The hardware and bandwidth for this mirror is donated by dogado GmbH, the Webhosting and Full Service-Cloud Provider. Check out our Wordpress Tutorial.
If you wish to report a bug, or if you are interested in having us mirror your free-software or open-source project, please feel free to contact us at mirror[@]dogado.de.
Recall that the likelihood of a model is the probability of the data set given the model (\(P(D|\theta)\)).
The deviance of a model is defined by
\[D(\theta,D) = 2(\log(P(D|\theta_s)) - \log(P(D|\theta)))\]
where \(\theta_s\) is the saturated model which is so named because it perfectly fits the data.
In the case of normally distributed errors the likelihood for a single prediction (\(\mu_i\)) and data point (\(y_i\)) is given by
\[P(y_i|\mu_i) = \frac{1}{\sigma\sqrt{2\pi}} \exp\bigg(-\frac{1}{2}\bigg(\frac{y_i - \mu_i}{\sigma}\bigg)^2\bigg)\] and the log-likelihood by
\[\log(P(y_i|\mu_i)) = -\log(\sigma) - \frac{1}{2}\big(\log(2\pi)\big) -\frac{1}{2}\bigg(\frac{y_i - \mu_i}{\sigma}\bigg)^2\]
The log-likelihood for the saturated model, which is when \(\mu_i = y_i\), is therefore simply
\[\log(P(y_i|\mu_{s_i})) = -\log(\sigma) - \frac{1}{2}\big(\log(2\pi)\big)\]
It follows that the unit deviance is
\[d_i = 2(\log(P(y_i|\mu_{s_i})) - \log(P(y_i|\mu_i)))\]
\[d_i = 2\bigg(\frac{1}{2}\bigg(\frac{y_i - \mu_i}{\sigma}\bigg)^2\bigg)\]
\[d_i = \bigg(\frac{y_i - \mu_i}{\sigma}\bigg)^2\]
As the deviance residual is the signed squared root of the unit deviance,
\[r_i = \text{sign}(y_i - \mu_i) \sqrt{d_i}\] in the case of normally distributed errors we arrive at \[r_i = \frac{y_i - \mu_i}{\sigma} \] which is the Pearson residual.
To confirm this consider a normal distribution with a \(\hat{\mu} = 2\) and \(\sigma = 0.5\) and a value of 1.
library(extras)
mu <- 2
sigma <- 0.5
y <- 1
(y - mu) / sigma
#> [1] -2
dev_norm(y, mu, sigma, res = TRUE)
#> [1] -2
sign(y - mu) * sqrt(dev_norm(y, mu, sigma))
#> [1] -2
sign(y - mu) * sqrt(2 * (log(dnorm(y, y, sigma)) - log(dnorm(y, mu, sigma))))
#> [1] -2These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.