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.
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.
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 R^{2} statistic. Usually, you interpret the p-values and the R^{2} statistic instead of the sums of squares.
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 s^{2}) is the variance around the fitted values.
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 R^{2} statistic. Usually, you interpret the p-values and the adjusted R^{2} statistic instead of the adjusted mean squares.
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.
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 R^{2} statistic based on the adjusted sums of squares.
Contribution displays the percentage that each source in the Analysis of Variance table contributes to the total sequential sums of squares (Seq SS).
Higher percentages indicate that the source accounts for more of the variation in the response.
An F-value appears for each test in the analysis of variance table.
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".
The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
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.
The p-value is a probability that measures 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.
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 the coefficient for a linear covariate term is zero when it is not.
Covariates can increase the multicollinearity in the model. Variance Inflation Factors (VIF) are a measure of multicollinearity. When you assess the statistical significance of terms for a model with covariates, consider the variance inflation factors (VIFs). For more information, go to Coefficients table for Analyze Definitive Screening Design and click VIF.
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?.
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.
The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
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.
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 group of two-way interactions is not statistically significant at the 0.05 level. Also, the tests for all the individual 2-way interactions are not statistically significant.
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.
The p-value is a probability that measures the evidence against the null hypothesis. Lower probabilities provide stronger evidence against the null hypothesis.
If the p-value is larger than the significance level, the test does not detect any lack-of-fit.