Analysis of Variance table for One-Way ANOVA

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

In This Topic

Welch's Test
DF
DF Num
DF Denom
Seq SS
Contribution
Adj SS
Adj MS
F-value
P-value

Welch's Test

Unlike the default one-way ANOVA procedure, Welch's test does not assume that all populations have equal variances. To have Minitab perform Welch's test for one-way ANOVA, deselect Assume equal variances in the Options sub-dialog box.

Interpretation

See "P-value" to determine how to interpret the results from Welch's test.

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.

If two conditions are met, then Minitab partitions the DF for error. The first condition is that there must be terms you can fit with the data that are not included in the current model. For example, if you have a continuous predictor with 3 or more distinct values, you can estimate a quadratic term for that predictor. If the model does not include the quadratic term, then a term that the data can fit is not included in the model and this condition is met.

The second condition is that the data contain replicates. Replicates are observations where each predictor has the same value. For example, if you have 3 observations where pressure is 5 and temperature is 25, then those 3 observations are replicates.

If the two conditions are met, then the two parts of the DF for error are lack-of-fit and pure error. The DF for lack-of-fit allow a test of whether the model form is adequate. The lack-of-fit test uses the degrees of freedom for lack-of-fit. The more DF for pure error, the greater the power of the lack-of-fit test.

DF Num

For Welch's ANOVA, Minitab uses the degrees of freedom for the numerator to calculate the probability of obtaining an F value that is at least as extreme as the observed F value.

Interpretation

Minitab uses the F-value to calculate the p-value. Typically, you should assess the p-value because it is easier to interpret.

DF Denom

For Welch's ANOVA, Minitab uses the degrees of freedom for the denominator to calculate the probability of obtaining an F value that is at least as extreme as the observed F value.

Interpretation

Minitab uses the F-value to calculate the p-value. Typically, you should assess the p-value because it is easier to interpret.

Seq SS

Sequential sums of squares are measures of variation for different components of the model. Unlike the adjusted sums of squares, the sequential sums of squares depend on the order the terms are entered into the model. For One-Way ANOVA, the sequential sums of squares always equal the adjusted sums of squares.

Contribution

Contribution displays the percentage that each source in the Analysis of Variance table contributes to the total sequential sums of squares (Seq SS).

Interpretation

Higher percentages indicate that the source accounts for more of the variation in the response.

Adj SS

Adjusted sums of squares are measures of variation for different components of 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 sums of squares into different components that describe the variation due to different sources.

Adj SS Term: The adjusted sum of squares for a term is the increase in the regression sum of squares compared to a model with only the other terms. It quantifies the amount of variation in the response data that is explained by each term in the model.
Adj 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.
Adj SS Total: The total sum of squares is the sum of the term 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.

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 the order they were entered. 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-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

An F-value appears for each term in the Analysis of Variance table:

F-value for the model or the 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 higher-order terms that include the predictors in the current model.

Interpretation

Minitab uses the F-value to calculate the p-value, which you use to make a decision about the statistical significance of the terms and 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 term or 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

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

Interpretation

Use the p-value in the ANOVA output to determine whether the differences between some of the means are statistically significant.

To determine whether any of the differences between the means are statistically significant, compare the p-value to your significance level to assess the null hypothesis. The null hypothesis states that the population means are all equal. 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 a difference exists when there is no actual difference.

P-value ≤ α: The differences between some of the means are statistically significant: If the p-value is less than or equal to the significance level, you reject the null hypothesis and conclude that not all population means are equal. Use your specialized knowledge to determine whether the differences are practically significant. For more information, go to Statistical and practical significance.
P-value > α: The differences between the means are not statistically significant: If the p-value is greater than the significance level, you do not have enough evidence to reject the null hypothesis that the population means are all equal. Verify that your test has enough power to detect a difference that is practically significant. For more information, go to Increase the power of a hypothesis test.