Analysis of variance table for Analyze Response Surface Design

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

In This Topic

DF
Adj SS
Adj MS
Seq SS
Contribution

F-value
P-Value – Model
P-value – Blocks
P-Value – Factors, interactions, 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.

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.

Adj SS

Adjusted sums of squares are measures of variation for different parts 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. 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 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 Error: The adjusted 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 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 and blocks.
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 parts 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.

Seq SS Model: The sequential sum of squares for the 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.
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 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 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 and blocks.
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 contributes to the total variation in the response.

Interpretation

Higher percentages indicate that the source accounts for more of the variation in the response variable. The percent contribution for the response surface model is the same as the R².

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 blocks and factor terms.
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, squared terms, 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 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 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 – 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 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.
If a quadratic term is significant, you can conclude that the response surface features curvature.

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.

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.

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 – 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.