Closed Emax shape and its normalized integrated response

The closed model has mean `a + delta * emax_shape(x, q, D)`, with an unrestricted baseline `a`, nonnegative endpoint effect `delta`, and shape `q` in `[0, 1]`. The normalized integrated excess response is `delta * emax_shape_area(q)`. The endpoint `q = 0` is linear; `q = 1` represents immediate saturation at every positive dose, with zero shape at dose zero. For `0 < q < 1`, ordinary Emax parameters satisfy `K = D * (1 - q) / q` and `b = delta / q`.

Usage

emax_shape(x, q, D)

emax_shape_area(q)

emax_response(x, a, delta, q, D)

emax_integrated_response(delta, q)

Arguments

x

Numeric nonempty dose vector in `[0, D]`.

q

Scalar shape in `[0, 1]`.

D

Positive dose-axis upper endpoint, in the same units as `x`.

a

Finite baseline response.

delta

Nonnegative endpoint excess response, in response units.

Value

`emax_shape()` returns a numeric vector. `emax_shape_area()` returns a scalar in `[1/2, 1]`. `emax_response()` returns a mean vector, and `emax_integrated_response()` a scalar in response units.

Details

Integration uses the specified linear dose axis. Arbitrary equally spaced codes do not recover an integral on unknown physical doses.

Examples

emax_shape(c(0, 0.5, 1), q = 0.75, D = 1)
#> [1] 0.0 0.8 1.0
emax_shape_area(0)
#> [1] 0.5
emax_integrated_response(delta = 2, q = 1)
#> [1] 2