Analysis of variance table for Analyze Mixture Design

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

In This Topic

DF
Seq SS
Adj SS
Adj MS
F-value
P-value – Regression
P-Value – Terms and groups of terms
P-value – Lack-of-fit

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. The DF for a term show how much information that term uses. Increasing your sample size provides more information about the population, which increases the total DF. Increasing the number of terms in your model uses more information, which decreases the DF available to estimate the variability of the parameter estimates.

Interpretation

The total DF depends on the number of observations. In a mixture design, the total DF is the number of observations number minus 1. The DF for a term is the number of coefficients estimated for that term. The residual error DF is whatever is left over after accounting for all model terms.

Seq SS

Sequential sums of squares are measures of variation for the different sources that are listed for the model. Unlike the adjusted sums of squares, the sequential sums of squares depend on the order that the terms are in the model. In the Analysis of Variance table, Minitab separates the sequential sums of squares into different sources as listed below.

Seq SS Regression: The sequential sum of squares for the entire model is the difference between the total sum of square and the error sum of squares. It is the sum of all of the sequential sums of squares for terms in the model.
Seq SS groups of terms: The sequential sum of squares for a group of terms in the model is the sum of the sequential sums of squares for all of the terms in the group. It quantifies the amount of variation in the response data that the group of terms explains.
Seq SS term: The sequential sum of squares for a term is the increase in the model sum of squares compared to a model with only the terms above it in the ANOVA table.
Seq SS Residual 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.
Seq SS Pure Error: The pure error sum of squares is part of the error sum of squares. The pure error sum of squares exists when the degrees of freedom for pure error exist. For more information, see DF in this topic. It quantifies the variation in the data for observations with the same factor values.
Seq SS Total: The total sum of squares is the sum of the model sum of squares and the error sum of squares. It quantifies the total variation in the data.

Interpretation

Minitab does not use the sequential sums of squares to calculate p-values when you analyze a design, but can use the sequential sums of squares when you use Fit Regression Model or Fit General Linear Model. Usually, you interpret the p-values and the R² statistic based on the adjusted sums of squares.

Adj SS

Adjusted sums of squares are measures of variation for the different sources that are listed for the model. The order of the predictors in the model does not affect the calculation of the adjusted sums of squares. In the Analysis of Variance table, Minitab separates the adjusted sums of squares into different sources as listed below.

Adj SS Regression: The adjusted sum of squares for the entire model is the difference between the total sum of square and the error sum of squares. It is the sum of all the adjusted sums of squares for terms in the model.
Adj SS groups of terms: The adjusted sum of squares for a group of terms in the model is the sum of the adjusted sum of squares for all of the terms in the group. It quantifies the amount of variation in the response data that the group of terms explains.
Adj SS term: The adjusted sum of squares for a term is the increase in the model sum of squares compared to a model with only the other terms. It quantifies the amount of variation in the response data that the term explains.
Adj SS Residual Error: The error sum of squares is the sum of the squared residuals. It quantifies the variation in the data that the model does not explain.
Adj SS Pure Error: The pure error sum of squares is part of the error sum of squares. The pure error sum of squares exists when the degrees of freedom for pure error exist. For more information, see DF in this topic. It quantifies the variation in the data for observations with the same factor values.
Adj SS Total: The total sum of squares is the sum of the model 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-values in the ANOVA table. 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.

Adj MS

Adjusted mean squares measure how much variation a term or a model explains, assuming that all other terms are in the model, regardless of their order in the model. Unlike the adjusted sums of squares, the adjusted mean squares consider the degrees of freedom.

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

Interpretation

Minitab uses the adjusted mean squares to calculate the p-values in the ANOVA table. 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

An F-value appears for each test in the analysis of variance table.

F-value for the model: The F-value is the test statistic used to determine whether any term in the model is associated with the response.
F-value for types of factor terms: The F-value is the test statistic used to determine whether a group of terms is associated with the response. Examples of groups of terms are linear effects and quadratic effects.
F-value for individual terms: The F-value is the test statistic used to determine whether the term is associated with the response.
F-value for the lack-of-fit test: The F-value is the test statistic used to determine whether the model is missing terms that include the components, process variables, and the amount in the experiment. If terms are removed from the model by a stepwise procedure, then the lack-of-fit test includes these terms also.

Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the test. 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 statistical significance.

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.

P-Value – Terms and groups of terms

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

Minitab does not display p-values for main effects in models for mixtures experiments because of the dependence between the components. Specifically, because the component proportions must sum to a fixed amount or proportion, changing a single component forces a change in the others. Additionally, the model for a mixtures experiment does not have an intercept term because the individual component terms behave like intercept terms.

Interpretation

If an item in the ANOVA table is statistically significant, the interpretation depends on the type of item. The interpretations are as follows:

If an interaction term that includes only components is statistically significant, you can conclude that the association between the blend of components and the response is statistically significant.
If an interaction term that includes components and a process variables is statistically significant, you can conclude that the effect of the components on the response variable depends on the process variables.
If a group of terms is statistically significant, you can conclude that at least one of the terms in the group has an effect on the response. When you use statistical significance to decide which terms to keep in a model, you usually do not remove entire groups of terms at the same time. The statistical significance of individual terms can change because of the terms in the model.

P-value – Lack-of-fit

The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis. Minitab automatically performs the pure error lack-of-fit test when your data contain replicates, which are multiple observations with identical x-values. Replicates represent "pure error" because only random variation can cause differences between the observed response values.

Interpretation

To determine whether the model correctly specifies the relationship between the response and the predictors, compare the p-value for the lack-of-fit test to your significance level to assess the null hypothesis. The null hypothesis for the lack-of-fit test is that the model correctly specifies the relationship between the response and the predictors. Usually, a significance level (denoted as alpha or α) of 0.05 works well. A significance level of 0.05 indicates a 5% risk of concluding that the model does not correctly specify the relationship between the response and the predictors when the model does specify the correct relationship.

P-value ≤ α: The lack-of-fit is statistically significant: If the p-value is less than or equal to the significance level, you conclude that the model does not correctly specify the relationship. To improve the model, you may need to add terms or transform your data.
P-value > α: The lack-of-fit is not statistically significant: If the p-value is larger than the significance level, the test does not detect any lack-of-fit.