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?

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.
