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85  Cronbach Alpha

85.1 Definition

Cronbach’s \(\alpha\) (Cronbach 1951) is a widely used measure of internal consistency. If \(K\) scored items \(X_i\) (i.e. questions in a survey) are used to generate the construct \(Y = \sum_{i=1}^{K} X_i\) then Cronbach’s \(\alpha\) is defined as

\[ \alpha = \frac{K}{K-1} \left( 1 - \frac{\sum_{i=1}^{K} \sigma_{X_i}^2}{\sigma_Y^2} \right) \]

where \(\alpha \leq 1\) (and can be negative when items are negatively correlated), \(\sigma_Y^2\) is the Variance of the sum of scored items, and \(\sigma_{X_i}^2\) is the variance of item \(i\).

The standardized Cronbach’s \(\alpha\) is obtained by the following expression

\[ \alpha_s = \frac{K \bar{r}}{(1 + (K-1)\bar{r})} \]

where \(\bar{r}\) represents the mean of the off-diagonal coefficients in the upper (or lower) triangular correlation matrix.

Since \(\alpha\) and \(\alpha_s\) are both dependent on the number of items \(K\) it is possible to assess the impact of “leaving one item out” by applying the following derived formula

\[ \alpha_{(i)} = \frac{K-1}{K-2} \left( 1 - \frac{\sum_{j=1}^{i-1} \sigma_{X_j}^2}{\sigma_{Y_{(i)}}^2} - \frac{\sum_{j=i+1}^{K} \sigma_{X_j}^2}{\sigma_{Y_{(i)}}^2}\right) \]

where \(\alpha_{(i)}\) is Cronbach’s \(\alpha\) without item \(i\), \(Y_{(i)} = \sum_{j=1}^{i-1} X_j + \sum_{j=i+1}^{K} X_j\), and \(\sigma_{Y_{(i)}}^2\) is the variance of \(Y_{(i)}\).

For any index \(i = 1, 2, …, K\) the values of \(\alpha_{(i)}\) can be compared with \(\alpha\). If \(\alpha_{(i)} > \alpha\) then it can be concluded that leaving out the \(i\)-th item improves the internal consistency of the construct.

85.2 R Module

85.2.1 Public website

The Cronbach’s \(\alpha\) R module is available on the public website:

85.2.2 RFC

The Cronbach’s \(\alpha\) module is also available in RFC under the “Descriptive / Multivariate Descriptive Statistics” menu item.

To compute the Cronbach’s \(\alpha\) on your local machine, the following script can be used in the R console:

library(psych)
x <- array(round(runif(241*20, 1, 5)), dim=c(241,20), dimnames=
list(1:241, c('A1','A2','A3','A4','A5','A6','A7','A8','A9','A10','A11','A12',
'A13','A14','A15','A16','A17','A18','A19','A20')))
par1 <- TRUE
r <- alpha(x, check.keys = par1)
Warning in alpha(x, check.keys = par1): Some items were negatively correlated with the first principal component and were automatically reversed.
 This is indicated by a negative sign for the variable name.
r$total
r$alpha.drop
 raw_alpha std.alpha   G6(smc) average_r       S/N        ase     mean
 0.2731866 0.2714251 0.3230631 0.0182865 0.3725424 0.06779127 2.971577
        sd median_r
 0.3179216  0.01377
     raw_alpha std.alpha   G6(smc)  average_r       S/N   alpha se       var.r
A1-  0.2884968 0.2855329 0.3327109 0.02060062 0.3996446 0.06647419 0.004213473
A2   0.2584879 0.2566118 0.3015165 0.01784382 0.3451922 0.06925998 0.003924048
A3   0.2554548 0.2535426 0.2998381 0.01756294 0.3396612 0.06952368 0.003915859
A4-  0.2433502 0.2405095 0.2918618 0.01639373 0.3166722 0.07064221 0.004229439
A5-  0.2535513 0.2509343 0.3033996 0.01732591 0.3349963 0.06968828 0.004409158
A6   0.2470958 0.2451414 0.2968811 0.01680495 0.3247514 0.07030088 0.004269636
A7-  0.2748498 0.2740723 0.3220841 0.01948377 0.3775476 0.06775832 0.004207752
A8-  0.2958610 0.2950696 0.3415620 0.02155563 0.4185797 0.06582894 0.004211779
A9   0.2677399 0.2662358 0.3134317 0.01873877 0.3628357 0.06836524 0.004107420
A10- 0.2917096 0.2902518 0.3359228 0.02107020 0.4089504 0.06620912 0.004206546
A11  0.2371661 0.2358408 0.2897223 0.01598394 0.3086279 0.07127118 0.004311509
A12  0.2500021 0.2493141 0.2995194 0.01717944 0.3321150 0.07006101 0.004176513
A13- 0.2379158 0.2361447 0.2852917 0.01601047 0.3091485 0.07117473 0.004087521
A14  0.2235427 0.2235387 0.2761547 0.01492616 0.2878942 0.07255839 0.004182863
A15  0.2781613 0.2767967 0.3218201 0.01974628 0.3827370 0.06737937 0.004026054
A16- 0.2697981 0.2682371 0.3196372 0.01892761 0.3665628 0.06821690 0.004438923
A17  0.2790207 0.2766017 0.3228968 0.01972744 0.3823644 0.06731999 0.004148344
A18  0.2770772 0.2755453 0.3240848 0.01962547 0.3803485 0.06750533 0.004275373
A19- 0.2511921 0.2487928 0.3005423 0.01713245 0.3311906 0.06992585 0.004261199
A20  0.2722933 0.2699537 0.3161361 0.01909036 0.3697761 0.06797105 0.004106019
          med.r
A1-  0.01559641
A2   0.01273992
A3   0.01325473
A4-  0.01325473
A5-  0.01273992
A6   0.01428526
A7-  0.01559641
A8-  0.01559641
A9   0.01428526
A10- 0.01559641
A11  0.01039292
A12  0.01325473
A13- 0.01272572
A14  0.01267270
A15  0.01544151
A16- 0.01559641
A17  0.01559641
A18  0.01325473
A19- 0.01272572
A20  0.01325473

To compute the Cronbach’s \(\alpha\), the R code uses the alpha function from the psych library. Note: check.keys = TRUE will automatically reverse-key negatively correlated items; inspect the output to confirm which items were reversed.

85.3 Purpose

Cronbach’s \(\alpha\) is an internal-consistency coefficient for the reliability of a construct based on survey scores; it is a lower bound on reliability only when model assumptions (tau-equivalence and uncorrelated errors) are met. A commonly cited rule of thumb is \(\alpha > 0.7\) for acceptable internal consistency (Nunnally 1978), though appropriate thresholds vary by research context and discipline.

85.4 Pros & Cons

85.4.1 Pros

The Cronbach’s \(\alpha\) has the following advantages:

  • It is widely known and understood by many readers.
  • It is relatively easy to interpret and many software packages are able to compute it.

85.4.2 Cons

The Cronbach’s \(\alpha\) has the following disadvantages:

  • It is a function of \(K\) and does not always truly reflect internal consistency (it should rather be regarded as an indicative degree of internal consistency). As an alternative, McDonald’s \(\omega\) is less sensitive to the number of items and can be computed with the omega function from the psych library.
  • In the context of confirmatory factor analysis, McDonald’s \(\omega\) is theoretically preferable because \(\alpha\) assumes tau-equivalence of items, an assumption that CFA can directly test and often rejects.

85.5 Example

Interactive Shiny app (click to load).
Open in new tab

The analysis shows (among others) the values of \(\alpha\), \(\alpha_s\), and \(\alpha_{(i)}\) for a set of four items that belong the construct IM.Know (a form of intrinsic, academic motivation). It can be concluded that leaving out any item does not improve \(\alpha\).