Cronbach’s alpha not equal to ICC 3,k I am calculating ICC over 16 items rated by 41 raters, using the ICC function from the psych package. The output is:
ICC 3,k = 0.99; number of subjects=16; number of judges=41

When I compute Cronbach’s alpha over the same dataset, using cronbach.alpha function from ltm package, I get an alpha equal to ICC 3,k, that is 0.99, but the output says
Items: 41
Sample units: 16

My interpretation is that items are the questions in my database and sample units are the raters. Thus, I need to transpose the dataframe, which I did.
The alpha of the transposed dataframe is 0.74, with 16 items and 41 sample units, thus being different from ICC 3,k.
How is this? What am I getting wrong? ICC 3,k and Cronbach alpha are supposed to be the same value?
Reproducible example
I ran this on anxiety data frame from the irr package on RStudio, which has 3 raters and 20 ratings.
When running ICC function (psych package), this is the output:
Call: ICC(x = anxiety, missing = FALSE, alpha = 0.05, lmer = TRUE, 
    check.keys = FALSE)

Intraclass correlation coefficients 
                         type  ICC   F df1 df2     p lower bound upper bound
Single_raters_absolute   ICC1 0.18 1.6  19  40 0.094      -0.077        0.48
Single_random_raters     ICC2 0.20 1.8  19  38 0.056      -0.039        0.49
Single_fixed_raters      ICC3 0.22 1.8  19  38 0.056      -0.046        0.52
Average_raters_absolute ICC1k 0.39 1.6  19  40 0.094      -0.275        0.74
Average_random_raters   ICC2k 0.43 1.8  19  38 0.056      -0.127        0.75
Average_fixed_raters    ICC3k 0.45 1.8  19  38 0.056      -0.153        0.77

 Number of subjects = 20     Number of Judges =  3

So, here ICC interprets my dataset as having 3 raters and 20 ratings, which is correct.
Now, using cronbach.alpha (ltm package) on exactly the same dataset, I get:
> cronbach.alpha(anxiety)

Cronbach's alpha for the 'anxiety' data-set

Items: 3
Sample units: 20
alpha: 0.453

Thus, alpha is equal to ICC(3,k), but the function interprets columns and rows differently, that is it should be Items: 20 and Sample units: 3.
Now, if I transpose the dataset and run cronbach.alpha on the transposed dataset, I get:
> cronbach.alpha(anxietyt)

Cronbach's alpha for the 'anxietyt' data-set

Items: 20
Sample units: 3
alpha: 0.699

Now, the interpretation of rows and columns is correct: I actually have 20 items and 3 sample units, but the alpha value is higher than ICC 3,k.
Now, as ICC 3,k and alpha are supposed to be equivalent, what is going wrong here?
 A: ICC 3,k gives the reliability of the mean over ratings of different raters. You have 3 raters and 20 ratees. ICC 3,k gives the reliability of the mean rating over the 3 raters, with one mean value computed for each of the 20 subjects.
Cronbach's alpha gives the reliability of a sum score (property of the ratees) over a set of items (raters). This corresponds to having 3 items and 20 sample units, not to having 20 items and 3 sample units.
A: I ran this on anxiety data frame from the irr package on RStudio, which has 3 raters and 20 ratings.
When running ICC function (psych package), this is the output:
Call: ICC(x = anxiety, missing = FALSE, alpha = 0.05, lmer = TRUE, 
    check.keys = FALSE)

Intraclass correlation coefficients 
                         type  ICC   F df1 df2     p lower bound upper bound
Single_raters_absolute   ICC1 0.18 1.6  19  40 0.094      -0.077        0.48
Single_random_raters     ICC2 0.20 1.8  19  38 0.056      -0.039        0.49
Single_fixed_raters      ICC3 0.22 1.8  19  38 0.056      -0.046        0.52
Average_raters_absolute ICC1k 0.39 1.6  19  40 0.094      -0.275        0.74
Average_random_raters   ICC2k 0.43 1.8  19  38 0.056      -0.127        0.75
Average_fixed_raters    ICC3k 0.45 1.8  19  38 0.056      -0.153        0.77

 Number of subjects = 20     Number of Judges =  3

So, here ICC interprets my dataset as having 3 raters and 20 ratings, which is correct.
Now, using cronbach.alpha (ltm package) on exactly the same dataset, I get:
> cronbach.alpha(anxiety)

Cronbach's alpha for the 'anxiety' data-set

Items: 3
Sample units: 20
alpha: 0.453

Thus, alpha is equal to ICC(3,k), but the function interprets columns and rows differently, that is it should be Items: 20 and Sample units: 3.
Now, if I transpose the dataset and run cronbach.alpha on the transposed dataset, I get:
> cronbach.alpha(anxietyt)

Cronbach's alpha for the 'anxietyt' data-set

Items: 20
Sample units: 3
alpha: 0.699

Now, the interpretation of rows and columns is correct: I actually have 20 items and 3 sample units, but the alpha value is higher than ICC 3,k.
Now, as ICC 3,k and alpha are supposed to be equivalent, what is going wrong here?
