The cocron Package

Description

Statistical Comparisons of Two or more Alpha Coefficients.

Details

Statistical tests for the comparison between two or more alpha coefficients based on either dependent or independent groups of individuals. A web interface is available at http://comparingcronbachalphas.org. A plugin for the R GUI and IDE RKWard is included. Please install RKWard from https://rkward.kde.org to use this feature. The respective R package 'rkward' cannot be installed directly from a repository, as it is a part of RKWard.

Author(s)

Birk Diedenhofen

Statistical comparisons of n alpha coefficients

Description

Performs a test of significance for the difference between n alpha coefficients (Cronbach, 1951). The function expects raw data input from which the alpha coefficients are calculated.

Usage

cocron(data, dep = FALSE, standardized = FALSE, los = 0.05,
  conf.level = 0.95)

Arguments

Details

To compare n dependent or independent alpha coefficients (Cronbach, 1951), the methods by Feldt, Woodruff, and Salih (1987) implemented in cocron.n.coefficients are used.

Value

Returns an object of the class "cocron.n.coefficients" (see cocron.n.coefficients).

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11, 93-103.

See Also

cocron.n.coefficients, cocron.two.coefficients

Examples

data("knowledge")

# independent alpha coefficients
cocron(knowledge, dep=FALSE)

# dependent alpha coefficients
cocron(knowledge, dep=TRUE)

Statistical comparisons of n alpha coefficients

Description

Performs a test of significance for the difference between n alpha coefficients (Cronbach, 1951). The function expects alpha coefficients as input.

Usage

cocron.n.coefficients(alpha, n, items = NULL, dep = FALSE, r = NULL,
  los = 0.05, conf.level = 0.95)

Arguments

Details

To compare n dependent or independent alpha coefficients (Cronbach, 1951), the methods by Feldt, Woodruff, and Salih (1987) are implemented.

Value

Returns an object of the class "cocron.n.coefficients" with the following slots:

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11, 93-103.

See Also

cocron, cocron.two.coefficients

Examples

# independent alpha coefficients
cocron.n.coefficients(alpha=c(.784,.875,.936), items=c(5,5,5), n=c(51,101,151),
dep=FALSE)

# dependent alpha coefficients
r <- rbind(
 c(1,.8,.6,.75),
 c(NA,1,.65,.7),
 c(NA,NA,1,.55),
 c(NA,NA,NA,1)
)
cocron.n.coefficients(alpha=c(.857,.875,.800,.833), items=c(50,40,35,25), n=100,
dep=TRUE, r=r)

Statistical comparisons of two alpha coefficients

Description

Performs a test of significance for the difference between two alpha coefficients (Cronbach, 1951). The function expects alpha coefficients as input.

Usage

cocron.two.coefficients(alpha, n, dep = FALSE, r = NULL, los = 0.05,
  alternative = "two.sided")

Arguments

Details

For comparing two dependent or independent alpha coefficients (Cronbach, 1951), the methods described in Charter and Feldt (1996) are available, which were originally introduced in Feldt (1969) and Feldt (1980).

Value

Returns an object of the class "cocron.two.coefficients" with the following slots:

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

Charter, R. A., & Feldt, L. S. (1996). Testing the equality of two alpha coefficients. Perceptual and Motor Skills, 82, 763-768.

Feldt, L. S. (1969). A test of the hypothesis that Cronbach's alpha or Kuder-Richardson coefficient twenty is the same for two tests. Psychomelrika, 34, 363-373.

Feldt, L. S. (1980). A test of the hypothesis that Cronbach's alpha reliability coefficient is the same for two tests administered to the same sample. Psychometrika, 45, 99-105.

See Also

cocron, cocron.n.coefficients

Examples

# independent alpha coefficients
cocron.two.coefficients(alpha=c(.78,.71), n=c(41,151), dep=FALSE)

# dependent alpha coefficients
cocron.two.coefficients(alpha=c(.82,.89), n=27,dep=TRUE, r=.74)

Cronbach's Alpha

Description

Calculates Cronbach's alpha (Cronbach, 1951), a coefficient of internal consistency. The coefficient typically serves as an estimate of the reliability of a psychometric test.

Usage

cronbach.alpha(x, standardized = FALSE)

Arguments

Details

For a test consisting of k items that measures a quantity X, Cronbach's alpha is defined as

\alpha = \frac{k}{k - 1}\left(1 - \frac{\sum_{i=1}^{k}{\sigma_Y}_i^2}{\sigma_X^2}\right)

with X = Y_1 + Y_2 + ... + Y_k. {\sigma_Y}_i^2 is the variance of item i, and \sigma_X^2 the variance of the total test score for a sample of individuals that completed the test.

The standardized Cronbach's alpha is defined as

\alpha_s = \frac{k\overline{r}}{\left(1 + (k - 1)\overline{r}\right)}

where k is the number of items and \overline{r} the mean correlation between the items.

Cases that have missing values on any of the items are excluded.

Value

Returns Cronbach's alpha as a numeric object.

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

See Also

cocron, cocron.n.coefficients, cocron.two.coefficients

Examples

data("knowledge")

cronbach.alpha(knowledge$test1)
cronbach.alpha(knowledge$test2)

Confidence interval for Cronbach's Alpha

Description

Calculates a confidence interval for Cronbach's alpha (Cronbach, 1951).

Usage

cronbach.alpha.CI(alpha, n, items, conf.level = 0.95)

Arguments

Details

The lower bound of a confidence interval for an \alpha that is based on the data of n individuals who responded to k items is defined as

L = 1 - \left((1 - \alpha) F(1 - c/2)\right)

where c is the level of confidence and F(1 - c/2) the 100(1 - c/2) percentile of the F-distribution with df_1 = n - 1 and df_2 = (n - 1)(k - 1) (Feldt, Woodruff, & Salih, 1987, p. 95, formula 6). The upper bound of the confidence interval is computed as

U = 1 - \left((1 - \alpha) F(c/2)\right)

(Feldt et al., 1987, p. 95, formula 7).

Value

Returns a confidence interval for Cronbach's alpha as a numeric vector.

References

Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16, 297-334.

Feldt, L. S., Woodruff, D. J., & Salih, F. A. (1987). Statistical inference for coefficient alpha. Applied Psychological Measurement, 11, 93-103.

See Also

cronbach.alpha

Examples

cronbach.alpha.CI(alpha=.83, n=100, items=20, conf.level=.95)

Sample dataset: knowledge

Description

Data of 312 testees who completed two tests on general knowledge consisting of 30 questions each.

Usage

data(knowledge)

Format

A list that contains a matrix for each of the two tests holding 312 observations (rows) on the 30 questions (columns) presented. The ones and zeros stand for correct and incorrect responses, respectively.

Examples

data(knowledge)