Student-t response family

student() defines a one-response Student-t distribution with formulas for location mu, residual scale sigma, and degrees of freedom nu.

Usage

student()

Value

A drm_family object.

Details

Here sigma is the Student-t scale, not the response standard deviation. The density is the location-scale t evaluated at z = (y - mu) / sigma, so the standard deviation of y is SD[y] = sigma * sqrt(nu / (nu - 2)) for nu > 2 and is strictly larger than sigma (about 73% larger at nu = 3, shrinking to sigma as nu -> Inf). This is the one implemented family whose public sigma is a scale rather than SD[y]: the location-scale t has no closed-form standard-deviation parameterization, and both drmTMB and its DRM.jl twin fit sigma as the scale.

The nu parameter uses a log link with a lower bound of 2: nu = 2 + exp(eta_nu). This keeps the fitted distribution in the finite-variance region (nu > 2) while still allowing heavy tails. The lower bound is a deliberate design choice, not a standard-deviation requirement: it guarantees a finite variance and a well-defined SD[y]. The model therefore cannot represent the very heavy tails of nu <= 2 (for example a Cauchy-like nu = 1); data that genuinely need nu <= 2 would require lifting the floor, which is not implemented. check_drm() warns when the fitted nu approaches the boundary at 2, where the slant of the likelihood in nu is weakly identified. Ordinary mu random intercepts and independent numeric slopes such as (1 | id) and (0 + x | id) are supported in the first Student-t mixed-model slice; correlated slopes, sigma random effects, and nu random effects remain separate planned gates.

Examples

student()
#> $name
#> [1] "student"
#> 
#> $family
#> [1] "student"
#> 
#> $n_response
#> [1] 1
#> 
#> $dpars
#> [1] "mu"    "sigma" "nu"   
#> 
#> $links
#>         mu      sigma         nu 
#> "identity"      "log"    "logm2" 
#> 
#> attr(,"class")
#> [1] "drm_family"