Calculations of scores - Example 2 - conf.detect()

[melindahiggins2000]

I am working with a polytomous dataset with a 5-level LiKert scaled response (1=strongly disagree to 5=strongly agree), and I'm trying to follow the logic of example 2 for the conf.detect() function. I am interested in running the polyDETECT analysis tests. However, I'm not following how the scores were calculated?

score <- stats::qnorm( ( rowMeans( dat )+.5 )  / ( 30 + 1 ) )

I'm not familiar with this equation. Why is 0.5 added to the row means? And why is this product then divided by 30+1. I'm guessing the 30 comes from the 30 columns of the dat dataset?

Please add some additional details for this example in the sirt::conf.detect() help pages and/or a reference on this equation for calculating scores using stats::qnorm() and how to adjust this equation for other datasets.

Thank you for your time and clarification!

====================================
Full example code:

## Not run: 
#############################################################################
# EXAMPLE 2: Big 5 data set (polytomous data)
#############################################################################

# attach Big5 Dataset
data(data.big5)

# select 6 items of each dimension
dat <- data.big5
dat <- dat[, 1:30]

# estimate person score by simply using a transformed sum score
score <- stats::qnorm( ( rowMeans( dat )+.5 )  / ( 30 + 1 ) )

# extract item cluster (Big 5 dimensions)
itemcluster <- substring( colnames(dat), 1, 1 )

# DETECT Item cluster
detect1 <- sirt::conf.detect( data=dat, score=score, itemcluster=itemcluster )
  ##        unweighted weighted
  ## DETECT      1.256    1.256
  ## ASSI        0.384    0.384
  ## RATIO       0.597    0.597

# Exploratory DETECT
detect5 <- sirt::expl.detect( data=dat, score=score,
                     nclusters=9, N.est=nrow(dat)  )
  ## DETECT (unweighted)
  ## Optimal Cluster Size is  6  (Maximum of DETECT Index)
  ##   N.Cluster N.items N.est N.val      size.cluster DETECT.est ASSI.est RATIO.est
  ## 1         2      30   500     0              6-24      1.073    0.246     0.510
  ## 2         3      30   500     0           6-10-14      1.578    0.457     0.750
  ## 3         4      30   500     0         6-10-11-3      1.532    0.444     0.729
  ## 4         5      30   500     0        6-8-11-2-3      1.591    0.462     0.757
  ## 5         6      30   500     0       6-8-6-2-5-3      1.610    0.499     0.766
  ## 6         7      30   500     0     6-3-6-2-5-5-3      1.557    0.476     0.740
  ## 7         8      30   500     0   6-3-3-2-3-5-5-3      1.540    0.462     0.732
  ## 8         9      30   500     0 6-3-3-2-3-5-3-3-2      1.522    0.444     0.724

# Plot Cluster solution
pl <- graphics::plot( detect5$clusterfit, main="Cluster solution" )
stats::rect.hclust(detect5$clusterfit, k=6, border="red")

[alexanderrobitzsch]

Yes, 30 refers to the number of items time the maximum number of categories (this, of course, only works without missing data). I will adapt this in the manual. The transformation of the mean score is made in order to define z scores for extreme cases (a raw score of 0 or 30).

[melindahiggins2000]

Thank you for the quick reply. The dataset in this example dat had 30 items from with responses ranging from 0 to 2 (which I think aligns with 0=neutral, agree, strongly agree (from the original big5 dataset), 1=disagree, and 2=strongly disagree?). So, the rowMeans will range from 0 to 2.

How does adding 0.5 to the rowMeans and then dividing by the number of items + 1, compute a probability?

I see how taking the probability (or area under the normal curve with mean=0, sd=1) = p, and then running stats::qnorm(p) results in a z-score.

I'm just not following how the equation for the "transformed sum score" computes an area under the Normal curve?

Thank you again for your time and help!!
@melindahiggins2000