R/ctr_pml_plrt.R

In lavaan: Latent Variable Analysis

ctr_pml_plrt <- function(lavobject = NULL, lavmodel = NULL, lavdata = NULL,
                         lavsamplestats = NULL, lavpartable = NULL,
                         lavpta = NULL,
                         lavoptions = NULL, x = NULL, VCOV = NULL,
                         lavcache = NULL) {

    if(!is.null(lavobject)) {
        lavmodel <- lavobject@Model
        lavdata <- lavobject@Data
        lavoptions <- lavobject@Options
        lavsamplestats <- lavobject@SampleStats
        lavcache <- lavobject@Cache
        lavpartable <- lavobject@ParTable
        lavpta <- lavobject@pta
    }
    if(is.null(lavpta)) {
        lavpta <- lav_partable_attributes(lavpartable)
    }


    if(is.null(x)) {
        # compute 'fx' = objective function value
        # (NOTE: since 0.5-18, NOT divided by N!!)
        fx <- lav_model_objective(lavmodel       = lavmodel,
                                  lavsamplestats = lavsamplestats,
                                  lavdata        = lavdata,
                                  lavcache       = lavcache)
        H0.fx <- as.numeric(fx)
        H0.fx.group <- attr(fx, "fx.group")
    } else {
        H0.fx <- attr(attr(x, "fx"), "fx.pml")
        H0.fx.group <- attr(attr(x, "fx"), "fx.group")
    }

    # fit a saturated model 'fittedSat'
    ModelSat <- lav_partable_unrestricted(lavobject      = NULL,
                                          lavdata        = lavdata,
                                          lavoptions     = lavoptions,
                                          lavpta         = lavpta,
                                          lavsamplestats = lavsamplestats)

    # FIXME: se="none", test="none"??
    Options <- lavoptions
    Options$verbose <- FALSE
    Options$se <- "none"
    Options$test <- "none"
    Options$baseline <- FALSE
    Options$h1 <- FALSE
    fittedSat <- lavaan(ModelSat, slotOptions = Options,
                        slotSampleStats = lavsamplestats,
                        slotData = lavdata, slotCache = lavcache)
    fx <- lav_model_objective(lavmodel = fittedSat@Model,
                              lavsamplestats = fittedSat@SampleStats,
                              lavdata = fittedSat@Data,
                              lavcache = fittedSat@Cache)
    SAT.fx <- as.numeric(fx)
    SAT.fx.group <- attr(fx, "fx.group")

    # we also need a `saturated model', but where the moments are based
    # on the model-implied sample statistics under H0
    ModelSat2 <-
        lav_partable_unrestricted(lavobject      = NULL,
                                  lavdata        = lavdata,
                                  lavoptions     = lavoptions,
                                  lavpta         = lavpta,
                                  lavsamplestats = NULL,
                                  sample.cov     = computeSigmaHat(lavmodel),
                                  sample.mean    = computeMuHat(lavmodel),
                                  sample.th      = computeTH(lavmodel),
                                  sample.th.idx  = lavsamplestats@th.idx)

    Options2 <- Options
    Options2$optim.method <- "none"
    Options2$optim.force.converged <- TRUE
    Options2$check.start <- FALSE
    Options2$check.gradient <- FALSE
    Options2$check.post <- FALSE
    Options2$check.vcov <- FALSE
    fittedSat2 <- lavaan(ModelSat2,
                         slotOptions = Options2,
                         slotSampleStats = lavsamplestats,
                         slotData = lavdata, slotCache = lavcache)

    # the code below was contributed by Myrsini Katsikatsou (Jan 2015)

# for now, only a single group is supported:
# g = 1L


########################### The code for PLRT for overall goodness of fit

# First define the number of non-redundant elements of the (fitted)
# covariance/correlation matrix of the underlying variables.
#nvar <- lavmodel@nvar[[g]]
#dSat <- nvar*(nvar-1)/2
#if(length(lavmodel@num.idx[[g]]) > 0L) {
#    dSat <- dSat + length(lavmodel@num.idx[[g]])
#}

# select `free' parameters (excluding thresholds) from fittedSat2 model
PT.Sat2 <- fittedSat2@ParTable
dSat.idx <- PT.Sat2$free[ PT.Sat2$free > 0L & PT.Sat2$op != "|" ] # remove thresholds

# Secondly, we need to specify the indices of the rows/columns of vcov(), hessian, and
# variability matrix that refer to all SEM parameters except thresholds.
PT <- lavpartable
index.par <- PT$free[PT$free > 0L & PT$op != "|"]

# Thirdly, specify the sample size.
# nsize <- lavdata@nobs[[g]]
nsize <- lavsamplestats@ntotal

# Now we can proceed to the computation of the quantities needed for PLRT.
# Briefly, to say that PLRT is equal to the difference of two quadratic forms.
# To compute the first and second moment adjusted PLRT we should compute
# the asymptotic mean and variance of each quadratic quantity as well as
# their asymptotic covariance.

##### Section 1. Compute the asymptotic mean and variance of the first quadratic quantity
# Below I assume that lavobject is the output of lavaan function. I guess
# vcov(lavobject) can be substituted by VCOV object insed lavaan function
# defined at lines 703 -708. But what is the object inside lavaan function
# for getHessian(lavobject)?
if(is.null(VCOV)) {
    lavoptions$se <- "robust.huber.white"
    VCOV <- lav_model_vcov(lavmodel       = lavmodel,
                           lavsamplestats = lavsamplestats,
                           lavoptions     = lavoptions,
                           lavdata        = lavdata,
                           lavpartable    = lavpartable,
                           lavcache       = lavcache)
}
InvG_to_psipsi_attheta0 <- (lavsamplestats@ntotal * VCOV )[index.par, index.par, drop = FALSE]  #G^psipsi(theta0)
#below the lavaan function getHessian is used
#Hattheta0 <- (-1) * H0.Hessian
#Hattheta0 <- H0.Hessian
#InvHattheta0 <- solve(Hattheta0)
InvHattheta0 <- attr(VCOV, "E.inv")
InvH_to_psipsi_attheta0 <- InvHattheta0[index.par, index.par, drop = FALSE]   #H^psipsi(theta0)
if(lavmodel@eq.constraints) {
    IN <- InvH_to_psipsi_attheta0
    IN.npar <- ncol(IN)

    # create `bordered' matrix
    if(nrow(lavmodel@con.jac) > 0L) {
        H <- lavmodel@con.jac[, index.par, drop = FALSE]
        inactive.idx <- attr(H, "inactive.idx")
        lambda <- lavmodel@con.lambda # lagrangean coefs
        if(length(inactive.idx) > 0L) {
            H <- H[-inactive.idx,,drop=FALSE]
            lambda <- lambda[-inactive.idx]
        }
        if(nrow(H) > 0L) {
            H0 <- matrix(0,nrow(H),nrow(H))
            H10 <- matrix(0, ncol(IN), nrow(H))
            DL <- 2*diag(lambda, nrow(H), nrow(H))
            # FIXME: better include inactive + slacks??
            E3 <- rbind( cbind(     IN,  H10, t(H)),
                         cbind( t(H10),   DL,  H0),
                         cbind(      H,   H0,  H0)  )
            Inv_of_InvH_to_psipsi_attheta0 <-
                MASS::ginv(IN)[1:IN.npar, 1:IN.npar, drop = FALSE]
        } else {
            Inv_of_InvH_to_psipsi_attheta0 <- solve(IN)
        }
    }
} else {
    # YR 26 June 2018: check for empty index.par (eg independence model)
    if(length(index.par) > 0L) {
        Inv_of_InvH_to_psipsi_attheta0 <-
            solve(InvH_to_psipsi_attheta0) #[H^psipsi(theta0)]^(-1)
    } else {
        Inv_of_InvH_to_psipsi_attheta0 <- matrix(0, 0, 0)
    }
}

H0tmp_prod1 <- Inv_of_InvH_to_psipsi_attheta0 %*% InvG_to_psipsi_attheta0
H0tmp_prod2 <- H0tmp_prod1 %*% H0tmp_prod1
E_tww <- sum(diag(H0tmp_prod1)) #expected mean of the first quadratic quantity
var_tww <- 2* sum(diag(H0tmp_prod2)) #variance of the first quadratic quantity

##### Section 2: Compute the asymptotic mean and variance of the second quadratic quantity.
# Now we need to evaluate the fitted (polychoric) correlation/ covariance matrix
# using the estimates of SEM parameters derived under the fitted model
# which is the model of the null hypothesis. We also need to compute the
# vcov matrix of these estimates (estimates of polychoric correlations)
# as well as the related hessian and variability matrix.
tmp.options <- fittedSat2@Options
tmp.options$se <- lavoptions$se
VCOV.Sat2 <- lav_model_vcov(lavmodel       = fittedSat2@Model,
                            lavsamplestats = fittedSat2@SampleStats,
                            lavoptions     = tmp.options,
                            lavdata        = fittedSat2@Data,
                            lavpartable    = fittedSat2@ParTable,
                            lavcache       = fittedSat2@Cache,
                            use.ginv       = TRUE)
InvG_to_sigmasigma_attheta0 <- lavsamplestats@ntotal * VCOV.Sat2[dSat.idx, dSat.idx, drop = FALSE]  #G^sigmasigma(theta0)
#Hattheta0 <- (-1)* getHessian(fittedSat2)
#Hattheta0 <- getHessian(fittedSat2)
#InvHattheta0 <- solve(Hattheta0)
InvHattheta0 <- attr(VCOV.Sat2, "E.inv")
InvH_to_sigmasigma_attheta0 <- InvHattheta0[dSat.idx, dSat.idx, drop = FALSE] #H^sigmasigma(theta0)
#Inv_of_InvH_to_sigmasigma_attheta0 <- solve(InvH_to_sigmasigma_attheta0) #[H^sigmasigma(theta0)]^(-1)
Inv_of_InvH_to_sigmasigma_attheta0 <- MASS::ginv(InvH_to_sigmasigma_attheta0,
                                               tol = .Machine$double.eps^(3/4))
H1tmp_prod1 <- Inv_of_InvH_to_sigmasigma_attheta0 %*% InvG_to_sigmasigma_attheta0
H1tmp_prod2 <- H1tmp_prod1 %*% H1tmp_prod1
E_tzz <- sum(diag(H1tmp_prod1))     #expected mean of the second quadratic quantity
var_tzz <- 2* sum(diag(H1tmp_prod2))#variance of the second quadratic quantity



##### Section 3: Compute the asymptotic covariance of the two quadratic quantities

drhodpsi_MAT <- vector("list", length = lavsamplestats@ngroups)
group.values <- lav_partable_group_values(fittedSat2@ParTable)
for(g in 1:lavsamplestats@ngroups) {

    #delta.g <- computeDelta(lavmodel)[[g]] # [[1]] to be substituted by g?
    # The above gives the derivatives of thresholds and polychoric correlations
    # with respect to SEM param (including thresholds) evaluated under H0.
    # From deltamat we need to exclude the rows and columns referring to thresholds.
    # For this:

    # order of the rows: first the thresholds, then the correlations
    # we need to map the rows of delta.g to the rows/cols of H_at_vartheta0
    # of H1

    PT <- fittedSat2@ParTable
    PT$label <- lav_partable_labels(PT)
    free.idx <- which(PT$free > 0 & PT$op != "|" & PT$group == group.values[g])
    PARLABEL <- PT$label[free.idx]

    # for now, we can assume that computeDelta will always return
    # the thresholds first, then the correlations
    #
    # later, we should add a (working) add.labels = TRUE option to
    # computeDelta
    #th.names <- lavobject@pta$vnames$th[[g]]
    #ov.names <- lavobject@pta$vnames$ov[[g]]
    #th.names <- lavNames(lavpartable, "th")
    #ov.names <- lavNames(lavpartable, "ov.nox")
    #ov.names.x <- lavNames(lavpartable, "ov.x")
    #tmp <- utils::combn(ov.names, 2)
    #cor.names <- paste(tmp[1,], "~~", tmp[2,], sep = "")

    # added by YR - 22 Okt 2017 #####################################
    #ov.names.x <- lavNames(lavpartable, "ov.x")
    #if(length(ov.names.x)) {
    #    slope.names <- apply(expand.grid(ov.names, ov.names.x), 1L,
    #                             paste, collapse = "~")
    #} else {
    #    slope.names <- character(0L)
    #}
    #################################################################

    #NAMES <- c(th.names, slope.names, cor.names)

    # added by YR - 26 April 2018, for 0.6-1
    # we now can get 'labelled' delta rownames
    delta.g <- lav_object_inspect_delta_internal(lavmodel = lavmodel,
                   lavdata = lavdata, lavpartable = lavpartable,
                   lavpta = lavpta, add.labels = TRUE, add.class = FALSE,
                   drop.list.single.group = FALSE)[[g]]
    NAMES <- rownames(delta.g)
    if(g > 1L) {
        NAMES <- paste(NAMES, ".g", g, sep = "")
    }

    par.idx <- match(PARLABEL, NAMES)
    if(any(is.na(par.idx))) {
        warning("lavaan WARNING: [ctr_pml_plrt] mismatch between DELTA labels and PAR labels!\n", "PARLABEL:\n", paste(PARLABEL, collapse = " "),
       "\nDELTA LABELS:\n", paste(NAMES, collapse = " "))
    }



    drhodpsi_MAT[[g]] <- delta.g[par.idx, index.par, drop = FALSE]
}
drhodpsi_mat <- do.call(rbind, drhodpsi_MAT)

tmp_prod <- t(drhodpsi_mat) %*% Inv_of_InvH_to_sigmasigma_attheta0 %*%
              drhodpsi_mat %*% InvG_to_psipsi_attheta0 %*% H0tmp_prod1
cov_tzztww <- 2* sum(diag(tmp_prod))

##### Section 4: compute the adjusted PLRT and its p-value
# PLRTH0Sat <- 2*nsize*(lavfit@fx - fittedSat@Fit@fx)
PLRTH0Sat <- 2*(H0.fx - SAT.fx)
PLRTH0Sat.group <- 2*(H0.fx.group - SAT.fx.group)
asym_mean_PLRTH0Sat <- E_tzz - E_tww
# catch zero value for asym_mean_PLRTH0Sat
if(asym_mean_PLRTH0Sat == 0) {
    asym_var_PLRTH0Sat <- 0
    scaling.factor <- as.numeric(NA)
    FSA_PLRT_SEM <- as.numeric(NA)
    adjusted_df  <- as.integer(NA)
    pvalue <- as.numeric(NA)
} else if(any(is.na(c(var_tzz, var_tww, cov_tzztww)))) {
    asym_var_PLRTH0Sat <- as.numeric(NA)
    scaling.factor <- as.numeric(NA)
    FSA_PLRT_SEM <- as.numeric(NA)
    adjusted_df  <- as.integer(NA)
    pvalue <- as.numeric(NA)
} else {
    asym_var_PLRTH0Sat  <- var_tzz + var_tww -2*cov_tzztww
    scaling.factor <- (asym_mean_PLRTH0Sat / (asym_var_PLRTH0Sat/2) )
    FSA_PLRT_SEM <- (asym_mean_PLRTH0Sat / (asym_var_PLRTH0Sat/2) )* PLRTH0Sat
    adjusted_df  <- (asym_mean_PLRTH0Sat*asym_mean_PLRTH0Sat) / (asym_var_PLRTH0Sat/2)
    # In some very few cases (simulations show very few cases in small
    # sample sizes) the adjusted_df is a negative number, we should then
    # print a warning like: "The adjusted df is computed to be a negative number
    # and for this the first and second moment adjusted PLRT is not computed."
    if(scaling.factor > 0) {
        pvalue <- 1-pchisq(FSA_PLRT_SEM, df=adjusted_df )
    } else {
        pvalue <- as.numeric(NA)
    }
}

list(PLRTH0Sat = PLRTH0Sat, PLRTH0Sat.group = PLRTH0Sat.group,
     stat = FSA_PLRT_SEM, df = adjusted_df, p.value = pvalue,
     scaling.factor = scaling.factor)
}
############################################################################


ctr_pml_aic_bic <- function(lavobject) {

    ########################## The code for PL version fo AIC and BIC
    # The following should be done because it is not the pl log-likelihood
    # that is maximized but a fit function that should be minimized. So, we
    # should find the value of log-PL at the estimated parameters through the
    # value of the fitted function.
    # The following may need to be updated if we change the fit function
    # so that it is correct for the case of missing values as well.

    logPL <- lavobject@optim$logl
    nsize <- lavobject@SampleStats@ntotal

    # inverted observed unit information
    H.inv <- lavTech(lavobject, "inverted.information.observed")

    # first order unit information
    J <- lavTech(lavobject, "information.first.order")

    # trace (J %*% H.inv) = sum (J * t(H.inv))
    dimTheta <- sum(J * H.inv)


    # computations of PL versions of AIC and BIC
    PL_AIC <- (-2)*logPL + 2*dimTheta
    PL_BIC <- (-2)*logPL + dimTheta *log(nsize)

    list(logPL = logPL, PL_AIC = PL_AIC, PL_BIC = PL_BIC)
}