14.6 Cronbach's alpha (\(\alpha\))
First, we will use the function alpha() from the package psych to estimate Cronbach's alpha coefficient (\(\alpha\)). Please, be aware that we are not considering the possibility of having two or more latent variables (i.e., factors). When using the function alpha() on all items, we are assuming that the scale is unidimensional (i.e., there is only one latent variable or factor).
The function alpha() provides a lot of useful information. However, the relevant coefficients and sections to inspect are raw_alpha, G6(smc), and the raw_alpha column under the section reliability if an item is dropped. In the Job Burnout Scale, we found a marginally optimal Cronbach's reliability coefficient (\(\alpha = .78\)). Although Guttman's \(\lambda_{6}\) (G6) was also optimal (\(\lambda_{6} = .81\)), be aware that it is usually considered an upper bound of Cronbach's alpha.
The alpha coefficients displayed in the column raw_alpha inform us about what could happen to the alpha coefficient of our scale (\(\alpha = .78\)) if we dropped the item under consideration. For example, the row related to item 11 (i.e., i11) shows that, if dropping that item, Cronbach's reliability coefficient (\(\alpha = .78\)) might increase by 1% (\(\alpha = .79\)). It is the only item of our scale that, if dropped, improves the reliability of our test.
psych::alpha(burnout, check.keys = TRUE)
##
## Reliability analysis
## Call: psych::alpha(x = burnout, check.keys = TRUE)
##
## raw_alpha std.alpha G6(smc) average_r S/N ase mean sd median_r
## 0.78 0.79 0.81 0.27 3.7 0.014 3.2 0.65 0.27
##
## lower alpha upper 95% confidence boundaries
## 0.75 0.78 0.81
##
## Reliability if an item is dropped:
## raw_alpha std.alpha G6(smc) average_r S/N alpha se var.r med.r
## i1 0.75 0.75 0.78 0.25 3.0 0.017 0.023 0.27
## i2 0.75 0.76 0.79 0.26 3.1 0.016 0.028 0.25
## i3 0.77 0.77 0.80 0.27 3.3 0.016 0.029 0.25
## i6 0.75 0.75 0.78 0.25 3.1 0.017 0.025 0.26
## i7 0.76 0.77 0.78 0.27 3.3 0.016 0.023 0.28
## i8 0.77 0.77 0.80 0.28 3.4 0.015 0.022 0.27
## i9 0.76 0.76 0.77 0.26 3.1 0.016 0.024 0.27
## i10 0.75 0.75 0.78 0.25 3.1 0.016 0.026 0.25
## i11 0.79 0.80 0.81 0.30 3.9 0.014 0.018 0.28
## i12 0.78 0.79 0.81 0.29 3.7 0.014 0.027 0.29
##
## Item statistics
## n raw.r std.r r.cor r.drop mean sd
## i1 510 0.68 0.68 0.65 0.57 2.7 1.12
## i2 510 0.65 0.65 0.60 0.54 3.4 1.08
## i3 510 0.61 0.58 0.51 0.45 2.9 1.36
## i6 510 0.68 0.67 0.63 0.56 2.5 1.15
## i7 510 0.59 0.60 0.57 0.47 3.3 1.11
## i8 510 0.55 0.54 0.47 0.40 2.8 1.19
## i9 510 0.63 0.64 0.62 0.52 3.6 1.03
## i10 510 0.65 0.67 0.62 0.55 3.6 0.99
## i11 510 0.36 0.37 0.26 0.20 3.7 1.12
## i12 510 0.43 0.44 0.32 0.29 3.6 1.04
##
## Non missing response frequency for each item
## 1 2 3 4 5 miss
## i1 0.12 0.36 0.23 0.23 0.06 0
## i2 0.04 0.21 0.23 0.39 0.13 0
## i3 0.17 0.30 0.12 0.25 0.15 0
## i6 0.21 0.38 0.17 0.19 0.04 0
## i7 0.05 0.19 0.25 0.35 0.15 0
## i8 0.12 0.38 0.17 0.23 0.09 0
## i9 0.04 0.11 0.24 0.42 0.20 0
## i10 0.02 0.15 0.22 0.46 0.15 0
## i11 0.05 0.12 0.17 0.40 0.26 0
## i12 0.03 0.13 0.20 0.45 0.19 0In the previous example, we used the function alpha() assuming that our scale was not unidimensional (i.e., all items reflect just one underlying latent variable or factor). However, we know that the burnout scale is not unidimensional, but bidimensional. We will create two datasets as a function of the EFA's factors previously extracted in the Chapter 13 (Exploratory Factor Analysis). Then, we will compute Cronbach's alpha per factor using the function alpha().
Please, note the difference of estimating Cronbach's alpha assuming unidimensionality or computing the same reliability coefficient of internal consistency when estimating alpha per factor. Estimating Cronbach's alpha on a non-unidimensional scale overestimates its reliability.
burnout %>%
select(c(i7, i9, i11)) %>%
psych::alpha(check.keys = TRUE)
##
## Reliability analysis
## Call: psych::alpha(x = ., check.keys = TRUE)
##
## raw_alpha std.alpha G6(smc) average_r S/N ase mean sd median_r
## 0.75 0.75 0.69 0.5 3 0.02 3.6 0.89 0.41
##
## lower alpha upper 95% confidence boundaries
## 0.71 0.75 0.79
##
## Reliability if an item is dropped:
## raw_alpha std.alpha G6(smc) average_r S/N alpha se var.r med.r
## i7 0.58 0.58 0.41 0.41 1.4 0.037 NA 0.41
## i9 0.58 0.58 0.41 0.41 1.4 0.037 NA 0.41
## i11 0.81 0.81 0.68 0.68 4.2 0.017 NA 0.68
##
## Item statistics
## n raw.r std.r r.cor r.drop mean sd
## i7 510 0.85 0.85 0.77 0.64 3.3 1.1
## i9 510 0.84 0.85 0.77 0.64 3.6 1.0
## i11 510 0.75 0.74 0.50 0.45 3.7 1.1
##
## Non missing response frequency for each item
## 1 2 3 4 5 miss
## i7 0.05 0.19 0.25 0.35 0.15 0
## i9 0.04 0.11 0.24 0.42 0.20 0
## i11 0.05 0.12 0.17 0.40 0.26 0When we computed Cronbach's reliability coefficient for the whole test (\(\alpha = .78\)), we concluded that the reliability was marginally optimal (\(\alpha > .80\) is usually optimal). A similar conclusion was reached with for Guttman's \(\lambda_{6}\) (G6, \(\lambda_{6} = .81\)).
However, when estimating Cronbach's reliability coefficient for the three items reflecting factor 1 (Affective), the reliability of the test reflecting this latent variable dropped (\(\alpha = .75\), \(\lambda_{6} = .69\)). We also found that item 11 (i.e., i11) was a case of concern because dropping this item would have improved greatly the reliability of the scale (\(\alpha = .81\)).
Below, we will compute Cronbach's alpha for the seven items reflecting factor 2 (Reward). We will set the argument check.keys = TRUE to detect negatively correlated items and reverse score them automatically if needed.
burnout %>%
select(-c(i7, i9, i11)) %>%
psych::alpha(check.keys = TRUE)
##
## Reliability analysis
## Call: psych::alpha(x = ., check.keys = TRUE)
##
## raw_alpha std.alpha G6(smc) average_r S/N ase mean sd median_r
## 0.78 0.78 0.77 0.34 3.6 0.015 3.1 0.75 0.33
##
## lower alpha upper 95% confidence boundaries
## 0.75 0.78 0.81
##
## Reliability if an item is dropped:
## raw_alpha std.alpha G6(smc) average_r S/N alpha se var.r med.r
## i1 0.73 0.73 0.71 0.31 2.7 0.019 0.0098 0.31
## i2 0.74 0.75 0.74 0.33 3.0 0.017 0.0162 0.31
## i3 0.77 0.77 0.76 0.36 3.4 0.015 0.0142 0.38
## i6 0.74 0.74 0.72 0.33 2.9 0.018 0.0111 0.31
## i8 0.75 0.75 0.74 0.34 3.1 0.017 0.0156 0.33
## i10 0.74 0.74 0.73 0.32 2.8 0.018 0.0156 0.31
## i12 0.79 0.79 0.78 0.39 3.9 0.015 0.0076 0.40
##
## Item statistics
## n raw.r std.r r.cor r.drop mean sd
## i1 510 0.75 0.76 0.73 0.63 2.7 1.12
## i2 510 0.68 0.69 0.61 0.54 3.4 1.08
## i3 510 0.62 0.58 0.47 0.41 2.9 1.36
## i6 510 0.71 0.70 0.66 0.57 2.5 1.15
## i8 510 0.68 0.67 0.59 0.52 2.8 1.19
## i10 510 0.70 0.72 0.66 0.59 3.6 0.99
## i12 510 0.47 0.49 0.35 0.30 3.6 1.04
##
## Non missing response frequency for each item
## 1 2 3 4 5 miss
## i1 0.12 0.36 0.23 0.23 0.06 0
## i2 0.04 0.21 0.23 0.39 0.13 0
## i3 0.17 0.30 0.12 0.25 0.15 0
## i6 0.21 0.38 0.17 0.19 0.04 0
## i8 0.12 0.38 0.17 0.23 0.09 0
## i10 0.02 0.15 0.22 0.46 0.15 0
## i12 0.03 0.13 0.20 0.45 0.19 0The previous results showed that Cronbach's reliability coefficient for the items reflecting factor 2 (Reward) was marginally optimal (\(\alpha = .78\), \(\lambda_{6} = .77\)). We found that, if we drop item 12 (i.e., i12), the reliability of the scale will be improved by 1% (\(\alpha = .79\)).