A lack of fit test is used to determine whether a full regression model fits a dataset significantly better than a reduced version of the model.

Consider the following regression model, which has four predictor variables.

Y = β0 + β1×1 + β2×2 + β3×3 + β4×4 + ε

A nested model is demonstrated by the following model, which contains only two of the original predictor variables.

Y = β0 + β1×1 + β2×2 + ε

We can use a Lack of Fit Test with the following null and alternative hypotheses to see if these two models differ significantly.

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