The model and its parameterisation

This vignette is the mathematical reference for freqTLS: the four-parameter logistic (4PL) thermal death-time curve, the direct CTmax/z parameterisation, the disjoint-bounds asymptote transform, the relative-versus-absolute survival threshold, and the exact algebraic bridge to bayesTLS. The package’s implementation contract is tested against these equations; this vignette mirrors it and checks the bridge identities numerically (no Stan required). If you already trust that freqTLS and bayesTLS target the same fitted curve, you can skip straight to the worked examples in vignette("freqTLS") and vignette("comparing-to-bayesTLS").

The 4PL thermal death-time curve

Let \(d\) be the exposure duration and \(\log d = \log_{10} d\). Survival probability follows the descending four-parameter logistic

\[ p = \mathrm{low} + \frac{\mathrm{up} - \mathrm{low}}{1 + \exp\!\big(k\,(\log_{10} d - \mathrm{mid})\big)}, \qquad k = \exp(\log k). \]

\(\mathrm{low}\) is the lower asymptote (survival at long exposures),
\(\mathrm{up}\) is the upper asymptote (survival at short exposures),
\(k > 0\) is the steepness, and
\(\mathrm{mid}\) is the midpoint on the \(\log_{10} d\) axis.

Survival is high at short durations and falls toward \(\mathrm{low}\) at long durations.

The direct `CTmax`/`z` parameterisation

freqTLS lets temperature enter the midpoint directly through the two headline quantities. With \(z_i = \exp(\eta_{\log z, i})\) and \(\mathrm{CTmax}_i\) a function of the design,

\[ \mathrm{mid}_i = \log_{10}(t_\mathrm{ref}) - \frac{T_i - \mathrm{CTmax}_i}{z_i}. \]

Two properties make this the right coordinate system for profiling:

At \(T_i = \mathrm{CTmax}_i\), the midpoint is exactly \(\log_{10}(t_\mathrm{ref})\) — i.e. CTmax is the temperature at which the threshold crossing occurs at the reference time \(t_\mathrm{ref}\).
The slope of the midpoint in temperature is \(\partial\,\mathrm{mid}/\partial T = -1/z\), so z is the change in temperature per decade (\(\log_{10}\) unit) of exposure duration — the thermal sensitivity (degrees Celsius per decade).

Because CTmax and z are model coordinates (not derived afterwards), each can be profiled directly. We can check the two properties on a fitted model:

set.seed(1)
dat <- simulate_tls(family = "binomial", CTmax = 36, z = 4, seed = 1)
fit <- fit_tls(dat, y = survived, n = total, time = duration, temp = temp,
               family = "binomial", tref = 1)
ct <- get_ctmax(fit)$estimate
zz <- get_z(fit)$estimate

# (1) midpoint at temp = CTmax equals log10(tref) = log10(1) = 0
predict(fit, data.frame(temp = ct, duration = 1), type = "midpoint")
#> [1] 0

# (2) midpoint slope in temperature is -1/z
mids <- predict(fit, data.frame(temp = c(ct, ct + 1), duration = 1), type = "midpoint")
c(slope = diff(mids), minus_one_over_z = -1 / zz)
#>            slope minus_one_over_z 
#>       -0.2501231       -0.2501231

Disjoint-bounds asymptotes

The asymptotes use the bayesTLS compute_4pl_bounds() parameterisation: the feasible band \([\ell, u]\) (default \([0, 1]\)) is split at its midpoint, and low and up each map an unconstrained coefficient onto one half, so \(\mathrm{up} > \mathrm{low}\) holds automatically:

\[ \mathrm{low} = \ell_{\min} + w_\mathrm{low}\,\mathrm{logit}^{-1}(\beta_\mathrm{low}), \qquad \mathrm{up} = u_{\min} + w_\mathrm{up}\,\mathrm{logit}^{-1}(\beta_\mathrm{up}). \]

low is confined to the lower half-band \([\ell_{\min}, \ell_{\max}]\) and up to the upper half-band \([u_{\min}, u_{\max}]\) (the bands meet, with a tiny separating gap, at the midpoint). Any \((\beta_\mathrm{low}, \beta_\mathrm{up})\) therefore gives a valid ordered pair \(\ell < \mathrm{low} < \mathrm{up} < u\). This is unconstrained and smooth, which keeps the optimiser and the profiles well-behaved, and it matches bayesTLS exactly so the two packages share the asymptote contract.

Under this parameterisation up has its own coordinate \(\beta_\mathrm{up}\), just like low. freqTLS nonetheless reports up with the delta-method Wald interval: its profile path is not yet wired for \(\beta_\mathrm{up}\) (the work is symmetric with low, simply not yet implemented), and it says so. Every other parameter is profiled on its single coordinate. The full table of coordinates and links:

Natural parameter	Internal coordinate	Link
`low`	`beta_low`	logit (onto the lower half-band)
`up`	`beta_up`	logit (onto the upper half-band)
`k`	`beta_logk`	log
`CTmax` (per group)	`beta_CT[g]`	identity
`z` (per group)	`beta_logz[g]`	log
`phi` (beta-binomial)	`log_phi`	log

Relative versus absolute thresholds

A “lethal time” such as the LT\(_{50}\) is the duration at which survival crosses a target probability. There are two conventions:

Relative (the freqTLS default): the target is interpreted relative to the fitted asymptotes, so \(p = 0.5\) means halfway between low and up — which is exactly the 4PL midpoint \(\mathrm{mid}\). This is the configuration that matches the bayesTLS target_surv = "relative" setting and is what the benchmark uses (see vignette("comparing-to-bayesTLS")).
Absolute: the target is an absolute survival probability, requiring \(p\) to lie strictly between low and up.

derive_lt() solves the 4PL for the crossing duration at a temperature, and plot_tdt_curve() plots the relative midpoint crossing against temperature. On the \(\log_{10} d\) axis the relative-threshold crossing is the line \(\log_{10} d = \mathrm{mid}(T) = \log_{10}(t_\mathrm{ref}) - (T - \mathrm{CTmax})/z\), whose slope is \(-1/z\) — the classic log-linear thermal death-time line.

# LT50 (relative midpoint) at three temperatures, in hours
derive_lt(fit, p = 0.5, temp = c(35, 36, 37))
#> [1] 1.6998018 0.9555978 0.5372198

The bridge to bayesTLS

bayesTLS fits the same 4PL but, in its constant-shape configuration (temp_effects = "mid"), parameterises the midpoint as a line in temperature,

\[ \mathrm{mid}(T) = b_{\mathrm{mid,Intercept}} + b_{\mathrm{mid},T_c}\,(T - \bar T), \]

and reads the thermal sensitivity and critical thermal maximum off that line:

\[ z = -\frac{1}{b_{\mathrm{mid},T_c}}, \qquad \mathrm{CTmax}(t_\mathrm{ref}) = \bar T + \frac{\log_{10}(t_\mathrm{ref}) - b_{\mathrm{mid,Intercept}}}{b_{\mathrm{mid},T_c}}. \]

Expanding the freqTLS midpoint as a line in \(T\) gives the inverse map:

\[ \beta_1 = -\frac{1}{z}, \qquad \beta_0 = \log_{10}(t_\mathrm{ref}) + \frac{\mathrm{CTmax} - \bar T}{z}, \]

\[ z = -\frac{1}{\beta_1}, \qquad \mathrm{CTmax} = \bar T + \frac{\log_{10}(t_\mathrm{ref}) - \beta_0}{\beta_1}. \]

These are exactly the bayesTLS identities. The reparameterisation shares (low, up, k) and is smooth and invertible (the Jacobian is nonsingular while z is finite), so the two models have the same likelihood, the same fitted curve, and the same maximum-likelihood estimate under the matched constant-shape, relative-threshold configuration. The only difference is that CTmax and z are coordinates in freqTLS, hence directly profile-able.

We can verify the bridge numerically without bayesTLS. Fit the midpoint line directly from the model’s own predicted midpoints (linear in \(T\)), then recover CTmax and z from the slope and intercept:

Tbar <- mean(c(35, 37))               # any centring temperature works
mids <- predict(fit, data.frame(temp = c(35, 37), duration = 1), type = "midpoint")
beta1 <- diff(mids) / 2               # slope in T  (= -1/z)
beta0 <- mids[1] - beta1 * (35 - Tbar) # intercept at T = Tbar

z_recovered     <- -1 / beta1
ctmax_recovered <- Tbar + (log10(1) - beta0) / beta1

rbind(
  fitted    = c(CTmax = ct, z = zz),
  recovered = c(CTmax = ctmax_recovered, z = z_recovered)
)
#>              CTmax        z
#> fitted    35.92586 3.998031
#> recovered 35.92586 3.998031