# Data generation Data generation for disjunctive, conjunctive and additive latent class probabilistic latent feature models. ```r gendatLCplfm(N,objpar,attpar,sizepar,maprule="disj",model=1) ``` ## Arguments - `N`: Number of replications (e.g. persons) for which binary object-attribute associations are generated. - `objpar`: True objectparameters. If `model`=1, `model`=3, `model`=4, or `model`=6 `objpar` is a **J X F X T** array, if `model`=2 or `model`=5 `objpar` is a **J X F** matrix. As object parameters are probabilities they should be between 0 and 1. - `attpar`: True attributeparameters. If `model`=2, `model`=3, `model`=5, or `model`=6 `attpar` is a **K X F X T** array, if `model`=1 or `model`=4 `attpar` is a **K X F** matrix. As attribute parameters are probabilities they should be between 0 and 1. - `sizepar`: A **T**-vector of true class size parameters. - `maprule`: Disjunctive (`maprule="disj"`), conjunctive (`maprule="conj"`) or additive (`maprule="add"`) mapping rule of the latent class probabilistic latent feature model. - `model`: The type of dependency and heterogeneity assumption included in the model. `model`=1, `model`=2, `model`=3 represent models with a constant object-feature classification per person and with, respectively, class-specific object parameters, class-specific attribute parameters, and class-specific object- and attribute parameters. `model`=4, `model`=5, `model`=6 represent models with a constant attribute-feature classification per person and with, respectively, class-specific object parameters, class-specific attribute parameters, and class-specific object- and attribute parameters. ## Details The function `gendatLCplfm` generates binary object-attribute associations for **N** replications according to a disjunctive, conjunctive or additive latent class probabilistic latent feature model of a specific model type. In addition, the function computes the **J X K** matrix of marginal object-attribute association probabilities and a **J X K X T** array of class-specific object-attribute association probabilities. To compute association probabilities the function **gendatLCplfm** uses a vector of class size parameters (`sizepar`) a matrix or array of object parameters (`objpar`) and a matrix or array of true attribute parameters (`attpar`) as input. According to the **disjunctive** probabilistic latent feature model, object **j** is associated to attribute **k** if the object and the attribute have at least one feature in common. More specifically, for `model=1` the class-specific object-attribute association probability in cell **(j,k)** for the disjunctive model can be computed as: $$ p(j,k|t)=1-\prod_f(1-objpar[j,f,t]*attpar[k,f]).p(j,k|t)=1-\prod_f(1-objpar[j,f,t]*attpar[k,f]). $$ According to the **conjunctive** probabilistic latent feature model, object **j** and attribute **k** are associated if object **j** has all the features that are linked to attribute **k**. In particular, for `model=1`,the class-specific object-attribute association probability in cell **(j,k)** for the conjunctive model can be computed as: $$ p(j,k|t)=\prod_f(1-(1-objpar[j,f,t])*attpar[k,f]).p(j,k|t)=\prod_f(1-(1-objpar[j,f,t])*attpar[k,f]). $$ According to the **additive** probabilistic latent feature model, an object and an attribute are more likely to be associated if they have more features in common. In particular, for `model=1`,the class-specific object-attribute association probability in cell **(j,k)** for the additive model can be computed as: $$ p(j,k|t)= \frac{1}{F} * \sum_f(objpar[j,f,t])*attpar[k,f]).p(j,k|t)= (1/F) * \sum_f(objpar[j,f,t])*attpar[k,f]). $$ The marginal object-attribute association probability can be computed as follows: $$ p(j,k)=\sum_t sizepar[t]*p(j,k|t).p(j,k)=\sum_t sizepar[t]*p(j,k|t). $$ ## Returns - **call**: Parameters used to call the function. - **data**: **I X J X K** matrix of association probabilities. - **class**: **I**-vector that contains latent class membership of each replication. - **condprob.JKT**: **J X K X T** array of class-specific conditional object-attribute association probabilities. - **margprob.JK**: **J X K** matrix of marginal object-attribute association probabilities. ## References Meulders, M., Tuerlinckx, F., and Vanpaemel, W. (2013). Constrained multilevel latent class models for the analysis of three-way three-mode binary data. **Journal of Classification, 30 (3)**, 306-337. ## Author(s) Michel Meulders ## See Also `LCplfm` ## Examples ```r ## Not run: # define constants I<-500 J<-10 K<-8 F<-2 T<-2 # model 1 # generate true parameters objpar<-array(runif(J*F*T),c(J,F,T)) attpar<-matrix(runif(K*F),c(K,F)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=1) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=1,maprule="conj") # model 2 # generate true parameters objpar<-matrix(runif(J*F),nrow=J) attpar<-array(runif(K*F*T),c(K,F,T)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=2) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=2,maprule="conj") # model 3 # generate true parameters objpar<-array(runif(J*F*T),c(J,F,T)) attpar<-array(runif(K*F*T),c(K,F,T)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=3) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=3,maprule="conj") # model 4 # generate true parameters objpar<-array(runif(J*F*T),c(J,F,T)) attpar<-matrix(runif(K*F),c(K,F)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=4) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=4,maprule="conj") # model 5 # generate true parameters objpar<-matrix(runif(J*F),nrow=J) attpar<-array(runif(K*F*T),c(K,F,T)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=5) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=5,maprule="conj") # model 6 # generate true parameters objpar<-array(runif(J*F*T),c(J,F,T)) attpar<-array(runif(K*F*T),c(K,F,T)) sizepar<-rep(1/T,T) # generate data d<-gendatLCplfm(N=I,objpar=objpar,attpar=attpar,sizepar=sizepar,maprule="conj",model=6) # estimate parameters of true model res<-LCplfm(data=d$data,F=2,T=2,model=6,maprule="conj") ## End(Not run) ```

gendatLCplfm function