Assessing the Fit of Regression Models

by Karen

A well-fitting regression model results in predicted values close to the observed data values. The mean model, which uses the mean for every predicted value, generally would be used if there were no informative predictor variables. The fit of a proposed regression model should therefore be better than the fit of the mean model.

Three statistics are used in Ordinary Least Squares (OLS) regression to evaluate model fit: R-squared, the overall F-test, and the Root Mean Square Error (RMSE). All three are based on two sums of squares: Sum of Squares Total (SST) and Sum of Squares Error (SSE). SST measures how far the data are from the mean and SSE measures how far the data are from the model’s predicted values. Different combinations of these two values provide different information about how the regression model compares to the mean model.

R-squared and Adjusted R-squared

The difference between SST and SSE is the improvement in prediction from the regression model, compared to the mean model. Dividing that difference by SST gives R-squared. It is the proportional improvement in prediction from the regression model, compared to the mean model. It indicates the goodness of fit of the model.

R-squared has the useful property that its scale is intuitive: it ranges from zero to one, with zero indicating that the proposed model does not improve prediction over the mean model and one indicating perfect prediction. Improvement in the regression model results in proportional increases in R-squared.

One pitfall of R-squared is that it can only increase as predictors are added to the regression model. This increase is artificial when predictors are not actually improving the model’s fit. To remedy this, a related statistic, Adjusted R-squared, incorporates the model’s degrees of freedom. Adjusted R-squared will decrease as predictors are added if the increase in model fit does not make up for the loss of degrees of freedom. Likewise, it will increase as predictors are added if the increase in model fit is worthwhile. Adjusted R-squared should always be used with models with more than one predictor variable. It is interpreted as the proportion of total variance that is explained by the model.

There are situations in which a high R-squared is not necessary or relevant. When the interest is in the relationship between variables, not in prediction, the R-square is less important. An example is a study on how religiosity affects health outcomes. A good result is a reliable relationship between religiosity and health. No one would expect that religion explains a high percentage of the variation in health, as health is affected by many other factors. Even if the model accounts for other variables known to affect health, such as income and age, an R-squared in the range of 0.10 to 0.15 is reasonable.

The F-test

The F-test evaluates the null hypothesis that all regression coefficients are equal to zero versus the alternative that at least one does not. An equivalent null hypothesis is that R-squared equals zero. A significant F-test indicates that the observed R-squared is reliable, and is not a spurious result of oddities in the data set. Thus, the F-test determines whether the proposed relationship between the response variable and the set of predictors is statistically reliable, and can be useful when the research objective is either prediction or explanation.

RMSE

The RMSE is the square root of the variance of the residuals. It indicates the absolute fit of the model to the data–how close the observed data points are to the model’s predicted values. Whereas R-squared is a relative measure of fit, RMSE is an absolute measure of fit. As the square root of a variance, RMSE can be interpreted as the standard deviation of the unexplained variance, and has the useful property of being in the same units as the response variable. Lower values of RMSE indicate better fit. RMSE is a good measure of how accurately the model predicts the response, and is the most important criterion for fit if the main purpose of the model is prediction.

The best measure of model fit depends on the researcher’s objectives, and more than one are often useful. The statistics discussed above are applicable to regression models that use OLS estimation. Many types of regression models, however, such as mixed models, generalized linear models, and event history models, use maximum likelihood estimation. These statistics are not available for such models.


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{ 9 comments… read them below or add one }

Grateful2U September 24, 2013 at 9:06 pm

Hi Karen,

Yet another great explanation.

Regarding the very last sentence – do you mean that easy-to-understand statistics such as RMSE are not acceptable or are incorrect in relation to e.g., Generalized Linear Models? Or just that most software prefer to present likelihood estimations when dealing with such models, but that realistically RMSE is still a valid option for these models too?

Thanks!!!

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Karen September 24, 2013 at 10:47 pm

Hi Grateful,

Hmm, that’s a great question. My initial response was it’s just not available–mean square error just isn’t calculated. But I’m not sure it can’t be. The residuals do still have a variance and there’s no reason to not take a square root. And AMOS definitely gives you RMSEA (root mean square error of approximation). Perhaps that’s the difference–it’s approximate. I will have to look that up tomorrow when I’m back in the office with my books. :)

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Grateful2U October 2, 2013 at 10:57 pm

Thanks, Karen. Looking forward to your insightful response.

There is lots of literature on pseudo R-square options, but it is hard to find something credible on RMSE in this regard, so very curious to see what your books say. :)

Thanks again.

Reply

Jane October 21, 2013 at 8:47 pm

Hi,
I wanna report the stats of my fit. if i fited 3 parameters, i shoud report them as: (FittedVarable1 +- sse), or (FittedVarable1, sse)
thanks

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roman April 3, 2014 at 11:47 am

I have read your page on RMSE (http://www.theanalysisfactor.com/assessing-the-fit-of-regression-models/) with interest. However there is another term that people associate with closeness of fit and that is the Relative average root mean square i.e. % RMS which = (RMS (=RMSE) /Mean of X values) x 100

However I am strugging to get my head around what this actually means . For example a set of regression data might give a RMS of +/- 0.52 units and a % RMS of 17.25%. I understand how to apply the RMS to a sample measurement, but what does %RMS relate to in real terms.?

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Karen April 4, 2014 at 9:16 am

Hi Roman,

I’ve never heard of that measure, but based on the equation, it seems very similar to the concept of coefficient of variation.

In this context, it’s telling you how much residual variation there is, in reference to the mean value. It’s trying to contextualize the residual variance. So a residual variance of .1 would seem much bigger if the means average to .005 than if they average to 1000. Just one way to get rid of the scaling, it seems.

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roman April 7, 2014 at 7:53 am

Hi Karen

I am not sure if I understood your explanation.
In view of this I always feel that an example goes a long way to describing a particular situation. In the example below, the column Xa consists if actual data values for different concentrations of a compound dissolved in water and the column Yo is the instrument response. The aim is to construct a regression curve that will predict the concentration of a compound in an unknown solution (for e.g. salt in water) Below is an example of a regression table consisting of actual data values, Xa and their response Yo. The column Xc is derived from the best fit line equation y=0.6142×-7.8042
As far as I understand the RMS value of 15.98 is the error from the regression (best filt line) for a measurement i.e. if the concentation of the compound in an unknown solution is measured against the best fit line, the value will equal Z +/- 15.98 (?). If this is correct, I am a little unsure what the %RMS actually measures. The % RMS = (RMS/ Mean of Xa)x100?
Any further guidance would be appreciated.

from
trendline
Actual Response equation
Xa Yo Xc, Calc Xc-Xa (Yo-Xa)2
1460 885.4 1454.3 -5.7 33.0
855.3 498.5 824.3 -31.0 962.3
60.1 36.0 71.3 11.2 125.3
298 175.5 298.4 0.4 0.1
53.4 22.4 49.2 -4.2 17.6
279 164.7 280.8 1.8 3.4
2780 1706.2 2790.6 10.6 112.5
233.2 145.7 249.9 16.7 278.0
Xm = 752.375 454.3 752.3 sum = 1532.2
SD.S = RMS=√(sum x residuals squared)/(N-2)= ±15.98
%rel RMS = (RMS/Xm)*100= ± 2.12

slope =0.6142
Y – intercept=-7.8042

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Shailen July 25, 2014 at 10:12 pm

Suppose I have the following data:

3 5 7 8 4 10 11 13 12

And I need to calculate corresponding “predicted values” for every data.
How do I do so? I need to calculate RMSE from above observed data and predicted value.

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ADIL August 24, 2014 at 7:56 pm

hi,
how method to calculat the RMSE, RMB
betweene 2 data Hp(10) et Hr(10)
thank you

Reply

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