Some changes to Chapter 7 - Chapter 7 The Two-Way Layout...

Chapter 7
The Two-Way Layout
Some Changes to pages 9 – 12:
7.4 One-sided Comparisons of Treatments with Control
Based on Friedman Rank Sums (Nemenyi, Wilcoxon – Wilcox, Miller)
Used after rejecting the null hypothesis of no difference between treatment effects. Compares
each treatment with a control group (call it treatment 1).
Hypotheses of interest are
• Ho: τ
u
= τ
1
vs.
Ha: τ
u
> τ
1
for all u = 2, 3, …, k with decision Rule: Decide τ
u
> τ
1
if (R
u
– R
1
) ≥ r*
α
.
• Or Ha: τ
u
> τ
1
for all u = 2, 3, …,k with decision rule τ
1
< τ
u
if (R
1
– R
u
) ≥ r*
α
.
Example: Use the hydrobromide data of the example on page 3 (copied below) to test whether
any of the hydrobromides produce a higher average number of hours of sleep per night that
does the “no hypnotic,” i.e., treatment A.
Patient A B C D
1 0.6 (1) 1.3 (2) 2.5 (4) 2.1 (3)
2 1.1 (1.5) 1.1 (1.5) 5.7 (3) 5.8 (4)
3 2.5 (1) 6.2 (2) 8 (3) 8.2 (4)
4 2.8 (1) 3.6 (2) 4.4 (4) 4.3 (3)
5 2.9 (1) 4.9 (2) 6.3 (3) 6.4 (4)
6 3.0 (2) 1.4 (1) 3.8 (3) 4.4 (4)
R
j
(7.5) (10.5) (20) (22)
We are interested in testing Ho: τ
u
= τ
1
vs. Ha: τ
u
> τ
1
, for all u = 2, 3, 4 simultaneously.
From Table A25, with n = 6, k = 4 and α = 0.0404, we find
Hence we will decide τ
u
> τ
1
if R
u
– R
1
≥ 10.
Show that, from the observed data we have R
1
= 7.5, R
2
= 10.5, R
3
= 20 and R
4
= 22 and hence
Hence we decide that Treatments C and D are better than control (Treatment A), the others are
not different from the control.