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In many models, there will be more than one predictor used in the regression, which complicates the relationships within them. As such, it can be challenging to determine what difference the introduction of an additional regressor variable makes, given it may influence other regressors or be affected by them in turn. Partial regression plots are intended to help address that problem by analyzing the responses of both the target predictor and the response variable against the other predictors.
Per Montgomery et al. (2012), both of the former are regressed against each other variable, and the residuals are plotted against each other, creating a set of partial regression plots. If the new predictor’s effects on the model are linear, then the plots will reflect that fact, if it is horizontal, there is no useful prediction information, and otherwise, a transformation of the new variable may be necessary.
By necessity, any model of a real-world process will omit some factors that have an influence on it, whether due to the difficulty of measuring it, the complexity that it introduces relative to its impact, or other reasons. However, the omission of these aspects creates the potential for errors that can render the model unsuitable for practical applications. Lack-of-fit tests are intended to demonstrate whether the model is applicable to real data. They rely on the existence of repeated measurements with the same predictor values, which will likely vary to some extent. Per Young (2018), the method involves separating the error sum of squares (SSE) into two figures: lack of fit and pure error.
The latter is determined by taking only replicate observations and determining the error sum of squares for them. Subtracting it from the SSE produces the lack of fit, which can be used to find the test statistic F0 and compare it to the expected distribution.
References
Montgomery, D. C., Peck, E. A., & Vining, G. G. (2012). Introduction to linear regression analysis (5th ed.). Wiley.
Young, D. S. (2018). Handbook of regression methods. CRC Press.
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