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I'm using GAMLSS to model a variable as normally distributed with mean and SD as linear functions of some parameters. Sometimes GAMLSS gives me negative global deviances, but in my limited understanding, a normal distribution can't have a negative deviance as the deviance is equivalent to the squared error. Should I be worried? I've done some searching and found that negative deviances can happen for some distributions, but nobody seems to explicitly mention the normal distribution.

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You can simulate some data to see what happens in a simple case:

n <- 10
x <- 1:n 
set.seed(1234) 
y <- 0.1 + 0.2 * x + rnorm(n, mean = 0, sd = 0.2)

library(gamlss) 
model <- gamlss(y ~ x, family = NO) 

summary(x) 

The global deviance reported by R is negative, the reason for this being the fact that the variability of the y observations about the fitted regression line is small.

If you increase the amount of this variability, the global deviance will become positive:

n <- 10
x <- 1:n 
set.seed(1234) 
y <- 0.1 + 0.2 * x + rnorm(n, mean = 0, sd = 2)

library(gamlss) 
model <- gamlss(y ~ x, family = NO) 

So it is possible to have a negative global deviance in this simple case if the model produces a low value for the residual sum of squares.

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    $\begingroup$ Thanks Isabella, that reassures me that I'm not making some basic blunder. I'm still puzzled however about how normal deviance can be negative at all. $\endgroup$ – Peter Land Jun 15 at 9:50
  • $\begingroup$ For the example I gave you, the global (fitted) deviance is defined as $n[log(2\pi * RSS/n) - 1]$, which can be shown to be negative if RSS is "small". This is how I was able to come up with an example where the residual sum of squares (RSS) is "small". See section 2.2.2 Testing between models of gamlss.com/wp-content/uploads/2013/01/book-2010-Athens1.pdf. $\endgroup$ – Isabella Ghement Jun 15 at 14:44

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