Analysis of variance table for Partial Least Squares Regression

Find definitions and interpretation guidance for every statistic in the Analysis of variance table.

In This Topic

DF
SS
MS
F-value
P-value – Regression

DF

The total degrees of freedom (DF) are the amount of information in your data. The analysis uses that information to estimate the values of unknown population parameters. The total DF is determined by the number of observations in your sample. Increasing your sample size provides more information about the population, which increases the total DF.

The DF for the regression shows how much information the components use. Increasing the number of components uses more information, which decreases the DF for residual error. The DF for residual error shows how much information is available to estimate the variability of the parameter estimates.

SS

The sum of squares (SS), which are the adjusted sums of squares, are measures of variation for different components of the model. Minitab separates the sums of squares into different components that describe the variation due to different sources.

SS Regression: The regression sum of squares is the sum of the squared deviations of the fitted response values from the mean response value. It quantifies the amount of variation in the response data that is explained by the model.
SS Error: The error sum of squares is the sum of the squared residuals. It quantifies the variation in the data that the predictors do not explain.
SS Total: The total sum of squares is the sum of the regression sum of squares and the error sum of squares. It quantifies the total variation in the data.

Interpretation

Minitab uses the adjusted sums of squares to calculate the p-value for a term. Minitab also uses the sums of squares to calculate the R² statistic. Usually, you interpret the p-values and the R² statistic instead of the sums of squares.

MS

The mean squares (MS), which are the adjusted mean squares, measure how much variation a term or a model explains, assuming that all other terms are in the model, regardless of the order they were entered. Unlike adjusted sums of squares, adjusted mean squares consider the degrees of freedom.

The adjusted mean square error (also called MSE or s²) is the variance around the fitted values.

Interpretation

Minitab uses the adjusted mean square to calculate the p-value for a term. Minitab also uses the adjusted mean squares to calculate the adjusted R² statistic. Usually, you interpret the p-values and the adjusted R² statistic instead of the adjusted mean squares.

F-value

The F-value is the test statistic used to determine whether the model is associated with the response.

Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the model. The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

A sufficiently large F-value indicates that the model is significant.

If you want to use the F-value to determine whether to reject the null hypothesis, compare the F-value to your critical value. You can calculate the critical value in Minitab or find the critical value from an F-distribution table in most statistics books. For more information on using Minitab to calculate the critical value, go to Using the inverse cumulative distribution function (ICDF) and click "Use the ICDF to calculate critical values".

P-value – Regression

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.

Interpretation

To determine whether the model explains variation in the response, compare the p-value for the model to your significance level to assess the null hypothesis. The null hypothesis for the overall regression is that the model does not explain any of the variation in 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 the model explains variation in the response when the model does not.

P-value ≤ α: The model explains variation in the response: If the p-value is less than or equal to the significance level, you conclude that the model explains variation in the response.
P-value > α: There is not enough evidence to conclude that the model explains variation in the response: If the p-value is greater than the significance level, you cannot conclude that the model explains variation in the response. You may want to fit a new model.