Review why it works

[seabbs]

I spoke to @parksw3 a bit about this and if interested a review of the why it works sections would be great.

[parksw3]

In order to reason upon the distribution of the censored delay time $T_c$, it is useful to consider the time from the end (right) point of the primary censoring interval to the secondary time as a random variable,

$$ S_+ = S - (t_P + w_P) = T - ((t_P + w_P) - P) = T - C_P. $$

Where $T$ is the delay distribution of interest and $C_P = (t_P + w_P) - P$ is interval between the end (right) point of the primary censoring window and the primary event time; note that by definition $C_P$ is not observed but we can relate its distribution to the distribution of $P$: $F_{C_P}(p) = Pr(C_P < p) = Pr(P > w_P - p)$.

I'm confused about the last equation. Maybe I'm missing something obvious here. Shouldn't the last equation be:

$F_{C_P}(p) = Pr(C_P < p) = Pr((t_P + w_P) - P < p) = Pr(P > t_P + w_P - p)$?

It looks like it's missing $t_P$?

Otherwise, this all looks great to me. I think it's worth specifying the implicit assumption that the delay distribution doesn't change over time (see below).

[parksw3]

The use of integration by part is very neat. And so I'm guessing we can do the same with the derivation in the long paper, which might give slightly simpler notation and consistency?

Starting from here (equations 12-14 in the long paper):

$$ \begin{aligned} \mathrm{P}(S_L < S < S_R | P_L < P < P_R) &= \frac{\mathrm{P}(P_L < P < P_R, S_L < S < S_R)}{\mathrm{P}(P_L < P < P_R)} \\ &= \frac{\int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x) f_x(y-x) dy dx}{\int_{P_L}^{P_R} g_P(x) dx}\\ &= \int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x|P_L, P_R) f_x(y-x)dy dx \end{aligned} $$

If we assume that the forward distribution doesn't vary over time (such that $f_x = f$), then

$$ \int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x|P_L, P_R) f_x(y-x)dy dx = \int_{P_L}^{P_R} g_P(x|P_L, P_R) \big[F(S_R - x) - F(S_L - x)\big] dx $$

Then, by using integration by parts, we get:

$$ \begin{aligned} \int_{P_L}^{P_R} g_P(x|P_L, P_R) \big[F(S_R - x) - F(S_L - x)\big] dx &= F(S_R - P_R) - F(S_L - P_R) - \int_{P_L}^{P_R} G_P(x|P_L, P_R) \big[f(S_L - x) - f(S_R - x)\big] dx \end{aligned} $$

which is same as equation 3.2 in the current vignette.

And we can use the same treatment when we account for the truncation+censoring... (writing more later)

[SamuelBrand1]

It looks like it's missing t P ?

Yes! Thats a typo (I think, I'll just double check myself)