Confidence Intervals for Coefficient Alpha and Standardized Alpha.
alphaci 
An R package for doing inference with coefficient alpha (Cronbach, 1951) and standardized alpha (Falk & Savalei, 2011). Many methods are supported, with special emphasis on small samples and non-normality.
Installation
Install from CRAN:
install.packages("alphaci")
Or use the development version.
# install.packages("remotes")
remotes::install_github("JonasMoss/alphaci")
Usage
Call the library function and load some data:
library("alphaci")
library("psychTools")
x <- bfi[, 1:5]
x[, 1] <- 7 - x[, 1] # Reverse-coded item.
head(x)
#> A1 A2 A3 A4 A5
#> 61617 5 4 3 4 4
#> 61618 5 4 5 2 5
#> 61620 2 4 5 4 4
#> 61621 3 4 6 5 5
#> 61622 5 3 3 4 5
#> 61623 1 6 5 6 5
Then calculate an asymptotically distribution-free confidence interval for ,
alphaci(x)
#> Call: alphaci(x = x)
#>
#> 95% confidence interval (n = 2709).
#> 0.025 0.975
#> 0.6828923 0.7246195
#>
#> Sample estimates.
#> alpha sd
#> 0.7037559 0.5536964
You can also calculate confidence intervals for standardized alpha
alphaci_std(x)
#> Call: alphaci_std(x = x)
#>
#> 95% confidence interval (n = 2709).
#> 0.025 0.975
#> 0.6938373 0.7331658
#>
#> Sample estimates.
#> alpha sd
#> 0.7135016 0.5218675
Supported techniques
alphaci supports three basic asymptotic confidence interval constructios. The asymptotically distribution-free interval of Maydeu-Olivares et al. 2007, the pseudo-elliptical construction of Yuan & Bentler (2002), and the normal method of van Zyl et al., (1999).
| Method | Description |
|---|---|
adf | The asymptotic distribution free method (Maydeu-Olivares et al. 2007). The method is asymptotically correct, but has poor small-sample performance. |
elliptical | The elliptical or pseudo-elliptical kurtosis correction (Yuan & Bentler, 2002). Uses the unbiased sample estimator of the common kurtosis (Joanes, 1998). Has better small-sample performance than adf and normal if the kurtosis is large and |
normal | Assumes normality of |
Standardized alpha, computed with alpha_std, support the same type arguments. Their formulas can be derived using the methods of Hayashi and Kamata (2005) and Neudecker (2007).
In addition, you may transform the intervals using one of four transforms:
- The Fisher transform, or
. Famously used in inference for the correlation coefficient.
- The
transform, where
. This is an asymptotic pivot under the elliptical model with parallel items.
- The identity transform. The default option.
- The
transform. This transform might fail when
is small, as negative values for
is possible, but
The option bootstrap does studentized bootstrapping Efron, B. (1987) with n_reps repetitions. If bootstrap = FALSE, an ordinary normal approximation will be used. The studentized bootstrap intervals are is a second-order correct, so its confidence intervals will be better than the normal approximation when is sufficiently large.
Finally, the option parallel = TRUE can be used, which is suitable if covariance matrix is compound symmetric. If the distribution is normal or (pseudo-)elliptic, it can be used to simplify the expression for the asymptotic variance of alpha and standardized alpha to
where is the common kurtosis parameter.
Similar software
There are several R packages that make confidence intervals for coefficient alpha, but not much support for standardized alpha. Most packages use some sort of normality assumption.
The alpha and alpha.ci functions of psych calculates confidence intervals for coefficient alpha following normal theory. semTools calculates numerous reliability coefficients with its reliability function. The Cronbach package provides confidence intervals based on normal theory, as does the alpha.CI function of psychometric. Confidence intervals for both alphas can, in principle, be calculated using structural equation modeling together with the delta method. Packages such as lavaan can be used for this purpose, but this is seldom done.
How to Contribute or Get Help
If you encounter a bug, have a feature request or need some help, open a Github issue. Create a pull requests to contribute.
References
- Falk, C. F., & Savalei, V. (2011). The relationship between unstandardized and standardized alpha, true reliability, and the underlying measurement model. Journal of Personality Assessment, 93(5), 445-453. https://doi.org/10.1080/00223891.2011.594129
- Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297-334. https://doi.org/10.1007/BF02310555#’
- Efron, B. (1987). Better Bootstrap Confidence Intervals. Journal of the American Statistical Association, 82(397), 171-185. https://doi.org/10.2307/2289144
- Maydeu-Olivares, A., Coffman, D. L., & Hartmann, W. M. (2007). Asymptotically distribution-free (ADF) interval estimation of coefficient alpha. Psychological Methods, 12(2), 157-176. https://doi.org/10.1037/1082-989X.12.2.157
- van Zyl, J. M., Neudecker, H., & Nel, D. G. (2000). On the distribution of the maximum likelihood estimator of Cronbach’s alpha. Psychometrika, 65(3), 271-280. https://doi.org/10.1007/BF02296146
- Yuan, K.-H., & Bentler, P. M. (2002). On robustness of the normal-theory based asymptotic distributions of three reliability coefficient estimates. Psychometrika, 67(2), 251-259. https://doi.org/10.1007/BF02294845
- Joanes, D. N., & Gill, C. A. (1998). Comparing measures of sample skewness and kurtosis. Journal of the Royal Statistical Society: Series D (The Statistician), 47(1), 183-189. https://doi.org/10.1111/1467-9884.00122
- Hayashi, K., & Kamata, A. (2005). A note on the estimator of the alpha coefficient for standardized variables under normality. Psychometrika, 70(3), 579-586. https://doi.org/10.1007/s11336-001-0888-1
- Neudecker, H. (2006). On the Asymptotic Distribution of the “Natural” Estimator of Cronbach’s Alpha with Standardised Variates under Nonnormality, Ellipticity and Normality. In P. Brown, S. Liu, & D. Sharma (Eds.), Contributions to Probability and Statistics: Applications and Challenges (pp. 167-171). World Scientific. https://doi.org/10.1142/9789812772466_0013