Which scale each method reports

Every quantity a fitted model reports lives on one of three scales, and nothing in the output says which. This page says it once, per method, so that a ported script can be read without guessing.

Value

This page documents a convention. It is not a function, so it returns no value.

Details

The three scales are:

Link. The linear predictor of one distributional parameter, before its inverse link is applied. This is where the coefficients live, where the standard errors are symmetric, and where a random effect has its variance.
Response. The units of the outcome itself.
Unitless. A standardized quantity, such as a Pearson or quantile residual, which has no units by construction.

What each method returns

method	returns	scale
`predict()`	a summary of the PREDICTIVE distribution	response
`frm_linpred()`, default	the `mu` linear predictor	link
`frm_linpred(type = "response")`	the conditional mean	response
`frm_linpred(dpar = "sigma")`	that predictor	link; `"response"` gives response
`fitted()`	a summary of the conditional mean	response
`conditional_effects()`	`estimate__` and its band	response
`residuals()`, default	observed minus `fitted()`	response
`residuals(type = "pearson")`	that, over the conditional SD	unitless
`residuals(type = "deviance")`	signed root unit deviance	unitless
`residuals(type = "osa")`	one-step-ahead quantile residual	unitless
`simulate()`	draws of the outcome	response
`coef()`, `fixef()`, `ranef()`	coefficients, modes	link, per dpar
`confint()`, `vcov()`, `summary()`	the same	link, per dpar
`VarCorr()`	SDs and correlations of a predictor	link, per dpar
`sigma()`	the residual SD, `NA` if it varies by row	response
`hypothesis()`	`Estimate` and its interval	link, per dpar
`posterior_summary()`	a summary of a draws MATRIX	that matrix's own
`pp_check()`	the outcome against replicates	response
`bayes_R2()`	refuses on a `frmtmb_fit`	neither
`logLik()`, `AIC()`, `BIC()`	the fitted likelihood	the data's own

Where this differs from brms

predict() used to be the first, and is no longer: since item 2.6d it is brms's, a summary of the predictive distribution on the response scale. The LINEAR PREDICTOR, which is what it returned before and what the glmmTMB type vocabulary reaches, moved to frm_linpred() unchanged. On a lognormal fit the two differ by about 8 against about 6,700, which is why they could not share a name. Measured with dev/generics-scale.R and dev/generics-scale-brms.R.

sigma is the one that remains. summary()'s Regression Coefficients block and fixef() report every coefficient on its own link, so a sigma WITH A FORMULA and the default log link is reported as log(sigma) and can be negative. A sigma nobody wrote a formula for is not a coefficient at all: it is in summary()'s Further Distributional Parameters block on its own response scale, as in brms, and sigma() returns that number.

What agrees with brms without any conversion

fitted(), residuals(), simulate(), ngrps(), ranef(), conditional_effects() and the coefficients of a linear predictor whose link is the identity.

For a lognormal fit with a CONSTANT sigma and no truncation, fitted() is exp(mu + sigma^2 / 2), the mean of the distribution rather than its median exp(mu); the ratio between the two is exp(sigma^2 / 2), which is the whole of the discrepancy a ported script sees. Both conditions are load-bearing, and neither is obvious from the formula:

With a distributional sigma (bf(y ~ x, sigma ~ x)), sigma() returns NA with a warning, because there is no one number to return, and the formula gives NA with it. Use frm_linpred(dpar = "sigma", type = "response"), which reproduces fitted(dpar = "sigma")[, "Estimate"] exactly (identical() is TRUE).
Under truncation, fitted() is the TRUNCATED mean and the formula answers a different question. With a lower bound lb, the truncated mean is exp(mu + sigma^2 / 2) * pnorm(sigma - a) / pnorm(-a) with a = (log(lb) - mu) / sigma, so the naive formula falls short by a relative 1 - pnorm(-a) / pnorm(sigma - a) in each row. That is an identity, not an estimate, and there is no single number for it: it depends on where the bound sits in each row's distribution, near zero for a row far above the bound and approaching one for a row whose mean is below it (dev/generics-trunc.R sweeps three bounds on one design).

Measured on one 400-row lognormal fit against a two-chain brms fit of the same model (dev/generics-scale-brms.R, seed 2026): the largest relative disagreement in fitted() over the 400 rows is 0.0106, which is 0.113 of brms's own posterior standard deviation for that row and 0.058 of it at the median. predict() disagrees by roughly 440 at the median row and 765 at row 1, because it is a different scale rather than a different answer. Those two are given loosely on purpose: brms's predict() summarizes fresh predictive draws, so it moves between calls on one fit, where the fitted() figures above are deterministic and repeat to the last digit.

Examples

set.seed(2026)
dd <- data.frame(x = rnorm(200))
dd$y <- exp(rnorm(200, 8 + 0.4 * dd$x, 0.4))
fit <- frm(bf(y ~ x) + lognormal(), data = dd)

# predict() summarizes the predictive distribution; frm_linpred()
# is the linear predictor and fitted() the expected response
head(predict(fit, ndraws = 200))
#>      Estimate Est.Error      Q2.5    Q97.5
#> [1,] 3832.300 1535.2687 1715.7392 7631.894
#> [2,] 2162.525  931.8458 1051.6008 4591.020
#> [3,] 3580.358 1627.6984 1459.8579 7605.905
#> [4,] 3114.912 1282.5725 1335.0592 6272.223
#> [5,] 2405.244  946.9348  979.7072 4669.094
#> [6,] 1219.641  509.4483  512.4535 2496.950
head(frm_linpred(fit))
#>        1        2        3        4        5        6 
#> 8.214705 7.569773 8.061016 7.970746 7.736238 6.990888 
head(fitted(fit))
#>      Estimate Est.Error     Q2.5    Q97.5
#> [1,] 3984.813 126.97345 3743.561 4241.611
#> [2,] 2090.826  87.35786 1926.431 2269.250
#> [3,] 3417.131  98.16226 3230.053 3615.045
#> [4,] 3122.182  89.39296 2951.800 3302.398
#> [5,] 2469.519  84.57864 2309.189 2640.981
#> [6,] 1171.956  89.39571 1009.212 1360.944

# the last two are related by the family's own mean
head(exp(frm_linpred(fit) + sigma(fit)^2 / 2) -
       fitted(fit)[, "Estimate"])
#> 1 2 3 4 5 6 
#> 0 0 0 0 0 0 

# a sigma nobody wrote a formula for is a distributional parameter
# on its own scale, not a coefficient on its link
summary(fit)$spec_pars
#>        Estimate  Est.Error  l-95% CI  u-95% CI
#> sigma 0.3886927 0.01943464 0.3524084 0.4287128
sigma(fit)
#> [1] 0.3886927