Use a Pareto chart of the effects to compare the relative magnitude and the statistical significance of main, square, and interaction effects. If the model does include an error term, the chart displays the absolute value of the standardized effects. If the model does not include an error term, Minitab does not create a Pareto chart.
Minitab plots the effects in the decreasing order of their absolute values. The reference line on the chart indicates which effects are significant. By default, Minitab uses a significance level of 0.05 to draw the reference line.
To determine whether the association between the response and each term in the model is statistically significant, compare the p-value for the term to your significance level to assess the null hypothesis. The null hypothesis is that the term's coefficient is equal to zero, which implies that there is no association between the term and the response. Usually, a significance level (denoted as α or alpha) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that an association exists when there is no actual association.
If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:
To determine how well the model fits your data, examine the goodness-of-fit statistics in the Model Summary table.
Use S to assess how well the model describes the response. Use S instead of the R^{2} statistics to compare the fit of models.
S is measured in the units of the response variable and represents the variation of how far the data values fall from the true response surface. The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.
The higher the R^{2} value, the better the model fits your data. R^{2} is always between 0% and 100%.
R^{2} always increases when you add additional predictors to a model. For example, the best five-predictor model will always have an R^{2} that is at least as high as the best four-predictor model. Therefore, R^{2} is most useful when you compare models of the same size.
Use adjusted R^{2} when you want to compare models that have different numbers of predictors. R^{2} always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R^{2} value incorporates the number of predictors in the model to help you choose the correct model.
Use predicted R^{2} to determine how well your model predicts the response for new observations. Models that have larger predicted R^{2} values have better predictive ability.
A predicted R^{2} that is substantially less than R^{2} may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the sample data and, therefore, may not be useful for making predictions about the population.
Predicted R^{2} can also be more useful than adjusted R^{2} for comparing models because it is calculated with observations that are not included in the model calculation.
Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results.
For more information on how to handle patterns in the residual plots, go to Residual plots for Analyze Factorial Design and click the name of the residual plot in the list at the top of the page.
Pattern | What the pattern may indicate |
---|---|
Fanning or uneven spreading of residuals across fitted values | Nonconstant variance |
Curvilinear | A missing higher-order term |
A point that is far away from zero | An outlier |
A point that is far away from the other points in the x-direction | An influential point |
Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.
Use the normal probability plot of the residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.
The patterns in the following table may indicate that the model does not meet the model assumptions.
Pattern | What the pattern may indicate |
---|---|
Not a straight line | Nonnormality |
A point that is far away from the line | An outlier |
Changing slope | An unidentified variable |