Use a Pareto chart of the effects to compare the relative magnitude and the statistical significance of the terms. The chart appears when the model leaves degrees of freedom for error.
Minitab plots the terms in decreasing order of their absolute values. The reference line on the chart indicates which terms are significant. By default, Minitab uses a significance level of 0.05 to draw the reference line.
Term | Coef | SE Coef | T-Value | P-Value | VIF |
---|---|---|---|---|---|
Constant | -0.756 | 0.736 | -1.03 | 0.314 | |
Conc | 0.1545 | 0.0633 | 2.44 | 0.022 | 1.03 |
Ratio | 0.2171 | 0.0316 | 6.86 | 0.000 | 1.02 |
Temp | 0.01081 | 0.00462 | 2.34 | 0.027 | 1.04 |
Time | 0.0946 | 0.0546 | 1.73 | 0.094 | 1.00 |
The predictors formaldehyde concentration, catalyst ratio, and temperature have p-values that are less than the significance level of 0.05. These results indicate that these predictors have relationships with wrinkle resistance that are statistically significant. For example, the coefficient for formaldehyde concentration estimates that the mean wrinkle resistance increases by 0.1545 units for each one-unit increase in concentration, while the other terms in the model are held constant.
The p-value for time is greater than 0.05, which indicates that there is not enough evidence to conclude that time is related to the response. The chemist may want to refit the model without this predictor.
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 that have no constant.
S is measured in the units of the response variable and represents how far the data values fall from the fitted values. 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.
Small samples do not provide a precise estimate of the strength of the relationship between the response and predictors. For example, if you need R^{2} to be more precise, you should use a larger sample (typically, 40 or more).
Goodness-of-fit statistics are just one measure of how well the model fits the data. Even when a model has a desirable value, you should check the residual plots to verify that the model meets the model assumptions.
S | R-sq | R-sq(adj) | R-sq(pred) |
---|---|---|---|
0.811840 | 72.92% | 68.90% | 62.81% |
In these results, the model explains approximately 73% of the variation in the response. For these data, the R^{2} value indicates the model provides an adequate fit to the data. If you fit additional models with different predictors, use the adjusted R^{2} values and the predicted R^{2} values to compare how well the models fit the data.
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 Fit Regression Model and click the name of the residual plot in the list at the top of the page.
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.
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 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.
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 |