Matakuliah Tahun Versi : I0134 – Metoda Statistika : 2005 : Revisi Pertemuan 19 Analisis Varians Klasifikasi Satu Arah 1 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : • Mahasiswa dapat menjelaskan analisis varians klasifikasi satu arah. 2 Outline Materi • • • • Hasil pengamatan Sumber keragaman Tabel ANOVA Pengambilan keputusan 3 • • • • • • Analysis of Variance and Experimental Design An Introduction to Analysis of Variance Analysis of Variance: Testing for the Equality of k Population Means Multiple Comparison Procedures An Introduction to Experimental Design Completely Randomized Designs Randomized Block Design 4 An Introduction to Analysis of Variance • Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means using data obtained from observational or experimental studies. • We want to use the sample results to test the following hypotheses. H0: 1 = 2 = 3 = . . . = k Ha: Not all population means are equal • If H0 is rejected, we cannot conclude that all population means are different. • Rejecting H0 means that at least two population means have different values. 5 Assumptions for Analysis of Variance • For each population, the response variable is normally distributed. • The variance of the response variable, denoted 2, is the same for all of the populations. • The observations must be independent. 6 Analysis of Variance: Testing for the Equality of K Population Means • Between-Samples Estimate of Population Variance • Within-Samples Estimate of Population Variance • Comparing the Variance Estimates: The F Test • The ANOVA Table 7 Between-Samples Estimate of Population Variance • A between-samples estimate of 2 is called the mean square_ between (MSB). = k MSB 2 n j ( x j x) 2 j1 k1 • The numerator of MSB is called the sum of squares between (SSB). • The denominator of MSB represents the degrees of freedom associated with SSB. 8 Within-Samples Estimate of Population Variance • The estimate of 2 based on the variation of the sample observations within each sample is called the mean square within (MSW). k MSW 2 (n j 1) s 2j j1 nT k • The numerator of MSW is called the sum of squares within (SSW). • The denominator of MSW represents the degrees of freedom associated with SSW. 9 Comparing the Variance Estimates: The F Test • If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSB/MSW is an F distribution with MSB d.f. equal to k - 1 and MSW d.f. equal to nT - k. • If the means of the k populations are not equal, the value of MSB/MSW will be inflated because MSB overestimates 2. • Hence, we will reject H0 if the resulting value of MSB/MSW appears to be too large to have been selected at random from the appropriate 10 F distribution. Test for the Equality of k Population Means • Hypotheses H0: 1 = 2 = 3 = . . . = k Ha: Not all population means are equal • Test Statistic F = MSB/MSW • Rejection Rule Reject H0 if F > F where the value of F is based on an F distribution with k - 1 numerator degrees of freedom and nT - 1 denominator degrees of freedom. 11 Sampling Distribution of MSTR/MSE • The figure below shows the rejection region associated with a level of significance equal to where F denotes the critical value. Do Not Reject H0 Reject H0 F Critical Value MSTR/MSE 12 The ANOVA Table Source of Sum of Variation Squares Treatment SSTR Error SSE Total SST Degrees of Freedom k-1 nT - k nT - 1 Mean Squares F MSTR MSTR/MSE MSE SST divided by its degrees of freedom nT - 1 is simply the overall sample variance that would be obtained if we treated the entire nT observations as one data set. k nj SST ( xij x) 2 SSTR SSE j 1 i 1 13 Fisher’s LSD Procedure • Hypotheses H0: i = j Ha: i j • Test Statistic xi x j t MSW( 1 n 1 n ) i j • Rejection Rule Reject H0 if t < -ta/2 or t > ta/2 where the value of ta/2 is based on a t distribution with nT - k degrees of freedom. 14 Fisher’s LSD Procedure_ _ Based on the Test Statistic xi - xj • Hypotheses • Test Statistic • Rejection Rule H0: i = j Ha: i j _ _ xi - xj _ _ Reject H0 if |xi - xj| > LSD where LSD t /2 MSW ( 1 n 1 n ) i j 15 ANOVA Table for a Completely Randomized Design Source of Variation F Sum of Squares Degrees of Freedom Treatments SSTR k-1 Error SSE nT - k Total SST nT - 1 Mean Squares SSTR MSTR k-1 MSE MSTR MSE SSE nT - k 16 • Selamat Belajar Semoga Sukses. 17