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Basic ANOVA

Basic ANOVA. ANALYSIS OF VARIANCE. Testing: Ho: μ 1 = μ 2 = μ 3 …… μ k k = no of exp groups or samples. Basic ANOVA. ANALYSIS OF VARIANCE. SS = sum of squared deviations, an estimate of variability, is always affected by sample size

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Basic ANOVA

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  1. Basic ANOVA ANALYSIS OF VARIANCE Testing: Ho: μ1 = μ2 = μ3…… μk k = no of exp groups or samples

  2. Basic ANOVA ANALYSIS OF VARIANCE SS = sum of squared deviations, an estimate of variability, is always affected by sample size Response variable= dependent variable = 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 Level= category of the factor Cell= within the level Alpha probability of F values – always calculated as one-tailed

  3. Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Q: Does smoking while pregnant affect birth weight? Birth weights of babies – 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

  4. Basic ANOVA EXAMPLE One-way (one-factor) ANOVA 12 babies weighed for each factor

  5. Basic ANOVA One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA Variability = property of being different. (It’s the opposite of being a constant)

  6. Basic ANOVA One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA cont. TOTAL = variability of all data points (from the grand mean)

  7. Basic ANOVA One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA cont. GROUP = variability of group (level) means. If group means far apart, more likely to reject Ho that means are equal

  8. Basic ANOVA One-way (one-factor) ANOVA Sources of variation in a one-way ANOVA cont. ERROR = variability within the levels. Measures variability caused by everything else other than smoking

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

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

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

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

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

  14. Total SS 8747374 Total DF 35 Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Total SS: total number of observations Degrees of Freedom (DF) = N-1 Total MS = 249924.9 = =

  15. Group SS 3131556 Group MS = 1565778 = = Group DF 2 Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Group Degrees of Freedom (DF) = k – 1, where k = number of groups

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

  17. Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Ho: μ1 = μ2 = μ3 Group MS - variance measuring how far apart group means are from each other Error MS – variance measuring random sampling error involved in estimating means

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

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

  20. Group SS 3127499 r2= = = 35.7% Error SS 8747373 Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Coefficient of determination

  21. Basic ANOVA EXAMPLE One-way (one-factor) ANOVA Understand the concepts surrounding sources of variation!

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