How to infer the analysts' priors from this figure?

data analysis bayes
5

What I’ve settled on after a lot of thought is the following:

Here’s a plot exhibiting registration of my digitization of the data, using R package digitize

image
image1551×1221 179 KB

The original figure came from p 192 of this FDA CDER Multidisciplinary Review, which states, “The relationship followed a bell-shaped curve and a binary logistic regression with quadratic function was used to evaluate the relationship.”

library(rms, quietly = TRUE)
fit <- lrm(y ~ poly(x,2), data = exposure_response)
xYplot(Cbind(yhat, lower, upper) ~ x
     , data = Predict(fit, x = 10*(0:40), fun = plogis, conf.int = 0.9)
     , type = 'l'
     , ylim = 0:1
       )

image
image1344×960 38.6 KB

FYI, here is a spline regression for comparison:

fit2 <- lrm(y ~ rcs(x,4), data = exposure_response)
xYplot(Cbind(yhat, lower, upper) ~ x
     , data = Predict(fit2, x = 10*(0:40), fun = plogis, conf.int = 0.9)
     , type = 'l'
     , ylim = 0:1
       )

image
image1344×960 35.4 KB

To evaluate the implied informational content of Figure 1B, I fitted Beta distributions to the estimated response probabilities using the method of moments — see e.g., BDA3 Eq (A.3), p583:

xout <- seq(0, 400, 25)
df <- data.frame(x = xout
               , μ = with(meancurve, spline(x, y, xout=xout)$y)
               , lcl = with(lclcurve, spline(x, y, xout=xout)$y)
               , ucl = with(uclcurve, spline(x, y, xout=xout)$y)
                 )

## Add method-of-moments estimates; see BDA3 Eq (A.3), p. 583.
df <- Hmisc::upData(df
                  , stderr = (ucl - μ)/qnorm(0.95)
                  , `α+β` = μ*(1-μ)/stderr^2 - 1
                  , α = `α+β`*μ
                  , β = `α+β` - α
                  , print = FALSE
                    )
x μ lcl ucl stderr α+β α β
0 0.0088 -0.0179 0.0336 0.0151 37.4802 0.3308 37.1494
25 0.0204 -0.0451 0.0780 0.0350 15.2738 0.3115 14.9623
50 0.0493 -0.0654 0.1651 0.0704 8.4431 0.4160 8.0270
75 0.1224 -0.0500 0.2772 0.0941 11.1325 1.3630 9.7696
100 0.2099 0.0403 0.3714 0.0982 16.2103 3.4027 12.8076
125 0.3052 0.1613 0.4507 0.0885 26.0998 7.9654 18.1344
150 0.4066 0.2618 0.5199 0.0689 49.8290 20.2603 29.5687
175 0.4588 0.3196 0.5865 0.0776 40.2055 18.4459 21.7596
200 0.4761 0.3233 0.6190 0.0869 32.0648 15.2671 16.7978
225 0.4543 0.3016 0.5939 0.0848 33.4422 15.1935 18.2487
250 0.3868 0.2449 0.5391 0.0926 26.6580 10.3107 16.3473
275 0.2939 0.1437 0.4727 0.1087 16.5680 4.8694 11.6986
300 0.1974 0.0079 0.3887 0.1163 10.7113 2.1140 8.5973
325 0.1097 -0.0649 0.2703 0.0976 9.2513 1.0152 8.2361
350 0.0488 -0.0598 0.1622 0.0690 8.7612 0.4276 8.3336
375 0.0189 -0.0405 0.0789 0.0365 12.9046 0.2435 12.6611
400 0.0091 -0.0168 0.0327 0.0143 42.8959 0.3905 42.5054

To the extent that the \mathrm{Beta}(\alpha, \beta) can be interpreted as equivalent to \alpha-1 successes and \beta-1 failures in \alpha+\beta-2 Bernoulli trials (cf. BDA3 p.35), these calculations suggest that the plotted curves embody strong prior assumptions that go far beyond the data collected, and indeed beyond any data that ever could be collected, since you can’t observe fewer than 0 successes. Thus, the form of the analysis seems to have ‘put a thumb on the scale’. How reasonable does this mode of interpretation seem to the Bayesians here?

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