Getting ready

If the packages are not ready installed on your computer use the function install.packages(). Then we include the libraries:

library(iarm)
library(corrplot)
library(readr)

Uni-dimensionality is a fundamental assumption in the Rasch model. In fact, it is central to all item response theory (IRT) models.

Example 2: SPADI data, correlation between residuals

The SPADI was shown to be two-dimensional in a Rasch analysis with the revised version of the disability sub scale including six polytomous items (Jerosch-Herold et al, 2018). Note that we remove person 86 because of missing values. We read in the data using the read_csv() function from the readr package.

myfile <- "https://raw.githubusercontent.com/ERRTG/ERRTG.github.io/master/SPADI.csv"
SPADI <- read_csv(myfile)

We create a new data frame SPADI.items0 containing the items.

SPADI.items0 <- SPADI[, c("P1", "P2", "P3", "P4", "P5", "D1", "D2", "D3", "D4", "D5",
    "D6", "D7", "D8")]

We remove persons with only missing values and create a new data frame SPADI.items.

ok <- as.logical((rowSums(is.na(SPADI.items0)) - ncol(SPADI.items0)))
SPADI.items <- SPADI.items0[ok, ]
which(!ok)
[1] 86

Before we analyse the residual correlations we have a look at the regular item correlations. The correlations between items are calculated using the cor() function. The default is the Pearson correlation coefficient (appropriate for dichotomous items). We specify use = "pairwise.complete.obs" such that the correlation between each pair of items is computed using all complete pairs of observations on those items.

SPADI.cor.items <- cor(SPADI.items, use = "pairwise.complete.obs")

To graphically displays the correlation matrix we use the corrplot() function.

corrplot(SPADI.cor.items)

Again there are high positive correlations between the items (because they measure the same thing).

We compute the parameter estimates of a partial credit model for polytomous item responses using the PCM() function from the eRm package.

PCM <- PCM(SPADI.items)

We estimate the person locations and compute the residuals.

SPADI.locations <- person.parameter(PCM)
SPADI.resid <- residuals(SPADI.locations)

Then, we look at the correlation between the residuals.

SPADI.cor.resid <- cor(SPADI.resid, use = "pairwise.complete.obs")
corrplot(SPADI.cor.resid)

From the correlation plot we see that there are negative values.

We can find the largest \(Q_3\) value (the largest correlation) and use the recommendations to evaluate if this constitutes an anomaly.

SPADI.Q3.max <- max(SPADI.cor.resid[(SPADI.cor.resid < 1)])
SPADI.Q3.average <- mean(SPADI.cor.resid[(SPADI.cor.resid < 1)])
SPADI.Q3.max
[1] 0.3926997
SPADI.Q3.star <- SPADI.Q3.max - SPADI.Q3.average
SPADI.Q3.star
[1] 0.4702648

We see that in this example the difference between the largest \(Q_3\) value and the average residual correlation is larger than .20, so there is evidence of local response dependence. This can be visualised using the corrplot() function by specifying p.mat and sig.level.

M <- SPADI.cor.resid
meanM <- mean(M[upper.tri(M)])
adM <- M - meanM
corrplot(M, p.mat = adM, sig.level = 0.2, insig = "pch", type = "upper", diag = FALSE,
    cl.pos = "n")

References

Christensen, K. B., Makransky, G., & Horton, M. (2017). Critical Values for Yen’s Q 3 : Identification of Local Dependence in the Rasch Model Using Residual Correlations. Applied Psychological Measurement, 41(3), 178-194. http://doi.org/10.1177/0146621616677520

Jerosch-Herold, C., Chester, R., Shepstone, L., Vincent, J. I., & MacDermid, J. C. (2017). An evaluation of the structural validity of the shoulder pain and disability index (SPADI) using the Rasch model. Quality of Life Research, 27(2), 389-400. http://doi.org/10.1007/s11136-017-1746-7

Kreiner, S., & Christensen, K. B. (2004). Analysis of Local Dependence and Multidimensionality in Graphical Loglinear Rasch Models. Communications in Statistics - Theory and Methods, 33(6), 1239-1276. http://doi.org/10.1081/STA-120030148

Marais, I. (2013). Local Dependence. In Rasch Models in Health (Vol. 44, pp. 111-130). Hoboken, NJ USA: John Wiley & Sons, Inc. http://doi.org/10.1002/9781118574454.ch7

Yen, W. M. (1984). Effects of Local Item Dependence on the Fit and Equating Performance of the Three-Parameter Logistic Model. Applied Psychological Measurement, 8(2), 125-145. http://doi.org/10.1177/014662168400800201