(blet yi jk be the response for the -th replication at level for factor and level for factor to assess which factors best explain the response variablewe use the anova model yi jk ai gi ei jk ( where ai gi gi define [ua ]give the corresponding model matrix (cnote that the parameters are linearly dependent in this case for examplea - and -( to retain only linearly independent variables consider the -dimensional parameter vector [ua ]find the matrix such that me (dgive the model matrix corresponding to show that for the birthweight data in section there is no significant decrease in birthweight for smoking mothers [hintcreate new variable nonsmoke -smokewhich reverses the encoding for the smoking and non-smoking mothers thenthe parameter in the original model is the same as the parameter in the model bwt age nonsmoke age nonsmoke now find for and see if it contains zero prove ( and ( in the tobit regression model with normally distributed errorsthe response is modeled aszi if ui zi yi (xbs in )ui if zi ui where the model matrix and the thresholds un are given typicallyui suppose we wish to estimate th :(bs via the expectation-maximization methodsimilar to the censored data example let [ yn ]be the vector of observed data (ashow that the likelihood of isy ( thphs (yi > bx ph((ui > )/ ) :yi >ui :yi =ui where ph is the cdf of the ( distribution and phs the pdf of the ( distribution (blet and be vectors that collect all yi ui and yi ui respectively denote the corresponding matrix of predictors by and xrespectively for each observation yi ui introduce latent variable zi and collect these into vector for the same indices collect the corresponding ui into vector show that the complete-data likelihood is given by ky xbk kz xbk { cg(yz thexp ( ps ) / tobit regression