(Rancangan Petak Terbagi)

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(Rancangan Petak Terbagi)
Split Plot Design (Rancangan Petak Terbagi) Kuswanto dan Rizali 2014

Split Plot Design: In RCB designs or Latin Squares, the experiment is broken up into homogenous blocks – every block gets every treatment combination The idea is to have the variation within each block as small as possible (reduces experimental error)

The greater the number of treatment combinations, the bigger each block must be - therefore the variation within each block increases To overcome this increased variation within large blocks, we may use the split plot design – but we pay a price We sacrifice precision in estimating the effects of one of the variables, but often gain precision in estimating effects of the second factor.

The split plot design examining irrigation (4 levels and variety (4 levels):
We begin by assigning levels of one factor randomly to blocks as in a RCB design: eg irrigation levels H L M Block 1 L H M Block 2 M L H Block 3 Each of these is referred to as a main plot

Next, we assign treatment levels of the second factor randomly to sub-plots within each main plot. eg variety H L M a c d b Block 1 L H M b d c a Block 2 M L H Block 3 d c a b

main-plot sub-plots H L M Block 1 L H M Block 2 M L H Block 3 a c d b
a c d b Block 1 L H M b d c a Block 2 M L H Block 3 d c a b

In a split plot design, we sacrifice precision for the factor assigned to main plots, but we gain precision for factor assigned to subplots (compared to RCB design) main-plot sub-plots H L M a c d b Block 1

error between main plots should be larger than error for sub-plots
a c d b Block 1

We may use a split plot when we are interested in testing the effect of one factor over a wider variety of conditions (in our example, test variety over a range of water conditions) Here we are more interested in the effect of variety than water.

We might also use a split plot design when we are interested in adding a factor to an existing experiment – ie blocks already set up, then add sub-plots. requires enough size in main-plots to be split into sub-plots

Another use of a split plot design is for treatments which require a certain area to be applied (eg types of harvesting equipment, pesticide applications etc.) These factors can be assigned to main plots, with a second factor assigned to sub-plots

Remember the idea is that variation between main plots is greater than within sub-plots.
Other possible situations include parts of plants – variation between plants may be greater than within leaves of the same plant so plants could be main plots with leaves as sub-plots Age of animals may add variation to response to a treatment – litters of animals could form main-plots, while individuals within a litter could form sub-plots

Analysis of a split plot design:
In a split plot design, we first look at the effect of main plot treatment, then sub-plot treatment H L M Block 1 L H M Block 2 M L H Block 3 so in our main plot, we have effect of water and block

so in our main plot, we have effect of water and block
Block 1 L H M Block 2 M L H Block 3 so in our main plot, we have effect of water and block GLM model for this is just like a RCB design: block irrigation block*irrigation (remainder)

But we must still deal with sub-plots:
Block 1 Block 2 Block 3 c a d b H L M But we must still deal with sub-plots: GLM for sub plots looks like: variety variety*irrigation error (really block*irrigation*variety)

So the whole model looks like:
block irrigation block*irrigation (remainder a) variety variety*irrigation error (remainder b) main plot effects sub plot effects remainder a is error term used to test main plot effects remainder b is error term to test sub plot effects

a generalized model looks like:
block main-plot treatment mp treat * block (remainder a) sub-plot treatment sp treat*mp treat error (remainder b)

Contoh soal

Perlakuan Ulangan Total 1 2 3
Sebuah penelitian menguji 4 (V) varietas kedele dengan dosis pengairan (P). Data dibawah adalah hasil pengamatan terhadap hasil biji (ku/ha) (Yitnosumarto, 1987) Perlakuan Ulangan Total 1 2 3 P1 V1 4,60 6,27 4,51 15,38 V2 2,63 2,26 1,67 6,56 V3 6,06 4,01 5,10 15,17 V4 6,44 6,35 4,26 17,05 P2 6,19 8,61 7,44 22,24 7,36 9,78 23,58 12,21 11,12 9,20 32,53 11,64 14,46 11,19 37,29 57,13 62,86 49,91 169,80

1. Analisis 2 arah untuk petak utama
Ulangan Total 1 2 3 P1 19,73 18,89 15,54 54,16 P2 37,70 43,97 43,37 115,64 57,03 62,86 49,91 169,80

2. Anova untuk petak utama
Dari tabel 2 arah antara petak utama dengan ulangan, dianalisis untuk menghitung jumlah kuadrat petak utama dan jumlah kuadrat ulangan. Perhatikan, tabel tersebut adalah tabel untuk data RAK faktor tunggal dengan 3 ulangan. Cara penghitungan jumlah kuadrat juga sama dengan RAK faktor tunggal, dan akan dihasilkan JK untuk ulangan, petak utama, galat percobaan (galat a) dan JK total petak utama. Hasil analisis ditunjukkan pada tabel anova petak utama.

Sumber Keragaman Db JK KT Fhit Ulangan 2 10,516 5,258 2,600 Petak Utama (P) 1 157,491 77,888* Galat a 4,044 2,022 Total PU 5 172,051

3. Tabel 2 arah interaksi PUxAP
Untuk menghitung JK anak petak dan interaksinya, dibuatkan tabel dua arah interaksi antara petak utama dengan anak petak. Petak Utama Anak petak Total V1 V2 V3 V4 P1 15,38 6,56 15,17 17,05 54,16 P2 22,24 23,58 32,53 37,29 115,64 37,62 30,58 47,70 54,34 169,80

4. Tabel anova lengkap Dari tabel tersebut dapat dihitung JK untuk anak petak (V), interaksi (PV), galat b. JK total dihitung sebagaimana rancangan yang lain, yaitu jumlah kuadrat dari semua data yang ada dikurangi Faktor Koreksi. Perhatikan, JK total berbeda dengan JK total petak utama. JK total adalah jumlah kuadrat keseluruhan data sedang JK total petak utama hanya dihitung dari nilai masing-masing petak utama.

Dengan demikian JK total akan digunakan untuk menghitung JK galat b
Dengan demikian JK total akan digunakan untuk menghitung JK galat b. Hasil perhitungan secara lengkap adalah sbb : Sumber Keragaman Db JK KT F hit T tabel 5% 1% Ulangan 2 10,516 5,258 2,600 Petak Utama (P) 1 157,491 77,888* Galat a 4,044 2,022 Anak Petak (V) 3 57,3 19,1 17,945** Interaksi PuxAP 17,137 5,712 5,372 Galat b 12 12,759 1,063 Total 23 259,248

Misal, petak utama dan anak petak memberikan pengaruh nyata dan sangat nyata, maka pembahasan ditekankan pada masing-masing faktor. Untuk uji perbandingan berganda masing-masing faktor menggunakan masing-masing KT galat. Apabila akan membedakan antar petak utama, digunakan KT galat a, sedang untu membedakan antar anak petak digunakan KT galat b.

Analisis ragam di r aov(Hasil ~ Blok+Pengairan*Varietas + Error(Pengairan:Blok), data=split.data)

Uji Tukey dari interaksi

Tugas Lakukan analisis data hasil penelitian dengan rancangan petak terbagi Petak utama adalah aplikasi pestisida Anak petak adalah varietas Lakukan analisis untuk menghitung anova dan menguji masing-masing perlakuannya Kumpulkan Minggu Depan

Soal Factor A (main plot): spraying at two levels
A1 = no insecticide A2 = insecticide Factor B (subplot): variety B1 =variety 1 B2 =variety 2 B3 =variety 3

... to be continue ...