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Analysis of Variance (ANOVA) Shibin Liu SAS Beijing R&D

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 2

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 3

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Lesson overview 4 μ μ0μ0 One sample t-Test

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Lesson overview 5 μ2μ2 Two-sample t-Test μ1μ1

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Lesson overview 6 μ3μ3 ANOVA μ1μ1 μ2μ2

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Lesson overview 7 ANOVA Predictor VariableResponse Variable Levels

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Lesson overview 8 ANOVA Predictor VariableResponse VariablePredictor Variable

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Lesson overview 9 ANOVA One sample t-Test Two-sample t-Test

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Lesson overview 10 What do you want to examine? The relationship between variables The difference between groups on one or more variables The location, spread, and shape of the data’s distribution Summary statistics or graphics? How many groups? Which kind of variables? SUMMARY STATISTICS DISTRIBUTION ANALYSIS TTEST LINEAR MODELS CORRELATIONS ONE-WAY FREQUENCIES & TABLE ANALYSIS LINEAR REGRESSION LOGISTIC REGRESSION Summary statistics Both Two Two or more Descriptive Statistics Descriptive Statistics, histogram, normal, probability plots Analysis of variance Continuous only Frequency tables, chi-square test Categorical response variable Descriptive Statistics Inferential Statistics Lesson 1Lesson 2Lesson 3 & 4Lesson 5

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 11

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The Two-Sample t-Test: Introduction 12 μ2μ2 μ1μ1

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The Two-Sample t-Test: Introduction 13 Salary ? Blood pressure? …

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The Two-Sample t-Test: Introduction 14 In this topic, we will learn to do the following: – Analyze differences between two population means using the t-Test task – Verify the assumption of and perform a two-sample t-Test – perform a one-sided t-Test

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The Two-Sample t-Test: Introduction 15 H 0 : μ 1 = μ 2 Two-Sample t-Test H 0 : μ 1 - μ 2 =0 Two-Sample t-Test

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Two-Sample t-Tests: Assumptions 16 Before we start the analysis, examine the data to verify that the statistical assumption are valid: – Independent observations: – Normally distributed data for each group – Equal variances for each group. No information provided by other observations No impact between the observations

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Two-Sample t-Tests: F-Test for Equality of Variance 17 F-Statistic When the null hypothesis is true, what value will the F-Statistic be close to?

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Two-Sample t-Tests: Examining the Equal Variance t-Test and p-Values 18 F-Test for Equal Variance > ?

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Two-Sample t-Tests: Examining the Equal Variance t-Test and p-Values 19 < H 0 : μ 1 - μ 2 =0

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Two-Sample t-Tests: Examining the Unequal Variance t-Test and p-Values 20 F-Test for Equal Variance < ?

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Two-Sample t-Tests: Examining the Unequal Variance t-Test and p-Values 21 < H 0 : μ 1 - μ 2 =0

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Two-Sample t-Tests: Demo 22 Scenario : compare two group’s means, girl’s and boy’s SAT scores Identify the data Classification variable? Continuous variable to analyze? TestScores

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Two-Sample t-Tests: Demo 23 Analyze > ANOVA > t Test

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Two-Sample t-Tests: Demo 24 Result interpreting:Normal?

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Two-Sample t-Tests: Demo 25 Result interpretingNormal?

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Two-Sample t-Tests: Demo 26 Result interpreting H 0 : μ 1 - μ 2 =0

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Two-Sample t-Tests: Demo 27 Result interpreting The confidence interval for the mean difference ( , 125.2) includes 0. this implies that you cannot say with 95% confidence that the difference between boys and girls is not zero. Therefore, it also implies that the p-value is greater than 0.05.

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Two-Sample t-Tests: One-Sided Tests 28 Two-sample t-Test New drug only have positive effect One-Sided Test

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Two-Sample t-Tests: One-Sided Tests 29 One-Sided Test One direction

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Two-Sample t-Tests: One-Sided Tests 30 H 0 : equality Critical region One-Sided Test

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Two-Sample t-Tests: One-Sided Tests 31 H 0 : equality One-Sided Test What is Power? Power is the probability when your test will reject the null hypothesis when the hypothesis is false. Or the probability when you detect a difference when the difference actually exists.

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Two-Sample t-Tests: One-Sided Tests/Scenario 32 SAT Scores Group 1 Group 2 one-sided upper-tailed t-test

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Two-Sample t-Tests: One-Sided Tests/Scenario 33 one-sided upper-tailed t-test t statistic

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Two-Sample t-Tests: One-Sided Tests/Scenario 34 Analyze > ANOVA > t Test > Two sample > Preview Code> Insert code

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Two-Sample t-Tests: One-Sided Tests/Scenario 35 0 is not in the interval < 0.05

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Two-Sample t-Tests 36 Question 1. What justifies the choice of a one-sided test versus a two-sided test a)The need for more statistical power b)Theoretical and subject-matter considerations c)The non-significance of a two-sided test d)The need for an unbiased test statistic Answer: b

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Two-Sample t-Tests 37 Question 2. Answer: a

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 38

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One-Way ANOVA: Introduction 39 μ3μ3 ANOVA μ1μ1 μ2μ2

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One-Way ANOVA: Objective Analyze difference between population means using the Linear Models task Verify the assumption of analysis variance. 40

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One-Way ANOVA: ANOVA overview 41 One-way ANOVA Predictor VariableResponse Variable Levels

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One-Way ANOVA: ANOVA overview 42 ProgrammerTeacher Programmer earns more than teacher? Response Variable Predictor Variable salary Job title Case 1

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One-Way ANOVA: ANOVA overview 43 ProgrammerTeacher Two-sample t-Test Predictor Variable Case 1

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One-Way ANOVA: ANOVA overview 44 ProgrammerTeacher Two-sample t-Test Predictor Variable One-way ANOVA = F statistic t statistic 2 Case 1

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One-Way ANOVA: ANOVA overview 45 Predictor Variable Case 2 Medication 1Medication 2Placebo Response Variable T-cell counts

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One-Way ANOVA: ANOVA overview 46 Case 2 Medication 1 Medication 2 Placebo Two-sample t-Test Medication 1 Medication 2 Placebo

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One-Way ANOVA: ANOVA overview 47 Case 2 Medication 1 Medication 2 Placebo One-way ANOVA

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One-Way ANOVA: The ANOVA Hypothesis 48 Small difference ANOVA

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One-Way ANOVA: The ANOVA Hypothesis 49 ANOVA Predictor Variable Medication 1 Medication 2 Placebo

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One-Way ANOVA: The ANOVA Hypothesis 50 ANOVA Predictor Variable Medication 1 Medication 2 Placebo different

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One-Way ANOVA: The ANOVA model 51 effect i: indexes treatments (three types of medication) K: observation number μ: overall population mean Error term τ 1 =μ 1 - μ τ 2 =μ 2 - μ τ 3 =μ 3 - μ One-way ANOVA

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One-Way ANOVA: Sums of Squares 52 Variability between groups Variability between groups Variability within groups Variability within groups By ratio >

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One-Way ANOVA: Sums of Squares 53

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One-Way ANOVA: Sums of Squares 54 Total variability Variability between groups Variability within groups SS T = SS M + SS E

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One-Way ANOVA: F statistic 55 In general, degree of freedom (DF) can be thought of as the number of independent pieces of information. 1.Model DF is the number of treatment minus 1. 2.Corrected total DF is the sample size minus 1. 3.Error DF is the sample size minus the number of treatments (or the difference between the corrected total DF and the Model DF)

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One-Way ANOVA: Coefficient of Determination 56

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One-Way ANOVA: Assumptions for ANOVA 57 Assumptions for ANOVA 1.Independent observations 2.Error terms are normally distributed 3.Error terms have equal variances Assessing ANOVA Assumptions 1.Good data collection methods help ensure the independence assumption. 2.Diagnostic plots can be used to verify the assumption that the error is approximately normally distributed 3.The Linear Models task can produce a test of equal variance. H0 for this hypothesis test is that the variances are equal for all populations.

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One-Way ANOVA: Predicted and Residual values 58 Estimates of the error term Residuals = Observation - Group mean

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One-Way ANOVA: Predicted and Residual values The predicted value in ANOVA is the group mean. A residual is the difference between the observed value of the response and the predicted value of the response variable.

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One-Way ANOVA: 60 Scenario: Comparing Group Means with One-Way ANOVA Predictor Variable Fertilizer Bulb Weight Response Variable Garlic

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One-Way ANOVA: 61 Scenario: Comparing Group Means with One-Way ANOVA ANOVA H a : at least one is different

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One-Way ANOVA: 62 Scenario: Comparing Group Means with One-Way ANOVA Description check of MGGARLIC > Summary Statistic

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One-Way ANOVA: 63 Question 1. a)-20 b)20 c)400 d)0 e)Need more information You have 20 observations in you ANOVA and you calculate the residuals. What will they sum to? Answer: d

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One-Way ANOVA: 64 Question 2. a)The variability between the groups b)The variability within the groups c)The variability explained by the error terms Which of the following phrases describes the model sums of squares, or SSM? Answer: a

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One-Way ANOVA: 65 Question 3. a) Match the null hypothesis to the correct SAS output b) a b

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One-Way ANOVA: 66 Scenario: Comparing Group Means with One-Way ANOVA Task> ANOVA>Linear Models, with MGGARLIC data

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One-Way ANOVA: 67 Scenario: Comparing Group Means with One-Way ANOVA Task> ANOVA>Linear Models

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One-Way ANOVA: Summary 68 Null Hypothesis: All means are equal Alternative Hypothesis: at least one mean is different 1.Produce descriptive statistics. 2.Verify assumptions. Independence Normality Equal variance 3.Examine the p-value in the ANOVA table. If the p-value is less than alpha, reject the null hypothesis.

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 69

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ANOVA with Data from a Randomized Block Design: Introduction 70 Brands age Medication 1 Medication 2 Placebo

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ANOVA with Data from a Randomized Block Design: Introduction 71 Under Over 50

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ANOVA with Data from a Randomized Block Design: Objective Recognize the difference between a completely randomized design and a randomized block design. Differentiate between observed data and designed experiments. Use the Linear Models task to analyze data from a randomized block design 72

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ANOVA with Data from a Randomized Block Design: Observational Studies 73 Groups can be naturally occurring. Gender and ethnicity Random assignment might be unethical or untenable Smoking or credit risk groups In Observational or Retrospective studies, the data values are observed as they occur, not affected by an experimental design.

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ANOVA with Data from a Randomized Block Design: Controlled Experiments 74 1.Random assignment might be desirable to eliminate selection bias. 2.You often wan to look at the outcome measure prospectively. 3.You can manipulate the factors of interest and can more reasonably claim causation. 4.You can design your experiment to control for other factors contributing to the outcome measure.

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ANOVA with Data from a Randomized Block Design 75 Question 3. a)Yes b)No Can you determine a cause-and–effect relationship in an observational study? Answer: b observational studycontrolled study In an observational study, you often examine what already occurred, and therefore have little control over factors contributing to the outcome. In a controlled experiment, you can manipulate the factors of interest and can more reasonably claim causation.

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ANOVA with Data from a Randomized Block Design: Nuisance Factors 76 Nuisance Factors are factors can affect the outcome but are not of interest in the experiment. T-Cell CountMedicationAge Randomized block design

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ANOVA with Data from a Randomized Block Design 77 Question 4. a)Sum of Squares Model b)Sum of Squares Error c)Degrees of Freedom Which part of the ANOVA tables contains the variation due to nuisance factors? Answer: b

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ANOVA with Data from a Randomized Block Design: Including a Blocking Variable in the Model 78 Age

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ANOVA with Data from a Randomized Block Design: Including a Blocking Variable in the Model 79 Age

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ANOVA with Data from a Randomized Block Design: More ANOVA Assumptions 80 ANOVA Independent observations Normally distributed data Equal variances Under Over 50 The treatment are randomly assigned to each block No interaction

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ANOVA with Data from a Randomized Block Design 81 Scenario: Creating a Randomized Block Design Garlic H a : at least one is different What’s the nuisance factors in this case Sun PH level of soil Rain

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ANOVA with Data from a Randomized Block Design 82 Scenario: Creating a Randomized Block Design Bulb Weightfertilizer PH of the soilSunRain

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ANOVA with Data from a Randomized Block Design 83 Scenario: Creating a Randomized Block Design Randomized block design

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ANOVA with Data from a Randomized Block Design 84 Scenario: Creating a Randomized Block Design

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ANOVA with Data from a Randomized Block Design 85 Question 5. a)Sum of Squares Model b)Sum of Squares Error c)Degrees of Freedom In a block design, Which part of the ANOVA tables contains the variation due to nuisance factors? Answer: a

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ANOVA with Data from a Randomized Block Design: Performing ANOVA with Blocking 86 Task> ANOVA>Linear Models, with MGGARLIC_BLOCK data

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ANOVA with Data from a Randomized Block Design: Performing ANOVA with Blocking 87 Task> ANOVA>Linear Models, with MGGARLIC_BLOCK data

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ANOVA with Data from a Randomized Block Design 88 My groups are different. What next? – The p-value for Fertilizer indicates you should reject the H 0 that all groups are the same. – From which pairs of fertilizers, are garlic bulb weights different from one another? – Should you go back and do several t-tests?

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 89

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ANOVA Post Hoc Tests: Introduction 90 One-way ANOVA Randomized block design μ1μ1 μ1μ1 μ2μ2 μ2μ2 μ3μ3 μ3μ3 μ?μ? μ?μ?

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ANOVA Post Hoc Tests: Introduction 91 μ1μ1 μ1μ1 μ2μ2 μ2μ2 μ3μ3 μ3μ3 μ?μ? μ?μ? Pairwise test H 0 : μ 1 = μ 3 H 0 : μ 2 = μ 3 P-value Type I error ANOVA Post Hoc Tests ANOVA Post Hoc Tests Multiple Comparison Method Type I error

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ANOVA Post Hoc Tests: Multiple Comparison Methods 92 Question 7. a)0.5 b)0.25 c)0.00 d)1.00 e)0.75 With a fair coin, your probability of getting heads on one flip is 0.5. if you flip a coin and got heads, what is the probability of getting heads on the second try? Answer: a

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ANOVA Post Hoc Tests: Multiple Comparison Methods 93 Question 8. a)0.5 b)0.25 c)0.00 d)1.00 e)0.75 With a fair coin, your probability of getting heads on one flip is 0.5. If you flip a coin twice, what is the probability of getting at least one head out of two? Answer: e

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ANOVA Post Hoc Tests: Multiple Comparison Methods 94 Pairwise t-test α=0.05 H 0 : μ 1 = μ 3 H 0 : μ 2 = μ 3 Type I Error

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ANOVA Post Hoc Tests: Multiple Comparison Methods 95 Pairwise t-test Type I Error Comparisonwise Error Rate Number of Comparisons Experimentwise Error Rate nc: Number of comparisons

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ANOVA Post Hoc Tests: Tukey's Multiple Comparison Method 96 EER Tukey Method H 0 : μ 1 = μ 3 H 0 : μ 2 = μ 3 Pairwise comparisons

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ANOVA Post Hoc Tests: Tukey's Multiple Comparison Method 97 EER<0.05 Tukey Method H 0 : μ 1 = μ 3 Pairwise comparisons EER=0.05

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ANOVA Post Hoc Tests: Tukey's Multiple Comparison Method 98 This method is appropriate when you consider pairwise comparisons only. The Experimentwise Error Rate is: – Equal to alpha when all pairwise comparisons are considered – Less than alpha when fewer than all pairwise comparisons are considered

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ANOVA Post Hoc Tests: scenario: determine which mean is different 99 Garlic Fertilizers: three organics, one control H a : at least one is different

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ANOVA Post Hoc Tests: Diffograms and the Tukey Method 100 least square mean by least square mean by least square mean Difference between the means Equality of the means

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ANOVA Post Hoc Tests: Diffograms and the Tukey Method 101 Is there the diff between the treatments 1 and 2? Can you identify the pairwise comparisons that do not have significant diff means?

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ANOVA Post Hoc Tests: Dunnett's Multiple Comparison Method 102 Special Case of Comparing to a Control Comparing to a control is appropriate when there is a natural reference group, such as a placebo group in a drug trial. – Experimentwise Error Rate is no greater than the stated alpha – Comparing to a control takes into account the correlations among tests – One-sided hypothesis test against a control group can be performed – Control comparison computes and tests k-1 GroupWise differences, where k is the number of levels of the classification variable. – An example is the Dunnett method

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ANOVA Post Hoc Tests: Control Plots and the Dunnett Method 103 Upper decision limit Lower decision limit

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ANOVA Post Hoc Tests: Performing a Post Hoc Tests 104

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ANOVA Post Hoc Tests: Performing a Post Hoc Tests: Turkey 105

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ANOVA Post Hoc Tests: Performing a Post Hoc Tests: Dunnett 106

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ANOVA Post Hoc Tests: Performing a Post Hoc Tests: t-test 107

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 108

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Two-Way ANOVA with Interactions: Introduction 109 μ3μ3 One-way ANOVA μ1μ1 μ2μ2

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Two-Way ANOVA with Interactions: Introduction 110 Predictor VariableResponse Variable Levels One-way ANOVA

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Two-Way ANOVA with Interactions: Introduction 111 ANOVA Response Variable Two-way ANOVA Predictor Variable Levels

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Two-Way ANOVA with Interactions: Introduction 112 Two-way ANOVA High alloy Low alloy Heat 1 Heat 2 Heat 3 Heat 4

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Two-Way ANOVA with Interactions: Objective Fit a two-way ANOVA Detect interactions between factors Analyze the treatments when there is a significant interaction 113

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Two-Way ANOVA with Interactions: n-Way ANOVA 114 One-way ANOVA Predictor VariableResponse Variable N-way ANOVA Response Variable Predictor Variable More than one Predictor Variable

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Two-Way ANOVA with Interactions: n-Way ANOVA 115 Randomized block design Randomized block design N-way ANOVA blocking factor interested factor

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Two-Way ANOVA with Interactions: interactions 116 No interaction Interaction

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Two-Way ANOVA with Interactions: interactions 117 Interactions Two-way ANOVA Alloys Heat setting ? ?

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Two-Way ANOVA with Interactions: The Two-Way ANOVA Model 118 Two-way ANOVA effect μ: overall population mean, Regardless of alloy and heating Error term α i =μ i - μ effect β j =μ j - μ Effect of interaction When non- significant

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Two-Way ANOVA with Interactions: Scenario: Using a Two-Way ANOVA 119 Two-way ANOVA Response VariablePredictor Variable Levels

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Two-Way ANOVA with Interactions: Scenario: Using a Two-Way ANOVA 120 Two-way ANOVA Blood pressure Response Variable Disease types Predictor Variable Drug doses Predictor Variable A, B, C 100ml, 200ml, 300ml, placebo

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Two-Way ANOVA with Interactions: Identify the data 121 Drug

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Two-Way ANOVA with Interactions: Applying the model 122 Two-way ANOVA assumptions: Independent observations Normally distributed data Equal variances

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Two-Way ANOVA with Interactions: The Two-Way ANOVA Model 123 Two-way ANOVA effect Error term α i =μ i - μ effect β j =μ j - μ Effect of interaction Observed BllodP for each patient Overall mean of BllodP

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Two-Way ANOVA with Interactions: The Two-Way ANOVA Model 124 Two-way ANOVA H 0 : ? H 0 : None are statistically Different H 0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 = μ 6 = μ 7 = μ 8 = μ 9 = μ 10 = μ 11 = μ 12 H 0 : μ 1 = μ 2 = μ 3 = μ 4 = μ 5 = μ 6 = μ 7 = μ 8 = μ 9 = μ 10 = μ 11 = μ 12

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Two-Way ANOVA with Interactions: Examining Your Data 125 /* Create format, Method I, via EG UI */ data drugdose; input dose $ 8. level; cards; Placebo 1 50 mg mg mg 4 ; run; /*Method II, by code*/ proc format library=work; value dosefmt 1='Placebo' 2='50 mg' 3='100 mg' 4='200 mg'; run;

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Two-Way ANOVA with Interactions: Examining Your Data 126 In which disease type does the drug dose appear to be most effective?

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Two-Way ANOVA with Interactions: Performing Two-Way ANOVA with Interactions 127

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Two-Way ANOVA with Interactions: Performing Two-Way ANOVA with Interactions 128

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Two-Way ANOVA with Interactions: Performing a Post Hoc Pairwise Comparison 129

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Two-Way ANOVA with Interactions: Performing a Post Hoc Pairwise Comparison 130

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Agenda 0. Lesson overview 1. Two-Sample t-Tests 2. One-Way ANOVA 3. ANOVA with Data from a Randomized Block Design 4. ANOVA Post Hoc Tests 5. Two-Way ANOVA with Interactions 6. Summary 131

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132 Question 9. a)One-Sample t-Tests b)One-Way ANOVA c)Two-Way ANOVA If you want to compare the average monthly spending for males versus females which statistical method should you choose? Answer: b

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Home Work: Exercise Using the t Test for Comparing Groups Elli Sageman, a Master of Education candidate in German Education at the University of North Carolina at Chapel Hill in 2000, collected data for a study: she looked at the effectiveness of a new type of foreign language teaching technique on grammar skills. She selected 30 students to receive tutoring; 15 received the new type of training during the tutorials and 15 received standard tutoring. Two students moved away from the district before completing the study. Scores on a standardized German grammar test were recorded immediately before the 12–week tutorials and then again 12 weeks later at the end of the trial. Sageman wanted to see the effect of the new technique on grammar skills. The data are in the GERMAN data set. Change change in grammar test scores Groupthe assigned treatment, coded Treatment and Control Assess whether the Treatment group changed the same amount as the Control group. Use a two- sided t-test. a.Analyze the data using the t Test task. Assess whether the Treatment group improved more than the Control group. b.Do the two groups appear to be approximately normally distributed? c.Do the two groups have approximately equal variance? d.Does the new teaching technique seem to result in significantly different change scores compared with the standard technique? 133

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Home Work: Exercise Analyzing Data in a Completely Randomized Design Consider an experiment to study four types of advertising: local newspaper ads, local radio ads, in-store salespeople, and in-store displays. The country is divided into 144 locations, and 36 locations are randomly assigned to each type of advertising. The level of sales is measured for each region in thousands of dollars. You want to see whether the average sales are significantly different for various types of advertising. The Ads data set contains data for these variables: Ad type of advertising Sales level of sales in thousands of dollars a.Examine the data. Use the Summary Statistics task. What information can you obtain from looking at the data? b.Test the hypothesis that the means are equal. Be sure to check that the assumptions of the analysis method that you choose are met. What conclusions can you reach at this point in your analysis? 134

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Home Work: Exercise Analyzing Data in a Randomized Block Design When you design the advertising experiment in the first question, you are concerned that there is variability caused by the area of the country. You are not particularly interested in what differences are caused by Area, but you are interested in isolating the variability due to this factor. The ads1 data set contains data for the following variables: Ad type of advertising Area area of the country Sales level of sales in thousands of dollars a.Test the hypothesis that the means are equal. Include all of the variables in your model. b.What can you conclude from your analysis? c.Was adding the blocking variable Area into the design and analysis detrimental to the test of Ad? 135

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Home Work: Exercise post Hoc Pairwise Comparisons Consider again the analysis of Ads1 data set. There was a statistically significant difference among means for sales for the different types of advertising. Perform a post hoc test to look at the individual differences among means for the advertising campaigns. a.Conduct pairwise comparisons with an experiments error rate of a=0.05. (use the Tukey adjustment) which types of advertising are significantly different? b.Use display (case sensitive ) as the control group and do a Dunnett comparison of all other advertising methods to see whether those methods resulted in significantly different amounts of sales compared with display advertising in stores? 136

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Home Work: Exercise Performing Two-Way ANOVA Consider an experiment to test three different brands of concrete and see whether an additive makes the cement in the concrete stronger. Thirty test plots are poured and the following features are recorded in the Concrete data set: Strength the measured strength of a concrete test plot Additive whether an additive was used in the test plot Brand the brand of concrete being tested a.Use the Summary Statistics task to examine the data, with Strength as the analysis variable and Additive and Brand as the classification variables. What information can you obtain from Looking at the data? b.Test the hypothesis that the means are equal, making sure to include an interaction term if the results from the Summary Statistics output indicate that would be advisable. What conclusions can you reach at this point in your analysis? c.Do the appropriate multiple comparisons test for statistically significant effects? 137

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