PREDICT
FTestTutorial.pdf

# FTestTutorial.pdf - Using the F-test to Compare Two Models...

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Using the F-test to Compare Two Models When fitting data using nonlinear regression there are often times when one must choose between two models that both appear to fit the data well. After plotting the residuals of each model and looking at the r 2 values for each model, both models may appear to fit the data. In this case, an F-test can be conducted to see which model is statistically better 1 . It gives a definitive answer and does not rely on arbitrary interpretation of an r 2 value or residual plot. An F-test follows an F-distribution and can be used to compare statistical models. The F-statistic is computed using one of two equations depending on the number of parameters in the models. If both models have the same number of parameters, the formula for the F statistic is F=SS 1 /SS 2 , where SS 1 is the residual sum of squares for the first model and SS 2 is the residual sum of squares for the second model. There are N - V degrees of freedom, where N is the number of data points and V is the number of parameters being estimated (one degree of freedom is lost per parameter estimated). The result- ing F statistic can then be compared to an F-table to extract the p-value. Alternatively, the p-value can be computed using MATLAB’s built in function ‘fcdf’. To do this simply type P=1-fdcf(F,df 1 ,df 2 ), where F is the computed F-statistic, and df 1 and df 2 are the degrees of freedom of each model equation. If the p-value is large (greater than α ) then the first model is statistically better than the second. If the p-value is small (less than 1- α ) then the second model is statistically better than the first. If the models have different numbers of parameters, the formula becomes: F = ( SS 1 - SS 2 ) / ( df 1 - df 2 ) SS 2 /df 2 The sum of squares for each model and the degrees of freedom for each model are calculated as before (note the models will have different degrees of freedom for this case). Additionally, the first model must be the one with fewer parameters (i.e. the simpler one). Once again, the F-statistic and degrees of freedom can be used to determine the p-value. Use df 1 -df 2 and df 2 degrees of freedom when finding the p-value. In this case a p-value less than α indicates that the more complex model (denominator of F-statistic) fits the data significantly better than the simpler model. 1 Note that the F-test tells you nothing about the physical significance of the model. It is ok to pick a model that does not fit the data as well if it has physical meaning. However, for illustration purposes the following assumes you do not know the physical laws governing the process.

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