Basic Analysis and Graphing - SAS

180 Performing Oneway Analysis Chapter 5
Statistical Details for the Oneway Platform
Figure 5.33 Geometric Relationship of t-test Statistics
t α
⋅ std( μˆ
– μˆ
)
-- 1 2
2
μˆ 1
t α
⋅ std( μˆ --- 1)
2
μˆ 2
t α
⋅ std( μˆ --- 2)
2
The radius of each circle is the length of the corresponding leg of the triangle, which is t α ⁄ 2
std( μˆ i)
.
The circles must intersect at the same right angle as the triangle legs, giving the following relationship:
• If the means differ exactly by their least significant difference, then the confidence interval circles around
each mean intersect at a right angle. That is, the angle of the tangents is a right angle.
Now, consider the way that these circles must intersect if the means are different by greater than or less than
the least significant difference:
• If the circles intersect so that the outside angle is greater than a right angle, then the means are not
significantly different. If the circles intersect so that the outside angle is less than a right angle, then the
means are significantly different. An outside angle of less than 90 degrees indicates that the means are
farther apart than the least significant difference.
• If the circles do not intersect, then they are significantly different. If they nest, they are not significantly
different. See Figure 5.12.
The same graphical technique works for many multiple-comparison tests, substituting a different
probability quantile value for the Student’s t.
Statistical Details for Power
To compute power, you make use of the noncentral F distribution. The formula (O’Brien and Lohr 1984) is
given as follows:
Power = Prob(F > F crit , ν 1 , ν 2 , nc)
where:
• F is distributed as the noncentral F(nc, ν 1 , ν 2 )and F crit = F (1 - α, ν1, ν2) is the 1 - α quantile of the F
distribution with ν 1 and ν 2 degrees of freedom.
• ν 1 = r -1 is the numerator df.
• ν 2 = r(n -1) is the denominator df.
• n is the number per group.
• r is the number of groups.