# Functions to transform and backtransform kinetic parameters for fitting

## Usage

transform_odeparms(parms, mkinmod, transform_rates = TRUE, transform_fractions = TRUE) backtransform_odeparms(transparms, mkinmod, transform_rates = TRUE, transform_fractions = TRUE)

## Arguments

parms
Parameters of kinetic models as used in the differential equations.
transparms
Transformed parameters of kinetic models as used in the fitting procedure.
mkinmod
The kinetic model of class mkinmod, containing the names of the model variables that are needed for grouping the formation fractions before ilr transformation, the parameter names and the information if the pathway to sink is included in the model.
transform_rates
Boolean specifying if kinetic rate constants should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. If TRUE, also alpha and beta parameters of the FOMC model are log-transformed, as well as k1 and k2 rate constants for the DFOP and HS models and the break point tb of the HS model.
transform_fractions
Boolean specifying if formation fractions constants should be transformed in the model specification used in the fitting for better compliance with the assumption of normal distribution of the estimator. The default (TRUE) is to do transformations. The g parameter of the DFOP and HS models are also transformed, as they can also be seen as compositional data. The transformation used for these transformations is the ilr transformation.

## Description

The transformations are intended to map parameters that should only take on restricted values to the full scale of real numbers. For kinetic rate constants and other paramters that can only take on positive values, a simple log transformation is used. For compositional parameters, such as the formations fractions that should always sum up to 1 and can not be negative, the ilr transformation is used.

The transformation of sets of formation fractions is fragile, as it supposes the same ordering of the components in forward and backward transformation. This is no problem for the internal use in mkinfit.

## Value

A vector of transformed or backtransformed parameters with the same names as the original parameters.

## Examples

SFO_SFO <- mkinmod( parent = list(type = "SFO", to = "m1", sink = TRUE), m1 = list(type = "SFO"))
Successfully compiled differential equation model from auto-generated C code.
# Fit the model to the FOCUS example dataset D using defaults fit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE) summary(fit, data=FALSE) # See transformed and backtransformed parameters
mkin version: 0.9.44 R version: 3.3.1 Date of fit: Tue Jun 28 10:25:37 2016 Date of summary: Tue Jun 28 10:25:37 2016 Equations: d_parent = - k_parent_sink * parent - k_parent_m1 * parent d_m1 = + k_parent_m1 * parent - k_m1_sink * m1 Model predictions using solution type deSolve Fitted with method Port using 153 model solutions performed in 1.788 s Weighting: none Starting values for parameters to be optimised: value type parent_0 100.7500 state k_parent_sink 0.1000 deparm k_parent_m1 0.1001 deparm k_m1_sink 0.1002 deparm Starting values for the transformed parameters actually optimised: value lower upper parent_0 100.750000 -Inf Inf log_k_parent_sink -2.302585 -Inf Inf log_k_parent_m1 -2.301586 -Inf Inf log_k_m1_sink -2.300587 -Inf Inf Fixed parameter values: value type m1_0 0 state Optimised, transformed parameters with symmetric confidence intervals: Estimate Std. Error Lower Upper parent_0 99.600 1.61400 96.330 102.900 log_k_parent_sink -3.038 0.07826 -3.197 -2.879 log_k_parent_m1 -2.980 0.04124 -3.064 -2.897 log_k_m1_sink -5.248 0.13610 -5.523 -4.972 Parameter correlation: parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink parent_0 1.00000 0.6075 -0.06625 -0.1701 log_k_parent_sink 0.60752 1.0000 -0.08740 -0.6253 log_k_parent_m1 -0.06625 -0.0874 1.00000 0.4716 log_k_m1_sink -0.17006 -0.6253 0.47163 1.0000 Residual standard error: 3.211 on 36 degrees of freedom Backtransformed parameters: Confidence intervals for internally transformed parameters are asymmetric. t-test (unrealistically) based on the assumption of normal distribution for estimators of untransformed parameters. Estimate t value Pr(>t) Lower Upper parent_0 99.600000 61.720 2.024e-38 96.330000 1.029e+02 k_parent_sink 0.047920 12.780 3.050e-15 0.040890 5.616e-02 k_parent_m1 0.050780 24.250 3.407e-24 0.046700 5.521e-02 k_m1_sink 0.005261 7.349 5.758e-09 0.003992 6.933e-03 Chi2 error levels in percent: err.min n.optim df All data 6.398 4 15 parent 6.827 3 6 m1 4.490 1 9 Resulting formation fractions: ff parent_sink 0.4855 parent_m1 0.5145 m1_sink 1.0000 Estimated disappearance times: DT50 DT90 parent 7.023 23.33 m1 131.761 437.70
## Not run: # fit.2 <- mkinfit(SFO_SFO, FOCUS_2006_D, transform_rates = FALSE) # summary(fit.2, data=FALSE) # ## End(Not run) initials <- fit$start$value names(initials) <- rownames(fit$start) transformed <- fit$start_transformed$value names(transformed) <- rownames(fit$start_transformed) transform_odeparms(initials, SFO_SFO)
parent_0 log_k_parent_sink log_k_parent_m1 log_k_m1_sink 100.750000 -2.302585 -2.301586 -2.300587
backtransform_odeparms(transformed, SFO_SFO)
parent_0 k_parent_sink k_parent_m1 k_m1_sink 100.7500 0.1000 0.1001 0.1002
## Not run: # # The case of formation fractions # SFO_SFO.ff <- mkinmod( # parent = list(type = "SFO", to = "m1", sink = TRUE), # m1 = list(type = "SFO"), # use_of_ff = "max") # # fit.ff <- mkinfit(SFO_SFO.ff, FOCUS_2006_D) # summary(fit.ff, data = FALSE) # initials <- c("f_parent_to_m1" = 0.5) # transformed <- transform_odeparms(initials, SFO_SFO.ff) # backtransform_odeparms(transformed, SFO_SFO.ff) # # # And without sink # SFO_SFO.ff.2 <- mkinmod( # parent = list(type = "SFO", to = "m1", sink = FALSE), # m1 = list(type = "SFO"), # use_of_ff = "max") # # # fit.ff.2 <- mkinfit(SFO_SFO.ff.2, FOCUS_2006_D) # summary(fit.ff.2, data = FALSE) # ## End(Not run)

Johannes Ranke