Fitting a reparameterised Gompertz growth model using R

Sun, 28 Apr 2019 21:13:14 -0500

Using the data set “gomp1.csv”, find the parameters of the reparameterised Gompertz model.

\[\begin{equation} y= y_0 + (y_{max} -y_0)*exp(-exp(k*(lag-x)/(y_{max}-y_0) + 1) ) \end{equation}\]

Import the data set.

dat <- read.csv("gomp1.csv", sep=";", header=T)
plot(dat$Time, dat$logN)

str(dat)
## 'data.frame':    13 obs. of  2 variables:
##  $ Time: int  0 2 4 6 8 10 12 14 16 18 ...
##  $ logN: num  -0.105 0.108 -0.111 0.734 2.453 ...

The next step is define the Gompertz function.

# Gompertz function
Gompertz <- function(x, y0, ymax, k, lag){
      result <- y0 + (ymax -y0)*exp(-exp(k*(lag-x)/(ymax-y0) + 1) )
      return(result)
      }

Check data by plot.

plot( logN ~ Time, data=dat, xlim=c(-1, 25), ylim=c(-1, 10))

Fit the model to data using the nls() funtion and extract the models parameters.

Gomp1 <-  nls(logN ~ Gompertz(Time, y0, ymax, k, lag),
               data = dat,
               start = list(y0=0.15, ymax=7, k=0.5, lag=3))
# ectract model parameters
coefs <- coef(Gomp1)
coefs
##          y0        ymax           k         lag 
## -0.02517334  9.60030027  2.78548012  5.88823263
y0   <- coefs[1]
y0
##          y0 
## -0.02517334
ymax <- coefs[2]
ymax
##   ymax 
## 9.6003
k   <- coefs[3]
k
##       k 
## 2.78548
lag <- coefs[4]
lag
##      lag 
## 5.888233

Now, plot the observed values against the fitted values.

time.levels <- seq(0, 24, by=0.1)
pred <- predict(Gomp1, list(x=time.levels) )
plot(logN ~ Time, data=dat, xlim=c(0,24), ylim=c(0, 10),
     main = "Fitted Gompertz model",
     sub = "Blue: fit; green: known")
lines(time.levels, Gompertz(time.levels,
                            y0 = y0, ymax = ymax, k = k, lag=lag), 
      lty = 1, col = "red")

Add the confidence interval to the line offitted values.

library(nlme)
library(twNlme)
Gomp11 <- gnls(logN ~ y0 + (ymax -y0)*exp(-exp(k*(lag-Time)/(ymax-y0) + 1) ),
              data = dat,
              start = list(y0=-0.025, ymax=9.5, k=2.7, lag=5.8))
Gomp11
## Generalized nonlinear least squares fit
##   Model: logN ~ y0 + (ymax - y0) * exp(-exp(k * (lag - Time)/(ymax - y0) +      1)) 
##   Data: dat 
##   Log-likelihood: -2.123991
## 
## Coefficients:
##          y0        ymax           k         lag 
## -0.02511771  9.60008391  2.78571200  5.88858202 
## 
## Degrees of freedom: 13 total; 9 residual
## Residual standard error: 0.3424286

# automatic derivation with accounting for residual variance model
nlmeFit <- attachVarPrep(Gomp11, fVarResidual=varResidPower)

newData <- data.frame( Time= seq(0,24,by=0.1))

resNlme <- varPredictNlmeGnls(Gomp11, newData)

# plotting prediction and standard errors
plot( logN ~ Time, dat, ylab="log CFU",
      xlim=c(0,24), ylim=c(-1, 10))
par(new=T)
plot( resNlme[,"fit"] ~ Time, newData, ylab="",
      type="l", xlim=c(0,24), ylim=c(-1, 10), lty="solid")
lines( resNlme[,"fit"]+resNlme[,"sdInd"] ~ Time, newData, 
       col="orange", lty="dashed"  )
lines( resNlme[,"fit"]-resNlme[,"sdInd"] ~ Time, newData, 
       col="orange", lty="dashed" )

Fitting a first-order or loglinear growth model using R

Thu, 25 Apr 2019 21:13:14 -0500

The data set “FirstOrder.csv” contains observations of microbial concentrations (log N) measured at different times (t) at a given environmental condition. Lets fit a first-order growth kinetics model \(log N = log N_0 + k \times t\) to the experimental data.

Let’s import the “FirstOrder.csv” dataset, and observe the first five lines.

dat <- read.csv("FirstOrder.csv", sep=";", header=TRUE)
dat
##    Time        N
## 1     0   37.298
## 2     1   56.149
## 3     2   81.580
## 4     3  108.758
## 5     4  147.276
## 6     5  197.160
## 7     6  273.014
## 8     7  368.254
## 9     8  490.822
## 10    9  673.304
## 11   10  909.398
## 12   11 1204.429
## 13   12 1640.532
## 14   13 2218.299
## 15   14 2983.926
## 16   15 4033.458

Plot N against Time, and observe that the relationship between the two variables is non-linear.

plot(dat$Time, dat$N, xlab="Time (hours)", ylab="N (cells/g)")

Take the logarithm of the variable N and add the new variable logN in the data set.

dat$logN <- log10(dat$N)
dat[1:5,]
##   Time       N     logN
## 1    0  37.298 1.571686
## 2    1  56.149 1.749342
## 3    2  81.580 1.911584
## 4    3 108.758 2.036461
## 5    4 147.276 2.168132

Now plot logN against Time, and observe that they have now a linear relationship.

plot(dat$Time, dat$logN, xlab="Time (hours)", ylab="log N (cels/g)")

The first-order kinetics model is defined as Nt = N0*exp(k*t), then the log-linear model becomes ln(Nt) = ln(N0) + k*t. Fit a linear model using the lm() (linear models) function.

lm1 <- lm(logN ~ Time, data=dat)

The results can be summarised using the summary() fucntion.

summary(lm1)
## 
## Call:
## lm(formula = logN ~ Time, data = dat)
## 
## Residuals:
##       Min        1Q    Median        3Q       Max 
## -0.052876 -0.006170  0.004664  0.011795  0.021331 
## 
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 1.6245612  0.0084968   191.2   <2e-16 ***
## Time        0.1328460  0.0009652   137.6   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.0178 on 14 degrees of freedom
## Multiple R-squared:  0.9993, Adjusted R-squared:  0.9992 
## F-statistic: 1.894e+04 on 1 and 14 DF,  p-value: < 2.2e-16

You can add fitted values to the original data set.

dat$fitted <- fitted(lm1)
dat[1:5,]
##   Time       N     logN   fitted
## 1    0  37.298 1.571686 1.624561
## 2    1  56.149 1.749342 1.757407
## 3    2  81.580 1.911584 1.890253
## 4    3 108.758 2.036461 2.023099
## 5    4 147.276 2.168132 2.155945

Plot the observations against time and superimpose a fitted line. First, extract the model’s parameters and use the model to draw a fitted line.

coefs <- coef(lm1)
coefs
## (Intercept)        Time 
##    1.624561    0.132846
logN0 <- coefs[1]
logN0
## (Intercept) 
##    1.624561
k <- coefs[2]
k
##     Time 
## 0.132846

Plot observed versus fitted values.

plot(dat$logN ~ dat$Time, data=dat,
     xlab="Time (hours)", ylab="log N (cells/g)")
time.levels <- seq(0, 20, by=1)
lines(time.levels, logN0 + k*time.levels, lty = 2, col = "blue")

regression | Vasco Cadavez

Fitting a reparameterised Gompertz growth model using R

Fitting a first-order or loglinear growth model using R