Basic Analysis and Graphing - SAS

Chapter 5 Performing Oneway Analysis 179
Statistical Details for the Oneway Platform
Statistical Details for Comparison Circles
One approach to comparing two means is to determine whether their actual difference is greater than their
least significant difference (LSD). This least significant difference is a Student’s t-statistic multiplied by the
standard error of the difference of the two means and is written as follows:
LSD = t α ⁄ 2
std( μˆ
1 – μˆ 2)
The standard error of the difference of two independent means is calculated from the following relationship:
[ std( μˆ
1 – μˆ 2)
] 2 = [ std( μˆ 1)
] 2 + [ std( μˆ 2)
] 2
When the means are un correlated, these quantities have the following relationship:
LSD 2 2
= t α ⁄ 2
std( ( μˆ
1 – μˆ 2)
) =
t α ⁄ 2
std( μˆ 1)
2
+
2
t α ⁄ 2
stdμˆ 2
These squared values form a Pythagorean relationship, illustrated graphically by the right triangle shown in
Figure 5.32.
Figure 5.32 Relationship of the Difference between Two Means
t α
⋅ std( μˆ
– μˆ
)
-- 1 2
2
t α
⋅ std( μˆ --- 1)
2
t α
⋅ std( μˆ --- 2)
2
The hypotenuse of this triangle is a measuring stick for comparing means. The means are significantly
different if and only if the actual difference is greater than the hypotenuse (LSD).
Suppose that you have two means that are exactly on the borderline, where the actual difference is the same
as the least significant difference. Draw the triangle with vertices at the means measured on a vertical scale.
Also, draw circles around each mean so that the diameter of each is equal to the confidence interval for that
mean.