On the proportionality assumption in the competing risk analysis

Chiang (1961, 1968) was the first to give a systematic account on the probability of death from specific causes in the presence of competing risks under what is called “proportionality assumption” i.e. µi (t)|µ(t) is independent of t for each i. He studied the relationship between the partial crude and the corresponding crude probabilities and developed their estimates. The theory of competing risks has also been applied to the statistical studies of life testing and medical follow-up. The proportionality assumption was later attacked by Kimball (1969) and defended by Chiang (1970). This paper examines the various statements made by Kimball and Chiang based on the proportionality assumption. These statements are modfied and mathematical proofs under more general assumptions are supplied. Also the relation between the variations in the crude probabilities and the variations in the intensity functions is analyzed.