Compare 2 regression models

I wonder if there is any tool to compare 2 regression models on the same sample pool. In general more terms you put into the model the closer the fit, but you risk over fitting. For example:

>> mdl=fitglm(FitZV,FitDataV,'linear')

mdl = 


Generalized linear regression model:
    y ~ 1 + x1
    Distribution = Normal

Estimated Coefficients:
                   Estimate        SE         tStat     pValue
                   ________    __________    _______    ______

    (Intercept)      81.101     0.0085111     9528.9      0   
    x1             -0.22506    0.00058189    -386.77      0   


9638 observations, 9636 error degrees of freedom
Estimated Dispersion: 0.698
F-statistic vs. constant model: 1.5e+05, p-value = 0
>> mdl2=fitglm(FitZV,FitDataV,'purequadratic')

mdl2 = 


Generalized linear regression model:
    y ~ 1 + x1 + x1^2
    Distribution = Normal

Estimated Coefficients:
                    Estimate          SE         tStat       pValue  
                   ___________    __________    _______    __________

    (Intercept)         81.286      0.012269     6625.6             0
    x1                -0.22447    0.00057029     -393.6             0
    x1^2           -0.00086632    4.2147e-05    -20.555    6.3993e-92


9638 observations, 9635 error degrees of freedom
Estimated Dispersion: 0.668
F-statistic vs. constant model: 7.83e+04, p-value = 0

Both mdl and mdl2 are statically better than a constant model, but does mdl2 explain the dataset significantly better than mdl (or the oppsite)? From what I have found devianceTest only test the model to constant, but couldn't find a function to compare 2 models? If anyone can point me to the right direction that would be appreciated.

Answer 1

[Ive J]

One way is to use likelihood ratio test (similar to R anova function). For more information see here.

y = randn(200, 1); % test response
x = randn(200, 1); % test predictor
model1 = fitlm(x, y)
model2 = fitlm([x, x.^2], y);
LR = 2*(model2.LogLikelihood - model1.LogLikelihood); % has a X2 distribution with a df equals to number of constrained parameters, here: 1
pval = 1 - chi2cdf(LR, 1);

[Yi-xiao Liu]

How was the degrees of freedom in chi2cdf determined?

In the example above it is 1, is it because model2 have one more coefficient than model1?

[Ive J]

Yes