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Chapter 14: One-Way Analysis of Variance (ANOVA) http://www.luchsinger-mathematics.ch/Var_Reduction.jpg

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Examples: ANOVA In each of the following situations, what is the factor and how many levels are there? 1)Do five different brands of gasoline have an effect on automobile efficiency? 2)Does the type of sugar solution (glucose, sucrose, fructose, mixture) have an effect on bacterial growth? 3)Does the hardwood concentration in pulp (%) have an effect on tensile strength of bags made from the pulp? 4)Does the resulting color density of a fabric depend on the amount of dye used?

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ANOVA: Graphical c)

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Examples: ANOVA 1)Do five different brands of gasoline have an effect on automobile efficiency? 2)Does the type of sugar solution (glucose, sucrose, fructose, mixture) have an effect on bacterial growth?

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ANOVA test statistic

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ANOVA test

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F Distribution http://www.vosesoftware.com/ModelRiskHelp/index.htm#Distributions/ Continuous_distributions/F_distribution.htm

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P-value for an upper-tailed F test shaded area=P- value = 0.05

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ANOVA Table: Formulas SourcedfSSMS (Mean Square) F Model (between) I – 1 Error (within) N – I TotalN – 1

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ANOVA Hypothesis test: Summary H 0 : μ 1 = μ 2 = = μ I H a : At least two μ i ’s are different Test statistic: P-value:P(F ≥ F test ) has a F ,dfm,dfe distribution

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ANOVA: Example A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels. The data is on the next slide. Does Paxil affect serotonin levels in healthy young men at a significance level of 0.05?

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ANOVA: Example (cont). Dose0 mg20 mg40 mg 48.6258.6068.59 49.8572.5278.28 64.2266.7282.77 62.8180.1276.53 62.5168.4472.33

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ANOVA: Example (cont) 1.Let 1 be the mean serotonin level for men receiving 0 mg of Paxil. Let 2 be the mean serotonin level for men receiving 20 mg of Paxil. Let 3 be the mean serotonin level for men receiving 40 mg of Paxil.

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ANOVA: Example (cont) 2.H 0 : 1 = 2 = 3 The mean serotonin levels are the same at all 3 dosage levels [or, The mean serotonin levels are unaffected by Paxil dose] H A : at least one I is different The mean serotonin levels of the three groups are not all equal. [or, The mean serotonin levels are affected by Paxil dose]

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ANOVA: Example (cont). (NOT USED) Dose0 mg20 mg40 mg 48.6258.6068.59 49.8572.5278.28 64.2266.7282.77 62.8180.1276.53 62.5168.4472.33overall nini 55515 x̅ i 57.6069.2875.7067.53 sisi 7.6787.8955.460 (n i -1)s i 2 235.78249.32119.24604.34 n i (x̅ i - x̅̅) 2 492.5615.36333.96841.88 X

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ANOVA: Example (cont). Dose0 mg20 mg40 mg 48.6258.6068.59 49.8572.5278.28 64.2266.7282.77 62.8180.1276.53 62.5168.4472.33overall nini 55515 x̅ i 57.6069.2875.7067.53 sisi 7.6787.8955.460

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ANOVA: Example (cont) SourcedfSSMSFP-Value Model2841.88420.94 Error12604.3450.36 Total141446.23 0.0053218.36

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Example: ANOVA (cont) 7. This data does give strong support (P = 0.005321) to the claim that there is a difference in serotonin levels among the groups of men taking 0, 20, and 40 mg of Paxil. This data does give strong support (P = 0.005321) to the claim that Paxil intake affects serotonin levels in young men.

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Problem with Multiple t tests

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Overall Risk of Type I Error in Using Repeated t Tests at = 0.05

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Procedure: Multiple Comparison 1.Perform the ANOVA test (obtain the ANOVA table); only continue if the results are statistically significant. 2.Select a family significance level, . 3.Select the multiple comparison methodology. 4.Calculate t**. 5.Calculate all of the confidence intervals required by the procedure. 6.Determine which ones are statistically significant. 7.Visually display the results. 8.Write a conclusion in the context of the problem.

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Example: Multiple Comparison A random sample of 15 healthy young men are split randomly into 3 groups of 5. They receive 0, 20, and 40 mg of the drug Paxil for one week. Then their serotonin levels are measured to determine whether Paxil affects serotonin levels. Which dosage would provide the largest change in serotonin levels?

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Example: Multiple Comparison (cont) SourcedfSSMSFP-Value Model2841.88420.948.360.005321 Error12604.3450.36 Total141446.23 Dose0 mg20 mg40 mg 48.6258.6068.59 49.8572.5278.28 64.2266.7282.77 62.8180.1276.53 62.5168.4472.33 x̅ i 57.6069.2875.70

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Example: Multiple Comparison (cont) Dunnett: Therefore, dosages of both 20 mg and 40 mg of Paxil does raise serotonin levels. Tukey: Therefore, 40 mg dosage of Paxil does raise serotonin levels, but a 20 mg dosage of Paxil does not raise serotonin levels.

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