Principles of Biostatistics
ANOVA
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
8
12
10
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
8
12
10
ANOVA (Analysis of Variance)
Assumptions:
• Samples are independent (within and among groups)
• Population variances are equal
• Populations are normally distributed
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
10.33
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Diet
Standard
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
Square Deviations (within groups)
(9-9.4)2
(10-9.4)2 (10-9.4)2 (10-9.4)2
(8-9.4)2
Junk Food (10-11.6)2 (10-11.6)2 (13-11.6)2 (13-11.6)2 (12-11.6)2
Organic
Diet
Standard
Junk Food
Organic
(9-10)2
(10-10)2
(12-10)2
(9-10)2
Square Deviations (within groups)
0.16
0.36
0.36
0.36
2.56
2.56
1.96
1.96
1
0
4
1
(10-10)2
1.96
0.16
0
9.4
11.6
10
10.33
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
Square Deviations (within groups)
0.16
0.36
0.36
0.36
2.56
2.56
1.96
1.96
1
0
4
1
8
12
10
1.96
0.16
0
Sum of Squares (within groups)  18.4
Degrees of Freedom  3(5  1)  12
Mean Square (within groups) 
18.4
 1.533
12
9.4
11.6
10
10.33
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
10.33
Diet
Square Deviations (between groups)
Standard
(9.4-10.33)2
Junk Food
(11.6-10.33)2
Organic
(10-10.33)2
Diet
Standard
Junk Food
Organic
Square Deviations (between groups)
0.87
1.60
0.11
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
Diet
Standard
Junk Food
Organic
8
12
10
9.4
11.6
10
10.33
Square Deviations (between groups)
0.87
1.60
0.11
Sum of Squares (between groups)  (0.87  1.60  0.11)(5)  12.9
Degrees of Freedom  3  1  2
Mean Square (between groups) 
12.9
 6.45
2
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Mean Square (between groups)
F
Mean Square (within groups)
6.45

 4.21
1.533
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gai (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
10.33
• Table shows weight gains for mice on 3 diets.
• Test the following hypothesis at the α = 0.05 sig level.
H0 : 1  2  3 vs Ha : 1  2 or 1  3 or 2  3
Mean Square (between groups)
F
Mean Square (within groups)
6.45

 4.21
1.533
F2,12 (0.95)  3.89
Reject H0
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gai (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
10.33
Calculating the p-value
  0.05
p-value  0.041
p-value  0.05
Since p-value   we Reject H0
Post-hoc Tests
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
8
12
10
H 0 : 1  2  3 vs H a : 1  2 or 1  3 or 2  3
Rejected H0
Post-hoc Tests
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
8
12
10
H 0 : 1  2  3 vs H a : 1  2 or 1  3 or 2  3
Rejected H0
Need to test each pair
H 0 : 1   2 vs H a : 1   2
H 0 : 1  3 vs H a : 1  3
H 0 : 2  3 vs H a :  2  3
Post-hoc Tests
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
8
12
10
H 0 : 1  2  3 vs H a : 1  2 or 1  3 or 2  3
Rejected H0
Need to test each pair
H 0 : 1   2 vs H a : 1   2
H 0 : 1  3 vs H a : 1  3
H 0 : 2  3 vs H a :  2  3
Assumptions
• Samples are independent
• Population variances are equal
• Populations are normally distributed
• Not paired samples
Estimating the Pooled Variance
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
10
13
9
(n1  1) s12  (n2  1) s22  (n3  1) s32
s 
n1  1  n2  1  n3  1
2
4·0.892  4·1.52 2  4·1.22 2

12
 1.53
s  1.53  1.24
Degrees of Freedom  12
8
12
10
St Dev
0.89
1.52
1.22
Back to the Post-hoc Tests
Diet
Standard
Junk Food
Organic
H0 : 1  2 vs Ha : 1  2
t

9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
t12,0.05/2  2.179
x1  x2
x x
 1 2
Standard Error s n1  n1
1
2
9.4  11.6
1.24
1
5
 15

2.2
 2.81
0.784
Reject H0
Post-hoc Confidence Interval
Diet
Standard
Junk Food
Organic
9
10
9
Find a 95% confidence interval for 1  2
( x1  x2 )  t /2 (Standard Error)
( x1  x2 )  t /2 s
1 1

n1 n2
(9.4  11.6)  2.179(1.24)
2.2  1.71
(3.91, 0.49)
1 1

5 5
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
Post-hoc Summary
Diet
Standard
Junk Food
Organic
9
10
9
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
t-statistic Decision
H 0 : 1  2 vs H a : 1  2
-2.81
Reject H0
H 0 : 1  3 vs H a : 1  3
-0.77
Retain H0
H 0 : 2  3 vs H a : 2  3
2.04
Retain H0
9.4
11.6
10
Confidence Interval
1  2
1  3
 2  3
(-3.91, -0.49)
(-2.31, 1.11)
(-0.11, 3.31)
SPSS Output
Bonferroni’s Correction
Overall error rate  (Pairwise error rate)(# of comparisons)
Example
• A study has 3 groups.
• 3 comparisons must be made.
• 1 to 2, 1 to 3, 2 to 3
• If the pairwise error rate is 5%,
• approximate the overall error rate
• Answer: 15%
Bonferroni’s Correction
Overall error rate  (Pairwise error rate)(# of comparisons)
Example
• A study has 4 groups.
• 6 comparisons must be made.
• 1 to 2, 1 to 3, 1 to 4,
2 to 3, 2 to 4, 3 to 4
• If the pairwise error rate is 5%,
• approximate the overall error rate
• Answer: 30%
Bonferroni’s Correction
Overall error rate  (Pairwise error rate)(# of comparisons)
Example
• A study has 3 groups.
• 3 comparisons must be made.
• 1 to 2, 1 to 3, 2 to 3
• If the overall error rate is 5%,
• approximate the pairwise error rate
• Answer: 5%  1.67%
3
Bonferroni’s Correction
Overall error rate  (Pairwise error rate)(# of comparisons)
Example
• A study has 4 groups.
• 6 comparisons must be made.
• 1 to 2, 1 to 3, 1 to 4,
2 to 3, 2 to 4, 3 to 4
• If the overall error rate is 5%,
• approximate the pairwise error rate
• Answer:
5%
 0.83%
6
Formula for # of Comparisons
m  # of groups
# of comparisons  m C2 
m!
2!(m  2)!
3 C2 
3!
3·2·1

3
2!(3  2)! 2·1(1)
4 C2 
4!
4·3·2·1

6
2!(4  2)! 2·1(2·1)
Post-hoc Confidence Intervals with
Bonferroni’s Correction
Want overall error rate  0.05
  0.05
# of comparisons  3
0.05
Pairwise error rate 
 0.0167
3
Post-hoc Confidence Intervals with
Bonferroni’s Correction
Want overall error rate  0.05
  0.05
# of comparisons  3
0.05
Pairwise error rate 
 0.0167
3
Instead of using t12,0.05/2  2.179
well use t12,0.0167/2  2.779
Post-hoc Confidence Intervals with
Bonferroni’s Correction
Diet
Standard
Junk Food
Organic
9
10
9
Find a 95% confidence interval for 1  2
( x1  x2 )  t /2 (Standard Error)
( x1  x2 )  t /2 s
1 1

n1 n2
(9.4  11.6)  2.779(1.24)
2.2  2.179
(4.38, 0.02)
1 1

5 5
Weight Gain (grams)
10
10
10
13
10
12
Mean
10
13
9
8
12
10
9.4
11.6
10
SPSS Output