Dose Response Formula Terms

library(Colossus)
library(data.table)

Dose Response Formula

As previously discussed, Colossus features a term composed of the sum of multiple linear and non-linear elements which can be used to define many dose-response curves used in radiation epidemiology. These terms are referred to as dose-response terms, but there is nothing prohibiting them from being used for non-dose covariates. The following formulae are available, reproduced from the starting description vignette.

$$\begin{aligned} S_{NL}=\sum_i (\alpha_i \times \exp(x_i \cdot \beta_i)) + \sum_i (\beta_i \cdot (x_i)^2) + \sum_i F_{LT} + \sum_i F_{STP} + \sum_i F_{LQ} + \sum_i F_{LEXP}\\ F_{LT} = \begin{cases} \alpha_i \cdot (x-\beta_i) & (x>\beta_i) \\ 0 &\text{else} \end{cases}\\ F_{STP} = \begin{cases} \alpha_i & (x>\beta_i) \\ 0 &\text{else} \end{cases}\\ F_{LQ} = \begin{cases} \beta_i \cdot x & (x>\alpha_i) \\ \lambda_i \cdot x^2 + \nu_i &\text{else} \end{cases}\\ F_{LEXP} = \begin{cases} \beta_i \cdot x & (x>\alpha_i) \\ \lambda_i - \exp{(\nu_i + \mu \cdot x)} &\text{else} \end{cases}\\ T_j=S_{LL,j} \times S_{L,j} \times S_{PL,j} \times S_{NL,j} \end{aligned}$$

For every subterm type, there are between 1 and 3 parameters that fully define the curve. The Linear-Quadratic and Linear-Exponential curves are continuously differentiable, so there are only 2-3 parameters that can be set.

$$\begin{aligned} \lambda_{LQ} = \beta_{LQ}/(2\alpha_{LQ})\\ \nu_{LQ} = (\beta_{LQ}*\alpha_{LQ})/2\\ \nu_{LEXP} = \ln(\beta_{LEXP})-\ln(\mu_{LEXP})+\mu_{LEXP}*\alpha_{LEXP}\\ \lambda_{LEXP} = \beta_{LEXP}*\alpha_{LEXP}+exp(\nu_{LEXP}-\mu_{LEXP}*\alpha_{LEXP}) \end{aligned}$$

Using The Different subterms

Subterm	tform Entry	Description
Exponential	loglin_top	parameter in the exponent of the term, $\beta_i$
Exponential	loglin_slope	parameter multiplied by the exponential assumed to be 1 if not given, $\alpha_i$
Linear Threshold	lin_slope	slope for the linear term, $\alpha_i$
Linear Threshold	lin_int	intercept for the linear term, $\beta_i$
Step Function	step_slope	step function value, $\alpha_i$
Step Function	step_int	step function intercept, $\beta_i$
Quadratic	quad_slope	parameter multiplied by the squared value, $\beta_i$
Linear-Exponential	lin_exp_slope	Linear slope term, $\beta_i$
Linear-Exponential	lin_exp_int	Intercept between linear to exponential, $\alpha_i$
Linear-Exponential	lin_exp_exp_slope	Slope term in the exponential, $\mu_i$
Linear-Quadratic	lin_quad_slope	Linear slope term, $\beta_i$
Linear-Quadratic	lin_quad_int	Intercept between linear to quadratic, $\alpha_i$

The linear-exponential and linear-quadratic curves must be either completely fixed or completely free. In contrast, the exponential, linear threshold, and step-function curves can be partially fixed. The exponential term can be provided with only the covariate in the exponent and assume the magnitude to be 1. The linear threshold and step functions can be provided a fixed threshold covariate, which can be used to define a linear-no-threshold model or a combination of linear and step functions with a known threshold.

One issue with these dose-response formulas is that the order of the subterms is important. For instance, loglin_slope should be listed before loglin_top if both are used. The slope for the Linear Threshold and Step Functions should be given before the intercepts. Slopes are listed before intercept for the Linear-Quadratic and Linear-Exponential models and lin_exp_exp_slope is listed last. The “Correct_Formula_Order” function has been provided to handle the correct order. Colossus automatically calls this function before regression, but the user should be aware that the subterms may be reordered when results are returned.

term_n <- c(0, 1, 1, 0, 0)
tform <- c("loglin", "quad_slope", "lin", "lin_int", "lin_slope")
keep_constant <- c(0, 0, 0, 1, 0)
a_n <- c(1, 2, 3, 4, 5)
names <- c("a", "a", "a", "a", "a")
val <- Correct_Formula_Order(term_n, tform, keep_constant, a_n, names)
term_n <- val$term_n
tform <- val$tform
keep_constant <- val$keep_constant
a_n <- val$a_n
names <- val$names