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Analysis of Variance (ANOVA). Penjelasan Umum Seringkali kita ingin menguji apakah tiga atau lebih populasi memiliki rata-rata yg sama. Contoh:  Apakah.

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Presentasi berjudul: "Analysis of Variance (ANOVA). Penjelasan Umum Seringkali kita ingin menguji apakah tiga atau lebih populasi memiliki rata-rata yg sama. Contoh:  Apakah."— Transcript presentasi:

1 Analysis of Variance (ANOVA)

2 Penjelasan Umum Seringkali kita ingin menguji apakah tiga atau lebih populasi memiliki rata-rata yg sama. Contoh:  Apakah bahan bakar/km yg digunakan untuk beberapa merek mobil sama?  Apakah pendapatan pekerja pada beberapa lapangan pekerjaan sama?  Atau apakah biaya produksi yg menggunakan beberapa proses yg berbeda adalah sama?

3 Penjelasan Umum Kita dapat menggunakan cara seperti yg lalu untuk menguji kesamaan rata-rata dua populasi, tetapi hal tersebut akan memakan waktu dan perhitungan yg lebih lama. Contoh: jika ada 5 pop, maka ada 5 C 2 cara/perhitungan yg harus dilakukan. Untuk itu kita dapat melakukan uji secara simultan /keseluruhan populasi tersebut dengan menggunakan distribusi F dan metoda yg disebut  ANOVA (Analysis of Variance)

4 Assumptions – Populations are normally distributed – Populations have equal variances – Samples are randomly and independently drawn One-Way Analysis of Variance

5 Hipotesis untuk One-Way ANOVA – Seluruh rata-rata populasi adalah sama – Artinya: Tidak ada efek treatment (tidak ada keragaman rata-rata dalam kelompok) – Minimal salah satu rata-rata populasi ada yang tidak sama – Artinya: Terdapat efek treatment (terdapat keragaman rata-rata dalam kelompok) – Tidak berarti bahwa semua rata-rata populasi tidak sama (beberapa pasang mungkin sama)

6 One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect)

7 One-Factor ANOVA At least one mean is different: The Null Hypothesis is NOT true (Treatment Effect is present) or (continued)

8 Partitioning the Variation

9 Total variation can be split into two parts: SST = Total Sum of Squares SSB = Sum of Squares Between SSW = Sum of Squares Within SST = SSB + SSW

10 Partitioning the Variation Total Variation = jumlah kuadrat total (SST), yang mengukur keragaman total dalam data Within-Sample Variation = kergaman di dalam masing-masing kelompok populasi (SSW) Between-Sample Variation = keragaman antar kelompok populasi (SSB) SST = SSB + SSW (continued)

11 Partition of Total Variation Variation Due to Factor (SSB) Variation Due to Random Sampling (SSW) Total Variation (SST) Commonly referred to as:  Sum of Squares Within  Sum of Squares Error  Sum of Squares Unexplained  Within Groups Variation Commonly referred to as:  Sum of Squares Between  Sum of Squares Among  Sum of Squares Explained  Among Groups Variation = +

12 Total Sum of Squares Dimana: SST = Total sum of squares (Jumlah kuadrat total) k = jumlah populasi (kelompok, level atau treatment) n i = sample size dari populasi ke-i x ij = pengamatan ke-j th dari populasi ke-i x = rata-rata total (rata-rata dari seluruh data) SST = SSB + SSW

13 Total Variation (continued)

14 Sum of Squares Between Dimana: SSB = Sum of squares between k = jumlah populasi (kelompok, level, atau treatment) n i = sample size dari populasi ke-i x i = rata-rata sample dari populasi ke-i x = rata-rata total (rata-rata dari seluruh data) SST = SSB + SSW

15 Between-Group Variation Variation Due to Differences Among Groups Mean Square Between = SSB/degrees of freedom

16 Between-Group Variation (continued)

17 Sum of Squares Within Dimana: SSW = Sum of squares within k = jumlah populasi (kelompok, level, atau treatment) n i = sample size dari populasi ke-i x i = rata-rata sample dari populasi ke-i x ij = pengamatan ke-j th dari populasi ke-i SST = SSB + SSW

18 Within-Group Variation Summing the variation within each group and then adding over all groups Mean Square Within = SSW/degrees of freedom

19 Within-Group Variation (continued)

20 One-Way ANOVA Table Source of Variation dfSSMS Between Samples SSBMSB = Within Samples N - kSSWMSW = TotalN - 1 SST = SSB+SSW k - 1 MSB MSW F ratio k = jumlah populasi (kelompok, level, atau treatment) N = jumlah seluruh pengamatan df = derajat bebas SSB k - 1 SSW N - k F =

21 One-Factor ANOVA F Test Statistic Test statistic MSB is mean squares between variances MSW is mean squares within variances Degrees of freedom – df 1 = k – 1 (k = number of populations) – df 2 = N – k (N = sum of sample sizes from all populations) H 0 : μ 1 = μ 2 = … = μ k H A : At least two population means are different

22 The F statistic is the ratio of the between estimate of variance and the within estimate of variance – The ratio must always be positive – df 1 = k -1 will typically be small – df 2 = N - k will typically be large The ratio should be close to 1 if H 0 : μ 1 = μ 2 = … = μ k is true The ratio will be larger than 1 if H 0 : μ 1 = μ 2 = … = μ k is false Interpreting One-Factor ANOVA F Statistic

23 One-Factor ANOVA F Test Example You want to see if three different golf clubs yield different distances. You randomly select five measurements from trials on an automated driving machine for each club. At the.05 significance level, is there a difference in mean distance? Club 1 Club 2 Club

24 One-Factor ANOVA Example: Scatter Diagram Distance Club 1 Club 2 Club Club 1 2 3

25 One-Factor ANOVA Example Computations Club 1 Club 2 Club x 1 = x 2 = x 3 = x = n 1 = 5 n 2 = 5 n 3 = 5 N = 15 k = 3 SSB = 5 [ (249.2 – 227) 2 + (226 – 227) 2 + (205.8 – 227) 2 ] = SSW = (254 – 249.2) 2 + (263 – 249.2) 2 +…+ (204 – 205.8) 2 = MSB = / (3-1) = MSW = / (15-3) = 93.3

26 F = One-Factor ANOVA Example Solution H 0 : μ 1 = μ 2 = μ 3 H A : μ i not all equal  =.05 df 1 = 2 df 2 = 12 Test Statistic: Decision: Conclusion: Reject H 0 at  = 0.05 There is evidence that at least one μ i differs from the rest 0  =.05 F.05 = Reject H 0 Do not reject H 0 Critical Value: F  = 3.885

27 Uji Wilayah Berganda Dari hasil pengujian kesamaan rata-rata populasi dgn ANOVA, jika keputusan adalah menolak Ho. Maka kita dapati kesimpulan bahwa tidak semua µ sama (paling sedikit ada dua yang tidak sama). Namun kita tidak tahu mana yang berbeda. Untuk mencari µ mana yang berbeda nyata → UJI WILAYAH BERGANDA DUNCAN DAN UJI TUKEY x μ 1 = μ 2 μ 3

28 Uji Duncan Prosedur: 1.Urutkan rata-rata sampel untuk masing- masing populasi (kelompok) dari yang terkecil hingga terbesar 2.Hitung wilayah nyata terpendek dari berbagai rata-rata

29 Uji Duncan 3.Kriteria pengujian Bandingkan selisih kedua rata-rata yang ingin dilihat perbedaannya dengan kriteria sbb:  x i – x j ≤ R p (Tidak berbeda nyata)  x i – x j > R p (Berbeda nyata)

30 Contoh: Uji Duncan 1. Urutkan rata-rata sampel: Club 1 Club 2 Club Hitung wilayah nyata terpendek dari berbagai rata-rata: α = 0.05, df = 12

31 Contoh: Uji Duncan 3.Bandingkan selisih rata-rata dengan R p :

32 Uji Tukey-Kramer Dimana: q  = Nilai dari standardized range table dengan df = k dan N - k MSW = Mean Square Within n i dan n j = Sample sizes dari populasi (kelompok) ke-i & ke-j

33 Contoh: Uji Tukey-Kramer 1. Compute absolute mean differences: Club 1 Club 2 Club Find the q value from the table Tukey with k and N - k degrees of freedom for the desired level of 

34 Contoh: Uji Tukey-Kramer 5. All of the absolute mean differences are greater than critical range. Therefore there is a significant difference between each pair of means at 5% level of significance. 3. Compute Critical Range: 4. Compare:


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