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Statistics 04

Statistics 04. ANOVA. Analysis of Variance (ANOVA). Z test or t test is used to test whether two sample means are sufficiently different to indicate the samples are from populations with different population means.

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Statistics 04

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  1. Statistics 04 ANOVA

  2. Analysis of Variance (ANOVA) • Z test or t test is used to test whether two sample means are sufficiently different to indicate the samples are from populations with different population means. • When more than two different groups are involved, we need to depend on ANOVA for the inference.

  3. Cases of more than two groups • Vocabulary test of candidates from four different regions (Europe, South America, North Africa, Far East) • Different parts of a test (listening, reading, vocabulary, Cloze, translation) • Different teaching methods (three textbooks)

  4. Problems with Z test • 1. Tedious computation: number of computation=N(N-1)/2 • Vocabulary test on regions: 4(4-1)/2=6 • Europe : South America • Europe : North Africa • Europe : Far East • South America : North Africa • South America : Far East • North Africa : Far East • 2. greater Type I error : αn

  5. Principles of ANOVA • Two kinds of differences in a test: systematic differences and random errors • Systematic differences are caused by different experimental conditions. • Random errors are caused by any factors other than experimental conditions. • The total variance between different groups represents systematic differences • The total variance within the group is random errors. • The ratio of these two variances follows the F distribution. • F=Sb2/Sw2 • Null hypothesis: Sb2 is not larger than Sw2 • Large values of the F statistic throw doubt on the validity of the null hypothesis.

  6. Principles of ANOVA • The ratio of these two variances follows the F distribution. • F=Sb2/Sw2 • Null hypothesis: Sb2 is not larger than Sw2 • Large values of the F statistic throw doubt on the validity of the null hypothesis.

  7. Types of ANOVA • One-way ANOVA: the comparison of the means of groups which are classified according to a single criterion variable. • Two-way ANOVA: when affected by more than one factor

  8. Calculation of ANOVA • F=Sb2/Sw2 • Sb2 (MSb): mean between-groups sum of squares • Sw2 (MSw): mean within-groups sum of squares • BSS (SSb): between-groups sum of squares • RSS (SSw): within-groups sum of squares or residual sum of squares • TSS (SSt): total sum of squares • TSS=BSS+RSS • Sb2 = SSb / dfb • Sw2 = SSw / dfw

  9. Example

  10. Total sum of Squares (TSS) • Need to compute: ΣX CF ΣX2 X-j

  11. Computation of ΣX • ΣX: the sum of the all observations • ΣX = X1,1+X2,1+ … X1,2+X2,2+ … + Xi,j • Example • ΣX = 10+12+ … +10+14+ … + 8+15 =180

  12. Computation of Correction Factor (CF) • CF=(ΣX)2/mn • m: the number of samples • n: the size of each sample • Example: • (ΣX)2=1802= 32400 • CF= 32400/(3*5)= 2160

  13. Computation of ΣX2 • ΣX2 : the sum of the squared observations • ΣX2 = X1,12+X2,12+ … +X1,22+X2,22+ … +Xij2 • Example: • ΣX2 = 102+122+ … +102+142+ … +82+152 =2352

  14. Computation of TSS • TSS=ΣX2－CF (Woods) • SSt=ΣX2－(ΣX)2/N (where: N=mn) (韩宝成) • The sum of all squared observations minus the correction factor • Example: • TSS=2352－2160=192

  15. Computation of BSS • BSS=ΣXj2/n－CF (Woods) • SSb=ΣT2/n－(ΣX)2/N (where: T=total of a group, N=mn) (韩宝成) • The sum of the totals of each group divided by the size of the sample (all samples are of the same size), then minus the correction factor • Example: • BSS=(502+552+752)/5－2160=11150/5－2160=2230－2160=70

  16. Computation of RSS • RSS=TSS-BSS • Example • RSS=192－70=122

  17. Computation of Degree of Freedom • dft: degree of freedom of the total • dft=mn－1 • product of the size of the sample and the number of the samples minus 1 • Example: dft=mn－1=3*5=15

  18. Computation of Degree of Freedom • dfb: degree of freedom of the between-groups • dfb=m-1 • the number of samples minus 1 • Example: dfb=m-1=3－1=2

  19. Computation of Degree of Freedom • dfw: degree of freedom of the within-group • dfw=m(n-1) • the number of the samples times the size of the sample minus 1 • Example: dfw=m(n-1)=3*(5-1)=12

  20. Computation of Sb2 and Sw2 (mean sums of squares 均方) • Sb2=BSS/ dfb • Example: Sb2=70/2=35

  21. Computation of Sb2 and Sw2 (mean sums of squares 均方) • Sw2=RSS/ dfw • Example: Sw2=122/12=10.17

  22. Computation of F-ratio • F=Sb2/Sw2 • Example: F=35/10.17= 3.44

  23. Inference for the significant difference • Look up for Fα(m-1,m(n-1)) in the Table of F-distribution • e.g.: F0.05(2,3*(10-1))= F0.05(2,27)=3.35 • (韩宝成：p.192, 分子：2，分母：27) • （Woods: p. 304, n1=2, n2=30）

  24. Compare F with Fα(m-1,m(n-1)) • e.g. F=3.44 • F0.05(2,27)=3.35 • F> F0.05(2,27) • Conclusion: p<0.05 (H0 rejected)

  25. ANOVA Table (English) Source df SS MSS F-ratio Confidence Level Between groups m-1 BSS Sb2 Sb2/ Sw2 p<α Within groups m(n-1) RSS Sw2 (residual) Total mn-1 TSS

  26. ANOVA Table (Chinese) 变异来源平方和自由度 均方 F 显著性水平 组间BSSm-1Sb2 Sb2/ Sw2 p<α 组内RSSm(n-1)Sw2 总变异TSSmn-1

  27. ANOVA Table (Example) Source df SS MSS F-ratio Confidence Level Between groups 2 70 35 3.44 p<0.05 Within groups 27 122 10.17 (residual) Total 14 192

  28. Steps of the Computation • 1.Computation of sums of squares: TSS, BSS, RSS • 2.Determination of degrees of freedom: dft, dfb, dfw • 3.Computation of mean sums of squares: Sb2, Sw2 • 4.F testing: F, Fα(m-1,m(n-1)) • 5. Output an ANOVA table

  29. Consistence of variances • Fmax=S2max/S2min • Check the Table of Critical Value of Fmax (韩宝成：p.198) • If Fmax > Fmaxα, there is inconsistency among the variances. • If Fmax < Fmaxα, there is no significant difference among the variances

  30. 完全随机化设计的方差分析（complete randomized design） • 随机区组实验设计的方差分析(randomized block design) • 多个平均数之间的比较

  31. 完全随机化设计的方差分析（complete randomized design） • 样本容量相同 • 样本容量不同

  32. 样本容量相同 • 5 steps • 1.  Compute for ΣX, (ΣX)2, ΣX2, k, n, N(N=mn) • 2. Compute for sum of squares (离差平方和) (total, between-groups, within-groups) SSt=ΣX2－(ΣX)2/N (where: N=mn) SSb=ΣT2/n－(ΣX)2/N (where: T=total of a group, N=mn) SSw=SSt－SSb

  33. 样本容量相同 3．Determine the degrees of freedom dft=N-1 dfb=k-1 dfw=dft-dfb 4. Compute for mean sum of squares Sb2=SSb/ dfb Sw2=SSw/ dfw 5. Compute for F ratio F=Sb2/Sw2

  34. 样本容量不同 • 5 steps • 1.  Compute for ΣX, (ΣX)2, ΣX2, k, n, N(N=mn) • 2. Compute for sum of squares (离差平方和) (total, between-groups, within-groups) SSt=ΣX2－(ΣX)2/N (where: N=mn) SSb=Σ(T2/n)－(ΣX)2/N (where: T=total of a group, N=mn) SSw=SSt－SSb

  35. 样本容量不同 3．Determine the degrees of freedom • dft=N-1 • dfb=k-1 • dfw=dft-dfb • 4. Compute for mean sum of squares • Sb2=SSb/ dfb • Sw2=SSw/ dfw • 5. Compute for F ratio • F=Sb2/Sw2

  36. Two-way ANOVA • Variations in the case of error gravity scores: • 1. Variation between m groups of judges (horizontal) • 2. Variation between n different errors (vertical) • 3. Residuals

  37. Procedure of Calculation • Calculations of TSS, ESS, GSS and Residual • Calculations of degrees of freedom: between errors, between groups of judges, residual • Calculation of mean sum of squares: Se2, Sg2, Sr2 • Calculation of F-ratio: Se2 / Sr2, Sg2/ Sr2 • Comparison of F and Fα

  38. Calculation of CF • CF=(ΣX)2/mn =24622/3*32 =63140.04

  39. Calculation of TSS, ESS, GSS • TSS=ΣYij2－CF • ESS: between errors sum of squares • ESS=ΣYi2/m－CF • GSS: between groups sum of squares • GSS=ΣYj2/n－CF • The divisor is the number of observations that have gone into each of the values being squared.

  40. Calculation of degree of freedom • dfbetween errors : n-1 • dfbetween groups: m-1 • dfresidual: (m-1)(n-1), or (mn-1)-(n-1)-(m-1) • dftotal: mn-1

  41. Calculations of MSS • MSSbetween errors = ESS/ dfbetween errors • MSSbetween groups = GSS / dfbetween groups • MSSresidual = RSS/ dfresidual

  42. Calculation of F-ratio • Fbetween errors = MSSbetween errors / MSSresidual • Degree of freedom: dfbetween errors, dfresidual • Fbetween groups = MSSbetween groups / MSSresidual • Degree of freedom: dfbetween groups, dfresidual

  43. ANOVA table Source df SS MSS F-ratio Confidence Level Between errorsn-1 ESS ESS/(n-1) EMSS/ RMSSp<α Between groupsm-1 GSS GSS/(m-1) GMSS/RMSS p<α Residual m(n-1) RSS RSS/m(n-1) Total mn-1 TSS

  44. Factorial analysis • Factors: Variants that affect the scores • Level of the factor: different values of each factor • Two null hypotheses in Two-way ANOVA e.g. 1. Mean scores are the same between geographical origins 2. Mean scores are the same between sexes

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