Chapter 1 Anava 2 jalan

1. Chapter 1Anava 2 jalan “Nikmat manakah yang kamu dustakan? “

2. Review Analisis Variansi dan Efek Utama • Analisis variansi dengan 1 efek utama dikenal sebagai analisis variansi satu jalan • Analisis variansi dengan 2 efek utama dikenal sebagai analisis variansi dua jalan • Analisis variansi dengan 3 efek utama dikenal sebagai analisis variansi tiga jalan • Dan demikian seterusnya

3. Analisis variansi satu jalan hanya terdiri atas satu faktor dengan dua atau lebih level • Analisis variansi dua jalan terdiri atas dua faktor, masing-masing dengan dua atau lebih level • Faktor menghasilkan efek utama sehingga di sini terdapat dua efek utama

4. FaktorUtamadanInteraksi • Dalamhallebihdarisatufaktor, faktoritudapatsajasalingmempengaruhiatautidaksalingmempengaruhi • Apabilafaktoritutidaksalingmempengaruhimakakitamemperolehduafaktorutamasaja • Apabilafaktoritusalingmempengaruhi, makaselainefekutama, kitamemperolehlagiinteraksipadasalingmempngaruhiitu • Dalamhalterdapatinteraksi, kitamemilikiefekutamadaninteraksi • Efekutama (denganperbedaanrerata) • Interaksi (denganinteraksidiantarafaktror)

5. Variansi dan Efek Utama Variansi sebelum ada efek Ada variansi dalam kelompok pada kelompok masing-masing Kelompok 1 (level 1) Kelompok 2 (level 2) Ada variansi antara kelompok Kelompok 3 (level 3) Variansiantarakelompok

6. Variansi dalam kelompok tidak berubah Variansi Sesudah Ada Efek Utama    Variansi antara kelompok menjadi besar: Ada efek, Paling sedikit ada satu pasang rerata yang beda       Variansiantarakelompok

7. Variansi Total  Dengan membuka batas semua kelompok, diperoleh variansi total         Variansi total

8. So …Sources of variance • When we take samples from each population, there will be two sources of variability • Within group variability - when we sample from a group there will be variability from person to person in the same group  Sesatan • We will always have this form of variability because it is sampling variability • Between group variability – the difference from group to group  Perlakuan • This form of variability will only exist if the groups are different • If the between group variability if large, the means of the two groups is likely not the same

9. We can use the two types of variability to determine if the means are likely different • How can we do this? • Look again at the picture • Blue arrow: within group, red arrow: between group

10. Types of Regression Models Rancangan Percobaan Random Lengkap RRL Blok Random RBRL Faktorial One-Way Anova (ANAVA 1 Jalan Two-Way Anova (ANAVA 2 Jalan

11. Rancangan Faktorial a x b • Eksperimen faktorial a x b melibatkan 2 faktor dimana terdapat a tingkat faktor A dan b tingkat faktor B, • Eksperimen diulang r kali pada tiap-tiap tingkat faktor kombinasi • Adanya replikasi inilah yang memungkinkan terjadinya interaksi antara faktor A dan B

12. Interaction • Occurs When Effects of One Factor Vary According to Levels of Other Factor • When Significant, Interpretation of Main Effects (A & B) Is Complicated • Can Be Detected In Data Table, Pattern of Cell Means in One Row Differs From Another Row In Graph of Cell Means, Lines Cross • The interaction between two factor A and B is the tendency for one factor to behave differently, depending on the particular level setting of the other variable. • Interaction describes the effect of one factor on the behavior of the other. If there is no interaction, the two factors behave independently.

13. Example • A drug manufacturer has three supervisors who work at each of three different shift times. Do outputs of the supervisors behave differently, depending on the particular shift they are working? Supervisor 1 always does better than 2, regardless of the shift. (No Interaction) Supervisor 1 does better earlier in the day, while supervisor 2 does better at night. (Interaction)

14. Graphs of Interaction Effects of Motivation (High or Low) & Training Method (A, B, C) on Mean Learning Time Interaction No Interaction Average Average High High Response Response Low Low A B C A B C

15. Interaksi X terhadap Y • Tanpa interaksi (dua efek utama) • Dengan interaksi (bentuk interaksi) X1 Y X2 Y X1 Y X2

16. Tanpa interaksi • Ada interaksi Y X1 X2 X interaksi Y X2 X1 X

17. Interaksi • Interaksi terjadi apabila perbedaan rerata pada satu level (misalnya level 1) tidak sama untuk dua level berbeda pada level 2 sehingga terjadi perpotongan Ada perpotongan karena tidak sama Level 1 Level 2

18. Two-Way ANOVA Assumptions • 1. Normality • Populations are Normally Distributed • 2. Homogeneity of Variance • Populations have Equal Variances • 3. Independence of Errors • Independent Random Samples are Drawn

19. Two-Way ANOVA Null Hypotheses 1. No Difference in Means Due to Factor A • H0: 1..= 2..=... = a.. 2. No Difference in Means Due to Factor B • H0: .1. = .2. =... = .b. 3. No Interaction of Factors A & B • H0: ABij = 0

20. The a x b Factorial Experiment • Let xijkbe the k-th replication at the i-th level of A and the j-th level of B. i = 1, 2, …,a j = 1, 2, …, b, k = 1, 2, …,r • The total variation in the experiment is measured by the total sum of squares:

21. ANAVA 2 JalanPartisiVariansi Total JKS JKA Variansi Total JKT Variansi A Variansi B JKB VariansiInteraksi VariansiSesatan JK(AB)

22. JKT dibagimenjadi 4 bagian : • JKA (JumlahKuadratfaktor A) : variansiantarafaktor A • JKB (JumlahKuadratfaktor B): variansiantarafaktor B • JK(AB) (JumlahKuadratInteraksi): variansiantarakombinasitingkatfaktorab • JKS (JumlahKuadratSesatan)

23. Tabel Data ANAVA 2 JalanSel sama Faktor Faktor B A 1 2 ... b Observation k X X ... X 1 111 121 1b1 Xijk X X ... X 112 122 1b2 X X ... X 2 211 221 2b1 X X ... X 212 222 2b2 Level i Factor A Level j Factor B : : : : : X X ... X a a11 a21 ab1 X X ... X a12 a22 ab2

24. Rumus-rumus

25. Contoh : Pabrik Obat Supervisor pabrikobatbekerjapada 3 shift yang berbedadanhasilproduksidihitungpada 3 hari yang dipilihsecara random a=2 b=3 r=3

26. Tabel ANAVA n –1 = abr - 1 a –1 RKA= JKA/(k-1) b –1 db Total = Rataan Kuadrat db Faktor A = db faktor B= db Interaksi = db Sesatan ? (a-1)(b-1) RKB = JKB/(b-1) RK(AB) = JK(AB)/(a-1)(b-1) RKS =JKS/ab(r-1) Dengan pengurangan

27. Two-way ANOVA: Output versus Supervisor, Shift Analysis of Variance for Output Source DF SS MS F P Supervis 1 19208 19208 26.68 0.000 Shift 2 247 124 0.17 0.844 Interaction 2 81127 40564 56.34 0.000 Error 12 8640 720 Total 17 109222

28. Tests for a Factorial Experiment • We can test for the significance of both factors and the interaction using F-tests from the ANOVA table. • Remember that s 2 is the common variance for all ab factor-level combinations. MSE is the best estimate of s 2, whether or not H 0 is true. • Other factor means will be judged to be significantly different if their mean square is large in comparison to MSE.

29. Tests for a Factorial Experiment • The interaction is tested first using F = MS(AB)/MSE. • If the interaction is not significant, the main effects A and B can be individually tested using F = MSA/MSE and F = MSB/MSE, respectively. • If the interaction is significant, the main effects are NOT tested, and we focus on the differences in the ab factor-level means.

30. Two-Way ANOVA Summary Table Source of Degrees of Sum of Mean F Variation Freedom Squares Square A a - 1 SS(A) MS(A) MS(A) (Row) MSE B b - 1 SS(B) MS(B) MS(B) (Column) MSE AB (a-1)(b-1) SS(AB) MS(AB) MS(AB) (Interaction) MSE Error n - ab SSE MSE Same as Other Designs Total n - 1 SS(Total)

31. Two-way ANOVA: Output versus Supervisor, Shift Analysis of Variance for Output Source DF SS MS F P Supervis 1 19208 19208 26.68 0.000 Shift 2 247 124 0.17 0.844 Interaction 2 81127 40564 56.34 0.000 Error 12 8640 720 Total 17 109222 The Drug Manufacturer The test statistic for the interaction is F = 56.34 with p-value = .000. The interaction is highly significant, and the main effects are not tested. We look at the interaction plot to see where the differences lie.

32. The Drug Manufacturer Supervisor 1 does better earlier in the day, while supervisor 2 does better at night.

33. Revisiting the ANOVA Assumptions • The observations within each population are normally distributed with a common variance • s 2. • 2. Assumptions regarding the sampling procedures are specified for each design. • Remember that ANOVA procedures are fairly robust when sample sizes are equal and when the data are fairly mound-shaped.

34. Diagnostic Tools • Many computer programs have graphics options that allow you to check the normality assumption and the assumption of equal variances. • Normal probability plot of residuals • 2. Plot of residuals versus fit or residuals versus variables

35. Residuals • The analysis of variance procedure takes the total variation in the experiment and partitions out amounts for several important factors. • The “leftover” variation in each data point is called the residual or experimental error. • If all assumptions have been met, these residuals should be normal, with mean 0 and variance s2.

36. Normal Probability Plot • If the normality assumption is valid, the plot should resemble a straight line, sloping upward to the right. • If not, you will often see the pattern fail in the tails of the graph.

37. Residuals versus Fits • If the equal variance assumption is valid, the plot should appear as a random scatter around the zero center line. • If not, you will see a pattern in the residuals.