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**S519: Evaluation of Information Systems**

Social Statistics Inferential Statistics Chapter 11: ANOVA

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**This week When to use F statstic How to compute and interpret**

Using FTEST and FDIST functions How to use the ANOVA

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**The problem with t-tests…**

We could compare three groups with multiple ttests: M1 vs. M2, M1 vs. M3, M2 vs. M3

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**What is ANOVA? “Analysis of Variance”**

A hypothesis-testing procedure used to evaluate mean differences between two or more treatments (or populations). Related to: t-tests using independent-measures or repeated- measures design. Advantages: 1) Can work with more than two samples. 2) Can work with more than one independent variable

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What is ANOVA? In ANOVA an independent or quasi-independent variable is called a factor. Factor = independent (or quasi-independent) variable. Levels = number of values used for the independent variable. One factor → “single-factor design” More than one factor → “factorial design”

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**What is ANOVA? An example of a single-factor design**

A example of a two-factor design

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**F value Variance between treatments can have two interpretations:**

Variance is due to differences between treatments. Variance is due to chance alone. This may be due to individual differences or experimental error.

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Three Types of ANOVA Independent measures design: Groups are samples of independent measurements (different people) Dependent measures design: Groups are samples of dependent measurements (usually same people at different times; also matched samples) “Repeated measures” Factorial ANOVA (more than one factor)

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**Excel: ANOVA Three different ANOVA: Anova: single factor - independent**

Anova: two factors with replication - factorial Anova: two factors without replication - dependent

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**Example (independent)**

Three groups of preschoolers and their language scores, whether they are overall different? Group 1 Scores Group 2 Scores Group 3 Scores 87 89 86 85 91 76 99 96 56 78 79 98 81 90 77 82 66 75 67 93

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**F test steps Step1: a statement of the null and research hypothesis**

One-tailed or two-tailed (there is no such thing in ANOVA)

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F test steps Step2: Setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 0.05

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**F test steps Step3: Selection of the appropriate test statistics**

See Figure 11.1 (S-p227) Simple ANOVA (independent)

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**F test steps Between-group degree of freedom=k-1**

k: number of groups Within-group degree of freedom=N-k N: total sample size

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F test steps Step4: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic Table B3 (S-p363) df for the denominator = n-k=30-3=27 df for the numerator = k-1=3-1=2

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F test steps Step5: comparison of the obtained value and the critical value If obtained value > the critical value, reject the null hypothesis If obtained value < the critical value, accept the null hypothesis 8.80 and 3.36

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**F test steps Step6 and 7: decision time What is your conclusion? Why?**

How do you interpret F(2, 27)=8.80, p<0.05

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Example (dependent) Five participants took a series of test on a new drug T1 T2 T3 T4 P1 3 4 6 7 P2 P3 2 1 5 P4 P5

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**F test steps Step1: a statement of the null and research hypothesis**

One-tailed or two-tailed (there is no such thing in ANOVA)

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F test steps Step2: Setting the level of risk (or the level of significance or Type I error) associated with the null hypothesis 0.05

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**F test steps Step3: Selection of the appropriate test statistics**

See Figure 11.1 (S-p227) Simple ANOVA (independent)

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**F test steps Between-group degree of freedom=k-1**

k: number of groups Within-group degree of freedom=N-k N: total sample size Between-subject degree of freedom=n-1 n: number of subjects Error degree of freedom=(N-k)-(n-1)

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F test steps Step4: determination of the value needed for rejection of the null hypothesis using the appropriate table of critical values for the particular statistic Table B3 (S-p363) df for the denominator = (N-k)-(n-1)=16-4=12 df for the numerator = k-1=4-1=3

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F test steps Step5: comparison of the obtained value and the critical value If obtained value > the critical value, reject the null hypothesis If obtained value < the critical value, accept the null hypothesis 24.88 and 3.49

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**F test steps Step6 and 7: decision time What is your conclusion? Why?**

How do you interpret F(3, 12)=24.88, p<0.05

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Factorial ANOAVA Next week

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