R/modelfit.cor.R

In CDM: Cognitive Diagnosis Modeling

## File Name: modelfit.cor.R
## File Version: 1.27

#############################################################################
modelfit.cor <-
function( data, posterior, probs ){
    K <- max( apply( data, 2, max, na.rm=TRUE ) )
    if ( K>1 ){ stop("modelfit.cor only allows for dichotomous data\n") }
    data.resp <- 1 - is.na(data)
    data[ is.na(data) ] <- 9
    data1 <- data*data.resp
    I <- ncol(data)
    # calculate counts (ignore weights here!!)
    n11 <- t(  ( data==1) * data.resp ) %*% ( ( data==1) * data.resp )
    n10 <- t(  ( data==1) * data.resp ) %*% ( ( data==0) * data.resp )
    n01 <- t(  ( data==0) * data.resp ) %*% ( ( data==1) * data.resp )
    n00 <- t(  ( data==0) * data.resp ) %*% ( ( data==0) * data.resp )

#    p1 <- colMeans(  ( data==1) * data.resp )
    # p0 <- colMeans(  ( data==0) * data.resp )

    # expected counts
#    exp1 <- rep(NA, I )
#    for (ii in 1:I){
        # ii <- 1
#        pr.ii1 <- matrix( probs[ii,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * posterior
#        exp1[ii] <- sum( rowSums( p3ii ) * data.resp[,ii ] ) / sum( data.resp[,ii] )
#         exp1[ii] <- sum( colSums( posterior * data.resp[,ii] ) * probs[ii,2,] ) / sum( data.resp[,ii] )
#                }

    #********************************
    # covariances

    ip <- itempairs <- t( combn(I,2 ) )
    colnames(itempairs) <- c("item1", "item2" )
    itempairs <- as.data.frame( itempairs )
    itempairs$n11 <- n11[ ip ]
    itempairs$n10 <- n10[ ip ]
    itempairs$n01 <- n01[ ip ]
    itempairs$n00 <- n00[ ip ]
    itempairs$n <- rowSums( itempairs[, c("n11","n10", "n01","n00") ] )
    itempairs$Exp00 <- itempairs$Exp01 <- itempairs$Exp10 <- itempairs$Exp11 <- NA
    itempairs$corExp <- itempairs$corObs <- NA

    m1 <- matrix( c(1,1,1,0,0,1,0,0), 4, 2, byrow=T )

    # define further quantities
    itempairs$X2 <- NA
#    itempairs$G2 <- NA
    itempairs$RESIDCOV <- NA
    itempairs$Q3 <- NA

    #***
    # calculate expected score for every person and every item
    exp.ii.jj <- posterior %*% t( probs[,2,] )
    #***

    for (ii in 1:(I-1) ){
        for (jj in (ii+1):I){
    # ii <- 1
    # jj <- 2
        diijj <- data.resp[,ii ]*data.resp[,jj ]
        ii1 <- which ( itempairs$item1==ii &  itempairs$item2==jj )
        ps.iijj <- colSums( posterior[ data.resp[,ii]*data.resp[,jj]>0, ] )

#        pr.ii1 <- matrix( probs[ii,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        pr.jj1 <- matrix( probs[jj,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * pr.jj1 * posterior
#        itempairs[ii1,"Exp11"] <- sum( rowSums( p3ii ) * diijj )
         itempairs[ii1,"Exp11"] <- sum( probs[ii,2,]*probs[jj,2,] * ps.iijj )

#        pr.ii1 <- matrix( probs[ii,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        pr.jj1 <- matrix( probs[jj,1,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * pr.jj1 * posterior
#        itempairs[ii1,"Exp10"] <- sum( rowSums( p3ii ) * diijj )
        itempairs[ii1,"Exp10"] <- sum( probs[ii,2,]*probs[jj,1,] * ps.iijj )

#        pr.ii1 <- matrix( probs[ii,1,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        pr.jj1 <- matrix( probs[jj,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * pr.jj1 * posterior
#        itempairs[ii1,"Exp01"] <- sum( rowSums( p3ii ) * diijj )
        itempairs[ii1,"Exp01"] <- sum( probs[ii,1,]*probs[jj,2,] * ps.iijj )

#        pr.ii1 <- matrix( probs[ii,1,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        pr.jj1 <- matrix( probs[jj,1,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * pr.jj1 * posterior
#        itempairs[ii1,"Exp00"] <- sum( rowSums( p3ii ) * diijj )
        itempairs[ii1,"Exp00"] <- sum( probs[ii,1,]*probs[jj,1,] * ps.iijj )

        itempairs[ii1, "corObs"]  <-   .corr.wt( x=m1[,1,drop=FALSE],  y=m1[,2,drop=FALSE],
            w=as.numeric( itempairs[ii1,c("n11","n10","n01","n00") ] ) )

        itempairs[ii1, "corExp"]  <-   .corr.wt( x=m1[,1,drop=FALSE],  y=m1[,2,drop=FALSE],
            w=as.numeric( itempairs[ii1,c("Exp11","Exp10","Exp01","Exp00") ] ) )
        #***
        # Q3 statistic
#        pr.ii1 <- matrix( probs[ii,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        pr.jj1 <- matrix( probs[jj,2,], nrow=nrow(data), ncol=dim(probs)[3], byrow=T )
#        p3ii <-  pr.ii1 * posterior
#        expii <- rowSums( p3ii )
#        p3jj <-  pr.jj1 * posterior
#        expjj <- rowSums( p3jj )
        # calculate residuals
        data.res <- data[, c(ii,jj) ] - exp.ii.jj[, c(ii,jj) ]
        data.res <- data.res[ diijj==1, ]
        itempairs[ii1,"Q3"] <- stats::cor(data.res)[1,2]
                            }
                }

    ##############################
    itempairs$X2 <- ( itempairs$n00 - itempairs$Exp00 )^2 / itempairs$Exp00 +
                    ( itempairs$n10 - itempairs$Exp10 )^2 / itempairs$Exp10    +
                    ( itempairs$n01 - itempairs$Exp01 )^2 / itempairs$Exp01    +
                    ( itempairs$n11 - itempairs$Exp11 )^2 / itempairs$Exp11
    # G2
#    itempairs$G2 <- itempairs$n00 * log( itempairs$Exp00 / max( itempairs$n00, .01 ) ) +
#                    itempairs$n01 * log( itempairs$Exp01 / max( itempairs$n01, .01 ) ) +
#                    itempairs$n10 * log( itempairs$Exp10 / max( itempairs$n10, .01 ) ) +
#                    itempairs$n11 * log( itempairs$Exp11 / max( itempairs$n11, .01 ) )
#    itempairs$G2 <- -2*itempairs$G2

    itempairs$RESIDCOV <- ( itempairs$n11 * itempairs$n00 - itempairs$n10 * itempairs$n01 ) / itempairs$n^2 -
                ( itempairs$Exp11 * itempairs$Exp00 - itempairs$Exp10 * itempairs$Exp01 ) / itempairs$n^2
    ##############################
    # labels
    itempairs$item1 <- colnames(data)[ itempairs$item1 ]
    itempairs$item2 <- colnames(data)[ itempairs$item2 ]

    # residual of correlation
    itempairs$fcor <- cdm_fisherz( itempairs$corObs ) - cdm_fisherz( itempairs$corExp )

    #----
    # p values and p value adjustments adjustments

    # X2 statistic
    itempairs$X2_df <- 1
    itempairs$X2_p <- 1 - stats::pchisq(itempairs$X2, df=1 )
    itempairs$X2_p.holm <- stats::p.adjust( itempairs$X2_p, method="holm")
    itempairs$X2_sig.holm <- 1 * ( itempairs$X2_p.holm < .05 )
    itempairs$X2_p.fdr <- stats::p.adjust( itempairs$X2_p, method="fdr")
    # fcor statistic
    itempairs$fcor_se <- ( itempairs$n - 3 )^(-1/2)
    itempairs$fcor_z <- itempairs$fcor / itempairs$fcor_se
    itempairs$fcor_p <- 1 - stats::pnorm( abs(itempairs$fcor_z ) )
    itempairs$fcor_p.holm <- stats::p.adjust( itempairs$fcor_p, method="holm")
    itempairs$fcor_p.fdr <- stats::p.adjust( itempairs$fcor_p, method="fdr")


    #**********************
    # model fit
    modelfit <- data.frame( "est"=c(
            mean( abs( itempairs$corObs - itempairs$corExp ), na.rm=TRUE),
            sqrt( mean( ( itempairs$corObs - itempairs$corExp )^2, na.rm=TRUE ) ),
            mean( itempairs$X2, na.rm=TRUE ), # mean( itempairs$G2),
            mean( 100*abs(itempairs$RESIDCOV ), na.rm=TRUE ),
            mean( abs( itempairs$Q3 ), na.rm=TRUE)
                        ) )
    rownames(modelfit) <- c("MADcor", "SRMSR", "MX2", # "MG2",
                "100*MADRESIDCOV", "MADQ3" )

#    "pfit" <- data.frame( "item"=colnames(data), "pObs"=p1, "pExp"=exp1 )

    #*****
    # summary statistics
    modelfit.test <- data.frame("type"=c("max(X2)","abs(fcor)"),
            "value"=c( max( itempairs$X2), max( abs(itempairs$fcor) )  ),
            "p"=c( min( itempairs$X2_p.holm), min( itempairs$fcor_p.holm)  )
                )

    #****
    # print results
#    print( round(modelfit,5), digits=3 )
#    cat("MAD Correlation (Observed minus Expected)", round( MADcor, 4 ), "\n" )
    res <- list( "modelfit.stat"=modelfit, "itempairs"=itempairs,
        "modelfit.test"=modelfit.test  )
    return(res)
    }

#######################################################################


.corr.wt <- function( x, y, w=rep(1,length(x))) {
#  stopifnot(length(x)==dim(y)[2] )
  w <- w / sum(w)
  # Center x and y, using the weighted means
  x <- x - sum(x * w)
  ty <- y - sum( y * w)
  # Compute the variance
  vx <- sum(w * x * x)
  vy <- sum(w * ty * ty)
  # Compute the covariance
  vxy <- sum(ty * x * w)
  # Compute the correlation
  vxy / sqrt(vx * vy)
}

## 00 n00
## 10 n10
## 01 n01
## 11 n11

## w <- w / sum(w)
## w <- nij / ( n00 + n01 + n10 + n11 )
## xm=sum( x * w )=( 0*n00  + 1*n10 + 0*n01 + 1*n11 ) / N=( n10 + n11 ) / N
## ym=sum( y*w)=(n01+n11) / N
## ---
## variance:  Because it is a binary variable, it is p(1-p)
##   if p denotes proportion
##  Cov( X, Y )=E(X*Y) - E(X)*E(Y)
## E(X*Y)=n11 / N