Repeated Measures ANOVA
Rogaine for Hair Growth in Women
V.H. Price and E. Menefee (1990). “Quantitative Estimation of Hair Growth I. Androgenic Alopecia in
Women: Effect of Minoxidil,” The journal of Investigative Dermatology, Vol.95, pp.683-687
Data Description
• Subjects: 8 Women with Androgenetic Alopecia
• Treatments: Minoxidil (Rogaine) and Placebo
(4 Women per treatment)
• Time Periods: Measurements made Pretreatment (0 weeks), 8, 16, 24, and 32 weeks
(will only consider post-treatment measures)
Data
Trt
1
1
1
1
0
0
0
0
Trt\Week
Rogaine
Placebo
Overall
Subject
1
2
3
4
5
6
7
8
Week0
216
130
206
106
142
178
189
180
8
189.75
179.75
184.75
Week8
290
146
193
130
154
161
219
185
16
227
183.75
205.375
Subject
1
2
3
4
5
6
7
8
Week16
340
206
218
144
145
170
197
223
24
217
193.25
205.125
Trt
Rogaine
Rogaine
Rogaine
Rogaine
Placebo
Placebo
Placebo
Placebo
Week24
275
220
223
150
160
194
218
201
32
225.5
175.5
200.5
Average
299.75
195.25
215
149.25
151.75
173.5
209.25
197.75
Week32
294
209
226
173
148
169
203
182
Overall
214.8125
183.0625
198.9375
Mean Hair Growth versus Week by Treatment
240
230
220
Hair Growth
210
200
Rogaine
Placebo
Overall
190
180
170
160
150
0
5
10
15
20
Week
25
30
35
Statistical Model
Yijk     i  b j ( i )   k  ( ) ik   ijk
where :
  Overall Mean Hair Growth
 i  Effect of Trt i i  1,2
b j ( i )  Effect of j th woman wit hin treatm ent i
 k  Effect of k th time period k  1,2,3,4
( )ik  Interactio n between Trt i and Time k
 ijk  Random error term
j  1,2,3,4
Sums of Squares
Total Sum of Squares (df  2(4)(4) - 1  31)


TSS  (290  198.94) 2  ...  (182  198.94) 2  73045.88
Treatment Sum of Squares (df  2 - 1  1) :


SSTRTS  4(4) (214.81  193.94) 2  (183.06  198.94) 2  8064.5
Subject Wi thin Treatment Sum of Squares (df  2(4 - 1)  6) :
SS Subjects(Trts) 


4 (299.75  214.81) 2  ...  (149.25  214.81) 2  (151.75  183.06) 2  ...  (197.75  183.06) 2 
 55475.88
Time Sum of Squares (df  4 - 1)  3 :


SSTime  2(4) (184.75  198.94) 2  (205.38  198.94) 2  (205.13  198.94) 2  (200.50  198.94) 2  2267.63
Treatment x Time Interactio n Sum of Squares (df  1(3)  3) :


SSTrtsxTime  4 (189.75  214.81  184.75  198.94) 2  ...  (175.5  183.06  200.50  198.94) 2  2004.75
Error2 Sum of Squares (Time x Subject(Tr t) Interactio n) (df  3(6)  18) :
SS Error2  5233.13 (By Subtractio n)
ANOVA Table/F-Tests
Source
Treatments
Subject(Trts)
Time
TrtxTime
Error2
Total
df
1
6
3
3
18
31
SS
8064.50
55475.88
2267.63
2004.75
5233.13
73045.88
MS
8064.50
9245.98
755.88
668.25
290.73
F
0.87
P-value
0.3864
2.60
2.30
0.0839
0.1120
Note: We cannot conclude any treatment or time effects or interaction at
the 0.05 significance level. The subject-to-subject variation is very large,
which makes finding a treatment effect very difficult in this setting (and
very few subjects as well)
Post-Hoc comparisons*
Comparing Treatments (1 Comparison : Rogaine - Placebo)
Y R  214.81 Y P  183.06 MS Subjects(Trt )  9246 df  6 t.05 / 2, 6  2.447
214.81  183.06  2.447
2(9246)
 31.75  83.19   51.44,114.94
4(4)
Comparing Time Means (6 Comparison s) :
Y ..1  184.75 Y ..2  205.38 Y ..3  205.13 Y ..4  200.5 MS Error2  290.73 t.05 /(2 x 6),18  2.963
BTime  2.963
2(290.73)
 25.26 No time means differ by more than 21.63
2(4)
* Bonferroni’s method for illustration only (no significant effects found from
Analysis of Variance)