Situation: A three part ability test with multiple-choice, essay, and oral components. I have scores for each of 110 test takers on each part and overall. The three parts are unequally weighted in computing the overall score. There is reason to think that the three parts measure different abilities and are not highly correlated.
Available information: internal-consistency reliability of the M/C test (.82), and interrater-reliability of the grading of the essay and oral components (both near .95). The two parts were graded by different graders.
Desired goal: Estimate test-retest reliability of overall score
Would the low reliability of .82 drive the reliability of the total score, or would the reliability of the total score be closer to a weighted average of the three reliabilities? Is there a formula for combining these reliabilities to get an estimate of the test reliability of the overall score? What else would I need to know or what assumptions would I need to make to use the formula? If this is a complex matter, can you refer me to a relevant textbook or journal article?
I found these two related questions on this site but the answers were not helpful to me: Reliability of Composite Variable Made of 4 Measures? How to assess the reliability of a composite scores?
I guess I could do a Monte Carlo study. My gut feeling is that the low reliability will drive the overall reliability. But is there an analytic approach that would give a more definitive answer?