Analysis of variancePetter Mostad2005.11.07 Comparing more than two groups

!p to now we have studied situations with"#ne o$servation per o$%ect

#ne group

&wo groups"&wo or more o$servations per o$%ect

'e will now study situations with one o$servation per o$%ect( and three or more groups of o$%ects

&he most important )uestion is as usual* +o the num$ers in the groups come from the same population( or from different populations, A-#.A

/f you have three groups( could plausi$ly do pairwise comparisons. 0ut if you have 10 groups, &oo many pairwise comparisons* 1ou would get too many false positives2

1ou would really li3e to compare a null hypothesis of all e)ual( against some difference

A-#.A* A-alysis #f .Ariance #ne4way A-#.A* 56ample

Assume 7treatment results7 from 18 patients visiting one of three doctors are given* "+octor A* 29(2:(81(27"+octor 0* 2;(81(80(8:(88"+octor C* 2;(27(89(2: ums of s)uares for two4way A-#.A21@ AKiiSSG H x x== 21@ AHjjSS K x x== 21 1@ AK Hij i ji jSS! x x x x = == +21 1@ AK Hiji jSST x x= == SSG SS SS! SST + + = A-#.A ta$le for two4way data>ource of variation>ums of s)uares+eg. of freedomMean s)uares E ratio0etween groups >>F B41 M>FG >>FL@B41A M>FLM>50etween $loc3s >>0 0G >>0L@0LM>55rror >>5 @B41A@5G >>5L@B41A@>& n41&est for $etween groups effect* compare to &est for $etween $loc3s effect* comparetoMSGMS!MSMS!1(@ 1A@ 1A K K HF 1(@ 1A@ 1A H K HF &wo4way A-#.A @with interactionA

&he setup a$ove assumes that the $loc3ing varia$le influences outcomes in the same way in all categories @and vice versaA

'e can chec3 if there is interaction $etween the $loc3ing varia$le and the categories $y e6tending the model with an interaction term >ums of s)uares for two4way A-#.A @with interactionA

Assume B categories( < $loc3s( and assume I o$servations 6i%1( 6i%2( J(6i%I for each category i and each $loc3 % $loc3( so we have nGBums of s)uares for two4way A-#.A @with interactionA21@ AKiiSSG H" x x== 21@ AHjjSS K" x x == 21 1@ AK Hij i ji jSS# " x x x x = == +21 1 1@ AK H "ijli j lSST x x= = == SSG SS SS# SS! SST + + + =21 1 1@ AK H "ijl iji j lSS! x x= = == A-#.A ta$le for two4way data @with interactionA>ource of variation>ums of s)uares+eg. of freedomMean s)uares E ratio0etween groups >>F B41 M>FG >>FL@B41A M>FLM>50etween $loc3s >>0 0G >>0L@0LM>5/nteraction >>/ @B41A@/G>>/L@B41A@/LM>55rror >>5 B5G >>5LB>& n41&est for interaction* compare M>/LM>5 with &est for $loc3 effect* compare M>0LM>5 with &est for group effect* compare M>FLM>5 with 1( @ 1A K KH"F 1( @ 1A H KH"F @ 1A@ 1A( @ 1A K H KH"F -otes on A-#.AAll analysis of variance @A-#.AA methods are $ased on the assumptions of normally distri$uted and independent errors&he same pro$lems can $e descri$ed using the regression framewor3. 'e get e6actly the same tests and results2 &here are many e6tensions $eyond those mentioned