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StatsToDo: Cronbach's Alpha
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Explanations and References
Historical notes :
Javascript Program
In 1937, Kuder and Richardson proposed a coefficient to evaluate the reliability of measurements that composed of multiple binary items. In 1941 Hoyt modified this coefficient, adjusting it for continuity, and name this the Kuder Richardson Hoyt Coefficient. Cronbach in 1951 showed that this coefficient can be used generally in all scaled measurements. As he intended this be a starting point to develop even better indices, he named it Coefficient Alpha. This index is now known as Cronbach's Alpha, and is a widely accepted measurement of internal consistency (reliability) of a multivariate measurement composing of correlated items. If Cronbach's Alpha is applied to binary data, the result is the same as the Kuder Richardson Coefficient (K-R 20). The initial Cronbach's Alpha, calculated from the covariance matrix, is now known as the Unstandardized Alpha. This value tends to be unstable, and influenced by the scalar measurements used. A better Alpha is considered to be the Standardized Alpha, calculated from the correlation matrix. This is thought to be better as all variables are standardized to a mean of 0 and Standard Deviation of 1, the resulting Alpha is independent of the scales used. Both indices can be used to measure the internal consistency of multiple-item measurements, representing the averaged correlation between the items. As multiple-item measurements are in theory repeated measurements of the same thing, these indices represents the reliability of the overall set of measurements. Indices of reliability are often used in the early stages of developing a multiple-items measurement, to ensure that all the items measures a common concept. Items are added, removed, and modified, according to whether the indices of reliability improves, and usually until Alpha is greater than 0.7. A recent development is the calculations of the Standard Error of Alpha, and from which the confidence interval. This algorithm, by Duhachek and Iacobucci, is now included in this page The development of the Standard Error measurement allows statistical comparison and significance testing. As well as the 95% confidence interval, z=Alpha / SE can be calculated, and the probability that this does not differ from zero follows the normal z distribution. Example Data entry and interpretation of results are best demonstrated using the default example data from the Javascript panel. In this example, we administered 4 multiple choice questions to 20 students, using 0 for wrong answer and 1 for correct answer. The data is therefore a table of 20 rows, each from a student, and 4 columns, each for one of the tests. We wish to know if the tests are similar in what they detect, that is, if the correct and incorect answers agree. The program first produces the covariance matrix, the diagonal of which is the variance of each measurement (test), and the off diagonal cells the covariance between the measurements (tests). Please note that the covariance matrix can also be used as a second option for data entry. The program then calculates the unstandardized Alpha, which is Unstandardized -8.60 n=20 SE=3.03 95%CI=-14.53 to -2.67 The program then converts the covariance matrix to a correlation matrix, from which the standardized Alpha is produced. Standardized Alpha = 0.60, n=20, SE=0.16, 95%CI=0.28 to 0.91 ReferencesCronbach LJ (1951) Coefficient Alpha and the Internal Structure of Tests. Psychometrika 16:297-334 Allinson PD (1975) A Simple Proof of the Spearman-Brown Formula for Continuous Test lengths. Psychometrika 40:4 135-136 Bland JM and Altman DG (1997) Statistics Notes: cronbach's Alpha. BMJ 314(7080) 572 Lopez M (2007) 20th Annual Conference of the National Advisory Committee on Computing Qualifications (NACCQ 2007), Nelson, New Zealand. (www.naccq.ac.nz) Duhachek A and Iacobucci D (2004) Alpha's Standard Error (ASE): An Accurate and Precise Confidence Interval Estimate. Journal of Applied Psychology vol 89 No. 5 p 792-808
Data Entry Using Covariance or Correlation Matrix :
The folowing is algorithm for Chronbach's Alpha and its 95% confidence interval, based on the correlation matrix of the data
R provides an easy calculation for alpha based on the psych or the ltm packages, but some of the dependencies will not load in some version of R. Because of this, the native R code is shown as follows
myDat = ("
0 1 0 1
1 0 1 0
1 1 0 0
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1 1 1
")
#install.packages("ltm") # subpackage lvnorm unavailable
#library(ltm)
myDataFrame <- read.table(textConnection(myDat),header=FALSE) # conversion to data frame
summary(myDataFrame) # check means and SDs
n = nrow(myDataFrame) # number of cases
g = ncol(myDataFrame) # number of variables
n # sample size
g # number of variables
corMx <- cor(myDataFrame)
corMx # correlation matrix
jtphij = sum(corMx) # trace
alpha = 1 - g / jtphij
alpha = (g / (g -1)) * alpha
alpha # Chronbach Alpha based on correlation matrix
mxSq <- corMx %*% corMx
trphisq = 0
for(i in 1:g) trphisq = trphisq + mxSq[i,i] #trace
trsqphi = g * g
jtphisqj = sum(mxSq)
omega = jtphij * (trphisq + trsqphi)
omega = omega - 2 * g * jtphisqj
omega = (2 / jtphij^3) * omega
s2 = g**2 / (g-1)^2 * omega
se = sqrt(s2 / n)
se # Standard Error
cimin95 = alpha - 1.96 * se
cimax95 = alpha + 1.96 * se
c(cimin95, cimax95) # 95% confidence interval
The results produced are
> n # sample size
[1] 20
> g # number of variables
[1] 4
> corMx # correlation matrix
V1 V2 V3 V4
V1 1.00000000 -0.05263158 0.68824720 -0.07647191
V2 -0.05263158 1.00000000 -0.07647191 0.68824720
V3 0.68824720 -0.07647191 1.00000000 0.44444444
V4 -0.07647191 0.68824720 0.44444444 1.00000000
> alpha # Chronbach Alpha based on correlation matrix
[1] 0.5957404
> se # Standard Error
[1] 0.1605844
> c(cimin95, cimax95) # 95% confidence interval
[1] 0.2809950 0.9104857
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