DOC PREVIEW
TAMU STAT 303 - Chapter 12

This preview shows page 1-2-23-24 out of 24 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 24 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

Analysis of VarianceComparing Several MeansExamplesAssumptions for ANOVAEstimating ANOVA ParametersSum of Squares DecompositionDegrees of Freedom and Mean SquaresR2 and pooled variance Sp2 in ANOVAComparing Several MeansANOVA TABLEExampleChapter 12 - One-Way Analysis of Variance(ANOVA)Anh DaoAugust 5th, 2009Chapter 12 - One-Way Analysis of Variance (ANOVA)Analysis of VarianceComparing Several MeansThe statistical methodology for comparing several means is calledanalysis of variance, or ANOVA.In this case, one variable is categorical: This variable forms the groupsto be compared.The response variable is numeric.This methodology is the extension of comparing two means.ANOVA HypothesisH0: µ1= µ2= ... = µI, where I denotes the number of groups tobe compared.Ha: not all the means µi’s are equal.Chapter 12 - One-Way Analysis of Variance (ANOVA)Analysis of VarianceExamples“An investigator is interested in studying the average number of daysrats live when fed diets that contain different amounts of fat. Threepopulations were studied, where rats in population 1 were fed a high-fatdiet, rats in population 2 were fed a medium-fat diet, and rats inpopulation 3 were fed a low-fat diet. The variable of interest is ‘Dayslived.’ ” (from Graybill, Iyer and Burdick, Applied Statistics, 1998).Chapter 12 - One-Way Analysis of Variance (ANOVA)Analysis of VarianceExamples“A state regulatory agency is studying the effects of secondhand smokein the workplace. All companies in the state that employ more than 15workers must file a report with the agency that describes the companyssmoking policy. In particular, each company must report whether (1)smoking is allowed (no restrictions), (2) smoking is allowed only inrestricted areas, or (3) smoking is banned. In order to determine theeffect of secondhand smoke, the state agency needs to measure thenicotine level at the work site. It is not possible to measure the nicotinelevel for every company that reports to the agency, and so a simplerandom sample of 25 companies is selected from each category ofsmoking policy.” (from Graybill, Iyer and Burdick, Applied Statistics,1998).Chapter 12 - One-Way Analysis of Variance (ANOVA)Analysis of VarianceAssumptions for ANOVAEach of the I population or group distributions is normal.Check with a Normal Quantile Plot (or boxplot) of each group.These distributions have identical variances (standard deviations).Check if largest standard deviation is greater 2 times smalleststandard deviation.Each of the I samples is a random sample.Each of the I samples is selected independently of one another.Chapter 12 - One-Way Analysis of Variance (ANOVA)Estimating ANOVA Parameters1We will estimate the population means µ1, µ2, ..., µIwith the samplemeans¯x1,¯x2, ...,¯xI.2Since we assume that the population variances are equal (say to σ2),we can estimate this σ2by pooled variance estimator s2p, given ass2p=(n1− 1)s21+ (n2− 1)s22+ .... + (nI− 1)s2In1+ n2+ ... + nI− Iwhere niand s2iare the sample sizes and the sample variances for theindividual samples, i = 1, 2, ..., I.Chapter 12 - One-Way Analysis of Variance (ANOVA)Sum of Squares DecompositionIn the following, i denotes the index for ith group, j denotes for the jth subjectwithin a group.1Variation between Groups SSG =PIi=1ni(¯xi−¯x)2, where¯xiis themean of the ithsample and¯x is the overall mean.2Variation within Group SSE =PIi=1(ni− 1)s2i3Total Variation SST =Pall data(xij−¯x)2=PIi=1Pnij=1(xij−¯x)24ANOVA Decomposition gives: SST = SSG + SSEChapter 12 - One-Way Analysis of Variance (ANOVA)Degrees of Freedom and Mean SquaresEach sum of squares has a degrees of freedom and mean squaresassociated with it.Degree of Freedom for:1Groups: dfG = I − 12Error: dfE = N − I3Total: dfT = N − 1, note that dfT = dfG + dfEWe denote by N the overall sample size and N = n1+ n2+ ... + nI.Mean Square=Sum of SquaresdfChapter 12 - One-Way Analysis of Variance (ANOVA)R2and pooled variance S2pin ANOVAFor ANOVA, the coefficient of determination isR2=SSGSSTIt shows the proportion of the total variation explained by the difference inmeans. It is easy to prove that the pooled variance estimator is equal to theMSE:s2p= MSEChapter 12 - One-Way Analysis of Variance (ANOVA)Comparing Several MeansStep 1: The null hypothesis for comparing several means isH0: µ1= µ2= . . . = µIwhere I is the number of populations to be compared.Step 2: The alternative hypothesis is Ha: not all of the µiare equal (atleast one of the means is different from the others)Step 3: State the significance levelStep 4: Calculate the F-statisticF =Mean Squares GroupMean Squares ErrorThis compares the variation between groups (group mean to groupmean) to the variation within groups (individual values to group means).Chapter 12 - One-Way Analysis of Variance (ANOVA)Comparing Several MeansStep 5: Find the p-valueThe p-value for an ANOVA F -test is always one-sided.The p-value isP(Fn1,n2> FCalculated)where n1= I − 1 (number of groups minus 1) and n2= N − 1(total sample size minus number of groups).Chapter 12 - One-Way Analysis of Variance (ANOVA)Comparing Several MeansStep 6: Reject or fail to reject H0based on the p-value.If the p-value is less than or equal to α, reject H0.It the p-value is greater than α, fail to reject H0.Step 7: State your conclusion.If H0is rejected, “There is significant statistical evidence that atleast one of the population means is different from another.”If H0is not rejected, “There is not significant statistical evidencethat at least one of the population means is different fromanother.”Chapter 12 - One-Way Analysis of Variance (ANOVA)Comparing Several MeansANOVA TABLESource DF Sum of Squares Mean Square F p-valueGroup I − 1Pni(¯xi−¯x)2SSGdfG= MSGMSGMSE= F P(F > Fobs)Error N − IP(ni− 1)s2iSSEdfE= MSETotal N − 1P(xij−¯x)2SSTdfT= MSTNote: MSE is the pooled sample variance (s2p), and SSG + SSE = SST.R2=SSGSSTis the proportion of the total variation explained by thedifference in means.Chapter 12 - One-Way Analysis of Variance (ANOVA)Comparing Several MeansANOVA TABLEProperties of F -DistributionA F -distribution is a probability distribution which is defined by 2parameters:degrees of freedom for numerator= I − 1degrees of freedom for denominator= N − IWe use the notation F (df 1, df 2) for the F-distribution with df1 degreesof freedom for the numerator and df2 is the degree of freedom fordenominator.The relation between t and F


View Full Document

TAMU STAT 303 - Chapter 12

Download Chapter 12
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Chapter 12 and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Chapter 12 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?