Comments on Section 2.3

[athowes]

• "There are also a range of trade offs that need consideration." Perhaps would be good to give a summary of these trade offs?
• It's a little unusual to list methods from least to most approximate, as discussion of more approximate methods paves the way for understanding of less approximate methods? Though here I see that having the full picture and then removing parts can also work.
• Under 2.3.2 why is the fact that it's daily reporting being introduced? Is this assumption used for all methods? If so, it should be in 2.3.1. If not, why is it just being set for 2.3.2?
  • Are the event times obviously written to be in days somewhere? Otherwise $\lfloor S \rfloor$ doesn't make sense
• In 2.3.2 you could say what the parameter vector $\boldsymbol{\theta}$ and data vector $\mathbf{Y}$ are. If the parameter vector changes between models, then perhaps just the data vector which I'd assume is the same?
  • $\mathbf{Y} = (P{L, 1}, \ldots, P{L, n}, P{R, 1}, \ldots, P{R, n}, S{L, 1}, \ldots, S{L, n}, S{R, 1}, \ldots, S{R, n})$?
• For Equation (22) it is the joint likelihood of the data, but it is following Equation (21) which is the conditional probability of the secondary event having observed the primary event. What about the probability of having observed the primary event? Isn't this being omitted in the joint likelihood?
• Equation (23) maybe should be $x_i \sim g_P(x_i | \cdots)$ rather than $x_i \sim g_P(x | \cdots)$?
• On line 662, perhaps write "we can equivalently write down the approximate latent variable model (Section X or Equation Y)". In general, there are lots of models with lots of names so when possible refer to things by section or equation I think
• For Equation (31) why not write $\text{Unif}(xi | p{L, i}, p_{R, i})$ rather than write the general version and say that it's the uniform case? It's already been specified to the uniform case in Equation (28) and (29). For both the primary and secondary events right?
• Line 732, you've already said about the parameter vector and data vector. Don't think need to say again.
• For Section 2.3.9, is there a nice way to have two sections: continuous time methods, and discrete time methods? It feels a bit odd to have the subsections of 2.3.9 be one level deeper than the continuous time methods. Could consider having 2.3 be continuous time and 2.4 being discrete time?
• In Equation (37) there is a lowercase $s$ for secondary event time and lowercase $p$ for primary event time (but in Equation [38] and Equation [39] say, they are uppercase). In general be consistent about $s, p, S, P$. In this case I think it's Equations 38 and 39 which are wrong. Also something like $d = S - P$ (line 689) seems suspicious, shouldn't it be $D = S - P$ and $d = s - p$?
• For Equation (34) why is this now called $\mathcal{L}_\text{cens+trunc}$ whereas previously it had the subscript "latent". I think I know that it's to set up calling the following things either "cens" or "trunc" only, but it seems a bit strange to change it half way through Section 2.3.4 for what I don't see as being a clear reason. What about changing away from the subscript "latent" at the start of Section 2.3.4 seeing as it's "without using latent variables"?

(I will edit this issue as I read and understand more)

Other misc things

• Why R version 4.2.2 on line 942 and then elsewhere R is not in unispaced font
• Line 953: "Rhat" and "R-hat"
• Line 975: why isn't https://github.com/epinowcast/epidist in unispace and linked?

[seabbs]

It's a little unusual to list methods from least to most approximate, as discussion of more approximate methods paves the way for understanding of less approximate methods? Though here I see that having the full picture and then removing parts can also work.

The full picture and then removing parts is the justification. If you don't do it this way its hard to reason on where the approximate methods are coming from and to see what the issues with them are.