1 / 30

ANALYSIS OF VARIANCE

ANALYSIS OF VARIANCE. ANOVA. Testing: Ho: μ 1 = μ 2 = μ 3 …… μ k k = no. of exp. groups or samples. ANOVA. SS = sum of squared deviations, an estimate of variability, always affected by sample size Response variable = dependent variable ( y ) = variable measured by each data point

lacota-cooper
Download Presentation

ANALYSIS OF VARIANCE

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. ANALYSIS OF VARIANCE

  2. ANOVA Testing: Ho: μ1 = μ2 = μ3…… μk k = no. of exp. groups or samples

  3. ANOVA • SS= sum of squared deviations, an estimate of variability, always affected by sample size • Response variable= dependent variable (y) = variable measured by each data point • Factor = grouping variable = categorical variable used to place each datum in a particular group. Factor is the independent variable (x) • Level = category of the factor

  4. One-way (one-factor) ANOVA Q: Does smoking while pregnant affect birth weight? • Birth weights of babies are grouped according to smoking status of mother • Response variable – birth weight • Factor – smoking status of mother • Levels – non-smoking, 1-pack/day, 1+ pack/day

  5. One-way (one-factor) ANOVA • 12 babies weighed for each factor

  6. One-way (one-factor) ANOVA • Key to understanding an ANOVA is to understand the sources of variation • VARIABILITY = property of being different

  7. One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA: TOTALVARIATION = variability of all data points (from the grand mean)

  8. One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA: GROUP VARIATION= variability of group means • How different is each group mean from the grand mean? • If group means are far apart, we are more likely to reject Ho that means are equal

  9. One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA: ERROR VARIATION= variability within groups • Measures variability caused by everything else other than smoking • How accurate are our group mean estimates?

  10. One-way (one-factor) ANOVA Partitioning of variability: so that we can test the null hypothesis TOTAL SS = GROUP SS + ERROR SS

  11. TotalSS =  (X – Xgrand)2 One-way (one-factor) ANOVA Partitioning of variability:

  12. GroupSS =  ngroup (Xgroup – Xgrand)2 One-way (one-factor) ANOVA Partitioning of variability:

  13. ErrorSS =  (Xgroup – Xgroup)2 One-way (one-factor) ANOVA Partitioning of variability:

  14. ANOVA table: Source SS DF MS Total Groups Error 8747373 3127499 5619874 35 2 33 249924.9 1563749.5 170299.2 One-way (one-factor) ANOVA Ho: μ1 = μ2 = μ3

  15. Total SS 8747374 Total DF 35 One-way (one-factor) ANOVA • Total SS: total number of observations • Degrees of Freedom = n-1 Total MS = 249924.9 = =

  16. Group SS 3131556 Group MS = 1565778 = = Group DF 2 One-way (one-factor) ANOVA • Group Degrees of Freedom = k – 1

  17. Error SS 5619878 Error MS = 170299.3 = = Error DF 33 One-way (one-factor) ANOVA • Error Degrees of Freedom = N – k

  18. One-way (one-factor) ANOVA Ho: μ1 = μ2 = μ3 Reject HO if group means are far apart relative to the amount of error involved in estimating them Group MS – variance measuring how far apart group means are from each other Error MS – variance measuring random sampling error involved in estimating mean

  19. REJECT THE NULL HYPOTHESIS One-way (one-factor) ANOVA IF: Group MS is significantly greater than Error MS

  20. Group MS 1565778 F = = = 9.194 Error MS 170299.3 One-way (one-factor) ANOVA DF numerator = group DF (k-1) = 2 DF denominator = error DF (n-k) = 33 F statistic is one-tailed

  21. One-way (one-factor) ANOVA From Table, probability of an F = 9.194 with 2 and 30 DF is 0.001>p>0.0005 Reject the Null Hypothesis Smoking does affect birth weight

  22. Group SS 3127499 r2= = = 35.7% Total SS 8747373 One-way (one-factor) ANOVA • Coefficient of determination (r2) • Explained variation (Group SS) as a proportion of total variation (Total SS) • Smoking explains about 36% of the variation in birth weight

  23. TotalSS =  (X – Xgrand)2 GroupSS =  ngroup (Xgroup – Xgrand)2 Example In a hypothetical experiment, Aspirin, Panado, and a placebo were tested to see how much pain relief each provides. Pain relief was rated on a five-point scale. Four subjects were tested per group. Is there a difference between the three groups?  (Xgroup – Xgroup)2 ErrorSS =

  24. SPSS One-Way ANOVA

  25. Analysis of Variance • ANOVA tests for a difference in the means of 3 or more groups • Dependent variable (y): Response • Independent variable (x): Predictor • As with the independent samples t-test, the data (y) must be coded with a group variable (x)

  26. Example A researcher has collected a species of lizard from three different island populations. Each island represents a different eco-zone. He collects 10 lizards from each island and measures their running speed Test whether lizards from the different islands differ in their running speeds

  27. One-Way ANOVA • Use the following for a simple one-way ANOVA • From the menus choose: Analyze Compare Means One-Way ANOVA •  Select one or more dependent variables •  Select a single independent grouping variable (factor) • Variables must be numeric

  28. ANOVA Results • Is there a significant difference?

  29. Post Hoc Tests • Multiple comparison tests • Once you know that differences exist among the means, post hoc tests can determine which means differ • These test the difference between each pair of means and yield a matrix where significantly different group means are indicated

  30. Tukey HSD • Tukey Honestly Significant Difference • In the One-Way ANOVA dialog box, click Post Hoc, and select Tukey

More Related