ANOVA- Analyisis of Variance

Transcription

1 ANOVA- Aalii of Variac CS 700 Comparig altrativ Comparig two altrativ u cofidc itrval Comparig mor tha two altrativ ANOVA Aali of Variac

2 Comparig Mor Tha Two Altrativ Naïv approach Compar cofidc itrval 3 O-Factor Aali of Variac (ANOVA) Vr gral tchiqu Loo at total variatio i a t of maurmt Divid ito maigful compot Alo calld O-wa claificatio O-factor xprimtal dig Itroduc baic cocpt with o-factor ANOVA Graliz latr with dig of xprimt 4

3 O-Factor Aali of Variac (ANOVA) Sparat total variatio obrvd i a t of maurmt ito:. Variatio withi o tm Du to radom maurmt rror. Variatio btw tm Du to ral diffrc + radom rror I variatio() tatiticall > variatio()? 5 ANOVA Ma maurmt of altrativ i ith maurmt o th altrativ Aum rror ar: Idpdt Gauia (ormal) 6 3

4 Maurmt for All Altrativ Altrativ Maurm t i i i i i Col ma.... Effct α α α α 7 Colum Ma Colum ma ar avrag valu of all maurmt withi a igl altrativ Avrag prformac of o altrativ. i i 8 4

5 Colum Ma Altrativ Maurm t i i i i i Col ma.... Effct α α α α 9 Dviatio From Colum Ma i i. + i dviatio of rror i maurmt i from colum ma 0 5

6 Error Dviatio From Colum Ma Altrativ Maurm t i i i i i Col ma.... Effct α α α α Ovrall Ma Avrag of all maurmt mad of all altrativ.. i i 6

7 Ovrall Ma Altrativ Maurm t i i i i i Col ma.... Effct α α α α 3 Dviatio From Ovrall Ma... + dviatio of ffct of colum ma from ovrall ma altrativ 4 7

8 Effct Dviatio From Ovrall Ma Altrativ Maurm t i i i i i Col ma.... Effct α α α α 5 Effct ad Error Effct i ditac from ovrall ma Horizotall acro altrativ Error i ditac from colum ma Vrticall withi o altrativ Error acro altrativ, too Idividual maurmt ar th: + + i.. i 6 8

9 Sum of Squar of Diffrc: SSE i i SSE. i + " i. ( ) ( ) i i ". i i 7 Sum of Squar of Diffrc: SSA #.... SSA + # " ( ) ( ) #. "

10 Sum of Squar of Diffrc: SST t i i SST #.. + # + i ( ) ( ) ti i ".. i + i i ".. i 9 Sum of Squar of Diffrc SSA SSE SST (. ".. ) ( i ". ) i ( ) i ".. i 0 0

11 Sum of Squar of Diffrc SST diffrc btw ach maurmt ad ovrall ma SSA variatio du to ffct of altrativ SSE variatio du to rror i maurmt SST SSA + SSE ANOVA Fudamtal Ida Sparat variatio i maurd valu ito:. Variatio du to ffct of altrativ SSA variatio acro colum. Variatio du to rror SSE variatio withi a igl colum If diffrc amog altrativ ar du to ral diffrc, SSA hould b tatiticall > SSE

12 Comparig SSE ad SSA Simpl approach SSA / SST fractio of total variatio xplaid b diffrc amog altrativ SSE / SST fractio of total variatio du to xprimtal rror But i it tatiticall igificat? 3 Statiticall Comparig SSE ad SSA Variac ma quar valu x total variatio dgr of frdom SSx df 4

13 Dgr of Frdom df(ssa), ic altrativ df(sse) ( ), ic altrativ, ach with ( ) df df(sst) df(ssa) + df(sse) - 5 Dgr of Frdom for Effct Altrativ Maurm t i i i i i Col ma.... Effct α α α α 6 3

14 Dgr of Frdom for Error Altrativ Maurm t i i i i i Col ma.... Effct α α α α 7 Dgr of Frdom for Error Altrativ Maurm t i i i i i Col ma.... Effct α α α α 8 4

15 Variac from Sum of Squar (Ma Squar Valu) a SSA SSE ( ) 9 Comparig Variac U F-tt to compar ratio of variac F F [" ; df ( um), df ( dom)] a tabulatd critical valu 30 5

16 F-tt If F computd > F tabl W hav ( α) * 00% cofidc that variatio du to actual diffrc i altrativ, SSA, i tatiticall gratr tha variatio du to rror, SSE. 3 ANOVA Summar Variatio Sum of quar Dg frdom Ma quar Computd F Tabulatd F Altrativ SSA a F SSA ( ) a [" ;( ), ( )] Error SSE ( ) SSE [ ( )] Total SST 3 6

17 7 33 ANOVA Exampl Effct Colum ma Ovrall ma 3 Maurmt Altrativ 34 ANOVA Exampl 3.89 Tabulatd Computd Ma quar 4 ) ( Dg frdom quar Sum of Total Error Altrativ Variatio [0.95;,] F F F SST SSE SSA a

18 Cocluio from xampl SSA/SST / % of total variatio i maurmt i du to diffrc amog altrativ SSE/SST / % of total variatio i maurmt i du to oi i maurmt Computd F tatitic > tabulatd F tatitic 95% cofidc that diffrc amog altrativ ar tatiticall igificat. 35 Cotrat ANOVA tll u that thr i a tatiticall igificat diffrc amog altrativ But it do ot tll u whr diffrc i U mthod of cotrat to compar ubt of altrativ A v B {A, B} v {C} Etc. 36 8

19 Cotrat Cotrat liar combiatio of ffct of altrativ c w w " 0 37 Cotrat E.g. Compar ffct of tm to ffct of tm w w w 3 " 0 c () + (" ) " + (0)

20 Cotruct cofidc itrval for cotrat Nd Etimat of variac Appropriat valu from t tabl Comput cofidc itrval a bfor If itrval iclud 0 Th o tatiticall igificat diffrc xit btw th altrativ icludd i th cotrat 39 Variac of radom variabl Rcall that, for idpdt radom variabl X ad X Var[ X + X Var[ ax ] ] Var[ X a Var[ X ] + Var[ X ] ] 40 0

21 4 Variac of a cotrat c w w w c ] Var[ ] Var[ )] ( Var[ ] Var[ " " " ) ( ) ( ) ( ) ( " df SSE w c c Aum variatio du to rror i quall ditributd amog total maurmt 4 Cofidc itrval for cotrat ) ( ) ( ), ( ) ( ; / " SSE w t c c c c c # m

22 Exampl 90% cofidc itrval for cotrat of [S- S] c " " 0.44 " [ ] c ( 0.44) % : ( c, c + ( ) + 0 3(5) ) ( ,0.096) 43 Summar U o-factor ANOVA to parat total variatio ito: Variatio withi o tm Du to radom rror Variatio btw tm Du to ral diffrc (+ radom rror) I th variatio du to ral diffrc tatiticall gratr tha th variatio du to rror? U cotrat to compar ffct of ubt of altrativ 44