Report

GROWTH MIXTURE MODELING 1 Shaunna L. Clark & Ryne Estabrook Advanced Genetic Epidemiology Statistical Workshop October 24, 2012 OUTLINE Growth Mixture Model Regime Switching Other Longitudinal Mixture Models OpenMx GMM How to extend GMM to FMM How to get individual class probabilities from OpenMx Exercise 2 HOMOGENEITY VS. HETEROGENEITY Previous session showed a growth model where everyone follows the same mean trajectory of use With some individual variations Is this an accurate representation of the development of substance abuse\dependence? Probably Not 25 Number of Drinks Per Week 20 15 10 5 0 12 14 16 18 Age 21 24 3 GROWTH MIXTURE MODELING (GMM) Muthén & Shedden, 1999; Muthén, 2001 Setting A single item measured repeatedly Hypothesized trajectory classes Example: Number of substances currently using Non-users; Early initiate; Late, but consistent use Individual trajectory variation within class Aims Estimate trajectory shapes Estimate trajectory class probabilities Linear, quadratic, etc. Proportion of sample in each trajectory class Estimate variation within class 4 LINEAR GROWTH MODEL DIAGRAM σ2 σ2Int,Slope Int I 1 mInt 1 xT1 σ2ε1 1 1 1 σ2Slope S mSlope 1 0 1 2 xT2 xT3 xT4 σ2ε2 σ2ε3 σ2ε4 3 4 xT5 σ2ε5 5 LINEAR GMM MODEL DIAGRAM C σ2 σ2Int,Slope Int I 1 mInt 1 xT1 σ2ε1 1 1 1 σ2Slope S mSlope 1 0 1 2 xT2 xT3 xT4 σ2ε2 σ2ε3 σ2ε4 3 4 xT5 σ2ε5 6 GMM EXAMPLE PROFILE PLOT 7 GMM EXAMPLE PROFILE PLOT 8 GROWTH MIXTURE MODEL EQUATIONS xitk = Interceptik + λtk*Slopeik + εitk Interceptik = α0k + ζ0ik Slopeik = α1k + ζ1ik for individual i at time t in class k εitk ~ N(0,σ) 9 LATENT CLASS GROWTH MODEL (LCGA) VS. GMM C σ2Int,Slope I 1 mInt 1 1 1 1 S mSlope 1 0 1 2 Same as GMM except no residual variance on growth factors 3 xT2 xT3 xT4 xT5 No individual variation within class 4 xT1 Nagin, 1999; Nagin & Tremblay, 1999 Everyone has the same trajectory LCGA is a special case of GMM 10 CLASS ENUMERATION Still cannot use LRT χ2 Information Criteria: AIC (Akaike, 1974), BIC (Schwartz,1978) Penalize for number of parameters and sample size Model with lowest value Interpretation and usefulness Profile plot Substantive theory Predictive validity Size of classes 11 ANALYSIS PLAN Determine growth function Determine number of classes Examine mean plots, with and without individual trajectories Determine if growth factor variances need: 1. 2. To be different from zero (GMM vs. LCGA) Should be held equal across classes Add covariates and distal outcomes 12 MODELING ZERO 13 HOW DO I MODEL ZEROS? Particularly relevant for substance abuse (or other outcome with floor effects) to model nonusers Some outcomes are right skewed so that there are many low values of the dependent variable However, some outcomes may have more zero’s than expected Example: Alcohol consumption; Individuals who never drink These individuals will always respond that consumed zero drinks 14 WHEN YOU HAVE MORE ZERO’S THAN EXPECTED In this case, zeros can be thought of coming from two populations Structural Zeros – zeros always occur in this population 1. Example: Never drinkers Others who produce zero with some probability at the time of measurement 2. Example: Occasional drinkers 15 ONE OPTION Identify those individuals in the two populations Structural zeros can then be eliminated Those who could potentially produce zeros are retained But it can very difficult to tell the difference between the two Or the population of interest is the entire population i.e. both drinkers and non-drinkers Stem issue 16 ZERO-CLASS Consider what you mean by a zero Only non-users who have not initiated use or those have initiated but only one try? Fix growth factor mean to zero Start not using, stay not using If only fix the means it will not be a pure zero-class Likely to pick up people that have tried once or twice, but have not moved to regular use Fix growth factor means and (co)variance to zero No variance in group Sometimes can cause computation issues 17 REGIME SWITCHING 18 IS GMM A GOOD MODEL FOR SUBSTANCE USE DEVELOPMENT? Maybe not Assumes that individuals remain in same trajectory over time Once a heavy smoker always a heavy smoker, even if you successfully quit for a period May not hold with many substance use outcomes Examples: Switching from moderate to heavy drinking, changing from daily smoker to nonsmoker 19 INDIVIDUAL TRAJECTORY PLOTS Dolan et al. (2005) presented the regime switching model (RSM) a way to get traction on this issue 20 DOLAN ET AL. REGIME SWITCHING MODEL (RSM) Regime = latent trajectory class Ex: habitual moderate drinkers, heavy drinkers Regime Switch = move from one regime to another Ex: A switch from moderate to heavy drinking Used latent markov modeling for normally distributed outcomes (Schmittmann et al., 2005) 21 RSM WITH ORDINAL DATA Dolan RSM model was designed to be used with normally distributed data Substance abuse measures are often: If continuous, not normally distributed Count Categorical Ex: # of drinking using days per month Ex: Do you use X substance? As we’ve seen in previous talks, can use the Mehta, Neale and Flay (2004) method when we have ordinal data 22 APPLICATION: ADOLESCENT DRINKING From Dolan et al. paper Data: National Longitudinal Survey Youth (NLSY97) Years 1998, 1999, 2000, 2001 737 white males and females Age 13 or 14 in 1998 Indicated the regularly drank alcohol Outcome: “In the past 30 days, on days you drank, how much did you drink?” Made ordinal: 0= 0-2 drinks; 1= 3 drinks; 2= 4-6 drinks, 3= 7+ drinks 23 MODEL SIMPLIFICATIONS FOR GMM & RSM APPLICATION Assumed linear model No correlation between intercept and slope Really quadratic Where you start drinking at the beginning of the study does not influence how your drinking develops during the study Transition probabilities equivalent across time Probability of drinking between age 12-13 are the same as 20-21 24 COMPARING GMM AND RSM Model -2*LL np AIC BIC saBIC 3-Class GMM -5077 18 995 -4199 -958 3-Class RSM -4589 26 990 -4183 -955 25 3-CLASS GMM PROFILE PLOT 12 10 8 Growing-72% 6 Moderate-18% Low-10% 4 2 0 -2 26 3-CLASS RSM PROFILE PLOT 12 10 8 6 Moderate-12% High-10% Low-77% 4 2 0 0 -2 1 2 3 27 GSM & RSM COMBINED PROFILE PLOT 12 10 8 RSW-Moderate 6 RSW-High RSW-Low GMM-Growing GMM-Moderate 4 GMM-Low 2 28 0 0 -2 1 2 3 RSM TRANSITION PROBABILITIES Class Low Moderate Heavy Low 0.74 0.01 0.04 Moderate 0.17 0.67 0.22 Heavy 0.09 0.32 0.74 Likely to stay in same class Low class unlikely to switch to other classes Most likely to switch between moderate and high drinking classes 29 OTHER LONGITUDINAL MIXTURE MODELS Longitudinal Latent Class Analysis Models patterns of change over time, rather than functional growth form Lanza & Collins, 2006; Feldman et al., 2009 LCA LCGA Binary item 35 11 3 category item 68 12 11 variables 3 Classes Quadratic 30 LATENT TRANSITION ANALYSIS •Models transition from one state to another over time •Unlike RSM, do not impost growth structure •Ex: Drinking alcohol or not over time •Graham et al., 1991; Nylund et al., 2006 •Script on the OpenMx forum C1 x1 x2 x3 x4 x5 Time 1 C2 x1 x2 x3 x4 x5 Time 2 31 OTHER LONGITUDINAL MIXTURE MODELS Survival Mixture Multiple latent classes of individuals with different survival functions Ex: Different groups based on age of initiation Kaplan, 2004; Masyn, 2003; Muthén & Masyn, 2005 32 OPENMX: GMM EXAMPLE 33 GMM_example.R 2 Classes Intercept and Slope MAKE OBJECTS FOR THINGS WE WILL REFERENCE THROUGHOUT THE SCRIPT #Number of measurement occasions nocc <- 4 #Number of growth factors (intercept, slope) nfac <- 2 #Number of classes nclass <- 2 #Number of thresholds; 1 minus categories of variable nthresh <- 3 #Function that will help us label our thresholds labFun <- function(name="matrix",nrow=1,ncol=1){matlab <matrix(paste(rep(name, each=nrow*ncol), rep(rep(1:nrow),ncol), rep(1:ncol,each=nrow),sep="_"))return(matlab)} 34 SETTING UP THE GROWTH PART OF THE MODEL #Factor Loadings lamda <- mxMatrix("Full", nrow = nocc, nco l= nfac, values = c(rep(1,nocc),0:(nocc-1)),name ="lambda") #Factor Variances phi <-mxMatrix("Diag", nrow = nfac, ncol = nfac, free = TRUE,labels = c("vi", "vs"), name ="phi") #Error terms theta <-mxMatrix("Diag", nrow = nocc, ncol = nocc, free = TRUE,labels = paste("theta",1:nocc,sep = ""), values = 1,name ="theta") #Factor Means alpha <- mxMatrix("Full", nrow= 1, ncol = nfac, free = TRUE, labels = c("mi", "ms"), name ="alpha") 35 GROWTH PART CONT’D #Item Thresholds thresh <- mxMatrix(type="Full", nrow=nthresh, ncol=nocc, free=rep(c(F,F,T),nocc), values=rep(c(0,1,1.1),nocc), lbound=.0001,labels=labFun("th",nthresh,nocc),name="thresh") cov <-mxAlgebra(lambda %*% phi %*% t(lambda) + theta, name="cov") mean <-mxAlgebra(alpha %*% t(lambda), name="mean” obj<-mxFIMLObjective("cov", "mean", dimnames=names(ordgsmsData), threshold="thresh",vector=TRUE) lgc <- mxModel("LGC", lamda, phi, theta, alpha, thresh, cov, mean, obj) 36 CLASS-SPECIFIC MODEL class1 <- mxModel(lgc, name ="Class1") class1 <- omxSetParameters(class1, labels = c("vi", "vs", "mi", "ms"), values = c(0.01, 0.05, 0.14, 0.32), newlabels = c("vi1", "vs1", "mi1", "ms1")) As in LCA, repeat for all your latent classes. Just make sure to change the class number and starting values accordingly. 37 CLASS PROPORTIONS #Fixing one probability to 1 classP <- mxMatrix("Full", nrow = nclass, ncol = 1, free = c(TRUE, FALSE), values = 1, lbound = 0.001, labels = c("p1", "p2"), name="Props") # rescale the class proportion matrix into a class probability matrix by dividing by their sum # (done with a kronecker product of the class proportions and 1/sum) classS <- mxAlgebra(Props%x%(1/sum(Props)), name ="classProbs") 38 CLASS-SPECIFIC OBJECTIVES # weighted by the class probabilities sumll<-mxAlgebra(-2*sum(log( classProbs[1,1]%x%Class1.objective + classProbs[2,1]%x%Class2.objective)), name = "sumll") # make an mxAlgebraObjective obj <- mxAlgebraObjective("sumll") 39 FINISH IT OFF # put it all in a model gmm <- mxModel("GMM 2 Class", mxData(observed = ordgsmsData, type ="raw”), classP, classS, sumll, obj) class1, class2, # run it gmmFit <- mxRun(gmm, unsafe = TRUE) # run it again using starting values from previous run summary(gmmFit2 <- mxRun(gmmFit)) 40 DIFFERENCE BETWEEN GMM AND FMM? C σ2 F 1 F σ2 C In t I 1 1 1 1 1 1 x1 x2 x3 x4 x5 xT1 xT2 xT3 xT4 xT5 41 Factor Mixture Model Intercept Only Growth Mixture Model GMM AND FMM The difference between the two models shown on the previous slide is that the factor loadings are restricted to 1 in the GMM where in the FMM they are freely estimated Adjust the script by having letting the values of the lambda matrix be freely estimated To run the FMM on the previous page, similar to factor analysis, need to fix a parameter so the model is identified Restrict the mean of two of the factors in two class to set the metric of the factor 42 FMM & MEASUREMENT INVARIANCE Clark et al. (In Press) In previous version, the threshold of the items were measurement invariant across classes Classes were differentiated based on difference in the mean and variances of the factor Can also have models where there are measurement non-invariant thresholds Classes arising because of difference in item thresholds Add thresholds to class-specific statements Need to restrict the factor mean to zero because can’t identify factor mean and item thresholds 43 HOW DO WE EXTRACT CLASS PROBABILITIES AND CALCULATE ENTROPY IN OPENMX 44 Ryne Estabrook OPEN MX EXERCISE\HOMEWORK Adjust the GMM_example.R script to include: A quadratic growth function A third class Run it Re-run it Interpret the output What are the classes? 45