1-FACTOR ANOVA (MOTIVATION) [DEVORE 10.1]

Transcription

1 1-FACTOR ANOVA (MOTIVATION) [DEVORE 10.1] Hgh varance between groups Low varance wthn groups s 2 between/s 2 wthn 1 Factor A clearly has a sgnfcant effect!! Low varance between groups Hgh varance wthn groups s 2 between/s 2 wthn 1 Factor A clearly has no sgnfcant effect! Low varance between groups Low varance wthn groups s 2 between/s 2 wthn 1 = Hard to tell f factor A has a sgnfcant effect... Hgh varance between groups Hgh varance wthn groups s 2 between/s 2 wthn 1 = Hard to tell f factor A has a sgnfcant effect...

2 1-FACTOR ANOVA (MOTIVATION) [DEVORE 10.1] 1-FACTOR ANOVA ASSUMPTIONS: Gven experment wth I > 2 groups, each of sze J. Let random varables X j j th measurement n the th group. Then the followng assumptons are requred n order to use 1-Factor ANOVA: X j d Normal(µ, σ 2 ) and All groups are ndependent of one another Here s another way to state the ANOVA assumptons: (RINSS acronym) (Randomzaton) All measurements are randomly selected. (Independence) All measurements are ndependent. (Normalty) All populatons are approxmately normally dstrbuted. (Same Spread) All populatons have approxmately same varance. 1-FACTOR ANOVA TEST STATISTIC: Gven an experment wth one factor and I > 2 groups. Moreover, suppose the ANOVA assumptons are all satsfed. Then, the F -test usng the followng test statstc value: f = s2 between s 2 wthn s the most-powerful test that prevents α-nflaton for hypotheses: H 0 : H A : s 2 between IN TERMS OF A MEAN SQUARE & SUM OF SQUARES: µ 1 = µ 2 = = µ I At least two of the µ s dffer The varance between groups, s 2 between, s the varance of the I group means x scaled by common group sze J: s 2 between MS A := SSA := J I 1 where the grand mean, x, s the mean of the I group means, x : x := 1 I = 1 IJ j xj A large varance between groups ndcates much of the observed varaton s explaned by the chosen Factor A. s 2 wthn IN TERMS OF A MEAN SQUARE & SUM OF SQUARES: The varance wthn the groups, s 2 wthn, s the mean of the varances of the I groups: s 2 wthn := 1 (J 1) I s2 = s2 := SSerr I(J 1) Effectvely, a large varance wthn the groups ndcates that much of the observed varaton s not explaned by the chosen Factor A. Therefore, the wthn varance s consdered unexplaned error n the experment. SUMS OF SQUARES AS A PARTITIONING OF VARIATION EXPLANATION: SS }{{ total } T otal V araton n Experment }{{} ν T otal dof s n Experment = SS }{{} A V araton due to F actor A = ν }{{} A Between Groups dof s + SS }{{ err } Unexplaned V araton + ν }{{} err W thn Groups dof s ν = IJ 1 = I 1 = I(J 1) F -TEST STATISTIC VALUE IN TERMS OF MEAN SQUARES: f A = s2 between s 2 wthn = MSA The test statstc value for 1-Factor ANOVA wll be denoted f A nstead of f. In terms of the F -test notaton n secton 9.5, f A s always f +.

3 1-FACTOR ANOVA (PROCEDURE) [DEVORE 10.1] 1-FACTOR ANOVA (GIVEN CELL MEANS x & STD. DEVIATION s s ): 2. Compute Grand Mean: x = 1 I 3. Compute SS err = (J 1) s2 4. Compute SS A = J 5. Compute Mean Squares: := SSerr 6. Compute Test Statstc Value: f A = MS A 7. Compute P-value: p A := P(F > f A) = 1 Φ F (f A;, ) 8. Render Decson: (by software) If p A α then reject H A 0 n favor of H A A, else accept H A 0. (by hand) If f A f,;α then reject H A 0 n favor of H A A, else accept H A 0. 1-FACTOR ANOVA (GIVEN CELL MEANS x & EST. STD. ERROR s σ x ): 2. Compute Grand Mean: x = 1 I 3. Compute Group Std. Dev s: s = J σ x 4. Compute SS err = (J 1) s2 5. Compute SS A = J 6. Compute Mean Squares: := SSerr 7. Compute Test Statstc Value: f A = MS A 8. Compute P-value: p A := P(F > f A) = 1 Φ F (f A;, ) 9. Render Decson: (by software) If p A α then reject H A 0 n favor of H A A, else accept H A 0. (by hand) If f A f,;α then reject H A 0 n favor of H A A, else accept H A 0. 1-FACTOR ANOVA (GIVEN OBSERVATIONS x j): 2. Compute Cell Means: x := 1 J 3. Compute Cell Varances: s 2 := 1 4. Compute Grand Mean: x = 1 I j xj 5. Compute SS err = (J 1) s2 6. Compute SS A = J for = 1,, I J 1 j (xj x )2 7. Compute Mean Squares: := SSerr 8. Compute Test Statstc Value: f A = MS A 9. Compute P-value: p A := P(F > f A) = 1 Φ F (f A;, ) 10. Render Decson: 1-FACTOR ANOVA TABLE: (by software) If p A α then reject H A 0 n favor of H A A, else accept H A 0. (by hand) If f A f,;α then reject H A 0 n favor of H A A, else accept H A 0. Varaton Source 1-Factor ANOVA Table (Sgnfcance Level α) Sum of Mean F Stat df P-value Squares Square Value Decson Factor A SS A MS A f A p A Acc/Rej H0 A Error SS err Total ν SS total

4 EX : The lfetmes of three lght bulb brands were measured and summarzed nto ths table: BULB BRAND (BULB LIFETIMES n yrs) Brand 1 (x 1 ) 9.22, 9.07, 8.95, 8.98, 9.54 Brand 2 (x 2 ) 8.92, 8.88, 9.10, 8.71, 8.85 Brand 3 (x 3 ) 9.08, 8.99, 9.06, 8.93, 9.02 An 1-Factor ANOVA at sgnfcance level α = 0.01 s to be performed. (a) Identfy factor A and ts levels. (b) Determne factor A s level count, I, and the common group sample sze, J. (c) Determne the correspondng degrees of freedom, &. (d) State the approprate null hypothess H A 0 & alternatve hypothess H A A. (e) Compute the cell means, x. (f) Compute the cell varances, s 2. (g) Compute the grand mean, x. (h) Compute the sums of squares, SS err & SS A. () Compute the square means, & MS A. (j) Compute the test F -statstc, f A. (k) By hand, lookup F cutoff value, f,;α. By software (SW), compute resultng P-value, p A. (l) Render the approprate decson. (m) Summarze everythng n an 1-Factor ANOVA table.

5 EX : Dentsts use resn compostes and ceramc fllngs among others for cavtes n teeth. The shear bond strengths of resn composte-ceramc bonds formed from three possble confguratons (conventonal, reversed, all-composte) were measured (n MPa) and summarzed n the followng table: GROUP: SAMPLE SIZE: MEAN: STD DEV: Conventonal 10 x 1 = s 1 = 1.99 Reversed 10 x 2 = s 2 = 2.52 All-Composte 10 x 3 = s 3 = 2.45 Ths table and all the detals regardng the experment can be found n the followng paper: A. Della Bona, R. van Noort, Shear vs. Tensle Bond Strength of Resn Composte Bonded to Ceramc, Journal of Dental Research, 74 (1995), An 1-Factor ANOVA at sgnfcance level α = 0.05 s to be performed. (a) Identfy factor A and ts levels. (b) Determne factor A s level count, I, and the common group sample sze, J. (c) Determne the correspondng degrees of freedom, &. (d) State the approprate null hypothess H A 0 & alternatve hypothess H A A. (e) Compute the grand mean, x. (f) Compute the sums of squares, SS err & SS A. (g) Compute the square means, & MS A. (h) Compute the test F -statstc, f A. () By hand, lookup F cutoff value, f,;α. By software (SW), compute resultng P-value, p A. (j) Render the approprate decson. (k) Summarze everythng n an 1-Factor ANOVA table.