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.

Install from CRAN:

Or use the development version.

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
```

`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 whenis small, as negative values for is possible, but do not accept them,

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

Finally, the option `parallel = TRUE`

can be used, which is suitable if covariance matrix

where

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.

If you encounter a bug, have a feature request or need some help, open a Github issue. Create a pull requests to contribute.

- 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