Source | DF | Adj SS | Adj MS | F-Value | P-Value |
---|---|---|---|---|---|
Paint | 3 | 281.7 | 93.90 | 6.02 | 0.004 |
Error | 20 | 312.1 | 15.60 | ||
Total | 23 | 593.8 |
In these results, the null hypothesis states that the mean hardness values of 4 different paints are equal. Because the p-value is less than the significance level of 0.05, you can reject the null hypothesis and conclude that some of the paints have different means.
Use the interval plot to display the mean and confidence interval for each group.
Interpret these intervals carefully because making multiple comparisons increases the type 1 error rate. That is, when you increase the number of comparisons, you also increase the probability that at least one comparison will incorrectly conclude that one of the observed differences is significantly different.
To assess the differences that appear on this plot, use the grouping information table and other comparisons output (shown in step 3).
If your one-way ANOVA p-value is less than your significance level, you know that some of the group means are different, but not which pairs of groups. Use the grouping information table and tests for differences of means to determine whether the mean difference between specific pairs of groups are statistically significant and to estimate by how much they are different.
For more information on comparison methods, go to Using multiple comparisons to assess the practical and statistical significance.
Use the grouping information table to quickly determine whether the mean difference between any pair of groups is statistically significant.
Groups that do not share a letter are significantly different.
Use the confidence intervals to determine likely ranges for the differences and to determine whether the differences are practically significant. The table displays a set of confidence intervals for the difference between pairs of means. The interval plot for differences of means displays the same information.
Confidence intervals that do not contain zero indicate a mean difference that is statistically significant.
Individual confidence level
The percentage of times that a single confidence interval includes the true difference between one pair of group means, if you repeat the study multiple times.
Simultaneous confidence level
The percentage of times that a set of confidence intervals includes the true differences for all group comparisons, if you repeat the study multiple times.
Controlling the simultaneous confidence level is particularly important when you perform multiple comparisons. If you do not control the simultaneous confidence level, the chance that at least one confidence interval does not contain the true difference increases with the number of comparisons.
For more information, go to Understanding individual and simultaneous confidence levels in multiple comparisons.
For more information about how to interpret the results for Hsu's MCB, go to What is Hsu's multiple comparisons with the best (MCB)?
Paint | N | Mean | Grouping | |
---|---|---|---|---|
Blend 4 | 6 | 18.07 | A | |
Blend 1 | 6 | 14.73 | A | B |
Blend 3 | 6 | 12.98 | A | B |
Blend 2 | 6 | 8.57 | B |
In these results, the table shows that group A contains Blends 1, 3, and 4, and group B contains Blends 1, 2, and 3. Blends 1 and 3 are in both groups. Differences between means that share a letter are not statistically significant. Blends 2 and 4 do not share a letter, which indicates that Blend 4 has a significantly higher mean than Blend 2.
Difference of Levels | Difference of Means | SE of Difference | 95% CI | T-Value | Adjusted P-Value |
---|---|---|---|---|---|
Blend 2 - Blend 1 | -6.17 | 2.28 | (-12.55, 0.22) | -2.70 | 0.061 |
Blend 3 - Blend 1 | -1.75 | 2.28 | (-8.14, 4.64) | -0.77 | 0.868 |
Blend 4 - Blend 1 | 3.33 | 2.28 | (-3.05, 9.72) | 1.46 | 0.478 |
Blend 3 - Blend 2 | 4.42 | 2.28 | (-1.97, 10.80) | 1.94 | 0.245 |
Blend 4 - Blend 2 | 9.50 | 2.28 | (3.11, 15.89) | 4.17 | 0.002 |
Blend 4 - Blend 3 | 5.08 | 2.28 | (-1.30, 11.47) | 2.23 | 0.150 |
To determine how well the model fits your data, examine the goodness-of-fit statistics in the Model Summary table.
S is measured in the units of the response variable and represents how far the data values fall from the fitted values. The lower the value of S, the better the model describes the response. However, a low S value by itself does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.
R^{2} is the percentage of variation in the response that is explained by the model. The higher the R^{2} value, the better the model fits your data. R^{2} is always between 0% and 100%.
A high R^{2} value does not indicate that the model meets the model assumptions. You should check the residual plots to verify the assumptions.
Use predicted R^{2} to determine how well your model predicts the response for new observations. Models that have larger predicted R^{2} values have better predictive ability.
A predicted R^{2} that is substantially less than R^{2} may indicate that the model is over-fit. An over-fit model occurs when you add terms for effects that are not important in the population. The model becomes tailored to the sample data and, therefore, may not be useful for making predictions about the population.
Predicted R^{2} can also be more useful than adjusted R^{2} for comparing models because it is calculated with observations that are not included in the model calculation.
S | R-sq | R-sq(adj) | R-sq(pred) |
---|---|---|---|
3.95012 | 47.44% | 39.56% | 24.32% |
In these results, the factor explains 47.44% of the variation in the response. S indicates that the standard deviation between the data points and the fitted values is approximately 3.95 units.
Use the residual plots to help you determine whether the model is adequate and meets the assumptions of the analysis. If the assumptions are not met, the model may not fit the data well and you should use caution when you interpret the results.
Use the residuals versus fits plot to verify the assumption that the residuals are randomly distributed and have constant variance. Ideally, the points should fall randomly on both sides of 0, with no recognizable patterns in the points.
Pattern | What the pattern may indicate |
---|---|
Fanning or uneven spreading of residuals across fitted values | Nonconstant variance |
A point that is far away from zero | An outlier |
Use the normal probability plot of the residuals to verify the assumption that the residuals are normally distributed. The normal probability plot of the residuals should approximately follow a straight line.
Pattern | What the pattern may indicate |
---|---|
Not a straight line | Nonnormality |
A point that is far away from the line | An outlier |
Changing slope | An unidentified variable |
If your one-way ANOVA design meets the guidelines for sample size, the results are not substantially affected by departures from normality.