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Analyses of Variance Review

Analyses of Variance Review. Simple Situation. Simple Situation. t -test. |x 1 -x 2 | 2[( 1 2 + 2 2 )/( n 1 + n 2 )]. t =. More than two treatments. Multiple t -tests.

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Analyses of Variance Review

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  1. Analyses of VarianceReview

  2. Simple Situation

  3. Simple Situation

  4. t-test |x1-x2| 2[(12+22)/(n1+n2)] t =

  5. More than two treatments

  6. Multiple t-tests • Brundage v Lambert; Brundage v Croft; Brundage v Stephens; Lambert v Croft; Lambert v Stephens; Croft v Stephens. • Problems? • If all tests were done at 95% significance level, and one difference was significant, we have done 6 tests and would expect 1/20 to be significant, at random.

  7. Analysis of Variance • Is an elegant and quicker way to calculate a pooled error term. • Analysis is simple in simple designs but can be complicated and lengthy in some designs (i.e. rectangular lattices). • In some experimental designs the ANOVA is the only method to estimate a pooled error term.

  8. Analysis of Variance • It can provide an F-test to tests specific hypotheses. (i.e. to test general differences between different treatments). • Can be an invaluable initial contribution to interpretation of experiments.

  9. Theory of Analysis of Variance ij(xij-x..)2 = ij[(xij-xi.) + (xi.-x..)]2 ij[(xij-xi.)2+2(xij-xi.)(xi.-x..)+(xi.-x..)2] ij(xij-x..)2 = ij(xij-xi.)2+ki(xi.-x..)2] ki(xi.-x..)2 = Between Treatment SS ij(xij-xi.)2 = Within Treatment SS

  10. Theory of Analysis of Variance BTMS ~ 2n-1 df: WTMS ~ 2nk-n df 2n-1 df2nk-n df ~ F Dist n-1,nk-n df

  11. Theory of Analysis of Variance [e2 + kt2]/e2 = 1, if kt2 = 0

  12. Assumptions behind the ANOVA • Assumption of data being normally distributed. • Homogeneity of error variance. • Additivity of variance effects. • Data collected from a properly randomized experiment.

  13. Analyses of CRB Designs Yij =  + ti + eij

  14. Analysis of Variance of CRB CF = [xij]2/jk

  15. Analyses of RCB Designs Yij =  + bi + tj + eij

  16. Analysis of Variance of RCB CF = [xij]2/rt

  17. Analyses of Latin Designs Yijk =  + ri + cj + tk(ij) +eijk

  18. Analysis of Variance of Latin CF = [xij]2/t2

  19. Efficiency of Latin Squares cw CRB Design [MSr + MSc + (t-1)EMS]/(t+1)EMS If value response is 325, then latin square in will increase precision by 225% over CRB and CRD would have need 2.25 x 4 = 9 replicates to be as accurate.

  20. Efficiency of Latin Squares cw RCB Design Row (RCB) = [MSr + (t-1)EMS]/(t+1)EMS Col(RCB) = [MSc + (t-1)EMS]/(t+1)EMS

  21. +266% -19% ☺

  22. -19% +226% ☺

  23. Analyses of Lattice Squares Yijk =  + ri + baj + tak +eijk

  24. Lattice Square ANOVA

  25. Efficiency of Lattice Design 100 x [Blk(adj)SS+Intra error SS]/k(k2-1)EMS 100 [11,382 + 14,533]/4(16)369 117% I II III IV V I II III IV V

  26. Dealing with Wrongful Data • It is usually assumed that the data collected is correct!. • Why would data not be correct? • Mis-recording, mis-classification, transcription errors, errors in data entry. • Outliers.

  27. Dealing with Wrongful Data • What things can help? • Keep detailed records, on each experimental unit. • Decide beforehand what values would arouse suspision.

  28. Dealing with Wrongful Data • What do you do with suspicios data? • If correct, and it is discarded, then valuable information is lost. This will bias the results. • If wrong and included, will bias results and may have extreme consequences.

  29. Checking ANOVA Accurucy • Coefficient of variation: [e/]x100. • CV=(√100.9/73.75)*100=13.6% • R2 value = {[TSS-ESS]/TSS}x100. • R2 = (1654/3654)*100 = 44.7%. • Compare the effect of blocking or sub-blocking (discussed later).

  30. Marvelous Marvin father of the Groom Alaskan Wedding Feast

  31. ANOVA of Factorial Designs

  32. Factorial AOV Example

  33. Factorial AOV Example

  34. Split-plot AOV

  35. Strip-plot AOV

  36. Fixed and Random Effects

  37. Expected Mean Squares • Dependant on whether factor effects are Fixed or Random. • Necessary to determine which F-tests are appropriate and which are not.

  38. Setting Expected Mean Squares • The expected mean square for a source of variation (say X) contains. • the error term. • a term in 2x. (or S2x ) • a variance term for other selected interactions involving the factor X.

  39. Coefficients for EMS Coefficient for error mean square is always 1 Coefficient of other expected mean squares is n times the product of factors levels that do not appear in the factor name.

  40. Expected Mean Squares • Which interactions to include in an EMS? • All the letter (i.e. A, B, C, …) appear in X. • All the other letters in the interaction (except those in X) are Random Effects.

  41. A and B Fixed Effects Model yield=A B A*B;

  42. A and B Random Effects Model yield=A B A*B; Test h = A B e=A*B;

  43. A Fixed and B Random Model yield=A B A*B; Test h = A e=A*B;

  44. Multiple Comparisons • Multiple Range Tests: • Tukey’s and Duncan’s. • Orthogonal Contrasts.

  45. Tukey’s Multiple Range Test W = q(p,f) x se[x] se[x] = (2/n) (94,773/4) = 153.9 W = 4.64 x 153.9 = 714.1

  46. Tukey’s Multiple Range Test

  47. Duncan’s Multiple Range Test

  48. Duncan’s Multiple Range Test

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