RAK Week 10/ / 47 Rancangan Acak Kelompok (RAK) Diterapkan pada percobaan yang dilakukan pada lingkungan tidak homogen (heterogen)

Struktur Data RAK Week 10/ / 47 Perlakuan Kelompok12…t 1x11x21xt1 2x12x22xt2 …………… …………… bx1bx2b xtb

Week 10/ / 47

Week 10/ / 47 ER  untuk memperoleh sensitivitas RAL yang sama dengan RAK maka ulangan yang digunakan dalam menerapkan RAL harus ER kali dari ulangan yang digunakan dalam RAK.

FAKTORIAL - RAL Week 10/ / 47 Dr. Ir. Rahmat Kurnia, M.Si

Week 10/ / 47 Two-Way ANOVA Examines the effect of Two factors of interest on the dependent variable e.g., Percent carbonation and line speed on soft drink bottling process Interaction between the different levels of these two factors e.g., Does the effect of one particular carbonation level depend on which level the line speed is set?

Week 10/ / 47 Two-Way ANOVA Assumptions Independent random samples are drawn Populations have equal variances Populations are normally distributed (continued)

Week 10/ / 47 Two-Way ANOVA Sources of Variation Two Factors of interest: A and B a = number of levels of factor A b = number of levels of factor B r = number of replications for each cell n = total number of observations in all cells (n = abr) X ijk = value of the k th observation of level i of factor A and level j of factor B

Week 10/ / 47 Two-Way ANOVA Sources of Variation SST Total Variation SSA Factor A Variation SSB Factor B Variation SSAB Variation due to interaction between A and B SSE Random variation (Error) Degrees of Freedom: a – 1 b – 1 (a – 1)(b – 1) ab(r – 1) n - 1 SST = SSA + SSB + SSAB + SSE (continued)

Week 10/ / 47 Two Factor ANOVA Equations Total Variation: Factor A Variation: Factor B Variation:

Week 10/ / 47 Two Factor ANOVA Equations Interaction Variation: Sum of Squares Error: (continued)

Week 10/ / 47 Two Factor ANOVA Equations where: r = number of levels of factor A c = number of levels of factor B n ’ = number of replications in each cell (continued)

Week 10/ / 47 Mean Square Calculations

Week 10/ / 47 Two-Way ANOVA: The F Test Statistic F Test for Factor B Effect F Test for Interaction Effect H 0 : μ 1.. = μ 2.. = μ 3.. = H 1 : Not all μ i.. are equal H 0 : the interaction of A and B is equal to zero H 1 : interaction of A and B is not zero F Test for Factor A Effect H 0 : μ.1. = μ.2. = μ.3. = H 1 : Not all μ.j. are equal Reject H 0 if F > F U

Week 10/ / 47 Two-Way ANOVA Summary Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Statistic Factor ASSAr – 1 MSA = SSA /(r – 1) MSA MSE Factor BSSBc – 1 MSB = SSB /(c – 1) MSB MSE AB (Interaction) SSAB(r – 1)(c – 1) MSAB = SSAB / (r – 1)(c – 1) MSAB MSE ErrorSSE rc(n ’ – 1) MSE = SSE/rc(n’ – 1) TotalSSTn – 1

Week 10/ / 47 Features of Two-Way ANOVA F Test Degrees of freedom always add up n-1 = rc(n ’ -1) + (r-1) + (c-1) + (r-1)(c-1) Total = error + factor A + factor B + interaction The denominator of the F Test is always the same but the numerator is different The sums of squares always add up SST = SSE + SSA + SSB + SSAB Total = error + factor A + factor B + interaction

Week 10/ / 47 Examples: Interaction vs. No Interaction No interaction: Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Factor B Level 1 Factor B Level 3 Factor B Level 2 Factor A Levels Mean Response Interaction is present:

Week 10/ / 47 Chapter Summary Described one-way analysis of variance The logic of ANOVA ANOVA assumptions F test for difference in c means The Tukey-Kramer procedure for multiple comparisons Described two-way analysis of variance Examined effects of multiple factors Examined interaction between factors