Questions of a Novice in Latent Markov Modelling*
Contents
1. The views expressed herein are those of the authors and do not necessarily reflect the policies of Statis tics Netherlands or the Canadian Institute for Health Information Correspondence can be addressed to F van de Pol home nl fpol cbs nl and Hmannan Qcihi ca 2 MPR Online 2002 Special Issue 2 The use of change models in addition to cross tables of change 2 1 Overview of functions of latent class analysis Latent class analysis has many faces It is a way of looking at categorical data that is frequency tables One may use it for data reduction of typically cross sectional data or for the analysis of repeated measurements using the assumption of a Markov chain Measurement errors and response uncertainty are inherent to survey data They are not only a normal phenomenon in opinion and attitude surveys but also occur when items refer to objective facts such as employment status Because of these errors asso ciations are usually underestimated and frequency distributions will be biased unless errors cancel out When several indicators of a concept are available a latent class mo del enables estimation of misclassification probabilities and also of frequency tables of the latent or hidden variables behind the measured indicators or items By taking the effect of measurement error into account we get a clearer view on the association between latent variables with generally stronger associations than2. In C C Clogg ed Sociological Methodology pp 213 247 Oxford Blackwell Van de Pol F J R Langeheine R amp de Jong W A M 1996 PANMARK 8 user s manual PANel analysis using MARKov chains a Latent Class Analysis program F van de Pol home nl Van de Pol F J R 1997 Educational mobility cohort and gender a latent class re analysis of the Ganzeboom and De Graaf data In J Rost amp R Langeheine eds Applications of latent trait and latent class models in the social sciences pp 412 419 Miinster New York Waxmann Vermunt J K Rodrigo M F amp Ato Garcia M 2001 Modeling Joint and Mar ginal Distributions in the Analysis of Categorical Panel Data
3. LR If the algorithm seems to be spinning there is an identification problem Either the model you specified is not identified for these data or the likelihood surface is almost flat thus making it almost impossible to find the optimum In the first case one or mo re eigenvalues will be zero or negative In the second case one or more eigenvalues will be very small in comparison to the biggest one In the first case you cannot compute standard errors of the parameters in the second case some standard errors will be very large The conclusion is that you should always check the identification of your model before asking for a bootstrap analysis The model should be identified and standard er rors should be reasonably low After all what can we conclude from an analysis that gives model parameters that are very inaccurately determined Finally even if the mo del is identified for the original sample you may have a special case that generates some bootstrap samples for which themodel is not well identified Moreover the option to have a large number of randomly chosen starting values is too time consuming to be applied in combination with bootstrapping Next best is to have estimation start with the estimates from the original sample Since there are fewer cells than cases I selected sampling from cells There are two ways to draw a non naive parametric bootstrap sample The most common approach is to draw a sample from the mult
4. answers 3 1 Getting started A scholar in Canada named Haider Mannan is preparing a paper to become a Ma ster of Science in Epidemiology and Biostatistics and decides to use latent Markov mo dels He obtains PANMARK and begins to pose questions At first some trivial questions arose like How can I manipulate a Windows shortcut to a DOS program When a program or its working directory are moved to a non default place a short cut to the program does not function anymore Experienced Windows users know that they need to check the shortcut s properties especially the program tab where the first line is for starting the exe file and where the second line should contain the directory where that program will start reading and writing files the working directory Next co mes a phase when the computer program is unfamiliar van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 7 I tried to fit a one chain manifest Markov model both transition and response probabilities time homogeneous by creating a restriction file and starting values file but the PANMARK output was wrong In Model Definition I selected Latent Markov chain since there is no manifest Markov chain option Specifying restrictions for a large number of parameters can easily be mixed up The present program identifies each parameter from the order in which it appears The user can copy this order from the file with parameter estima
5. in the observed fre quency tables In order to get accurate estimates it is useful but not always necessary to have more than one indicator for the same latent concept When measurements at successive occa sions are available measurement error and true change can be separated using the assumption of a Markov process Population heterogeneity is often shown by cross classifying a target variable with background characteristics like age sex or region However an important part of the variation in the target variable may not be explained by the characteristics that have been measured The unexplained part may be labelled latent heterogeneity In the pre sent paper we aim at explaining heterogeneity in change characteristics A classical ex ample of latent heterogeneity is the difference between people who frequently move to a new house the movers and people who will not move at all the stayers With the van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 3 mover stayer model one can estimate from panel data the proportion movers and the mover s turnover Blumen Kogan and McCarthy 1955 Also more sophisticated mixtu res of Markov chains may be estimated which are used among others in marketing rese arch for modelling brand loyalty Poulsen 1990 For completeness we also mention localisation of types homogeneous clusters of ob jects as a third function of latent class analysis Using
6. the recruitment probabilities a by product of latent class analysis these types may be related to background characteri stics One may for instance measure a dozen sorts of youth criminality A latent class analysis can reveal types like aggressive behaviour property motivated behaviour severe criminality and no criminality and their occurrence Next relations between these types and for instance gender or drinking and smoking habits may be analysed Reduction of tables is another type of data reduction In this approach tables descri bing a large number of sub samples are reduced to a small number of latent tables Clogg and Goodman 1984 De Leeuw et al 1990 Loadings of the sub samples on these latent tables are generated For instance a cross table describing educational mo bility may be available for a large number of cohorts By simultaneous analysis of these cohorts one may estimate for instance two latent tables one with high mobility and one with low mobility Older cohorts will load higher on the low mobility table than younger cohorts Van de Pol 1997 Instead of a cross table also a one way frequency distributi on may be analysed representing for instance time budgets Apart from this descriptive sort of application simultaneous analysis of sub samples may also be used to obtain a maximum likelihood estimate of one table in a panel survey including all sub samples that are created by panel
7. Methods of Psychological Research Online 2002 Special Issue Institute for Science Education Internet http www mpr online de 2002 IPN Kiel Questions of a Novice in Latent Markov Modelling Frank van de Pol Statistics Netherlands Haider Mannan Canadian Institute for Health Information 1 Introduction Change is the main issue when panel data are collected This paper focuses on latent class analysis as a means to represent the regularities in such repeated measurements of the same objects It does so without adding any novelty to the extensive methodological literature The emphasis is rather on facilitating the application of these methods We will elaborate on the questions of a student Mannan 2001 who recently finalised a paper on smoking behaviour that relies heavily on latent class analysis Before addressing these questions an overview of the uses of latent class analysis will be given in the next section followed by a brief introduction to the general model that is used by Mannan for the analysis of his panel data This model is at the basis of the computer program PANMARK which the first author developed in collaboration with Rolf Langeheine In fact the present paper is written at the occasion of his retirement And that is a good reason for writing a paper because the many quotations of his work by various authors show that the views of Rolf Langeheine have shaped the methods for categorical data in discrete time
8. a gt by T sh s la 1b 1c 1 11 21 22 32 33 Ish Pijash T blash P slosh T elbsh Prlbsh Extension to multiple indicators is straightforward Langeheine and Van de Pol 1993 1994 For two indicators at each time point with subscripts i and i for time point 1 subscripts j and J for time point 2 subscripts k and k for time point 3 we would get 6 MPR Online 2002 Special Issue S A BC 3 1 11 21 21 32 42 32 53 63 Prii ijt ak Va 2 gt D Top bash Pijash Pitash T hash P josh Pitbsh T elbsh Prlesh Phish s 1 a 1 b 1 c In a panel sample an estimate Phijk is found or in the case of two indicators for each occasion Pit jj hk From these estimates one can compute maximum likelihood parameter estimates for the y 7 6 p and 7 for instance with the computer program PANMARK provided that identifying restrictions are applied It uses a version of the EM algorithm Van de Pol and De Leeuw 1986 Van de Pol and Langeheine 1990 The present notation in terms of conditional probabilities can be replaced by a nota tion in terms of log linear parameters Vermunt 2001 An advantage of the latter no tation is that it allows generalization to more complex models On the other hand probabilities that are on the boundary of the parameter space 0 or 1 are not easily es timated in this approach since they correspond to log linear parameters that go to in finity and computers set a limit to any number 3 Questions and
9. annan had the intention to explain latent parame ters like the proportion who stopped smoking For analysing a covariate do we only need the covariate being measured at the first time point say the covariate being measured at grade 6 for transitions of all students from grade 6 to 12 is sufficient for analysing such a covariate If that is so then we are assuming that the covariate is time independent Can we handle a time dependent covariate and how Can we handle continuous covariates in latent Markov models in any way There are two schools of thought with respect to incorporating explanatory exoge nous variables into a measurement model One school emphasizes the importance of si multaneous estimation of all parameters involved Van der Heijden et al 1996 Muthen 2002 On the other hand it has been argued that explanatory variables should not have any influence on the measurement model An explanatory variable that associates stron gly with only one of the indicators in the measurement model will easily give an unreali stic boost to the reliability estimate of that indicator and the reliability of the other in dicators will seems smaller than they really are In the latter approach you can use the recruitment probabilities which are almost the same as grades of membership GOM s to get information on class membership for van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 15 every res
10. ansition probabilities should be fixed to 0 25 in the present case of four response categories Less restrictive and well known from log linear models is a model that assumes independence between occasions In the present context independence is only assumed for one chain It means that for instance 86 86 bias i A 6 Sahat Ti yes irrespective of the originating category 7 Denoting transition probabilities that are equal with the same number these equality restrictions are written 1234 1234 1234 1234 with rows indicating grade 6 category and columns the grade 8 category 3 4 Second order Markov models Mannan also fitted a second order Markov chain This models allows for the initial grade 6 situation making a difference for the change between the two occasions that followed from grade 8 to rade 11 Could you please explain the rules for arranging frequencies in the dataset for 2nd and higher order Markov models I don t understand when and how zero frequen cies appear In among others Langeheine and Van de Pol 2000 frequency data are arranged in such a way that a program for first order Markov chains can fit second order Markov chains The trick is that each cross table of consecutive measurements for instance oc casion 1 times occasion 2 is presented to the program as one variable The observed frequencies are interspersed with zero frequencies at positions that are impossible like 1 10 MPR Online 2002 Spe
11. attrition 2 2 Description of data and the models involved An extensive description of the models involved is given by Langeheine 1994 2002 The questions that will be addressed in the next section are typical for someone who has read these publications but lacks application experience What follows is a brief formal 4 MPR Online 2002 Special Issue description of Mixed Markov Latent Class MMLC models based on the user manual of PANMARK Van de Pol and Langeheine 1996 In the simplest case one polytomous variable x that is measured at one or more T consecutive occasions al a T is analysed A realization in a specific category is denoted t for the first occasion and j k m for consecutive time points This variable which is measured at several occasions can also be viewed as an indicator for a latent variable also named an indirectly measured or hidden variable For model identification it is convenient if the development in time of this latent variables is described as a Mar kov chain In fact more than one type of development can be modelled a mixture of Markov chains In order to have a well identified model also with a latent mixture of Markov chains another generalisation is useful more than one indicator can refer to the same latent variable Finally a categorical exogenous variable may be introduced by making compartments in the model for several subpopulations Analysis with MMLC models focuses on th
12. by allowing the response probabilities of the first occasion to be different you will have to assume stationary transition probabilities Finally Mannan tried to fit a latent second order Markov chain 12 MPR Online 2002 Special Issue I have reanalyzed 2nd order Markov model by creating a totally different starting values file which I gave you The transition matrix which I created in this file has 16 rows and 16 columns But still I am not getting the correct result Mannan forgot to include a line with delta s in the starting values file So there is no initial distribution defined Moreover there is another problem Despite the fact that PANMARK says it can handle more than 200 categories in fact the limit is 15 at the moment The problem is that internally a 16x16 transition matrix has 256 entries and that is just one too many for the routine that sets equality restrictions 3 5 The bootstrap for assessing model fit The likelihood ratio is a model fit criterion that can be evaluated with the chi square distribution as long as sample size is large enough in every cell of the contingency table Another requirement for using this theoretical distribution is that there is no doubt about the degrees of freedom Sometimes parameters get fixed during estimation to one of the bounds of the parameter space 0 or 1 The effect of such boundary values on the degrees of freedom is uncertain Finally equality restrictions do stri
13. cial Issue and 1 at occasions 1 and 2 respectively in combination with 2 and 1 at occasions 2 and 3 respectively Occasion 2 cannot have score 1 and 2 at the same time With data on three occasions only the second order Markov chain is not a restrictive model It is merely a reparametrization of the data a saturated model For testing the fit of mixed Markov latent class models one needs two or more indicators at each occa sion Langeheine and Van de Pol 1990 Measurements on more than three occasions are also helpful Mannan laid his hands on measurements on four occasions grades 6 8 10 and 12 With data on four occasions Langeheine s partially latent mized Markov model could be estimated Langeheine and Van de Pol 1992 This is a mover stayer model with response error in the mover type only The assumption that stayers don t make response errors is both plausible and practical Plausible because it is more easy to produce the correct answer if one s position is stable Practical because data on one indicator measu red at four occasions are not informative enough to estimate a more complex model Hope you are well Presently I am analyzing the second and last data set There are 256 4x4x4x4 cells i e there are 4 indicators smoking each having 4 cate gories Many of the cells are 0 I successfully fitted manifest Markov and latent Markov model with stationary and non stationary transition probabilities and re sponse
14. cripts indicating the occasions grades 6 8 and 11 Mannan had taken a file with estimates and altered these near zero parameters into exactly zero Starting the estimation of a parameter with 0 has the same effect as fixing it to 0 because the pre sent program uses the EM algorithm Iterations in this algorithm involve multiplicative adjustment of estimates thus leaving zeroes at 0 until convergence 8 MPR Online 2002 Special Issue Mannan got confused because the program warned that Row ending lt some num ber gt does not add up to 1 If someone is in smoking state 2 for instance he or she must show up in any of the possible states next time supposing there was no drop out This property summation to one holds for any set of conditional probabilities be it Pia response probabilities given latent state a 2 proportions belonging to any chain Sor r jl probability in a set is fixed to zero the starting values for the remaining probabilities of 6 ns eae At S manifest transitions originating from one original status 7 If one the same row must be increased to the extent that they add up to 1 3 2 Time homogeneity Stationarity or time homogeneity is a recurring issue in latent Markov models It is the assumption that the probability to move from state a to state b is the same for all time intervals It is attractive to make this assumption since the accuracy of the esti mates is much bett
15. ctly spoken prohibit the use of the simple theoretical chi square distribution since a more complicated dis tribution applies with equalities If one of these conditions is not met the distribution of the fit statistic can be gene rated with repeated parametric bootstrap samples Langeheine et al 1996 Now that fast computers offer the computational possibilities researchers will test the fit of a mo del using the bootstrap even if sample size is large Hope you received my last email I am having problems with bootstrap analysis I ran a latent Markov model with unrestricted parameters last night at around 9 and until 2 30 am today the run was still going on i e for 17 5 hours with only 200 bootstrap samples being drawn The data set has only 64 frequencies with 16 being zero So I really lost my patience and stopped the run The stopping crite rion was set at 1e 8 the default It is likely that the algorithm for the program is not efficient or the algorithm is spinning because of the tight criterion for conver gence In case the algorithm spins then a larger criterion should be used What stopping criterion do you recommend For mixed Markov models including mo van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 13 ver stayer black and white etc what stopping criterion do you suggest Note that for my run I selected the option the EM algorithm will stop if bootstrap LR is less than the original
16. den P G M amp Verboon P 1990 A latent time budget model Statistica Neerlandica 44 pp 1 22 Langeheine R 1994 Latent Variable Markov Models In A von Eye amp C C Clogg eds Latent Variables Analysis Applications for Developmental Research pp 373 395 Thousand Oaks California Sage Langeheine R 2002 Latent Markov Chains In A L McCutcheon amp J A Hagenaars eds Advances in Latent Class Analysis Cambridge University Press Langeheine R amp van de Pol F 1990 A unifying framework for Markov model ing in discrete space and discrete time Sociological Methods amp Research 18 pp 416 441 Langeheine R amp van de Pol F 1993 Multiple Indicator Markov models In R Steyer K F Wender amp K F Widaman eds Psychometric Methodology Pro ceedings of the 7th European Meeting of the Psychometric Society in Trier 1993 pp 248 252 Stuttgart New York Fischer Langeheine R amp van de Pol F 1994 Discrete time mixed Markov latent class models In A Dale amp R Davies eds Analyzing social amp political change a Case book of Methods pp 167 197 London Sage Langeheine R Stern E amp van de Pol F 1994 State mastery learning dy namic models for longitudinal data Applied Psychological Measurement 18 pp 277 291 Langeheine R Pannekoek J amp van de Pol F 1996 Bootstrapping Goodness of Fit Measures in Categorical Data Analysis Sociol
17. e types of change that exist in these sub populations The general MMLC model assumes that each subject belongs to one sub population like gender or birth cohort Membership of subpopulation h h 1 H is assumed constant in time for all indicators In the sequel we will refer to manifest measured variables as indicators in contrast to the latent variables The proportion that belongs to subpopulation h is denoted Th All other parameters that will be descri bed below are considered conditional on subpopulation h i e all or some parameters may be different in each subpopulation Each member of subpopulation h belongs to one latent or manifest chain of people having the same dynamics A proportion T p in subpopulation h belongs to chain s Ih Hence the proportion in subpopulation h and chain s is Yn Tah A member of subpopulation h and chain s is assumed to belong to one of A classes The proportion in class a a 1 A with variable 1 for subpopulation h and chain s van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 5 is denoted Oih Hence the proportion in subpopulation h chain s and class a for vari able 1 is y 7 6 sih alsh The probability to answer 2 for indicator 1 given h s and a the response probability La Piash is assumed the same for all subjects in subpopulation h chain s and class a Hence the proportion in subpopulation h chain s c
18. ent probability 2 Score pattern 2 2 1 1 class 4 recruitment probability 1 Don t use multiple groups for this with older versions of PANMARK The GOM s are only correct with 1 g y group 3 A bit more complicated is to focus on the transition from smoker to ex smoker This would involve 681012 681012 summation of the Gaiei ijk OVE latent indices c and d In order to reduce the size of the latent cross table also the number of latent states at occasions 1 and 2 should be reduced 16 MPR Online 2002 Special Issue And so on for all score patterns in the analysis Each record represents partial membership of one category of the latent variable at occasion 2 Then the fourfold data file can be matched with these recruitment probabilities using the score pattern and the latent class as matching keys Finally the recruitment probabilities should be used as weights in the analyses that follow Note that every case still has weight one being the sum of all four recruitment probabilities of the relevant score pattern With the resul ting data file you can examine membership of a specific cell of the latent variable in a logit or probit analysis with as many covariates as you like The advocates of simultaneous estimation full information maximum likelihood FIML can argue that there is no evidence the measurement model holds for the whole population If for instance reliability is lower for male smokers than for female smoker
19. er if the assumption is correct i e standard errors are smaller For Mannan s data this assumption is questionable The Latent Markov model may require equally spaced measurements for fitting a totally constrained model Since my smoking measurements are unequally spaced in time do you think that it will be inappropriate to fit the above men tioned model Possibly more turn over between latent states takes place in a longer time interval 86 a a for the two year period Therefore the off diagonal transition probabilities 7 should be allowed to be smaller than the corresponding ones in the three year period that followed Of course estimates will deviate somewhat from this expectation due to sampling error and as a consequence some off diagonal probabilities will suggest more change in the shorter period contrary to our expectation Suppose the only exception is 86 118 2 86 118 Tya gt Taja gt then I would suggest to re estimate the model with assumption Taja Taja van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 9 3 3 Restrictive mixed Markov models In his search for a good fitting model Mannan tested among other things whether part of the respondents responded randomly It is not clear to me how black and white and mover random response models can be fitted by PANMARK The manual does not give examples of these models In a random response chain the all tr
20. inomial distribution with the po pulation proportions that were estimated after fitting the model In PANMARK this is Latent class models may have more than one optimum in the likelihood surface Of course the most likely one is to be preferred but the problem is that an optimisation routine may end up in a local opti mum Therefore it is recommended to start the algorithm with several iterations from many points of the likelihood surface and to continue with the most promising locations The more parameters are involved the more starting points are needed The relationship is not linear but seems rather exponential Dou bling the number of independent parameters one should take the square number of starting points In practice this is not feasible though 14 MPR Online 2002 Special Issue implemented by drawing successive binomial samples and referred to as sampling from cells For models that can be written in terms of conditional probabilities there is an alternative approach that is based on drawing each case according to the model proba bilities This is referred to as sampling from cases Langeheine et al 1996 Sampling from cases is to be preferred always Sampling from cells has a small bias in the last cell I found recently at least in my implementation I will eliminate the sampling from cells option from new program versions 3 6 Exogeneous variables covariates At the beginning of his analysis M
21. lass a with variable 1 and category t 11 for indicator 1 is WT sth bra Pi If the variables of the model are not latent but al ash manifest the latent classes a for variable 1 b for variable 2 etc coincide with the manifest categories 7 for indicator 1 j for indicator 2 etc Moreover the response 11 probabilities p 3 11 ilash AL superfluous in a manifest model Pijash 1 fora a i and else 0 If the model does not only involve a latent variable at time 1 but also one at time 2 then for subpopulation h each member of chain s is assumed to behave according to 21 the same transition probabilities Thash from class a for time 1 to class b for time 2 As for the time 1 indicator also for the time 2 indicator the probability to answer j given 22 being in class b chain s and subpopulation h P ish is assumed to be the same for all subjects in subpopulation h chain s and class b Hence P saibi the proportion in sub population h chain s class a for latent variable 1 category i for indicator 1 class b for 11 21 22 eT 1 latent variable 2 and category j for indicator 2 is Vn Tah Osh Pijash T yjash P josh An expression for the observable population table Prag of two manifest indicators not the latent variables is obtained by summing over the latent indices s a and b The model equation for the latent mixed Markov chain for three time points is written as S A BOC i Frij Yh
22. ogical Methods amp Research 24 pp 492 516 Langeheine R amp van de Pol F 2000 Fitting Higher Order Markov Chains Methods of Psychological Research 5 www mpr online de Pabst Science Publishers Mannan H R 2001 Latent Markov Modelling of Smoking Transitions London Canada University of Western Ontario 18 MPR Online 2002 Special Issue 14 15 16 17 18 19 20 21 22 Muthen B 2002 Beyond SEM General Latent Variable Modeling http www statmodel com Behaviormetrikapaper1 pdf Poulsen C S 1990 Mixed Markov and latent Markov modelling applied to brand choice behaviour International Journal of Research in Marketing 7 pp 5 19 Van der Heijden P Dessens J amp Bockenholt U 1996 Estimating the Con comitant Variable Latent Class Model with the EM Algorithm Journal of Educa tional and Behavioral Statistics 21 pp 215 229 Van der Heijden P t Hart H amp Dessens J 1997 A Parametric Bootstrap Procedure to Perform Statistical Tests in a LCA of Anti Social Behaviour In J Rost amp R Langeheine eds Applications of Latent Trait and Latent Class Models in the Social Sciences pp 196 208 Miinster Waxmann Van de Pol F J R amp de Leeuw J 1986 A latent Markov model to correct for measurement error Sociological Methods amp Research 15 pp 118 141 Van de Pol F J R amp Langeheine R 1990 Mixed Markov latent class models
23. ot all parameters of the PLMS model can be identified This can be seen from the information matrix which is not positive definite Not all Eigenvalues are positive When you take a look at your data you will see that only 9 respondents remain in category 4 never smoker and few respondents leave this condition So there is almost no information to separate reliability and stability for this group We have to make more assumptions Would it be reasonable to assume that class 3 experimenting with smoking is empty with stayers After all you cannot continue experimenting for ever It does not make sense to be experimenting during 6 years especially because stayers are assumed to respond with perfect reliability Because only one eigenvalue is 0 I have tried to estimate this model As the output shows you will have to make more assumptions in addition I suggest you make some stationarity assumption You don t have to assume the complete transi tion parameter matrices tlxt2 t2xt3 and t3xt4 to be equal It may be enough to set only some of the content of these matrices equal I cannot say which parameters can safely be assumed to be stationary without messing up your main hypothesis I suppo se you want to assess the effect of some experiment between time points Another thing to keep in mind is that the first wave of a panel often is measured a little less reliably than the occasions that follow If you want to assess this phenomenon
24. ponse pattern that you have in your sample This information is a probability 681012681012 I distribution of latent variables for each combination of the indicators a bedlijkl the presence of a latent table with theoretically 4 256 cells you will probably aggre gate the T by summation over most of the latent variables focussing on for instance the latent proportions at occasion 2 Let us have a look at two score patterns and the accompanying recruitment probabilities for class b 1 2 3 and 4 respectively Score pattern 1 1 2 2 recruitment probabilities 2 1 4 3 Score pattern 2 2 1 1 recruitment probabilities 4 3 2 1 Next you create a new variable in your data file called latent class Make four co pies of your data file with latent class 1 2 3 and 4 respectively Stack these four files below each other in one new file This new file has four times as many cases as the ori ginal one In order to match this new data file to these recruitment probabilities they should be rewritten as follows Score pattern 1 1 2 2 class 1 recruitment probability 2 Score pattern 1 1 2 2 class 2 recruitment probability 1 Score pattern 1 1 2 2 class 3 recruitment probability 4 Score pattern 1 1 2 2 class 4 recruitment probability 3 Score pattern 2 2 1 1 class 1 recruitment probability 4 Score pattern 2 2 1 1 class 2 recruitment probability 3 Score pattern 2 2 1 1 class 3 recruitm
25. probabilities But for latent mover stayer model with time dependent transition probabilities all parameters are not identified This model has consi derably more degrees of freedom than the of parameters I tightened the stop ping criterion to 10 But still all parameters are not identified I don t under stand why this is happening I fixed the response probabilities for stayers to 0 and 1 but still all parameters are not identified Fitting a latent mover stayer model generally is asking too much from the data Even if you have four or more panel waves standard errors are very large and for some data sets the algorithm won t even converge Therefore I have assumed that the stayers know what they are talking about i e the reliability of their answers is perfect the partially latent mover stayer model as Rolf Langeheine calls it Because you don t want to assu me stationary transition probabilities even that restricted PLMS model has some pa van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 11 rameters that are hard to estimate While running the analysis a first indication of this is the slow convergence of the algorithm For each iteration only a small improvement in fit is obtained This usually means that some parameters are highly correlated and hence their standard deviations are large Sometimes this can only be mended by adding some restrictions to the model For your dataset n
26. s and this difference is ignored in the measurement model the influence of gender on smoking behaviour will be estimated with a bias This type of heterogeneity can be tested with simultaneous estimation of the measurement model in subgroups of the sample However sample size sets limits to this type of corroboration of results 4 Conclusion The readers who have accompanied us to this point might want see an example For this we can refer to the literature in the present volume for instance Or better even we should refer to the reader s own data that can be analysed with latent class models References 1 Blumen I Kogan M amp McCarthy P J 1966 Probability models for mobility In P F Lazarsfeld amp N W Henry eds Readings in mathematical social science pp 318 334 reprint from The industrial mobility of labor as a probability process Cornell Univ Press Ithaca N Y idem 1955 Cambridge Massachusetts MIT Press van de Pol amp Mannan Questions of a Novice in Latent Markov Modelling 17 2 10 11 12 13 Clogg C C amp Goodman L A 1984 Latent structure analysis of a set of multi dimensional contingency tables Journal of the American Statistical Association 79 pp 762 771 Collins L M amp Wugalter S E 1992 Latent class models for stage sequential dynamic latent variables Multivariate Behavioral Research 27 pp 131 157 De Leeuw J van der Heij
27. tes but apparently this is an error prone procedure However a single manifest Markov chain can also be estimated in a much simpler way that is by selecting a standard analysis a mixed Markov model with only 1 chain This does not require any extra input with respect to restrictions or starting values Soon the use of starting values becomes inevitable for the present data because some transitions are impossible at the latent level In an educational setting such models in volve irreversible steps in learning Collins et al 1992 Langeheine et al 1994 0 u I have smoking data for grade 6 grade 8 and grade 11 students My su pervisor wants me to fit the models by restricting three parameters to 0 which are transition probabilities from state current smoker 1 to never smoker 4 from ex smoker 2 to never smoker and from experimental smoker 3 to never smo ker I did not get the correct result for latent Markov model when I fi zed 3 transition probabilities from state 1 to 4 2 to 4 and 3 to 4 to 0 Transition from any of the smoking conditions to never smoked 4 is impossible if people make no errors in responding The latent Markov model can be used to separate response errors from true change Doing so the latent matrix of transition probabili 86 118 ties should have zeroes for these impossible transitions i e T 0 dla Tabb gt 0 a b 4 with supers
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