1Stat 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health SciencesInstructor: Ivo Dinov, Asst. Prof. of Statistics and NeurologyTeaching Assistants:Fred Phoa, Anwer Khan, Ming Zheng & Matilda HsiehUniversity of California, Los Angeles, Fall 2005http://www.stat.ucla.edu/~dinov/courses_students.htmlStat 13, UCLA, Ivo DinovSlide 2Chapter 11 Analysis of Variance - ANOVAStat 13, UCLA, Ivo DinovSlide 3Comparing the Means of I Independent Samplesz In Chapter 7 we considered the comparisons of two independent group means using the independent t testz We need to expand our thinking to compare I independent samplesz The procedure we will use is called Analysis of Variance (ANOVA)Stat 13, UCLA, Ivo DinovSlide 4Comparing the Means of I Independent SamplesExample: 5 varieties of peas are currently being tested by a large agribusiness cooperative to determine which is best suited for production. A field was divided into 20 plots, with each variety of peas planted in four plots. The yields (in bushels of peas) produced from each plot are shown in the table below:Variety of PeaA B C D E 26.2 29.2 29.1 21.3 20.124.3 28.1 30.8 22.4 19.321.8 27.3 33.9 24.3 19.928.1 31.2 32.8 21.8 22.1 Stat 13, UCLA, Ivo DinovSlide 5Comparing the Means of I Independent Samplesz In applying ANOVA, the data are regarded as random samples from k populationsz Notation (let sub-indices1 = A, 2 = B, etc…): Population means: µ1, µ2, µ3, µ4, µ5 Population standard deviations: σ1, σ2, σ3, σ4, σ5Stat 13, UCLA, Ivo DinovSlide 6Issues in ANOVAz We have five group means to comparez Why not just carry out a bunch of t tests? Repeated t tests would mean:Ho: µ1= µ2Ho: µ2= µ3Ho: µ3= µ4Ho: µ4= µ5Ho: µ2= µ4Ho: µ2= µ5Etc…z We would have to make comparisonsz What is so bad about that? 1025=⎟⎟⎠⎞⎜⎜⎝⎛2Stat 13, UCLA, Ivo DinovSlide 7z Each test is carried out at α= 0.05, so a type I error is 5% for eachz The overall risk of a type I error is larger than 0.05 and gets larger as the number of groups (I) gets largerz SOLUTION: Need to make multiple comparisons with an overall error of α= 0.05 (or whichever level is specified).Issues in ANOVAStat 13, UCLA, Ivo DinovSlide 8z There are other positive aspects of using ANOVA: Can see if there is a trend within the I groups; low to high Estimation of the standard deviation Global sharing of information of all data yields precision in the analysisz The main idea behind ANOVA is that we need to know how much inherent variability there is in the data before we can judge whether there is a difference in the sample meansIssues in ANOVAStat 13, UCLA, Ivo DinovSlide 9z To make an inference about means we compare two types of variability: variability between sample means variability within each groupz It is very important that we keep these two types of variability in mind as we work through the following formulasz It is our goal to come up with a numeric quantity that describes each of these variability’sIssues in ANOVAStat 13, UCLA, Ivo DinovSlide 10varietyyieldEDCBA3432302826242220Individual Value Plot of yield vs varietybetweenwithinIssues in ANOVAStat 13, UCLA, Ivo DinovSlide 11betweenwithinIssues in ANOVAStat 13, UCLA, Ivo DinovSlide 12The Basic ANOVAz Because we now have I groups each with it’s own observations, we need to modify our notation Notation: yij= group i observation j z For the pea example:y11= 26.2y12= 24.3…y21= 29.2…y54= 22.13Stat 13, UCLA, Ivo DinovSlide 13z More notation: I = number of groups ni= number of observations in group i n*= total number of observations = n1+ n2+ …+ niThe Basic ANOVAStat 13, UCLA, Ivo DinovSlide 14z Formulae:The group mean for group i is: The grand mean is:To compute the difference between the means we will compare each group mean to the grand mean *11..nyyIinjiji∑∑===injijinyyi∑==1.The Basic ANOVAStat 13, UCLA, Ivo DinovSlide 15Variation Between Groupsz Goal #1 is to describe the variation between the groups meansz RECALL: For the independent t test we described the difference between two group means as z In ANOVA we describe the difference between I means as sums of squares between:SS(between) = Can be though of as the difference between each group mean and the grand mean – look at the formula21yy −()∑=−Iiiiyyn12...Stat 13, UCLA, Ivo DinovSlide 16z As our other measures of variation have used in the past is degrees of freedom, SS(between) also has degrees of freedomdf (between) = I – 1z Finally our measure of between group variability is mean square between:MS(between) =This measures variability between the sample means)()(betweendfbetweenSSVariation Between GroupsStat 13, UCLA, Ivo DinovSlide 17Variation Within Groupsz Goal #2 is to describe the variation within the groupsz RECALL: To measure the variability within a single sample we used: z In ANOVA to describe the combined variation within I groups we use sums of squares within:SS(within) = Can be though of as the combination of variation within the I groups 1)(2−−=∑nyysi()∑∑==−Iinjiijiyy112.Stat 13, UCLA, Ivo DinovSlide 18zSS(within) also has degrees of freedomdf (within) = n* - Iz Finally our measure of within variability is mean square within:MS(within) =This is a measure of variability within the groups)()(withindfwithinSSVariation Within Groups4Stat 13, UCLA, Ivo DinovSlide 19More on MS (within)z The quantity for MS(within) is a measure of variability within the groupsz If there were only one group with n observations, then SS(within) = df(within) = n* - 1MS(within) = ()∑=−njjyy12()112−−∑=nyynjjThis was s2from chapter 2!Stat 13, UCLA, Ivo DinovSlide 20More on MS (within)z ANOVA deals with several groups simultaneously. MS(within) is a combination of the variances of the groupsz It is pooling together measurements of variability from the different groupsz With similar logic MS(within) for two groups can be transformed into the pooled standard deviation remember our talk in chapter 7 about the pooled and unpooled methods?Spooled= )(withinMSStat 13, UCLA, Ivo DinovSlide 21A Fundamental Relationship of ANOVAz The last formula based discussion we need to have is regarding the total variability in the data()()()...... yyyyyyiiijij−+−=−Deviation of an observation from the grand mean=Total variabilitywithin betweenStat 13, UCLA, Ivo DinovSlide 22z This also corresponds to the sums of
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