Two_Way_ANOVA.doc

3.3: TWO-WAY ANALYSIS OF VARIANCE The analysis of variance technique shown previously is called a one-way analysis of variance since there is only one independent variable. The two-way analysis of variance is an extension of the one-way analysis of variance; it involves two independent variables. The independent variables are also called factors. Using the two-way analysis of variance, the researcher is able to test the effects of two independent variables or factors on one dependent variable. It is an analysis method for randomized block design. The benefit of considering other factors is that we can reduce the error variance. A second treatment variable than when included in the ANOVA analysis will have the effect of reducing the SSE term, is referred to as a blocking variable. The purpose of designing a randomized block experiment is to reduce the within treatment variation to more easily detect differences between the treatment means.Conditions or Assumptions for the Two-Way ANOVA1. The populations from which the samples were obtained must be normally or approximately normally distributed.2. The samples must be independent.3. The variances of the populations from which the samples were selected must be equal.4. The groups must be equal in sample size.The Sum of SquaresThe total variation can be partition into three sources of variation:SST = SSA + SSB + SSEThe treatments sum of squares, SSA:The blocks sum of squares, SSB:The total sum of squares, SST:The error sum of squares, SSE:SSE = SST - SSA SSBMean Squares for Treatment, MSA: Mean Squares for Block, MSB:Mean Squares for Error, MSE:Test Statistic:1) 2) Hypothesis StatementTo test the treatment means: A two-way ANOVA has several null hypotheses. There is one for each independent variable. H0: m1 = m2 = m3 = = mk H0: There is no different between treatment means H1: There is a different between treatment meansTo test the block means:H0: There is no different between block meansH1: There is a different between block means.Analysis of Variance Summary TableSource Sum of df MeanFSquares Squares Treatments SSA a-1 MSA Blocks SSB b-1 MSB Error SSE (a-1)(b-1) MSE _ Total SST n-1Example: A randomized block experiment produced the following statistics:a=5 b=12 SSA=1500 SSB=1000 SST=3500Test to determine whether the treatment means differ. Use a=0.01.Exercise:1. In a preliminary study, 20 people who were more than 50 pounds overweight were recruited to compare four diets. The people were matched by age. The oldest four became block 1, the next oldest four became block 2 and so on. The number of pounds that each person lost is listed in the table. Can we infer at the 1% significance level that there are differences between the four diets? BlockDiet 1 2 3 412345 5 2 6 8 4 7 8 10 6 12 9 2 7 11 16 7 9 8 15 142. In recent years the irradiation of food to reduce bacteria and preserve the food longer has become more common. A company that performs this service has developed four different methods of irradiating food. To determine which is best, it conducts an experiment where different foods are irradiated and the bacteria count is measured. As part of the experiment the following foods are irradiated: beef, chicken, turkey, eggs, and milk. The results are shown below. Bacteria CountFoodMethod 1Method 2Method 3Method 4Beef47533668Chicken53614875Turkey68855545Eggs25242027Milk44483846Set up the ANOVA Table. Use = 0.01 to determine the critical values. Can the company infer at the 1% significance level that differences in the bacteria count exist among the four irradiation methods?3. A partial ANOVA table in a randomized block design is shown below, where the treatments refer to different high blood pressure drugs, and the blocks refer to different groups of men with high blood pressure.Source of VariationSSdfMSFTreatments*4*Blocks3,1206*Error*115Total12,60034Fill in the missing values (identified by asterisks) in the above ANOVA Table. Can we infer at the 5% significance level that the treatment means differ?SPSS for Two-Way ANOVARefer to Exercise #1.Univariate Analysis of VarianceRefer to Exercise #2Univariate Analysis of Variance3.4 Multiple Mean Paired Comparison When we reject the null hypothesis that the means are equal, we may want to know which treatment means differ. Several procedures has been developed to determine where the significant differences in the means lie after the ANOVA procedure has been performed. Among the most commonly used tests are Scheff test, Tukey test and Bonferroni test. All these test compare the means two at a time using all possible combinations of means. Example: If there are three means, the following comparisons must be done:Refer to Exercise #1: One Way ANOVA From the hypothesis testing, we have concluded at 5% significance level that there are different in the number of job offers between the three MBA majors. So which pair is different? Finance and Marketing?Finance and Management?Marketing and Management? Stated before that the F test can only show whether or not a difference exists but not where the difference lies. So, to obtain which pair is different, we need to do the multiple mean paired comparison.Exercise:Many people in Country DD suffer from high levels of cholesterol, which can lead to heart attacks. For those with very high levels (over 280), doctors prescribe drugs to reduce cholesterol levels. A pharmaceutical company has recently developed four such drugs. To determine whether any differences exist in their benefits, an experiment was organized. The company selected 25 groups of four men and treated as block. Each of the men has ch