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Framing the Question The model supposes that when $n$ subjects are given a dose $x \gt 0$ and independently develop toxic responses, the count of those responses $Y(x)$ has a Binomial distribution with count parameter $n$ and probability parameter p(x;a,b) = \frac{1}{1 + 10^{(\log(a) - \log(x))/b}} = \frac{1}{1 + \exp\left(\beta_0 + \beta_1 \log(x)\...

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It depends on what the error estimates represent: errors in the measurements going into the model, or expected errors in predictions from a fitted model. For terminology, it's simplest to discuss in terms of the error variance estimates (which for a given study size bear a one-to-one relationship with the confidence-interval widths). Standard linear ...

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A few options: Fit a Poisson Generalized Linear Model with the Poisson family and only an intercept (this will estimate the mean, but on the log scale): tmp <- data.frame(y = c(10, 15, 12)) fit <- glm(y ~ 1, data=tmp, family=poisson()) fit summary(fit) confint(fit) exp(confint(fit)) Or use the fact that Poisson values are summable, so pass the sum ...

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If $\sigma$ is known, you use Gaussian distribution. When it is unknown, to account for the uncertainty over possible $\sigma$'s, you use student-t distribution. So, the issue is the unknown variance. When the student-t distribution is well approximated by Gaussian with large samples, the problem with uncertainty over $\sigma$ decreases and the two ...

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As @whuber notes in a comment, you do need to deal first with what seem to be incorrect premises in your approach. Most important, if your samples aren't truly random then "you can't use any of these methods to make inferences," as he put it. Fix that first. In terms of mean versus median as a measure of central tendency, the choice is yours based on your ...

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I think you might make progress by asking your audience to assume that these values are distributed on the range [0,5] in the set {(0:10)/2} with a beta-binomial distribution. The beta-binomial distribution arose from a different process than your situation but it is an ordered discrete distribution. Ben Bolker has a nice discussion of simulation using the ...

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One approach is to use bootstrapping: library(simpleboot) library(boot) set1 <- as.data.frame(c(3,3,2.5,2.5,4.5,3,2,4,3,3.5,3.5,2.5,3,3,3.5,3,3,4,3.5,3.5,4,3.5,3.5,4,3.5)) colnames(set1) <- "numbers" set1.boot = one.boot(set1\$numbers, mean, R=10^4) ## hist(set1.boot) boot.ci(set1.boot, type="bca") ## BOOTSTRAP CONFIDENCE INTERVAL CALCULATIONS ## ...

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Asymptotic confidence intervals on 'small' samples can have wider than nominal coverage, narrower than nominal coverage, or a mixture of both… sometimes a complicated mixture of both wider and narrower coverage. (This is a broad question, so this answer is similarly broad.) The particulars of wider vs narrower vs a mixture of both will depend on the ...

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In this answer I aim to describe the difference between confidence intervals and credible intervals in an intuitive way. I hope that this may help to understand: why/how credible intervals are better than confidence intervals. on which conditions the credible interval depends and when they are not always better. Credible intervals and confidence ...

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