Least Significant Difference (LSD)
The one factor and interaction graphs can have least significant difference (LSD) bars around the predicted means . The height of the bars is determined by the design, model, confidence level and unexplained variation. If the ANOVA shows a significant result for the overall model test, these bars can be used test for a significant difference in the predictions.
The default view is the average LSD I-beam around the predictions, which are good for quick comparisons. If the I-beams do not overlap then the predictions may be significantly different. Exact pairwise comparisons produced when a prediction (square, triangle, any shape that is not a circle) is clicked. A horizontal line, called the prediction line, is drawn from the selected prediction. Vertical LSD bars are drawn from the other displayed predictions. If the horizontal prediction line is outside an LSD bar, the two predictions are significantly different.
The formula for the LSD is as follows:
\(LSD_i = t_{(\alpha/2, residual\, df)} \cdot \frac{\displaystyle\sum_{i\neq{j}}^{k} \sqrt{(x_i - x_j)(X^TV^{-1}X)^{-1}(x_i - x_j)^T}}{k - 1}\)
\(i\) |
Point of interest determined by Factors Tool settings and selected treatment on graph. |
\(j\) |
Arbitrary reference IDs for each displayed prediction point. |
\(k\) |
The total number of displayed prediction points. |
\(t\) |
The Student’s t critical value |
\(α\) |
alpha risk = 1 - confidence level |
residual df |
residual degrees of freedom found on the ANOVA. |
\(x_i, x_j\) |
Expanded point vector for the point of interest or displayed prediction point. |
\(X\) |
The expanded model matrix. |
\(V\) |
The variance matrix |
The expanded model matrix (X) has one row for each run in the design (n) and one column (p) for each term in the model. The coded form of the X matrix can be seen by going to the ANOVA report, and selecting the View menu, Show X Matrix.
The variance matrix (V) is n by n. For randomized designs V is the identity matrix times the mean squared error (residual mean square on the ANOVA). For a Split-plot design the V matrix can be obtained by going to the ANOVA report and selecting the View menu, Show V Matrix.
An expanded point vector (\(x_i, x_j\)) has similar form to a row of the X matrix. The linear term’s elements are the settings for the factors; higher-order terms are the product of the linear term’s elements.
The protected Fisher’s least significant difference test does not completely control the family-wise type I (false positive) error rate. Stat-Ease, Inc. chooses to maximize the power of detecting true effects while allowing the family-wise type I error rate to grow slightly larger than the alpha significance cutoff. If decisions are based on a falsely positive effect, future experiments and production outcomes will likely show no change based on that effect. If true differences are missed due to low power and their factors allowed to vary across the tested range, at best, production will be plagued by unexplained variation, and at worst an opportunity to optimize the process will have been missed.