RCBD (Randomized Complete Block Design)

Transkripsi

1 RCBD (Randomized Complete Block Design) Randomized Block Design Rancangan Acak Kelompok (RAK)

2 Types of Experimental Designs Experimental Designs Completely Randomized Randomized Block Factorial One-Way Anova Two-Way Anova EPI809/Spring

3 Kondisi Percobaan Yang sesungguhnya: -Ada nuisance factor (pengganggu), homogenitas materi terganggu : -(Data HETEROGEN) Misalnya: Pengaruh ransum terhadap ADG (kg) Umur juga berpengaruh terhadap ADG sehingga : umur mrpk faktor pengganggu Pilihan: 1 Umur juga diteliti : RAL Pola Faktorial, umur sebagai faktor perlakuan juga 2 menggunakan umur untuk pengelompokan (sebagai BLOK): Mengeluarkan variasi yang bersumber pada umur dari variasi error percob Asumsi TIDAK ADA interaksi antar perlakuan Catatan: jika ragu-ragu dengan Asumsi Sebaiknya faktor pengganggu dijadikan perlakuan, gunakan RAL Faktorial

4 Graphs of Interaction Occurs When Effects of One Factor Vary According to Levels of Other Factor Effects of Gender (Jantan-Betina) & dietary group (Rendah,Sedang,Tinggi) energi terhadap pertumbuhan Interaction No Interaction Average Response male Average Response male female female RDH SDG TINGGI RDH SDG TINGGI Detected : In Graph, Lines Cross EPI809/Spring

5 Persyaratan RAK :

6 Keuntungan; Kerugian;

7 Randomized Block Design 1Experimental Units (Subjects) Are Assigned Randomly within Blocks Blocks are Assumed Homogeneous 2One Factor or Independent Variable of Interest 2 or More Treatment Levels or Classifications 3 One Blocking Factor EPI809/Spring

8 The Blocking Principle Blocking is a technique for dealing with nuisance factors A nuisance factor is a factor that probably has some effect on the response, but it is of no interest to the experimenter however, the variability it transmits to the response needs to be minimized Typical nuisance factors include batches of raw material, operators, pieces of test equipment, time (shifts, days, etc), different experimental units Many industrial experiments involve blocking (or should) Failure to block is a common flaw in designing an experiment (consequences?)

9 The Blocking Principle If the nuisance variable is known and controllable, we use blocking If the nuisance factor is known and uncontrollable, sometimes we can use the analysis of covariance to statistically remove the effect of the nuisance factor from the analysis If the nuisance factor is unknown and uncontrollable, we hope that randomization balances out its impact across the experiment Sometimes several sources of variability are combined in a block, so the block becomes an aggregate variable

10 Randomized Complete Block Design An experimental design in which there is one independent variable, and a second variable known as a blocking variable, that is used to control for confounding or concomitant variables It is used when the experimental unit or material are heterogeneous There is a way to block the experimental units or materials to keep the variability among within a block as small as possible and to maximize differences among block The block (group) should consists units or materials which are as uniform as possible

11 A Randomized Block Design Single Independent Variable Blocking Variable MST MSE Individual observations

12 Randomized Block Design Factor Levels: (Treatments) A, B, C, D Experimental Units Treatments are randomly assigned within blocks Block 1 A C D B Block 2 C D B A Block 3 B A D C Block D C A B EPI809/Spring

13 Randomized Block F-Test Hypotheses H 0 : 1 = 2 = = p All Population Means are Equal No Treatment Effect H a : Not All j Are Equal At Least 1 Pop Mean is Different Treatment Effect 1 2 p Is wrong f(x) f(x) 1 = 2 = 3 1 = 2 3 X X EPI809/Spring

14 Random Assignment Block by Gender Random Assignment Randomized Block Design 50 Women New Medication- 25 subjects Old Medication-25 subjects Compare level of pain relief as reported by subjects 100 Subjects 50 Men New Medication- 25 subjects Old Medication-25 subjects Compare level of pain relief as reported by subjects

15 Randomized Block F-Test Test Statistic 1 Test Statistic F = MST / MSE MST Is Mean Square for Treatment MSE Is Mean Square for Error 2 Degrees of Freedom 1 = p -1 2 = n b p +1 p = # Treatments, b = # Blocks, n = Total Sample Size EPI809/Spring

16 Partitioning the Total Sum of Squares in the Randomized Block Design SStotal (total sum of squares) SSE (error sum of squares) SST (treatment sum of squares) SSB (sum of squares blocks) SSE (sum of squares error)

17 ANOVA Table for a Randomized Block Design Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Treatments SST t 1 SST/t-1 MST/MSE Blocks SSB r - 1 Error SSE (t - 1)(r - 1) SSE/(t-1)(r-1) Total SSTot tr - 1

18 Contoh: Percobaan mengetahui efek Level lemak (L1L2L3) terhadap pertambahan BB Bloking dilakukan terhadap BB sbb Perla kuan Blok L L L SSY = 356,44 SSP =248,44 SSB = SSE = Tabel ANOVA: Sumber variasi df SS MS F-stat F Tabel Perlakuan ,2,1 0=756 Blok Error 10 25, Total ,444 F- stat lebih besar dari F-Tab Kesimpulan: terdapat perbedaan efek Lemak (P<0,01)

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22 Extension of the ANOVA to the RCBD ANOVA partitioning of total variability: t 1 i r 1 j 2 j i ij r 1 j 2 j t 1 i 2 i t 1 i r 1 j 2 j i ij j i t 1 i r 1 j 2 ij ) y y y (y ) y (y t ) y (y r ) y y y (y ) y (y ) y (y ) y (y E Blocks Treatments T SS SS SS SS

23 Extension of the ANOVA to the RCBD The degrees of freedom for the sums of squares in SS T SS Treatments SS Blocks SS E are as follows: tr 1 ( t 1) ( r 1) [( t 1)( r 1)] Ratios of sums of squares to their degrees of freedom result in mean squares, and The ratio of the mean square for treatments to the error mean square is an F statistic used to test the hypothesis of equal treatment means

24 ANOVA Procedure The ANOVA procedure for the randomized block design requires us to partition the sum of squares total (SST) into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error The formula for this partitioning is SSTot = SST + SSB + SSE The total degrees of freedom, n T - 1, are partitioned such that k - 1 degrees of freedom go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term

25 Example: Eastern Oil Co Automobile Type of Gasoline (Treatment) Blocks (Block) Blend X Blend Y Blend Z Means Treatment Means