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5. Process Improvement
5.5. Advanced topics
5.5.9. An EDA approach to experimental design Cumulative residual standard deviation plot

Motivation: How do we Construct a Goodness-of-fit Metric for a Model?

Motivation This question deals with the issue of how to construct a metric, a statistic, that may be used to ascertain the quality of the fitted model. The statistic should be such that for one range of values, the implication is that the model is good, whereas for another range of values, the implication is that the model gives a poor fit.
Sum of absolute residuals Since a model's adequacy is inversely related to the size of its residuals, one obvious statistic is the sum of the absolute residuals. \[ \mbox{AR} = \sum_{i=1}^{n}{|r_{i}|} \] Clearly, for a fixed n,the smaller this sum is, the smaller are the residuals, which implies the closer the predicted values are to the raw data Y, and hence the better the fitted model. The primary disadvantage of this statistic is that it may grow larger simply as the sample size n grows larger.
Average absolute residual A better metric that does not change (much) with increasing sample size is the average absolute residual: \[ \mbox{AAR} = \frac{\sum_{i=1}^{n}{|r_{i}|}} {n} \] with n denoting the number of response values. Again, small values for this statistic imply better-fitting models.
Square root of the average squared residual An alternative, but similar, metric that has better statistical properties is the square root of the average squared residual. \[ \sqrt{ \frac{\sum_{i=1}^{n}{r_{i}^{2}}} {n}} \] As with the previous statistic, the smaller this statistic, the better the model.
Residual standard deviation Our final metric, which is used directly in inferential statistics, is the residual standard deviation \[ s_{res} = \sqrt{ \frac{\sum_{i=1}^{n}{r_{i}^{2}}} {n-p}} \] with p denoting the number of fitted coefficients in the model. This statistic is the standard deviation of the residuals from a given model. The smaller is this residual standard deviation, the better fitting is the model. We shall use the residual standard deviation as our metric of choice for evaluating and comparing various proposed models.
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