# How to adjust SSE or RMSE for the number of free parameters in the model?

How do I adjust SSE (sum of squared errors) or RMSE (root-mean-square errors) for the number of free parameters in the model?

Is there an "adjusted" RMSD metric similar to the adjusted r-squared metric?

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please don't assume that we know what RMSD stands for. The more effort you invest in carefully phrasing your question, the more motivated people will feel to answer it. –  Artem Kaznatcheev Feb 8 '13 at 5:04
This sounds more like a question better suited for stats.stackexchange.com . Is there an aspect of this that relates particularly to cognitive modelling? –  Jeromy Anglim Feb 8 '13 at 5:13
Not sure to understand what you mean by "adjusted" here ? Do you mean adjusted as for "goodness of fit" or as correction in regards to some parameters ? –  Cheatboy2 Feb 8 '13 at 14:30
@Cheatboy2 I believe he is referring to a correction in the number of parameters, as in adjusted $r^2$ –  Jeff Feb 8 '13 at 23:17

There are at least 3 ways to discount SSE (or RMSE) by the number of free params:

$$\text{adjusted RMSE} = \sqrt{\frac{SSE}{n - k}}$$

$$AIC = n \times ln\left(\frac{SSE}{n}\right) - k \times ln(n)$$

$$BIC = n \times ln\left(\frac{SSE}{n}\right) - 2 \times k$$

or in computer code style:

k = number of free params
n = number of DV's
SSE = sum of squared errors

adjusted RMSE = sqrt ( SSE / n - k )
AIC = n * ln(SSE/n) - k * ln(n)
BIC = n * ln(SSE/n) - 2 * k

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RMSD, unlike $R^2$, isn't typically used to compare models across the literature. $R^2$ represents the proportion of variance explained by the model, a construct which translates well across different experimental designs. Adjusted $R^2$ distorts this by accounting for the number of parameters in your model, but is a better estimate of the proportion of variance in the population explained by your model.