Hypothesis testing: ANOVA Test of the equality of means among c groups. Flowchart


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1 Hypothesis testig: ANOVA Test of the equality of meas amog groups February 6, 200 Moez Hababou() Flowhart Desribig iformatios Drawig olusios Foreastig Improve busiess proesses Data Colletio Probability & Probability Distributio Regressio Aalysis Timeseries Aalysis Summary tables ad harts estimatio Hypothesis testig Simple Regressio model Multiple Regressio model Desriptive Statistis Oe sample Two samples Two or more samples February 6, 200 Moez Hababou() 2
2 Proedure for hypothesis testig for the mea differee betwee two samples State ull hypothesis H o : µ =µ 2 = =µ State alterative hypothesis H A : at least oe µ is differet Determie test statistis F=(SSA/)/(SSW/) Determie reetio regios Choose ofidee level Make olusios February 6, 200 Moez Hababou() 3 Illustratig ull ad alterative hypotheses µ µ2 µ3 Η Α : µ=µ3<µ2 µ µ2 µ3 Η ο : µ=µ3=µ2 February 6, 200 Moez Hababou() 4 2
3 Deomposig total variatio Total Variatio (SST) df= Amog group variatio (SSA) df=  withi group variatio (SSW) df=  February 6, 200 Moez Hababou() 5 SST = Deomposig variatio: Total variatio SST = = umber of observatio s i group# = total umber of observatio s i all group ombied = i= = i= + 2 ( ) = grad mea = observatio #i i group# i = umber of groups i i February 6, 200 Moez Hababou() 6 3
4 SSA = Amoggroup variatio: SSA = = = umber = total umber = + = = i= i= = ( 2 ) grad mea for group# = umber of groups i i mea of observatio s i group# of observatio s i all group 2 ombied February 6, 200 Moez Hababou() 7 Withigroup variatio: SSW SSW = i = = = umber = total umber = + ( mea for group# observatio #ii group# i= = = i= = umber of groups i i ) of observatio s i group# of observatio s i all group 2 ombied February 6, 200 Moez Hababou() 8 4
5 Calulatig the mea squares ad the oeway ANOVA Ftest statisti SSA MSA = SSW MSW = SST MST = MSA F = = MSW SSA SSW Mea squared sum amog groups Mea squared sum withi groups Total mea squared sum oeway ANOVA Ftest statisti February 6, 200 Moez Hababou() 9 ANOVA summary table Soure of variatio Degrees of freedom Sum of squares Mea square F Pvalue Amog groups  SSA MSA=SSA/() F=MSA/MS W Withi groups  SSW MSW=SSW/() Total  SST=SSA+ SSW Exerise: LKB 7.42 pp394 February 6, 200 Moez Hababou() 0 5
6 Deisio rule F=MSA/MSW follows Fdistributio with ,  degree of freedom At a level of sigifiae α, we would Reet Ho: µ =µ 2 =... =µ if F>F U(α,,) I that ase olude that at least oe group mea µ is statistially differet from the other group meas Do ot Reet Ho: µ =µ 2 =... =µ if F<F U(α,,) I that ase olude that we a ot ifer at least o µ is statistially differet from the other group meas February 6, 200 Moez Hababou() Reetio ad aeptae regio a Do ot Reet Ho Reet Ho a F U,α February 6, 200 Moez Hababou() 2 6
7 Assumptios Radomess ad idepedee betwee samples (samples draw from idepedet radom populatios) Pseudoormality of eah sampled group Homogeeity of variae σ 2 =σ2 2 =... =σ2 February 6, 200 Moez Hababou() 3 Example# It is widely believed that eduatio ad holdig additioal degrees irease salary. Verify this statemet, a sample of 44 employees from ompay A were radomly seleted. They a either hold o degree, or a ollege, or a uiversity degree. Is it true, at the 5% level, that all three groups of employees have equal average salaries at ompay A? February 6, 200 Moez Hababou() 4 7
8 ANOVA: umerial appliatio Salary Degree ibarbar (ibarbar) 2 (ibar) (ibar) 2 23,2533,284 0,782,244 ,82,398,99 26, ,943,964 3,855,5 9,8766,66 44,364,0284,559 20,788,626 23,6022,935 8,62, ,646 2,8644,673 2,833,4972,57 6,62,378 9,337,224 52,80,8705,22 26,239,54 22,983,556 2,642,524 ,454 2,5,438 25,0 ,427 2,035, ,02 30,83 4,294 8,44,590 6,396 40,903,002 30,657 4,20 6,977,426 6,222 38,707,628 24,904 ,633 2,665, ,535 26, , ,846 2,69 24,98 24,83,24,50 20,262,79 23, ,935 8,62,0692,58 6,342,444 22, ,090 6,725,0963,673 3,494,02 23, ,935 8,62,0692,58 6,342,444 23,43 23,24 9,757,082,707 7,330,28 2, ,082 25,822,994,665 2,766,53 25, ,465 2,45,49 ,048,099,86 2, ,868 23,693,8484,45 9,85,48 24, ,797 3,227,889 ,380,905,562 23, ,935 8,64,0062,59 6,344,06 25, , ,79 26, , , ,087 9,532,4262,67 7,35,582 29, ,06 9,37,970 3,478 2,093,556 33, ,38 50,956,287 7,555 57,07,665 27, ,899,009,07,232 32,70 2 6,64 37,999,424 6,58 43,304,02 3, ,9 26,950,294 5,608 3,444,943 29,87 2 2,650 7,024,447 3,067 9,403,907 3, ,9 26,950,294 5,608 3,444,943 3, ,9 26,950,294 2,537 6,435,946 34,6 3 7,624 58,30,976 4,970 24,700,072 30,00 3 3,473 2,064, ,625 33, ,38 50,956,287 4,484 20,05,509 35,33 3 8,596 73,897,530 5,942 35,306,374 22, ,640 3,246,9266,294 39,65,485 24, ,087 4,354,0364,74 22,477,87 34,34 3 7,597 57,79,989 4,943 24,432,425 24, ,765 3,3,9294,49 9,528,298 28, ,238 5,00, ,25 24, ,797 3,227,8894,45 9,82,43 25, ,4333,373,377,69 SST= 80,9,555 SSW= 664,363,935 February 6, 200 Moez Hababou() 5 Computig SST, SSW, SSA SST=80,9,555 groups divided by degree group = No degree bar=24,435 group2 = ollege degree 2bar= 26,20 group3 = uiversity degree 3bar= 29,9 SSW = = i= ( i ) 2 = 664,363,935 SST=SSA+SSW => SSA=SSTSSW = February 6, 200 Moez Hababou() 6 8
9 Alterative way to ompute SSA umber of groups =3 total sample size =44 SSA bar barbarbar (barbarbar) 2 *(barbarbar)2 group 24,4352,0 4,44, group2 2 26, , group3 2 29,9 2,654 7,046, barbar= 26,537 SSA February 6, 200 Moez Hababou() 7 Hypothesis testig Ho: µ=µ2=µ3 Ha: at least oe µ i is differet Soure of variatio degree of freedom sum of squares mea squared variae F amog groups betwee groups 4 664,363, total 43 80,9,555 reet Ho if F>F U(α, ,) F U(α, 3,443) = Sie F=4.2> F α,, Colusio: Reet Ho, at least oe group of employees has a salary that is statistially differet from the others February 6, 200 Moez Hababou() 8 9
10 Example#2 The retailig maager would like to kow whether produt pakagig has ay effet o the sale of offee paks. He observed reetly the five more reet daily sales (give ai umber of uits sold) for three types of pakages ad wishes if you a help him i this exerises? Day Day 2 Day 3 Day 4 Day 5 sober shiy February 6, 200 Moez Hababou() vitage ANOVA table ANOVA Soure SS df MS F Pvalue Amog Withi Total F.05,2,2 =3.88 F=5.22>F.05,2,2 =3.88 Reet Ho, at least oe group has average daily sales that is statistially differet from the other groups February 6, 200 Moez Hababou() 20 0
11 ANOVA versus 2 sample Ttest ANOVA ust idiates that oe group mea is differet from the other ad does ot tell whether sigifiatly smaller (equivalet to twotail test). 2 sample Ttest a be a prohibitive exerise whe dealig with umerous groups (umber of pairs to ompare = ()/2. Ttests are usually easier to reet. February 6, 200 Moez Hababou() 2 Sharp proedure for ANOVA (STAT) (TEST) (IputList)(ENTER)(ENTER) Eter data i L, L2,, L (=umber of groups) For this partiular exerise, use L=23, 22, 6, 5, 24 L2=26, 24, 25, 28, 3 L3=33, 2, 24, 29, 30 Hit 2dF (Matrix) Selet ANOVA Number of meas C= (eter 3 for this exerise) February 6, 200 Moez Hababou() 22
12 ANOVA proedure output SST SSW SSA , 94, 68 MSA NDF() MSW DDF() 84.4, 2, 6.6, 2 F Pvalue Group meas ANOM H LDL GRANDMEAN UDL February 6, 200 Moez Hababou() 23 Hypothesis testig usig the pvalue approah Deisio Rule: Reet Ho if pvalue<α February 6, 200 Moez Hababou() 24 2
13 ANOM The Aalysis of Mea (ANOM) allows us to idetify whih group has a mea that is sigifiatly differet from the others. It also speifies whether it is sigifiatly smaller or sigifiatly larger. It is based o derivig a upper ad lower limits (UDL ad LDL). If bar (mea group ) is larger tha UDL, we olude that it is sigifiatly larger tha the other groups average. If bar (mea group ) is less tha LDL, we olude that it is sigifiatly larger tha the other groups average. February 6, 200 Moez Hababou() 25 LDL ad UDL LDL=barbar  h α,, * square root(msw*()/) UDL=barbar + h α,, * square root(msw*()/) barbar = grad mea h α,, = Nelso s statisti at,  degree of freedom = total umber of observatios MSW = Mea squared sum withi groups MSW = SSW/ February 6, 200 Moez Hababou() 26 3
14 Sharp 9600CL output Group meas ANOM H LDL GRANDMEAN UDL February 6, 200 Moez Hababou() Graphial represetatio of ANOM UDL 3bar 2bar barbar LDL 20 bar February 6, 200 Moez Hababou() 28 4
15 Suggested exerises LKB 7.42,7.4, 7.44, 7.47 February 6, 200 Moez Hababou() 29 5
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