# Interpret all statistics and graphs for Fully Nested ANOVA

Find definitions and interpretation guidance for every statistic and graph that is provided with fully nested ANOVA.

## 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.

## 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. In the Analysis of Variance table, Minitab separates the sequential sums of squares into different components that describe the variation due to different sources.

Seq SS Term
The sequential sum of squares for a term is the unique portion of the variation explained by a term that is not explained by the previously entered terms. It quantifies the amount of variation in the response data that is explained by each term as it is sequentially added to the model.
Seq 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.
Seq 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 sequential sums of squares to calculate the p-value for a term. Usually, you interpret the p-values instead of the sums of squares.

## MS

Sequential mean squares measure how much variation a term or a model explains. The sequential mean squares depend on the order the terms enter the model. Unlike the sequential sums of squares, the sequential mean squares consider the degrees of freedom.

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

### Interpretation

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

## F-value

An F-value appears for each term in the Analysis of Variance table. The F-value is the test statistic used to determine whether the term 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 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

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 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.

All factors in a fully nested ANOVA model are random. Consequently, a factor that is statistically significant indicates that it contributes to the amount of variation in the response.

###### Note

If your fully nested design is not balanced, Minitab does not calculate the F and P-values.

## Var Comp

Variance components estimate the amount of variation in the response that is attributable to each random term in an ANOVA table.

### Interpretation

Use to assess how much of the variation in the study can be attributed to each random term. Higher values indicate that the term contributes more variability to the response.

## % of Total (Variance)

The % of Total estimates the percentage of the total variance that is contributed by each random term in the model. It is calculated as the variance for each source divided by the total variation, then multiplied by 100 to express as a percentage.

If a variance component estimate is less than zero, Minitab displays zero for the percent of total variability.

### Interpretation

Use the percentage of the total variance to assess the variation from each source.

## StDev

StDev is the standard deviation for each random term in the Variance Components table. The standard deviation is equal to the square root of the variance for that source.

The standard deviation is a convenient measure of variation because it has the same units of measurement as the response variable.

## Expected mean squares

In models that include random terms, expected mean squares describe how each source of variation consists of a linear combination of variances.

### Interpretation

Minitab uses the linear combinations to solve for the variance components and the error term for synthesized tests. Usually, you interpret the variance components and the p-values from the synthesized tests instead of the expected mean squares.

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