are assumed with different models holding in each class variable labels positions and sample frequencies n of rec start end categories no label cats ord col col 0 1 2 3 N a yt a Er oS 01 item01 4 1 3 3 781 509 350 361 2001 02 item02 4 1 4 4 798 476 379 348 2001 03 item03 4 1 5 5 742 497 402 360 2001 04 item04 1 6 6 691 551 388 371 2001 05 item05 4 1 7 7 697 545 394 365 2001 06 item06 4 1 8 8 616 566 451 368 2001 07 item07 4 1 9 9 601 557 436 407 2001 08 item08 4 1 10 10 590 539 434 438 2001 09 item09 4 1 11 11 571 566 427 437 2001 10 item10 4 1 12 12 534 546 453 468 2001 number of cases with invalid data 0 The table above shows the label and the position in the datafile for each WINMIRA 2001 88 Example Output file variable in the scale as defined by the user The data file may have more than one row per observation so that both the record row and the start and end columns of the variables have to be specified saturated likelihood 14815 1431
56. r polychotomous item responses The British Journal for Mathematical and Statistical Psychology 1991 44 75 92 Rost J Carstensen C amp Davier von M An Application of the Mixed Rasch Model to Personality Questionaires In Rost J amp Langeheine R Eds Applications of Latent Trait and Latent Class Models Proceedings of the IPN Symposium in Sankelmark 1994 WINMIRA 2001 References 113 Rost J amp Davier von M A conditional Item Fit Index for Rasch Models Applied Psychological Measurement 1994 Rost J amp Davier von M Mixture Distribution Rasch Models Chapter 14 in Fischer G amp Molenaar I Eds Rasch Models Foundations Recent Developments and Applications New York Springer 1995 Rost J amp Georg W Alternative Skalierungsm glichkeiten zur klassischen Testtheorie am Beispiel der Skala Jugendzentrismus Zentral Archiv Informationen 28 52 74 1991 Rost J amp Langeheine R A guide through latent structure models for categorical data In Rost J amp Langeheine R Eds Applications of Latent Trait and Latent Class Models Proceedings of the IPN Symposium in Sankelmark 1994 Tarnai C Rost J Identifying aberrant response patterns in the Rasch model The Q index Soz wiss Forschungsdokumentationen Munster 1990 Warm T A Weighted likelihood estimation of ability in item response models Psychometrica 1989 54 427 450 Wright B D After
57. rmation on model selection WINMIRA 2001 The Mixed Rasch Model The Mixed Rasch Model extends the Rasch model to a discrete mixture model The main goal of applying this model is to classify a possible inhomogeneous sample into Rasch homogenous subsamples The Mixed Rasch model can be used for very different tasks e g e for testing model fit of the Rasch Model by comparing the one class and the two class solution e for identifying a Rasch scaleable subpopulation or separating a class of unscaleables respectively e for analyzing rating data when different subsamples have different response sets e for measuring a latent ability when different people apply different solution strategies for solving the items or e for profile analysis of questionnaire items with ordinal response formats WINMIRA 32 can be applied to dichotomous and polytomous data All characteristics of the Rasch Model are preserved within the latent classes so that the program can also be used for ordinary Rasch Analyses by WINMIRA 2001 62 The dichotomous Mixed Rasch model computing the one class solution The dichotomous Mixed Rasch model The model equation of the MRM for dichotomous item responses xvi 0 1 is P xvi Lg Ing exp xvi Ove oig 1 exp Ovg oig 3 where Ovs is the person parameter of subject v in latent class g and Ois is the item difficulty of item 7 in latent class g Rost 1990 The laten
58. sage window in order to make sure that parameter constraints should be used by expert users only and all computations carried out with parameter constraints should be examined carefully 92 This will open the parameter constraints window Please use this option carefully S Constraining item parameters can affect the algorithm s convergence in unexpected ways so that results may be invalid if inappropriate constraints were imposed Winmira offers three types of parameter constraints WINMIRA 2001 38 Probability constraints e either category probability constraints e or constrain item parameter and threshold distances e where both can have simultanous constraints imposed on class sizes Probability constraints WINMIRA 32 pro provides the means to impose both parameter fixations and equality constraints on the class specific response probabilities The picture below shows the different ways to enter fixations or equality constraints in WINMIRA 32 even though the constraints seen below might not make too much sense the primary reason for entering the numbers and letters in this way is to show what can be done in one single screen shot WINMIRA 2001 Probability constraints 39 Parameter constraints x b f f i VARI2 VARI3 0 25000 a 0 25000 0 25000 B 0 25000 a a 0 25000 6 0 25000 a 0 25000 6 0 25000 a Parameter fixations are entered as numerical values between 0 000 and 1 000 in the constraints dialog E
59. scrimination i e a Guttman pattern and one indicating perfect anti discrimination A value of 0 5 indicates no relationship between the individual parameter and the reaction to the item The Zq column is a transformation of the Q index that is approximative normally distributed if the Rasch model holds for the respective item High positive values indicate that the item discrimination is lower than assumed by the Rasch model under fit negative values indicate higher discrimination than assumed over fit In this example all items seem to fit fairly well Misfitting items can be detected by examining the Zq value in the table above High positive values indicate lower discrimination than expected The Zq transform of Q index for items detects very small deviations of the item characteristic with increasing power i e sample size Therefore an item should be removed from the scale only after examining the items content and additional information from the estimated model e g strange category frequencies or non monotone threshold parameters WINMIRA 2001 96 Example Output file Final Estimates in CLASS 2 of 3 with size 0 28155 MIRA Mixed Rasch Model according to the rating scale model Following now is the output for latent class 2 This class is expected to include about 30 of the sample The model assumed in this class is the rating scale model again Therefore a detailed comment is omitted in
60. sed for gives examples illustrating the capabilities of the mixture distribution models WINMIRA 2001 Parameter estimation in MDM 45 Parameter estimation in MDM Parameter estimation in discrete Mixture Distribution Models is quite complicated and time consuming An iterative algorithm the EM algorithm E stimation M aximization or iterative proportional fitting has to be employed because the latent classes are not known beforehand The EM algorithm works as follows Within each E step the expected frequencies of the sufficient statistics for the model parameters are computed for each subpopulation This is usually done by computing posterior probabilities given the current parameter estimates Within each M step Maximum Likelihood estimates in each subpopulation are computed by means of some standard procedure like Newton Raphson given the sufficient statistics from the previous E step The iteration procedure is based on the first and second order partial derivatives of the likelihood function L of the complete crosstable i e WINMIRA 2001 46 Parameter estimation in MDM the observed crosstable extended by the latent class variable From the matrix of the second order derivatives only the diagonal elements are used so that the iteration rule for a model parameter u is pt u i WI WI where L u is the first partial derivative L u the second order derivative with respect to the parameter u and
61. st restrictive model i e the mixed Rating Scale Model ixs is decomposed as follows WINMIRA 2001 64 The polytomous Mixed Rasch model aixg pig Axg for all g and x 1 m with the condition x Axg 0 to avoid indeterminacies The category parameters Axg parameterizes the easiness of threshold x in class g In the program output the combined parameters Oixg are printed for each model so that different models can be compared more easily The second model i e the mixed Equidistance Model has the decomposition oixg pig m 1 2x dig for all g and x 1 m The third model i e the mixed Dispersion Model has both equidistance and threshold parameters and is decomposed as follows oixg pig Axg m 1 2x dig for all g x 1 m In the fourth model i e the mixed Ordinal or Partial Credit Model all ixg are estimated separately there is no restriction except the normalizing condition WINMIRA 2001 The Rating Scale Model 65 The Rating Scale Model oixe Hig Axg The Rating Scale Model assumes that all threshold distances are constant acrosss the items which is indicated by the missing item index i for the threshold parameters xg The figure below shows the category characteristic curves CCCs for two items as an example where the difference between threshold 1 and 2 is 2 0 for both items the distance between threshold 2 and 3 is approx 3 5 for both items WINMIRA 2
62. t classes are identified by means of an EM Algorithm and the item or threshold parameters are computed by means of conditional maximum Likelihood CML estimation within each M step The CML estimation requires the latent score distributionslatentscore i e the distributions of test scores in each latent class to be estimated in order to condition out the person parameters in the CML procedure Then the probability of a response pattern_x can be written as P x LGe me me exp LXi xi oig ye 4 WINMIRA 2001 The polytomous Mixed Rasch model 63 with response pattern x x1 xk latent score distribution parameters Tg the probability of score r in class g and symmetric function y s of order r in class g The polytomous Mixed Rasch model WINMIRA can be used to estimate model parameters for different polytomous ordinal Rasch models the rating scale model Andrich 1978 the equidistance model Andrich 1982 the dispersion model Rost 1988 and the partial credit model Masters 1982 as well as their mixture generalizations compare Rost 1991 von Davier amp Rost 1995 Generalized to mixture distribution models the polytomous Rasch models can be written as follows P x Ss Ing sus exp LNi 1s lotixg yrg 5 with response pattern x X xk Xi E 0 m mrg probability of score r in class g and symmetric function Yrg of order r in class g In case of the mo
63. tances 0 393 0 558 This table shows the item parameter estimates for the Rating Scale model WINMIRA 2001 94 Example Output file in class one These threshold parameters should be ordered 1 e the parameters should decrease from threshold to threshold whenever an ordered response format is assumed A decreasing easiness of the threshold parameters indicates that every response category is representative for an interval of the individual parameter dimension The last column is an overall easiness parameter computed by summing up all threshold parameters Again item and item seem to be very difficult that is they have a high difficulty parameter item fit assessed by the Q index itemlabel Q index Zq p X gt Zq je _ Ze item01 0 1099 O4 PLO4 30476281 Sr item02 0 1241 0 82 90 05 298183 Qt item03 0 1173 0 4272 0 6654 Fre De item04 11 87 9 26 74 039459 re item05 0 1026 0 9459 1 1 02 7071 9s ll Seal 20 item06 0 1014 909200 049207 See Ode item07 0 0995 0 1398 0 44439 Q 4 item08 0 0927 SOAS 05979213 li Sate tL Owe item09 0 1052 On 972 170516534 as item10 0 0886 01952 0 42261 Furl ean 9 lt 0 08 422B gt 0 95 WINMIRA 2001 Example Output file 95 p lt 0 01 p gt 0 99 The table above shows the Q Index a class specific item fit measure for Rasch models The Q index lies between zero indicating perfect di
64. ter parameters may be interpreted as dispersion parameters since the dispersion of the probability WINMIRA 2001 54 The threshold approach in the LCA cont distribution of a variable depends on the widths of its threshold intervals the smaller dig is the closer the thresholds and the greater the dispersion of that variable This relationship is more easily understood if the opposite case is imagined a large distance between the left and the right threshold of a category means that it is easy to get in but hard to get out of this category Hence the probability distribution has a peak over this category and therefore a smaller dispersion in general So far all models of the Latent Class Module of WINMIRA 32 have been specified It must be noted however that they are usually written in a slightly different way This different notation is obtained if the fsix terms are cumulated as indicated in equation 1 Since s 1 uig x is and I s I m 1 2s x m x model 2 may be written as 2 pvix Vers exp xpig x m x di Lxexp xpigs x m x di In order to get rid of the summation symbol when the fgix parameters have to be summed up cumulative threshold parameters WINMIRA 2001 What does the LCA part ofthe program 55 agix Xs 1 fgis may be defined This notation is helpful for deriving equations for parameter estimation but they have no direct interpretation The parameters provided in the output f
65. three types of restrictive assumptions leads to a system of 8 models for ordinal variables WINMIRA 2001 What can LCA be used for 59 Beginning with the latter the distinction of category distances which hold for all persons and class specific distances divides the system in two groups of four models each i e class independent distances and class specific distances The assumption that all manifest variables have the same distances between categories gives models 1 and 5 called rating scale LCA in the model selection menu The so called equidistance assumption i e all categories of a manifest variable have the same distance yields models 2 and 6 called equidistance LCA A scaling concept which goes back to Thurstone s method of successive intervals assumes that the categories have their own distances as a characteristic of the response format but the variables have their own dispersions If no assumption about category distances at all is made models 4 and 8 are obtained where 8 is the unrestricted latent class model and 4 makes the assumption that the same distances hold for all persons whatever they are for a particular variable and a particular category It can be decided empirically which of these assumptions is most WINMIRA 2001 60 What can LCA be used for appropriate to the data Please refer to section on the Bootstrap or to the section describing an Example Output file for further info
66. tio defined above is referred to as a threshold probability because it is the conditional probability of choosing x if only x or x are considered This can be interpreted as passing the threshold from the lower category to reach the higher category The term on the left hand side varies between O and 1 and denotes a WINMIRA 2001 50 The threshold approach in the ordinal LCA probability namely the conditional probability of responding in category x if the response is either in x or in x 1 This conditional probability is usually referred to as threshold probability because it can be interpreted as the probability of passing a threshold between two response categories It equals 0 5 if both categories have the same probability 0 0 if nobody passes and 1 0 if all people pass the threshold to x The latter form of equation 11 shows that the threshold probabilities depend on the model parameters fgix and their dependency is defined by the smooth logistic curve known from the Rasch model Hence the model parameters fgix can be interpreted as defining the location of threshold x on a latent continuum This was outlined in some detail in order to stress the point that a threshold assumption is not necessary to derive the model but it helps to interpret the parameters The model itself only requires knowledge of which categories are adjacent and in that sense requires ordered categories more WINMIRA 2001 The threshold appro
67. tional Maximum Likelihood so that the WLE person parameters are only used for the extreme rawscore groups Start Run Choose Start gt Start Job or function key F9 in order to make WINMIRA start computations with the current settings on Job Definition Start Graphs Window Help Start Job Start Multiple Jobs vs je jr e e vo 0 0 0 0 0 0 0 0 0 0 0 0 WINMIRA 2001 36 Start Run This will minimize WINMIRA s main window and will open the run dialog which is used to display some information about the computational process Winmira 32 running Module running LCA Iteration Jer Message 24883 4037545305 TTT Finally the main window of WINMIRA will reappear and the programs output file will be shown in a second child window within the main form The output file shown in the active child window contains descriptive statistiscs of the dataset model parameters estimates and information about the model data fit This file is already saved usually in the same directory where the dataset is found unless some other directory and filename was specified in the filename dialog WINMIRA 2001 Parameter constraints The Job Definition gt Parameter constraints menu entry Jecihcation Job Definition Start Graphs Window N of Classes Select Model Parameter constraints Bootstrap GoF Filenames Edit Defaults 1 o 2 0791 0 6215 10 90 15 opens a confirmation mes
68. tional function as their basis If you use WINMIRA for the first time please do not change anything Clicking OK with the default settings will choose the most common Mixed Rasch Model which is suitable both for dichotomous and polytomous data The remaining options impose restrictions on the model parameters of the Mixed Rasch Model More information on restricted Rasch models can WINMIRA 2001 Open data 17 also be found in the references e Change Output Options if necessary e choose Bootstrap Fit Statistics if your data are sparse e Start Run e Display Graphical Output Open data Open SPSS Data Files Please select the menu entry File gt Open gt Open Spss Data as shown in the screenshot below WINMIRA 32 professional File Edit Search Data Specification Job Definition WINMIRA 2001 18 Import ASCII Data Select the data file in the file selection dialog Open SPSS OO Seen Sf ese ca J nerven dir UNBENANNT J test2 sav EJ Daten Say Dateiname Daten sav tme Dateityp SPSS Files In this example the file named Daten sav was chosen Click the OK button in order to confirm the selection This will load the datafile Import ASCII Data If you choose a file with an extension different from SAV it is assumed that ASCII data are imported Alternatively you may also choose Load ASCII Data from the submenu of the File gt Open menu entry WINMIRA 2001 Open SPSS Im
69. word in Rasch G Probabilistic models for some intelligence and attainment test Chicago 1980 Wright B D Masters G Rating scale analysis Chicago MESA Press 1982 WINMIRA 2001 114 References Wright B D Stone M Best test design Chicago MESA Press 1979 Yamamoto K A Hybrid model of IRT and latent class models ETS Research Report RR 89 41 Princeton NJ Educational Testing Service 1989 Yamamoto K amp Everson H Detecting Speededness using a Hybrid IRT Latent Class Model In Rost J amp Langeheine R Eds Applications of Latent Trait and Latent Class Models Proceedings of the IPN Symposium in Sankelmark 1994 WINMIRA 2001
70. x log likelihood of the observed item pattern It can be applied to any unidimensional Rasch model like the dichotomous model the rating scale model Andrich 1978 the equidistance model Andrich 1982 the partial credit model Masters 1982 or dispersion model Rost 1988 The fit of an item is evaluated with regard to the conditional probability of its observed item response vector i e p xi ni0 nim exp x xviBiv t B ni0 nim where the denominator is given by the symmetric functions of order ni0 nim of the person parameters This conditional pattern probability is standardized twice First it is divided by the maximum probability a pattern with a particular score distribution can reach i e the probability of the_optimum pattern x OPT or Guttman pattern The Likelihood ratio LRi OPT p xi nix p x OPT nix then approximates 1 one for an increasing pattern probability The logarithm of this Likelihood ratio is standardized again 1 e it is divided by the smallest possible Likelihood ratio LRPESS OPT p x PESS nix p xOPT nix WINMIRA 2001 The Q Index 77 where x PESS denotes the pessimum pattern or the anti Guttman pattern which is the pattern with lowest probability The ratio of both log likelihood ratios is Qi In PCXi nix P XOPT nix In P XPESS nix P XOPT nix Div xvi x V OPT BY V xv PESS xV OPT BY which is a very simple function

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