It's a Friday night, so a good time to derive the likelihood function of a simple model to estimate the death rate from an illness that kills quickly but takes a long time to cure. Buckle up!
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By the time the illness has passed, we could estimate its death rate given infection as (reported deaths + unrep'd deaths)/(rep'd deaths + unrep'd deaths + rep'd recoveries + unrep'd recs). 2 problems for us: unreported X isn't known, and deaths might arrive sooner than recovery
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For the moment, let's set reporting rates to 1. Why is the arrival time of death vs recovery an issue? Imagine the death rate is 20%. Deaths take 1 week on average, but rec takes 3 weeks. You infect 1000 people and wait two weeks; 100 die, and 100 are cured. What's your estimate?
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You wouldn't say that it's 100/200= .5. There are all these sick people who haven't died or ben cleared yet. Their survival so far contains useful information for our estimate. So how do we use it?
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Let's call f(d|t) the density of death arrival times t. f(c|t) is the density of dates that you get cleared. F() indicates CDF. θ is the death rate to be estimated. The data we need is each patient's state s (dead, sick, cleared), and days since infection, t.
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So what's the likelihood of observing this data for a patient, p(s,t|θ)? If they're dead, it's θ*f(d|t), so log likelihood contribution is log(θ) + log(f(d|t)) If they're cleared, it's (1-θ)*f(c|t); ll contribution = log(1-θ) + log(f(c|t)).
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If they're neither dead nor alive, a bit more tricky. Then it's the probability they're going to die but haven't yet, plus the probability they'll live but haven't been cleared yet. So θ*(1 - F(d|t))+ (1 -θ)*(1-F(c|t)). Taking logs of that is a bit nasty, so we use log_sum_exp
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log_sum_exp(a, b) = log(exp(a) + exp(b)). So log(θ*(1 - F(d|t))+ (1 -θ)*(1-F(c|t))) = log_sum_exp(log(θ) + log(1 - F(d|t)), log(1-θ) + log(1 - F(c|t))). Beautiful! Two more challenges: what distributions for f(d|t) and f(c|t); and what about low reporting?
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On dist choice: one nice fact is that a mixture of normal densities has a mixture of normal CDF. So you can use mixtures of normals for your f()s to capture most of the weirdness going on in unusual arrival time distributions, while staying analytically convenient.
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On reporting rates--this is a real challenge! I'm sure there are clever people who do this for a living, but I'd use plug-in rates for sensitivity checking. It just doesn't seem like a easy thing to estimate. Have a great weekend!
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