ANOVA for Independent Samples from Normal Distributions: One-Way & Two-Way, Study notes of Statistics

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D Assumption :

k independent random samples from k normal distributions Nuj. 6

) for j

• • l (^).. - - , k with (^) Gmmonvarianu6# Ho :^ U, =^.^.^ -^ = My Hii ( Ha^ ) ' 0 One - way ANNA

• I continuous dependent variable

/ categorical independent variable^ (^ factory treatment^ )

k different^ levels^ of^ the^ categorical variable^ C^ k different^ groups ) statis-hzsofone-wayhwoVAJ.TK#ampemanlofjth group ) Fj.

hjnxij dverallsamplemean-x-tn.EE#j--hEEniFi Total variation IEEE ( (^) Xij-^ It^

with Ln- y (^) degrees of (^) freedom Bevar WthYhhn^

B

EE, nifxj^

• TH ' w - - EEE!^ cxii-x.jp with Lk- D degrees of^ freedom with^ =k
• H degrees of^ freedom E

ja Cnj-^ H

÷÷÷÷i÷÷÷÷÷:.÷i÷÷÷aai.^

.÷÷÷* H H Factor level §!Nj

• H (^) Sj '

0 Estimator of 6

of

S

• • Fn - k

get confidence^ interval^ for^ nj

: T.j-to.rs , n-^ t^ × ¥ If the^ 10011-211^ confidence^ interval^ for Mi^ ,^ Ma^

• -. Mn do not overlap ,^ Ho is not^ supported. (^0) Review (^) of one - way ANNA (^) Model ①

Variation of the^ Xijs is driven

by a (^) facer at^ different^ levels Mi (^) ,^ -^ -^ - i Uk in addition^ to^ ②^ random (^) fluctuations ( random^ errors I ① ②^

Xij = (^) Nt ⑧ t (^) i- I (^)..... n j

• • (^) I (^). - n , k
• Individual

effect Eij -^ Mo^ ,^62 ) Independent

Avg effect re

factor (^ Treatment^ )^ effect Pj

EE Pj^

• (^) o

Null Hypothesis

Ho : f. =^.^.^.

= for = (^) o NO^ treatment effect Ecxij )^

Ecu)'t^ Elfj)

t Eceij )^ = (^) at

Pj to

Varlxij )^ = (^) Var (^) ( Utfjt (^) Eij )^ = Var ( Eij )^ = 6 ' 6 Two-way ANOVA Xij = n El (^).^ -^

, r j

-^ - I (^) , - , c n : Average effect Pj : C different treatment ( (^) column (^) ) levels y;^ i (^) r (^) different block ( (^) row (^) ) (^) levels Eijn Nco,^64 are^ independent

E.fi =o EE, Pj^

• (^) o Hypotheses : No column^ effect^ no^

f,^ a -^ -^ -^ = fo =o No row^ effect^ Ho^ :^ Yo =^

• • • (^) = yr =D StatntrsofTvo-wayAN# Sample mean^ at^ ith^ block (^) level (^) Crow) Sample

mean at jth-heatmen.tl#adumm)

- -^ -

Xi.^ - EE,^ Xijfc^ Xj^

• EE, Xijfr dveraksampk-meani-x-EE.li/n-otavana-hbn : Total SS = Ee, Ei (^ Xij
• FI

with Irc^ -11 (^) degrees of (^) freedom Between - blocks Crow (^) ) variation : BEfXi

a (^) wither- y degrees of^ freedom Be¥¥umns) (^) variation : Bad = r. ft.

( Xj - Fj

with CC^ - D degrees of (^) freedom Res¥enor ) (^) variation : Residual (^) SS = FETE, Nij - Fi.^ -

Ty

. (^) t (^) FT with Cr- 1) (^) Co - t (^) )

degrees of^ freedom

Two way ANOVA^ Decomposition : Total (^) SS =^ Brow^ t^ Bad t Residual SS g Between - blocks variation^ →^ tow (^) factor level Between - treatment Variation^ → Column (^) factor level Residual Variation^ →^ NOT explained (^) by row^ or^ column^ factors

(^6) Two - way ANOVA Table

F-ssmsfp-valuefa7.wr-ibmw-B-I.ws#

titties(^ r^ -^ 1)^ (^ C^ -^ t^ )

Titus (^0) Residuals Xij

• (^) ut ti tpjt Eij

p.nte.E.IT

÷ +' +

txii-xni-x.ir# Bj Eij = Xij

• Fi.^ - Fj TX^ ~^ Nco^ , 62 ) Estimator of 62=5--4%^ = Residual Ms