# Examples - Maximum likelihood

**Example:**Consider maximum likelihood estimation of the mean of the gamma density.

**Example:** Consider a model for the growth of fish.

The data set at http://notendur.hi.is/gunnar/kennsla/alsm/data/set121.dat contains measurements of individual fish, collected by the Marine Research Institute (http://www.hafro.is). The data include a column (aldur) containing the age of fish and the column (le) containing the length of the same fish.

The von Bertalanffy growth model can be fitted using the R commands

dat<-read.table("http://notendur.hi.is/~gunnar/kennsla/alsm/data/set121.dat",header=T) le<-dat$le a<-dat$aldur fm<-nls(le~Linf*(1-exp(-K*(a-t0))),start=list(t0=0,Linf=80,K=0.25)) summary(fm)

Once the above commands have been issued, the summary command can be used:

> summary(fm) Formula: le ~ Linf * (1 - exp(-K * (a - t0))) Parameters: Estimate Std. Error t value Pr(>|t|) t0 -0.23160 0.23739 -0.976 0.331683 Linf 91.22292 14.47924 6.300 8.72e-09 *** K 0.15672 0.04414 3.550 0.000595 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 2.788 on 97 degrees of freedom Number of iterations to convergence: 3 Achieved convergence tolerance: 6.375e-07

A different test can also be used to investigate whether :

> fmR<-nls(le~Linf*(1-exp(-K*(a))),start=list(Linf=80,K=0.25)) > anova(fm,fmR) Analysis of Variance Table Model 1: le ~ Linf * (1 - exp(-K * (a - t0))) Model 2: le ~ Linf * (1 - exp(-K * (a))) Res.Df Res.Sum Sq Df Sum Sq F value Pr(>F) 1 97 753.78 2 98 762.33 -1 -8.557 1.1012 0.2966Note that the F-test and t-test are not the same in the nonlinear case. Both depend on linearity assumptions but in different ways.