Dose Response Formula

As previously discussed, Colossus features a term composed of the sum of multiple linear and non-linear term 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 formula 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 which fully define the curve. The Linear-Quadratic and Linear-Exponential curves are continuously-differentiable, so there are only 2-3 parameters which 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 is important. For instance, loglin_slope should be listed prior to loglin_top if both are used. The slope for the Linear Threshold and Step Functions should be given prior to the intercepts. Slopes are listed prior to 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 prior to regression.

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
der_iden <- val$der_iden
names <- val$names