Model selection and validation table for Partial Least Squares Regression

The components represent the number of components included in each model. Minitab displays the model with 1 component, with 2 components, and so on, until it includes the number of components you specified. If you didn't specify the number of components to include in the model, Minitab shows the model selection statistics for 10 components, or the number of predictors, whichever is less.

The x-variance is the amount of variance in the terms that is explained by the model. The x-variance value is between 0 and 1.

The closer the x-variance value is to 1, the better the components represent the original set of terms. If you have more than 1 response, the x-variance value is the same for all responses.

The error is the error sum of squares, which is the sum of the squared residuals. It quantifies the variation in the data that the model does not explain. For the final model, the error matches the SS for residual error in the ANOVA table for the final model.

R² is the percentage of variation in the response that is explained by the model. It is calculated as 1 minus the ratio of the error sum of squares (which is the variation that is not explained by model) to the total sum of squares (which is the total variation in the model).

Interpretation

Use R² to determine how well the model fits your data. The higher the R² value, the better the model fits your data. R² is always between 0% and 100%.

You can use a fitted line plot to graphically illustrate different R² values. The first plot illustrates a simple regression model that explains 85.5% of the variation in the response. The second plot illustrates a model that explains 22.6% of the variation in the response. The more variation that is explained by the model, the closer the data points fall to the fitted regression line. Theoretically, if a model could explain 100% of the variation, the fitted values would always equal the observed values and all of the data points would fall on the fitted line. However, even if R² is 100%, the model does not necessarily predict new observations well.

Consider the following issues when interpreting the R² value:

• 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.

• Small samples do not provide a precise estimate of the strength of the relationship between the response and predictors. For example, if you need R² 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.

Predicted R² indicates how well each calculated model predicts the response and is only calculated when you perform cross-validation. If one response variable is in the data, Minitab selects the PLS model with the highest predicted R². If multiple response variables are in the data, Minitab selects the PLS model with the highest mean predicted R² for all of the response variables. Predicted R² is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. The value of predicted R² ranges between 0% and 100%. (While the calculations for predicted R² can produce negative values, Minitab displays zero for these cases.)

The prediction error sum of squares (PRESS) is a measure of the deviation between the fitted values and the observed values. PRESS is similar to the sum of squares of the residual error (SSE), which is the summation of the squared residuals. However, PRESS uses a different calculation for the residuals. The formula used to calculate PRESS is equivalent to a process of systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation.