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Dose Response Formula Terms

Source: vignettes/Dose_Formula_Inputs.Rmd
Dose_Formula_Inputs.Rmd
library(Colossus)
#> Note: From versions 1.3.1 to 1.4.1 the expected inputs changed. Regressions are now run with CoxRun and PoisRun and formula inputs. Please see the 'Unified Equation Representation' vignette for more details.
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.

SNL=∑i(αi×exp(xi⋅βi))+∑i(βi⋅(xi)2)+∑iFLT+∑iFSTP+∑iFLQ+∑iFLEXPFLT={αi⋅(x−βi)(x>βi)0elseFSTP={αi(x>βi)0elseFLQ={βi⋅x(x>αi)λi⋅x2+νielseFLEXP={βi⋅x(x>αi)λi−exp(νi+μ⋅x)elseTj=SLL,j×SL,j×SPL,j×SNL,j \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.

λLQ=βLQ/(2αLQ)νLQ=(βLQ*αLQ)/2νLEXP=ln(βLEXP)−ln(μLEXP)+μLEXP*αLEXPλLEXP=βLEXP*αLEXP+exp(νLEXP−μLEXP*αLEXP) \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, βi\beta_i
Exponential loglin_slope parameter multiplied by the exponential assumed to be 1 if not given, αi\alpha_i
Linear Threshold lin_slope slope for the linear term, αi\alpha_i
Linear Threshold lin_int intercept for the linear term, βi\beta_i
Step Function step_slope step function value, αi\alpha_i
Step Function step_int step function intercept, βi\beta_i
Quadratic quad_slope parameter multiplied by the squared value, βi\beta_i
Linear-Exponential lin_exp_slope Linear slope term, βi\beta_i
Linear-Exponential lin_exp_int Intercept between linear to exponential, αi\alpha_i
Linear-Exponential lin_exp_exp_slope Slope term in the exponential, μi\mu_i
Linear-Quadratic lin_quad_slope Linear slope term, βi\beta_i
Linear-Quadratic lin_quad_int Intercept between linear to quadratic, αi\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.