Chapter 5 Student Lecture Notes 5-1

Transcription

1 Chapter 5 Student Lecture Notes 5- Chapter Goals QM5: Busness Statstcs Chapter 5 Analyss of Varance () After completng ths chapter, you should be able to: Recognze stuatons n whch to use analyss of varance Understand dfferent analyss of varance desgns Perform a sngle-factor hypothess test and nterpret results Conduct and nterpret post-analyss of varance parwse comparsons procedures Set up and perform randomzed blocs analyss Chapter Overvew Logc of Analyss of Varance One-Way F-test Tuey- Kramer test Analyss of Varance () Randomzed Complete Bloc F-test Fsher s Least Sgnfcant Dfference test Investgator controls one or more ndependent varables Called factors (or treatment varables) Each factor contans two or more levels (or categores/classfcatons) Observe effects on dependent varable Response to levels of ndependent varable Epermental desgn: the plan used to test hypothess Completely Randomzed Desgn One-Way Analyss of Varance Epermental unts (subjects) are assgned randomly to treatments Only one factor or ndependent varable Wth two or more treatment levels Analyzed by One-factor analyss of varance (one-way ) Called a Balanced Desgn f all factor levels have equal sample sze Evaluate the dfference among the means of three or more populatons Eamples: Accdent rates for st, nd, and rd shft Epected mleage for fve brands of tres Assumptons Populatons are normally dstrbuted Populatons have equal varances Samples are randomly and ndependently drawn Data s measurement level s nterval or rato

2 Chapter 5 Student Lecture Notes 5- ypotheses of One-Way 0 : μ μ μ μ All populaton means are equal A.e., no treatment effect (no varaton n means among groups) : Not all of the populaton means are the same At least one populaton mean s dfferent.e., there s a treatment effect Does not mean that all populaton means are dfferent (some pars may be the same) μ One-Factor 0 : μ μ μ μ A μ μ : Not all μ are the same All Means are the same: The Null ypothess s True (No Treatment Effect) One-Factor 0 : μ μ μ μ A : Not all μ are the same At least one mean s dfferent: The Null ypothess s NOT true (Treatment Effect s present) or Parttonng the Varaton Total varaton can be splt nto two parts: SST = Total Sum of Squares (total varaton) SSB = Sum of Squares Between (varaton between samples) SSW = Sum of Squares Wthn (wthn each factor level) μ μ μ μ μ μ Parttonng the Varaton Partton of Total Varaton Total Varaton (SST) Total Varaton (SST) = the aggregate dsperson of the ndvdual data values across the varous factor levels Between-Sample Varaton (SSB) = dsperson among the factor sample means Wthn-Sample Varaton (SSW) = dsperson that ests among the data values wthn a partcular factor level Varaton Due to Factor (SSB) = + Commonly referred to as: Sum of Squares Between Sum of Squares Among Sum of Squares Eplaned Among Groups Varaton Varaton Due to Random Samplng (SSW) Commonly referred to as: Sum of Squares Wthn Sum of Squares Error Sum of Squares Uneplaned Wthn Groups Varaton

3 Chapter 5 Student Lecture Notes 5- Where: Total Sum of Squares SST n j ( ) SST = Total sum of squares = number of populatons (levels or treatments) n = sample sze from populaton j = j th measurement from populaton = grand mean (mean of all data values) j SST ( Total Varaton Response, X Group Group Group ) ( )... (n ) Sum of Squares Between Between-Group Varaton Where: SSB n ( ) SSB = Sum of squares between = number of populatons n = sample sze from populaton = sample mean from populaton = grand mean (mean of all data values) SSB n ( ) Varaton Due to Dfferences Among Groups j SSB Mean Square Between = SSB/degrees of freedom SSB n Between-Group Varaton Response, X Group Group Group ( ) n( )... n( ) Sum of Squares Wthn nj SSW (j ) j Where: SSW = Sum of squares wthn = number of populatons n = sample sze from populaton = sample mean from populaton j = j th measurement from populaton

4 Chapter 5 Student Lecture Notes 5-4 Wthn-Group Varaton Wthn-Group Varaton SSW nj j ( ) Summng the varaton wthn each group and then addng over all groups j SSW n T Mean Square Wthn = SSW/degrees of freedom SSW ( ) ( )... (n ) Response, X Group Group Group One-Way Table One-Factor F Test Statstc Source of Varaton Between Samples Wthn Samples SS df MS SSB SSB - = - SSW SSW n T - = n T - Total SST = n T - SSB+SSW F rato F = = number of populatons n T = sum of the sample szes from all populatons df = degrees of freedom 0 : μ = μ = = μ A : At least two populaton means are dfferent Test statstc s mean squares between varances s mean squares wthn varances Degrees of freedom F df = ( = number of populatons) df = n T (n T = sum of sample szes from all populatons) Interpretng One-Factor F Statstc Steps The F statstc s the rato of the between estmate of varance and the wthn estmate of varance The rato must always be postve df = - wll typcally be small df = n T - wll typcally be large The rato should be close to f 0 : μ = μ = = μ s true The rato wll be larger than f 0 : μ = μ = = μ s false. Specfy parameter of nterest. Formulate hypotheses. Specfy the sgnfcance level, α 4. Select ndependent, random samples Compute sample means and grand mean 5. Determne the decson rule 6. Verfy the normalty and equal varance assumptons have been satsfed 7. Create table 8. Reach a decson and draw a concluson

5 Chapter 5 Student Lecture Notes 5-5 One-Factor F Test Eample One-Factor Eample: Scatter Dagram You want to see f three dfferent golf clubs yeld dfferent dstances. You randomly select fve measurements from trals on an automated drvng machne for each club. At the 0.05 sgnfcance level, s there a dfference n mean dstance? Club Club Club Club Club Club Dstance Club One-Factor Eample Computatons Club Club Club = 49. = 6.0 = 05.8 = 7.0 n = 5 n = 5 n = 5 n T = 5 = SSB = 5 [ (49. 7) + (6 7) + (05.8 7) ] = SSW = (54 49.) + (6 49.) + + ( ) = 9.6 = / (-) = 58. = 9.6 / (5-) = F One-Factor Eample Soluton 0 : μ = μ = μ A : μ not all equal = 0.05 df = df = Do not reject 0 Crtcal Value: F =.885 = 0.05 Reject 0 F 0.05 =.885 F = 5.75 F Test Statstc: Decson: Reject 0 at = 0.05 Concluson: There s evdence that at least one μ dffers from the rest SUMMARY Groups Count Sum Average Varance Club Club Club Source of Varaton Between Groups Wthn Groups -- Sngle Factor: Ecel Output EXCEL: tools data analyss : sngle factor SS df MS F P-value F crt E Total The Tuey-Kramer Procedure Tells whch populaton means are sgnfcantly dfferent e.g.: μ = μ μ Done after rejecton of equal means n Allows par-wse comparsons Compare absolute mean dfferences wth crtcal range μ = μ μ

6 Chapter 5 Student Lecture Notes 5-6 Tuey-Kramer Crtcal Range Crtcal Range n n j q where: q - = Value from standardzed range table wth and n T - degrees of freedom for the desred level of = Mean Square Wthn n and n j = Sample szes from populatons (levels) and j The Tuey-Kramer Procedure: Eample. Compute absolute mean Club Club Club dfferences: Fnd the q value from the table n append J wth and n T - degrees of freedom for the desred level of q-α.77 The Tuey-Kramer Procedure: Eample. Compute Crtcal Range: 9. Crtcal Range q-α n n j All of the absolute mean dfferences are greater than crtcal range. Therefore there s a sgnfcant dfference between each par of means at 5% level of sgnfcance. 4. Compare: Randomzed Complete Bloc Le One-Way, we test for equal populaton means (for dfferent factor levels, for eample)......but we want to control for possble varaton from a second factor (wth two or more levels) Used when more than one factor may nfluence the value of the dependent varable, but only one s of ey nterest Levels of the secondary factor are called blocs Randomzed Complete Bloc Assumptons Populatons are normally dstrbuted Populatons have equal varances The observatons wthn samples are ndependent The date measurement must be nterval or rato Applcaton eamples Testng 5 routes to a destnaton through dfferent cab companes to see f dfferences est Determnng the best tranng program (out of 4 choces) for varous departments wthn a company Parttonng the Varaton Total varaton can now be splt nto three parts: SST = SSB + SSBL + SSW SST = Total sum of squares SSB = Sum of squares between factor levels SSBL = Sum of squares between blocs SSW = Sum of squares wthn levels

7 Chapter 5 Student Lecture Notes 5-7 Sum of Squares for Blocng SST = SSB + SSBL + SSW Parttonng the Varaton Total varaton can now be splt nto three parts: SSBL ( j ) j Where: = number of levels for ths factor b = number of blocs j = sample mean from the j th bloc = grand mean (mean of all data values) b SST = SSB + SSBL + SSW SST and SSB are computed as they were n One-Way SSW = SST (SSB + SSBL) Mean Squares SSBL L Mean square blocng b SSB Mean square between SSW Mean square wthn ( )(b ) Source of Varaton Between Blocs Between Samples Wthn Samples Randomzed Bloc Table SS SSB SSW df - ( )(b-) Total SST n T - = number of populatons b = number of blocs MS SSBL b - L F rato L n T = sum of the sample szes from all populatons df = degrees of freedom Blocng Test 0 : μb μb μb... A : Not all bloc means are equal Man Factor Test 0 : μ μ μ... μ A : Not all populaton means are equal F = L Blocng test: df = b - df = ( )(b ) F = Man Factor test: df = - df = ( )(b ) Reject 0 f F > F Reject 0 f F > F

8 Chapter 5 Student Lecture Notes 5-8 Fsher s Least Sgnfcant Dfference Test To test whch populaton means are sgnfcantly dfferent e.g.: μ = μ μ Done after rejecton of equal means n randomzed bloc desgn Allows par-wse comparsons Compare absolute mean dfferences wth crtcal range = Fsher s Least Sgnfcant Dfference (LSD) Test LSD t/ b where: t / = Upper-taled value from Student s t-dstrbuton for / and ( - )(b - ) degrees of freedom = Mean square wthn from table b = number of blocs = number of levels of the man factor NOTE: Ths s a smlar process as Tuey-Kramer Is Fsher s Least Sgnfcant Dfference (LSD) Test j LSD t/ LSD? If the absolute mean dfference s greater than LSD then there s a sgnfcant dfference between that par of means at the chosen level of sgnfcance b Compare: etc... Chapter Summary Descrbed one-way analyss of varance The logc of assumptons F test for dfference n means The Tuey-Kramer procedure for multple comparsons Descrbed randomzed complete bloc desgns F test Fsher s least sgnfcant dfference test for multple comparsons