Interpret the key results for Analyze Response Surface Design

Use a Pareto chart of the standardized 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 standardized 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.

P-value ≤ α: The association is statistically significant

If the p-value is less than or equal to the significance level, you can conclude that there is a statistically significant association between the response variable and the term.

P-value > α: The association is not statistically significant

If the p-value is greater than the significance level, you cannot conclude that there is a statistically significant association between the response variable and the term. You may want to refit the model without the term.

If there are multiple predictors without a statistically significant association with the response, you can reduce the model by removing terms one at a time. For more information on removing terms from the model, go to Model reduction.

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:

• If a coefficient for a factor is significant, you can conclude that not all level means are equal.
• If a coefficient for a squared term is significant, you can conclude that the relationship between the factor and the response follows a curved line.
• If a coefficient for an interaction term is significant, the relationship between a factor and the response depends on the other factors in the term. In this case, you should not interpret the main effects without considering the interaction effect.

To determine how well the model fits your data, examine the goodness-of-fit statistics in the Model Summary table.

S

Use S to assess how well the model describes the response. Use S instead of the R² 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.

R-sq

The higher the R² value, the better the model fits your data. R² is always between 0% and 100%.

R² always increases when you add additional predictors to a model. For example, the best five-predictor model will always have an R² that is at least as high as the best four-predictor model. Therefore, R² is most useful when you compare models of the same size.

R-sq (adj)

Use adjusted R² when you want to compare models that have different numbers of predictors. R² always increases when you add a predictor to the model, even when there is no real improvement to the model. The adjusted R² value incorporates the number of predictors in the model to help you choose the correct model.

R-sq (pred)

Use predicted R² to determine how well your model predicts the response for new observations. Models that have larger predicted R² values have better predictive ability.

A predicted R² that is substantially less than R² 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 R² can also be more useful than adjusted R² for comparing models because it is calculated with observations that are not included in the model calculation.

AICc and BIC

When you show the details for each step of a stepwise method or when you show the expanded results of the analysis, Minitab shows two more statistics. These statistics are the corrected Akaike’s Information Criterion (AICc) and the Bayesian Information Criterion (BIC). Use these statistics to compare different models. For each statistic, smaller values are desirable.

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.

Residuals versus order plot

Use the residuals versus order plot to verify the assumption that the residuals are independent from one another. Independent residuals show no trends or patterns when displayed in time order. Patterns in the points may indicate that residuals near each other may be correlated, and thus, not independent. Ideally, the residuals on the plot should fall randomly around the center line:

If you see a pattern, investigate the cause. The following types of patterns may indicate that the residuals are dependent.

Trend

Shift

Cycle