extcoef_logistic {growthrates}

Extended Set of Coefficients of a Logistic Growth Model

Description

Estimate model-specific derived parameters of the logistic growth model

Usage

extcoef_logistic(object, quantile = 0.95, time = NULL, ...)

Arguments

object

model object fited by fit_growthmodel

quantile

fraction of the capacity parameter (K) for the quantile method

time

2-valued vector of the search interval for the independent variable (time). Note: this needs to be set this manually if saturation is not reached within the observation time period taken from the data.

...

reserved for future extensions

Details

This function returns the estimated parameters of a logistic growth model (y0, mumax, K) and a series of estimates for the time of approximate saturation. The estimates are defined as follows:

• turnpoint: time of turnpoint (50% saturation)

• sat1: time of the minimum of the 2nd derivative

• sat2: time of the intercept between the steepest increase (the tangent at mumax) and the carrying capacity K

• sat3: time when a quantile of K (default 0.95) is reached

This function is normally not directly called by the user. It is usually called indirectly from coef or results if extended=TRUE.

Value

vector that contains the fitted parameters and some derived characteristics (extended parameters) of the logistic function.

Note

The estimates for the turnpoint and the time of approximate saturation (sat1, sat2, sat3) may be unreliable, if saturation is not reached within the observation time period. See example below. A set of extended parameters exists currently only for the standard logistic growth model (grow_logistic). The code and naming of the parameters is preliminary and may change in future versions.

Examples

## =========================================================================
## The 'extended parameters' are usually derived
## =========================================================================

data(antibiotic)

## fit a logistic model to a single data set
dat <- subset(antibiotic, conc==0.078 & repl=="R4")

parms <- c(y0=0.01, mumax=0.2, K=0.5)
fit <- fit_growthmodel(grow_logistic, parms, dat$time, dat$value)
coef(fit, extended=TRUE)

## fit the logistic to all data sets
myData <- subset(antibiotic, repl=="R3")
parms <- c(y0=0.01, mumax=0.2, K=0.5)
all <- all_growthmodels(value ~ time | conc,
                         data = myData, FUN=grow_logistic,
                         p = parms, ncores = 2)


par(mfrow=c(3,4))
plot(all)
results(all, extended=TRUE)
## we see that the the last 3 series (10...12) do not go into saturation
## within the observation time period.

## We can try to extend the search range:
results(all[10:12], extended=TRUE, time=c(0, 5000))


## =========================================================================
## visualisation how the 'extended parameters' are derived
## =========================================================================

# Derivatives of the logistic:
#   The 1st and 2nd derivatives are internal functions of the package.
#   They are used here for the visualisation of the algorithm.

deriv1 <- function(time, y0, mumax, K) {
  ret <- (K*mumax*y0*(K - y0)*exp(mumax * time))/
    ((K + y0 * (exp(mumax * time) - 1))^2)
  unname(ret)
}

deriv2 <- function(time, y0, mumax, K) {
  ret <- -(K * mumax^2 * y0 * (K - y0) * exp(mumax * time) *
             (-K + y0 * exp(mumax * time) + y0))/
    (K + y0 * (exp(mumax * time) - 1))^3
  unname(ret)
}
## =========================================================================

data(bactgrowth)
## extract one growth experiment by name
dat <- multisplit(bactgrowth, c("strain", "conc", "replicate"))[["D:0:1"]]


## unconstraied fitting
p <- c(y0 = 0.01, mumax = 0.2, K = 0.1) # start parameters
fit1 <- fit_growthmodel(FUN = grow_logistic, p = p, dat$time, dat$value)
summary(fit1)
p <- coef(fit1, extended=TRUE)

## copy parameters to separate variables to improve readability ------------
y0 <-    p["y0"]
mumax <- p["mumax"]
K  <-    p["K"]
turnpoint <- p["turnpoint"]
sat1 <-  p["sat1"]  # 2nd derivative
sat2 <-  p["sat2"]  # intercept between steepest increase and K
sat3 <-  p["sat3"]  # a given quantile of K, default 95\%

## show saturation values in growth curve and 1st and 2nd derivatives ------
opar <- par(no.readonly=TRUE)
par(mfrow=c(3, 1), mar=c(4,4,0.2,0))
plot(fit1)

## 95% saturation
abline(h=0.95*K, col="magenta", lty="dashed")

## Intercept between steepest increase and 100% saturation
b <- deriv1(turnpoint, y0, mumax, K)
a <- K/2 - b*turnpoint
abline(a=a, b=b, col="orange", lty="dashed")
abline(h=K, col="orange", lty="dashed")
points(sat2, K, pch=16, col="orange")
points(turnpoint, K/2, pch=16, col="blue")

## sat2 is the minimum of the 2nd derivative
abline(v=c(turnpoint, sat1, sat2, sat3),
       col=c("blue", "grey", "orange", "magenta"), lty="dashed")

## plot the derivatives
with(dat, plot(time, deriv1(time, y0, mumax, K), type="l", ylab="y'"))
abline(v=c(turnpoint, sat1), col=c("blue", "grey"), lty="dashed")

with(dat, plot(time, deriv2(time, y0, mumax, K), type="l",  ylab="y''"))
abline(v=sat1, col="grey", lty="dashed")
par(opar)