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The Kuder Richardson Coefficient of reliability (K-R 20) is used to test the reliability of binary measurements,
to see if the items within the instruments obtained the same binary (no/yes, right/wrong) results
over a population of testing subjects.
Javascript Program
The formula for the coefficient can easily be obtained from Wikipedia on the Internet.
## ExampleThe example data is artificially created to demonstrate the procedures, and not real. It purports to be a study of 4 no/yes questions in an exam. The questions are tested on 5 students.Please note This data set is deliberately small to make the demonstration easier to interpret. In a real study, many more questions (many tens) and students (serveral hundreds) would be required to ensure the results are reproducible.
We have 4 multiple choice questions (T1 to T4), administered to 5 students, as shown in the table on the left.
The number 0 is used to represent the wrong answer, and 1 the correct answer. The data set to be analysed is therefore as shown in the table to the right, and the results are K-R 20 = 0.75 The interpretation of the K-R 20 value is similar to that of Kappa. A K-R 20 of <0.2 is considered poor agreement, 0.21-0.4 fair, 0.41-0.6 moderate, 0.61-0.8 strong, and more than 0.8 near complete agreement. The original descriptions of K-R 20 provided no test of statistical significance or confidence interval, although these can be obtained using the Cronbach's Alpha algorithm. ## ReferencesKuder, G. F. ; M. W. Richardson (1937)The theory of the estimation of test reliability. Psychometrika, 2: 151-60 https://en.wikipedia.org/wiki/Kuder%E2%80%93Richardson_formulas Kuder Richardson Coefficient on Wikipedia
the following is the algorithm for Kuder Richardson Coefficient of Reliability
dat = (" 0 1 1 0 1 1 1 1 0 1 0 0 0 0 1 0 1 1 1 1 ") mx = read.table(textConnection(dat),header=FALSE) rows = nrow(mx) cols = ncol(mx) SumPQ = 0 # sum of pq for each item or col for(j in 1:cols) { p = 0 for(i in 1:rows) { if(mx[i,j]==1) { p = p + 1 } } p = p / rows SumPQ = SumPQ + p * (1.0 - p) } ex = 0 exx = 0 for(i in 1:rows) { p = 0 for(j in 1:cols) { if(mx[i,j]==1) { p = p + 1 } } ex = ex + p exx = exx + p * p } v = (exx - ex*ex/rows) / (rows) # variance between subjects kr_20 = (1.0 * cols / (cols - 1.0)) * ((v - SumPQ) / v) kr_20 # Kuder Richardson Coefficient of ReliabilityThe result is > kr_20 # Kuder Richardson Coefficient of Reliability [1] 0.7536232 |