This post is one of those ‘explain to myself how things work’ documents, which are not necessarily completely correct but are close enough to facilitate understanding.

## Background

Let’s assume that we are working with a fairly simple linear model, where we only have a response variable (say tree stem diameter in cm). If we want to ‘guess’ the diameter for a tree (y_{i}) our best bet is the average (μ) and we will have a residual (ε_{i}). The model equation then looks like:

\(y_i = mu + \varepsilon_i\)

We want to estimate both the overall mean (\(\mu\)) and the error variance (\(\sigma_{\varepsilon}^2\)), for which we could use least squares. However, we will use an alternative method (maximum likelihood) because that is the point of this post. A likelihood function expresses the probability of obtaining the observed sample from a population given a set of model parameters. Thus, it answers the question *What is the probability of observing the current dataset, when I assume a given set of parameters for my linear model?* To do so, we have to assume a distribution, say normal, for which the probability density function (p.d.f) is:

\(p.d.f. (y) = \frac{1}{\sqrt{2 \pi} \sigma} e^{-1/2 \frac{(y- \mu)^2}{\sigma^2}}\)

with n independent samples the likelihood (*L*) is:

\(L = \prod_{i=1}^n \frac{1}{\sqrt{2 \pi} \sigma} e^{-1/2 \frac{(y- \mu)^2}{\sigma^2}}\)

where \(\prod\) is a multiplication operator, analogous to the summation operator \(\sum\). Maximizing *L* or its natural logarithm (*LogL*) is equivalent, then:

\(LogL = \sum_{i=1}^n \log\left (\frac{1}{\sqrt{2 \pi} \sigma} e^{-1/2 \frac{(y- \mu)^2}{\sigma^2}}

\right ) \\

LogL= \sum_{i=1}^n \left [ \log \left ( \frac{1}{sqrt{2 \pi} \sigma}

\right ) + \log \left ( e^{-1/2 \frac{(y- \mu)^2}{\sigma^2}}

\right )

\right ] \\

LogL= \sum_{i=1}^n \log(2 \pi)^{-1/2} + \sum_{i=1}^n \log \left (\frac{1}{\sigma}

\right ) + \sum_{i=1}^n \left ( -1/2 \frac{(y- \mu)^2} {sigma^2}

\right ) \\

LogL= -\frac{n}{2} \log(2 \pi) – n \log(\sigma) – \frac{1}{2 \sigma^2} \sum_{i=1}^n ( y – \mu)^2\)

Considering only \(\mu\), *LogL* is maximum when \(\sum_{i=1}^n ( y – \mu)^2\) is a minimum, i.e. for: \(\mu = \frac{\sum_{i=1}^n y_i}{n}\)

Now considering σ:

\(\frac{\delta LogL}{\delta \sigma} = -\frac{n}{\sigma} – (-2) \frac{1}{2} \sum_{i=1}^n (y – \mu)^2 \sigma^{-3}\)

\(\frac{\delta LogL}{\delta \sigma} = -\frac{n}{\sigma} + \frac{1}{\sigma^3} \sum_{i=1}^n (y – \mu)^2\)

and setting the last expression equal to 0:

\(\frac{n}{\sigma} = \frac{1}{\sigma^3} \sum_{i=1}^n (y – \mu)^2\)

\(\sigma^2 = \frac{\sum_{i=1}^n (y – \mu)^2}{n}\)

We can obtain an estimate of the error variance (\(\hat{\sigma}\), notice the hat) by replacing \(\mu\) by our previous estimate:

\(\hat{\sigma}^2 = \frac{sum_{i=1}^n (y – \bar{y})^2}{n}\)

which is biased, because the denominator is n rather than (n – 1) (the typical denominator for sample variance). This bias arises because maximum likelihood estimates do not take into account the loss of degrees of freedom when estimating fixed effects.

## Playing in R with an example

We have data for stem diameters (in mm) for twelve 10 year-old radiata pine (*Pinus radiata* D. Don) trees:

diams <- c(150, 124, 100, 107, 170, 144, 113, 108, 92, 129, 123, 118) # Obtain typical mean and standard deviation mean(diams) [1] 123.1667 sd(diams) [1] 22.38438 # A simple function to calculate likelihood. # Values quickly become very small like <- function(data, mu, sigma) { like <- 1 for(obs in data){ like <- like * 1/(sqrt(2*pi)*sigma) * exp(-1/2 * (obs - mu)^2/(sigma^2)) } return(like) } # For example, likelihood of data if mu=120 and sigma=20 like(diams, 120, 20) [1] 3.476384e-24 # A simple function to calculate log-likelihood. # Values will be easier to manage loglike <- function(data, mu, sigma) { loglike <- 0 for(obs in data){ loglike <- loglike + log(1/(sqrt(2*pi)*sigma) * exp(-1/2 * (obs - mu)^2/(sigma^2))) } return(loglike) } # Example, log-likelihood of data if mu=120 and sigma=20 loglike(diams, 122, 20) [1] -53.88605 # Let's try some combinations of parameters and # plot the results params <- expand.grid(mu = seq(50, 200, 1), sigma <- seq(10, 40, 1)) params$logL <- with(params, loglike(diams, mu, sigma)) summary(params) library(lattice) contourplot(logL ~ mu*sigma, data = params, cuts = 20)

It is difficult to see the maximum likelihood in this plot, so we will zoom-in by generating a smaller grid around the typical mean and standard deviation estimates.

# Zooming in params <- expand.grid(mu = seq(120, 126, 0.01), sigma <- seq(18, 25, 0.1)) params$logL <- with(params, loglike(diams, mu, sigma)) summary(params) contourplot(logL ~ mu*sigma, data = params, cuts = 10)

We can now check what would be actual results using any of the functions to fit models using maximum likelihood, for example `gls`

:

library(nlme) m1 <- gls(diams ~ 1, method='ML') summary(m1) Generalized least squares fit by maximum likelihood Model: diams ~ 1 Data: NULL AIC BIC logLik 111.6111 112.5809 -53.80556 Coefficients: Value Std.Error t-value p-value (Intercept) 123.1667 6.461815 19.06069 0 Residual standard error: 21.43142 Degrees of freedom: 12 total; 11 residual

In real life, software will use an iterative process to find the combination of parameters that maximizes the log-likelihood value. If we go back to our function and use `loglike(diams, 123.1667, 21.43142)`

we will obtain -53.80556; exactly the same value calculated by `gls`

. In addition, we can see that the estimated standard deviation (21.43) is slightly lower than the one calculated by the function `sd()`

(22.38), because our biased estimate divides the sum of squared deviations by n rather than by n-1.

P.S. The code for the likelihood and log-likelihood functions is far from being optimal (the loops could be vectorized). However, the loops are easier to understand for many people.