Principles of BiostatisticsANOVA

Diet Weight Gain (grams)Standard 9 10 10 10 8Junk Food 10 10 13 13 12

Organic 9 10 12 9 10

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams)Standard 9 10 10 10 8Junk Food 10 10 13 13 12

Organic 9 10 12 9 10

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

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.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Diet Square Deviations (within groups)Standard (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 (9-10)2 (10-10)2 (12-10)2 (9-10)2 (10-10)2

Diet Square Deviations (within groups)Standard 0.16 0.36 0.36 0.36 1.96Junk Food 2.56 2.56 1.96 1.96 0.16

Organic 1 0 4 1 0

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Diet Square Deviations (within groups)Standard 0.16 0.36 0.36 0.36 1.96Junk Food 2.56 2.56 1.96 1.96 0.16

Organic 1 0 4 1 0

Sum of Squares (within groups) 18.4Degrees of Freedom 3(5 1) 12

18.4Mean Square (within groups) 1.53312

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Diet Square Deviations (between groups)Standard (9.4-10.33)2

Junk Food (11.6-10.33)2

Organic (10-10.33)2

Diet Square Deviations (between groups)Standard 0.87Junk Food 1.60

Organic 0.11

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Diet Square Deviations (between groups)Standard 0.87Junk Food 1.60

Organic 0.11

Sum of Squares (between groups) (0.87 1.60 0.11)(5) 12.9Degrees of Freedom 3 1 2

12.9Mean Square (between groups) 6.452

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gai (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Mean Square (between groups)Mean Square (within groups)6.45 4.21

1.533

F

• Table shows weight gains for mice on 3 diets.• Test the following hypothesis at the α = 0.05 sig level.

0 1 2 3 1 2 1 3 2 3H : vs H : or or a

Diet Weight Gai (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 1010.33

Mean Square (between groups)Mean Square (within groups)6.45 4.21

1.533

F

2,12 (0.95) 3.89F

0Reject H

Calculating the p-value

0.05 -value 0.041p

0

-value 0.05

Since -value we Reject H

p

p

Post-hoc TestsDiet Weight Gain (grams)

Standard 9 10 10 10 8Junk Food 10 10 13 13 12

Organic 9 10 12 9 10

0 1 2 3 1 2 1 3 2

0

3H : vs H : or Rejec

orted

H

a

Post-hoc TestsDiet Weight Gain (grams)

Standard 9 10 10 10 8Junk Food 10 10 13 13 12

Organic 9 10 12 9 10

0 1 2 3 1 2 1 3 2

0

3H : vs H : or Rejec

orted

H

a

0 1 2 1 2

0 1 3 1 3

0 2 3 2 3

Need to test each pairH : vs H :

H : vs H :

H : vs H :

a

a

a

Post-hoc TestsDiet Weight Gain (grams)

Standard 9 10 10 10 8Junk Food 10 10 13 13 12

Organic 9 10 12 9 10

0 1 2 3 1 2 1 3 2

0

3H : vs H : or Rejec

orted

H

a

0 1 2 1 2

0 1 3 1 3

0 2 3 2 3

Need to test each pairH : vs H :

H : vs H :

H : vs H :

a

a

a

Assumptions• Samples are independent • Population variances are equal• Populations are normally distributed

• Not paired samples

Estimating the Pooled Variance

Diet Weight Gain (grams) St DevStandard 9 10 10 10 8 0.89Junk Food 10 10 13 13 12 1.52

Organic 9 10 12 9 10 1.22

2 2 22 1 1 2 2 3 3

1 2 3

2 2 2

1) 1) 1)1 1 1

4·0.89 ·1.52 ·1.22

1.53

1.53 1.2

( ( (

4 41

4

2

n n nsns s s

n n

s

Degrees of Freedom 12

Back to the Post-hoc Tests

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 10

0 1 2 1 2H : vs H :a

1 2

1 2 1 21 1

1 15 5

Standard Error

2.2 2.810.7841.24

9.4 11.6

n n

x x x xts

12,0.05/2 2.179t

0Reject H

Post-hoc Confidence Interval

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 10

1 2

1 2 /2

1 2 /21 2

( (

(

Find a 95% confidence interval for

) Standard Error)

1 1)

1 1(9.4 11.6) 2.179(1.24)5 5

2.2 1.71( 3.91, 0.49)

x x t

x x t sn n

Post-hoc Summary

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 10

t-statistic Decision

-2.81 Reject H0

-0.77 Retain H0

2.04 Retain H0

0 1 2 1 2

0 1 3 1 3

0 2 3 2 3

H : vs H :H : vs H :H : vs H :

a

a

a

1 2

1 3

2 3

Confidence Interval

(-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

2

3 2

4 2

# of groups

!# of comparisons2!( 2)!

3! ·2·1 32!(3 2)! ·1(1)

4! · ·2·1 62!(4

32

42)! ·1( ·1

32 2 )

mmCm

C

C

m

Post-hoc Confidence Intervals with Bonferroni’s Correction

Want overall error rate 0.050.05

# of comparisons 3

0.05Pairwise error rate 0.01673

Post-hoc Confidence Intervals with Bonferroni’s Correction

Want overall error rate 0.050.05

# of comparisons 3

0.05Pairwise error rate 0.01673

12,0.05/2

12,0.0167/2

2.1Instead of using

we ll us

79

2. 7e 7 9

t

t

Post-hoc Confidence Intervals with Bonferroni’s Correction

Diet Weight Gain (grams) MeanStandard 9 10 10 10 8 9.4Junk Food 10 10 13 13 12 11.6

Organic 9 10 12 9 10 10

1 2

1 2 /2

1 2 /21 2

Find a 95% confidence interval for

) Standard Error)

1 1)

1 1(9.4 11.6) 2.779(1.24)5 5

2.2 2.179( 4.38, 0.

( (

02)

(

x x t

x x t sn n

SPSS Output