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To give you an example: let's suppose I have 3 subscales A, B and C, each with 5 item, and I know

  1. The Cronbach's alpha of each subscale (let's suppose: 0.80 for A, 0.85 for B and 0.90 for C),

  2. The Cronbach's alpha of the global (15-item) scale (let's suppose: 0.90 for A+B+C=S).

If I assume that the three between-subscale correlations (AB, BC and AC) are equal, is there a formula to estimate such parameter?

Given that Cronbach's alpha derives from mean inter-item correlation (https://stats.stackexchange.com/a/339775/159259), then I think the question can be restated as:

Can we derive the between-subscale correlation (assuming it is constant across pairs) from the mean inter-item correlation of each subscale and of the global scale?

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