.- 8.2 F test

We want to compare treatment groups.

Fishing line t = 3 groups

Stren Trilene XL Trilene XT11.1 11.5 11.611.1 11.3 11.8

11.1 11.4 11.7

Treatment Groupi=l i=2 i-3

j = 1 Yll Y21 Y31j = 2 Y12 Y22 Y32

MeanYl,nlYl.

Y3,nsY3.

Y2,n2

th.

A dot (.) means sum or average over index"nl- - L.Jj=lYlj

Yl. - nl

y.. = overall mean

- - Ei Ej Yij - EtniiJi.Y.. - E ni - ni

= weighted average of Yi.'s2-

For most of the semester, we are assuming normal populations with equal variances...: <r/

\ //~L\

}A. 3

~(j"

t;fA~ /v.-1-

-

Population Sample SampleMean Size Mean

P,l nl Yl.J.t2 n2 Y2.

P,3 n3 Y3.

- - - - - - - - - - - -- - - - - -

To estimate (72,we pool sample variance:

82 - (nl-l)s~+(n2-1)s~+(n3-1)s~w - (nl-l)+(n2-1)+(na-l)

a weighted average (w stands for within group variance).We want to test Ho : J..Ll= J..L2 = J..L3 = .. .= J.Ltwhere t = # of treatment groups.Assuming all measurements are independent as in a completely randomized design.

Tyec.l + , T r~o-.t ~- 1 reo--t- ~

'I XI.>' 0 c) ~ -)t7'- ~

"'" )< 0 }(:. 0 .x * () ~ ~.- -_._- '---'--' ' - - '.---'---'--------

\-JI.

I

~~-

I

dJ~How much evidence we have for different J.L'Sdepends on how spread out (variable)the sample means are relative to the variability of individual points.

V ar(y' 8) = r:(~s.y,)2

y. = mean of all y values.

If J..L'Sare equal, V ar(y's) ~ ~ where n is the number of replicates in each group.If the J..Li'sare not equal, E(Var(y's)) > 0-2

Regardless of the J..Li'S,E( 8~) = 0-2.

F - nVar(y's)- ~- 2 - 2Sw Sw

F ~ 1 if J..Li'sare equal and~ > 1 if J..Li'Sare not equal

If the J..Li'sare equal, F has a Fisher's F distribution with t -1 df in the numeratorand r:(ni - 1) df in the denominator.

F test compares two variances, the variance of the sample means and the withingroup variance.ANOVA= Analysis of Variance.

For unequal ni

(- -

)2 0'2Y. - Y ~ -, .. niF = Eni(Yi-y_,)2 / 82t-l W

- ~- ----

.. Th~ terms in the F test are called mean squares.MSB = Eni(jJi-y..)2 = SSB

t-l . dfB

MSW - ~ni-l)8~ - ssw- (ni-l) - dfwSSW = ~ (n' - 1)~(Yij-yiJ2 = ~ ~ (y" - Y-. )

2~ ni-l ~3~.

This is an error sum of squares. error = Yij-Yi.. where Yij = jth value from the ith group

SSB + SSW = TSS (Total Sum of Squares)

TSS = ~ ~(Yij - y..)2= ~ ~ [(Yij - Yi.)+ (Yi.- y..)]2

= ~ ~(Yij - Yi.)2+ ~ ~(Yi. - y..)2+ ~ ~ (Yij- Yi.)(Yi.- y..)~ ~(Yi. - y..)2= ~ ni(Yi. - y..)2= SSB

~ ~ (Yij - Yd (Yi.- y..)= ~(Yi. - y..)~(Yij - Yi.)= ~(Yi. - y..) X 0

TSS = SSB + SSW

dfTotal = df B + df w

= (t - 1) + ~~=l(ni- 1)

= t -1 + ~ni - t

= (~ni) - 1

ANOVA Table

Source df sa MS FTreatments t - 1 SSB MSB MSB/MSWWithin ~(ni -1) SSW MSW = s~Total (~ni) - 1 TSS

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