Figure 1
Graphical proof that q A (a) + δ = q F (a) + q I (a) + q S (a) - q F (a + 1) The seven lines represent all possible situations in which cancer is diagnosed over the interval (a, a + 1). Therefore, the sum of all their probabilities represents the incidene of cancer over that interval. Each row corresponds to a unique potential history of cancer detection, so the probabilities on the left and right side of each row are identical. The labels on each side are different because the realized events differ depending on whether screening started at time 0 (left side) or time 1 (right side). Defining q d (a) = ∑ j q dj (a) and setting the left side sum of probabilities equal to the right side sum gives the formula on top. Under Assumption 1, that once a subject is detectable on screening, the subject will always be detectable on screening, δ = 0.