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S519: Evaluation of Information Systems Social Statistics Inferential Statistics Chapter 11: ANOVA This week When to use F statstic How to compute and interpret Using FTEST and FDIST functions How to use the ANOVA The problem with t-tests… We could compare three groups with multiple ttests: M1 vs. M2, M1 vs. M3, M2 vs. M3 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 What is ANOVA? In ANOVA an independent or quasiindependent 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” What is ANOVA? An example of a single-factor design A example of a two-factor design 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. 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) Excel: ANOVA Three different ANOVA: Anova: single factor - independent Anova: two factors with replication - factorial Anova: two factors without replication - dependent Example (independent) Three groups of preschoolers and their language scores, whether they are overall different? Group 1 Scores Group 2 Scores 87 86 76 56 78 98 77 66 75 67 Group 3 Scores 87 89 85 91 99 96 85 87 79 89 81 90 82 89 78 96 85 96 91 93 F test steps Step1: a statement of the null and research hypothesis One-tailed or two-tailed (there is no such thing in ANOVA) H 0 : 1 2 3 H1 : at least one is different 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 F test steps Step3: Selection of the appropriate test statistics See Figure 11.1 (S-p227) Simple ANOVA (independent) 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 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 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 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 Example (dependent) Five participants took a series of test on a new drug T1 T2 T3 T4 P1 3 4 6 7 P2 0 3 3 6 P3 2 1 4 5 P4 0 1 3 4 P5 0 1 4 3 F test steps Step1: a statement of the null and research hypothesis One-tailed or two-tailed (there is no such thing in ANOVA) H 0 : 1 2 3 4 H1 : at least one is different 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 F test steps Step3: Selection of the appropriate test statistics See Figure 11.1 (S-p227) Simple ANOVA (independent) F test steps Between-group degree of freedom=k-1 Within-group degree of freedom=N-k N: total sample size Between-subject degree of freedom=n-1 k: number of groups n: number of subjects Error degree of freedom=(N-k)-(n-1) 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 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 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 Factorial ANOAVA Next week