conducting anova’s. i.why? a. more than two groups to compare

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Conducting ANOVA’s

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Page 1: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

Page 2: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compare

Page 3: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compare

What’s the prob?

D. putrida low densityD. putrida high densityD. putrida with D. tripuncatata

Page 4: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compare

What’s the prob?

D. putrida low densityD. putrida high densityD. putrida with D. tripuncatata

What was our solution?

Page 5: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compare

What’s the prob?

D. putrida low densityD. putrida high densityD. putrida with D. tripuncatata

What was our solution?

U

U

U

Page 6: Conducting ANOVA’s. I.Why? A. more than two groups to compare

What’s the prob?

D. putrida low densityD. putrida high densityD. putrida with D. tripuncatata

Tested each contrast at p = 0.05

Probability of being correct in rejecting each Ho:

1 2 30.95 0.95 0.95

U

U

U

Page 7: Conducting ANOVA’s. I.Why? A. more than two groups to compare

What’s the prob?

D. putrida low densityD. putrida high densityD. putrida with D. tripuncatata

Tested each contrast at p = 0.05

Probability of being correct in rejecting all Ho:

1 2 30.95 x 0.95 x 0.95 = 0.86

So, Type I error rate has increased from 0.05 to 0.14

U

U

U

Page 8: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Probability of being correct in rejecting all Ho:

1 2 30.95 x 0.95 x 0.95 = 0.86

So, Type I error rate has increased from 0.05 to 0.14

Hmmmm….. What can we do to maintain a 0.05 level across all contrasts?

Page 9: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Probability of being correct in rejecting all Ho:

1 2 30.95 x 0.95 x 0.95 = 0.86

So, Type I error rate has increased from 0.05 to 0.14

Hmmmm….. What can we do to maintain a 0.05 level across all contrasts?

Right. Adjust the comparison-wise error rate.

Page 10: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Probability of being correct in rejecting all Ho:

1 2 30.95 x 0.95 x 0.95 = 0.86

So, Type I error rate has increased from 0.05 to 0.14

Simplest: Bonferroni correction:

Comparison-wise p = experiment-wise p/nWhere n = number of contrasts.Experient-wise = 0.05Comparison-wise = 0.05/3 = 0.0167

Page 11: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Probability of being correct in rejecting all Ho:

1 2 30.95 x 0.95 x 0.95 = 0.86

So, Type I error rate has increased from 0.05 to 0.14

Simplest: Bonferroni correction:

Comparison-wise p = experiment-wise p/nWhere n = number of contrasts.Experient-wise = 0.05Comparison-wise = 0.05/3 = 0.0167So, confidence = 0.983

Page 12: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Probability of being correct in rejecting all Ho:

1 2 30.983 x 0.983 x 0.983 = 0.95

So, Type I error rate is now 0.95

Simplest: Bonferroni correction:

Comparison-wise p = experiment-wise p/nWhere n = number of contrasts.Experient-wise = 0.05Comparison-wise = 0.05/3 = 0.0167So, confidence = 0.983

Page 13: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compare

What’s the prob? - multiple comparisons reduce

experiment-wide alpha level.

- Bonferroni adjustments assume contrasts as independent; but they are both part of the same experiment…

Page 14: Conducting ANOVA’s. I.Why? A. more than two groups to compare

- Bonferroni adjustments assume contrasts as independent; but they are both part of the same experiment…

Our consideration of 1 vs. 2 might consider the variation in all treatments that were part of this experiment; especially if we are interpreting the differences between mean comparisons as meaningful.

1 vs. 2 – not significant1 vs. 3 – significant. So, interspecific competition is more important than intraspecific competition

Page 15: Conducting ANOVA’s. I.Why? A. more than two groups to compare

- Bonferroni adjustments assume contrasts as independent; but they are both part of the same experiment…

Our consideration of 1 vs. 2 might consider the variation in all treatments that were part of this experiment; especially if we are interpreting the differences between mean comparisons as meaningful.

Page 16: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?

A. more than two groups to compareB. complex design with multiple factors

- blocks - nested terms - interaction effects - correlated variables (covariates) - multiple responses

Page 17: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?II. How?

A. Variance Redux

Of a population Of a sample

Page 18: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Sum of squaresn - 1S2 =

Page 19: Conducting ANOVA’s. I.Why? A. more than two groups to compare

“Sum of squares” = SSn - 1S2 =

= SS(S x2) - (Sx)2 n

Page 20: Conducting ANOVA’s. I.Why? A. more than two groups to compare

“Sum of squares” = SSn - 1S2 =

= SS(S x2) - (Sx)2 n

n - 1MS =

Page 21: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?II. How?

A. Variance ReduxB. The ANOVA Table

Source of Variation df SS MS F p

Page 22: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Group A Group B Group C(Control) (Junk Food) (Health Food)10.8 12.7 9.811 13.9 8.69.7 11.8 810.1 13 7.511.2 11 99.8 10.9 1010.5 13.6 8.19.5 10.9 7.810 11.5 7.910.2 12.8 9.1

Weight gain in mice fed different diets

Group sumsSxSx2

(Sx)2/n

Page 23: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Group A Group B Group C(Control) (Junk Food) (Health Food)10.8 12.7 9.811 13.9 8.69.7 11.8 810.1 13 7.511.2 11 99.8 10.9 1010.5 13.6 8.19.5 10.9 7.810 11.5 7.910.2 12.8 9.1

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

Page 24: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Group A Group B Group C(Control) (Junk Food) (Health Food)10.8 12.7 9.811 13.9 8.69.7 11.8 810.1 13 7.511.2 11 99.8 10.9 1010.5 13.6 8.19.5 10.9 7.810 11.5 7.910.2 12.8 9.1

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

Correction term = (SSx)2/N = (310.7)2/30

Page 25: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Group A Group B Group C(Control) (Junk Food) (Health Food)10.8 12.7 9.811 13.9 8.69.7 11.8 810.1 13 7.511.2 11 99.8 10.9 1010.5 13.6 8.19.5 10.9 7.810 11.5 7.910.2 12.8 9.1

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Page 26: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29

SStotal = 3305.09 – 3217.816 = 87.274

= SS(S x2) - (Sx)2 n

Page 27: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274

SStotal = 3305.09 – 3217.816 = 87.274

= SS(S x2) - (Sx)2 n

Page 28: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973

SSgroup = 3283.789 – 3217.816 = 65.973

= SS(S x2) - (Sx)2 n

Page 29: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986

MSgroup = 65.973/2 = 32.986

(S x2) - (Sx)2 n

n - 1MS =

Page 30: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986“ERROR” (within) 27 21.301

Page 31: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986“ERROR” (within) 27 21.301 0.789

MSerror = 21.301/27 = 0.789

Page 32: Conducting ANOVA’s. I.Why? A. more than two groups to compare

GOOD GRIEF !!!

Page 33: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986“ERROR” (within) 27 21.301 0.789

Variance (MS) between groupsVariance (MS) within groupsF =

Page 34: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986 41.81“ERROR” (within) 27 21.301 0.789

32.9860.789F = = 41.81

Page 35: Conducting ANOVA’s. I.Why? A. more than two groups to compare
Page 36: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Weight gain in mice fed different diets

Group sums TotalsSx 102.8 122.1 85.8 310.7Sx2 1059.76 1502.41 742.92 3305.09(Sx)2/n 1056.784 1490.841 736.164 3283.789

3217.816 = CT

Source of Variation df SS MS F pTOTAL 29 87.274GROUP 2 65.973 32.986 41.81 < 0.05“ERROR” (within) 27 21.301 0.789

32.9860.789F = = 41.81

Page 37: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Conducting ANOVA’s

I. Why?II. How?III. Comparing Means “post-hoc mean comparison tests – after ANOVA

TUKEY – CV = q MSerror

n

Q from table A.7 = 3.53n = n per group (10)

= 0.9915

Page 38: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Means:

Health Food 8.58Control 10.29Junk Food 12.25

H – C = 1.70J – C = 1.93H – J = 3.67

All greater than 0.9915, so all mean comparisions are significantly different at an experiment-wide error rate of 0.05.

Page 39: Conducting ANOVA’s. I.Why? A. more than two groups to compare

Means:

Health Food 8.58 aControl 10.29 bJunk Food 12.25 c

H – C = 1.70J – C = 1.93H – J = 3.67

All greater than 0.9915, so all mean comparisions are significantly different at an experiment-wide error rate of 0.05.