Analysis of variance table for Analyze Factorial Design

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

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.

DF for curvature
If a design has center points, then one DF is for the test for curvature. If the term for center points is in the model, the row for curvature is part of the model. If the term for center points is not in the model, the row for curvature is part of the error that is used to test terms that are in the model.
DF for error
If two conditions are met, then Minitab partitions the DF for error that are not for curvature. The first condition is that there must be terms that you can fit with the data that are not in the current model. For example, if you the design has blocks but the blocks are not in the model. The center point term is always for curvature, so the center point term does not count as a term that you can fit with the data that is not in the current model.
The second condition is that the data contain replicates. Replicates are observations where each predictor has the same value, including blocks and covariates if those terms are in the model or if a stepwise procedure excluded those terms. For example, if the design has two center points in the same block and no covariate, the center points are replicated points. If the center points are in different blocks and the term for blocks is in the model, the center points are not replicated points.
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.

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 Model
The adjusted sum of squares for the model is the difference between the total sum of square and the error sum of squares in the model compared to a model that uses only the mean of the response. It is the sum of all the sequential 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 sequential 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 each term in the model explains.
Adj SS Error
The adjusted 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 Curvature
The adjusted sum of squares for curvature can be part of the model sum of squares or part of the error sum of squares. It quantifies the amount of variation in the response data explained by the center point term. This variation represents the combined effect of one or more quadratic terms.
Adj SS Pure Error
The adjusted 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, go to the section on Degrees of Freedom (DF). It quantifies the variation in the data for observations with the same values of the factors, blocks, and covariates.
Adj SS Total
The adjusted 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.

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 that the terms are in 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 Model
The sequential sum of squares for the model is difference between the total sum of square and the error sum of squares. It is the sum of all of the 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 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. It quantifies the increase in the model sum of squares when that term is added to a model with the terms above it.
Seq SS Error
The sequential 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 Curvature
The sequential sum of squares for curvature can be part of the model sum of squares or part of the error sum of squares. It quantifies the amount of variation in the response data explained by the center point term. This variation represents the combined effect of one or more quadratic terms.
Seq SS Pure Error
The sequential 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, go to the section on Degrees of Freedom (DF). It quantifies the variation in the data for observations with the same values of the factors, blocks, and covariates.
Seq SS Total
The sequential 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.

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.

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, including covariates, blocks, factor terms, and curvature.
F-value for covariates as a group
The F-value is the test statistic used to determine whether any of the covariates are associated with the response simultaneously.
F-value for individual covariates
The F-value is the test statistic used to determine whether an individual covariate is associated the response.
F-value for blocks
The F-value is the test statistic used to determine whether different conditions among blocks are 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 2-way interactions.
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 curvature
The F-value is the test statistic used to determine whether any of the factors have a curved relationship 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 factors in the experiment. If blocks or covariates 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 – Model

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 model 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 – Covariates

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

In a designed experiment, covariates account for variables that are measurable, but difficult to control. For example, members of a quality team at a hospital network design an experiment to study length of stay for patients admitted for total knee replacement surgery. For the experiment, the team can control factors like the format of pre-surgical instructions. To avoid bias, the team records data on covariates that they cannot control, such as the age of the patient.

Interpretation

To determine whether the association between the response and a covariate is statistically significant, compare the p-value for the covariate to your significance level to assess the null hypothesis. The null hypothesis is that the coefficient for the covariate is zero, which implies that there is no association between the covariate 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 different conditions between runs change the response when the conditions do not.

When you assess the statistical significance of terms for a model with covariates, consider the variance inflation factors (VIFs).

P-value ≤ α: The association is statistically significant
If the p-value is less than or equal to the significance level, you conclude that the association between the response and the covariate is statistically significant.
P-value > α: The association is not statistically significant
If the p-value is greater than the significance level, you cannot conclude that the association between the response and the covariate is statistically significant. You may want to fit a model without the covariate.
Note

All the VIF values are 1 in most factorial designs, which simplifies the determination of statistical significance. The inclusion of covariates in the model and the occurrence of botched runs during data collection are two common ways that VIF values increase, which complicates the interpretation of statistical significance. VIF values are in the Coefficients table. For more information, go to Coefficients table for Analyze Factorial Design and click VIF.

P-value – Blocks

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

Blocks account for the differences that might occur between runs that are performed under different conditions. For example, an engineer designs an experiment to study welding and cannot collect all of the data on the same day. Weld quality is affected by several variables that change from day-to-day that the engineer cannot control, such as relative humidity. To account for these uncontrollable variables, the engineer groups the runs performed each day into separate blocks. The blocks account for the variation from the uncontrollable variables so that these effects are not confused with the effects of the factors the engineer wants to study. For more information on how Minitab assigns runs to blocks, go to What is a block?.

Interpretation

To determine whether different conditions between runs change the response, compare the p-value for the blocks to your significance level to assess the null hypothesis. The null hypothesis is that different conditions do not change 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 different conditions between runs change the response when the conditions do not.

P-value ≤ α: The different conditions change the response
If the p-value is less than or equal to the significance level, you conclude that the different conditions change the response.
P-value > α: There is not enough evidence to conclude that the different conditions change the response
If the p-value is greater than the significance level, you cannot conclude that the different conditions change the response. You may want to fit a model without blocks.

P-Value – Factors, interactions, 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.

Interpretation

If a model term is statistically significant, the interpretation depends on the type of term. The interpretations are as follows:
  • If a covariate is significant, you can conclude that the coefficient for the covariate does not equal zero.
  • If a categorical factor is significant, you can conclude that not all the level means are equal.
  • If an interaction term is significant, you can conclude that the relationship between a factor and the response depends on the other factors in the term.

Tests of groups of terms

If a group of terms is statistically significant, then 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.

Factorial Regression: Strength versus Material, InjPress, InjTemp, CoolTemp

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Model 10 447.766 44.777 17.61 0.003 Linear 4 428.937 107.234 42.18 0.000 Material 1 181.151 181.151 71.25 0.000 InjPress 1 112.648 112.648 44.31 0.001 InjTemp 1 73.725 73.725 29.00 0.003 CoolTemp 1 61.412 61.412 24.15 0.004 2-Way Interactions 6 18.828 3.138 1.23 0.418 Material*InjPress 1 0.342 0.342 0.13 0.729 Material*InjTemp 1 0.778 0.778 0.31 0.604 Material*CoolTemp 1 4.565 4.565 1.80 0.238 InjPress*InjTemp 1 0.002 0.002 0.00 0.978 InjPress*CoolTemp 1 0.039 0.039 0.02 0.906 InjTemp*CoolTemp 1 13.101 13.101 5.15 0.072 Error 5 12.712 2.542 Total 15 460.478

In this model, the test for the two-way interactions is not statistically significant at the 0.05 level. Also, the tests for all the 2-way interactions are not statistically significant.

Factorial Regression: Strength versus Material, InjPress, InjTemp, CoolTemp

Analysis of Variance Source DF Adj SS Adj MS F-Value P-Value Model 5 442.04 88.408 47.95 0.000 Linear 4 428.94 107.234 58.16 0.000 Material 1 181.15 181.151 98.24 0.000 InjPress 1 112.65 112.648 61.09 0.000 InjTemp 1 73.73 73.725 39.98 0.000 CoolTemp 1 61.41 61.412 33.31 0.000 2-Way Interactions 1 13.10 13.101 7.11 0.024 InjTemp*CoolTemp 1 13.10 13.101 7.11 0.024 Error 10 18.44 1.844 Total 15 460.48

If you reduce the model one term at a time, beginning with the 2-way interaction with the highest p-value, then the last 2-way interaction is statistically significant at the 0.05 level.

P-value – Curvature

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

Minitab tests for curvature when the design has center points. The test looks at the fitted mean of the response at the center points relative to the expected mean if the relationships between the model terms and the response are linear. To visualize the curvature, use factorial plots.

The center points are far away from the line that joins the means of the corner points, which suggests a curved relationship. Use the p-value to verify that the curvature is statistically significant.

The center points are near the line that joins the means of the corner points. The curvature is probably not statistically significant.

Interpretation

To determine whether at least one of the factors have a curved relationship with the response, compare the p-value for curvature to your significance level to assess the null hypothesis. The null hypothesis is that all the relationships between the factors and the response are linear.

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 different conditions between runs change the response when the conditions do not.

P-value ≤ α: At least one factor has a curved relationship with the response
If the p-value is less than or equal to the significance level, you conclude that at least one of the factors has a curved relationship with the response. You may want to add axial points to the design so that you can model the curvature.
P-value > α: There is not enough evidence to conclude that any factors have a curved relationship with the response
If the p-value is greater than the significance level, you cannot conclude that any of the factors have a curved relationship with the response. If the curvature is part of the model, you may want to refit the model without a term for center points so that the curvature is part of the error.
Note

Usually, if the curvature is not statistically significant, you remove the center point term. If you leave the center points in the model, Minitab assumes that the model contains curvature that the factorial design cannot fit. Due to the inadequate fit, Contour Plot, Surface Plot, and Overlaid Contour Plot are not available. Also, Minitab does not interpolate between the factor levels in the design with Response Optimizer. For more information on ways to use the model, go to Stored model overview.

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.

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.

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