Tests For More Than Two Means

 

 

 

1) Analysis of Variance (ANOVA) for Completely Randomized Design

 

 

Simplification of test statistics

 

 

 

Sample

 

 

Data

 

Total

Mean

1

y11

y12

y13

y14

T1

Y1

2

y21

y22

y23

y24

T2

Y2

3

y31

y32

y33

y34

T3

Y3

 

 

 

n= The total sample size = Sni

 

G= The sum of all sample observations = S Ti

 

Y = The average of all sample observations = G / n

 

 

 

 

 

 

 

 

 

 

 

Total sum of squares TSS = S yij2 – G2/n

 

Within-sample of squares SSW = S (yij – Yi )2

 

Between-sample sum of squares SSB = S (Ti2/ni) – G2/n

 

 

TSS = SSW + SSB

 

 

 

The analysis of variance (ANOVA) table is as follows

 

Source

Sum of Squares

Degrees of Freedom

Mean Sum of Squares

F Test

B samples

SSB

T – 1

SSB / (t-1)

MSB/MSW

W samples

SSW

n – t

SSW / (n-t)

 

Totals

TSS

n – 1

 

 

 

 

Ho: m1 = m2= m3= ……

 

Ha: At least one of the means is different from the rest of means

 

 

Reject Ho if calculated F test > F (t-1), (n-t) a

 

 

 

 

 

 

 

2) ANOVA for Randomized Block Design

 

 

 

 

 

Block

 

 

 

Treatment

1

2

….

b

Total

Mean

1

y11

y12

 

y1b

T1

Y1

2

y21

y22

 

y2b

T2

Y2

:

:

:

 

:

:

:

t

yt1

yt2

 

ytb

Tt

Yt

 

 

 

 

 

 

 

Total

B1

B2

 

Bb

G

 

 

b1

b2

 

bb

 

Ÿ

 

yij = The observation for treatment i and block j

 

t = The number of treatments

 

b = The number of blocks

 

n = The total number of sample measurements = bt

 

Ti = The total of all observations receiving treatment i

 

Bj = The total of all observations in block j

 

G = The total of all sample observations

 

Yi = The sample mean for treatment i = Ti/b

 

bj = The sample mean fro block j = Bj/t

 

Ÿ = The overall sample mean = G/n

 

 

 

 

 

TSS = S (yij – Ÿ)2 Total sum of squares

 

SST = b S (Yi – Ÿ)2 Between treatment sum of squares

 

SSB = t S (bj – Ÿ)2 between block sum of squares

 

SSE = TSS – SST – SSB Sum of squares of error

 

Shortcut formulas

 

TSS = S yij2 – G2 / n

 

SST = S Ti2 / b – G2 / n

 

SSB = S Bj2 / t – G2 / n

 

SSE = TSS – SST – SSB

 

 

Source

SS

df

Ms

F

Treatment

SST

t-1

SST/t-1

MST/MSE

Block

SSB

b-1

SSB/b-1

MSB/MSE

Error

SSE

(b-1)(t-1)

SSE/(b-1)(t-1)

 

Totals

TSS

bt-1

 

 

 

 

Reject Ho if Treatment F > Fa, t-1, (b-1)(t-1)

 

Reject Ho if Block F > Fa, b-1, (b-1)(t-1)

 

 

 

 

 

3) ANOVA for t x t Latin Square Design

 

 

 

Column

 

 

Row

1

2

3

4

1

I

II

III

IV

2

II

III

IV

I

3

III

IV

I

II

4

IV

I

II

III

 

yijk = the response on treatment i in row j and column k

 

t = the number of treatments, also the number of rows and columns

 

n = the total number of samples

 

Ti = the total for all observations receiving treatment i

 

Yi = the sample mean for treatment i (Ti/t)

 

Rj = the total for all observations in row j

 

Âj = the sample mean for row j (Rj/t)

 

Ck = the total for all observations in column k

 

Çk = the sample mean for column k (Ck/t)

 

G = the total for all sample measurements

 

Ÿ = the overall sample mean (G/n)

 

TSS = SST + SSR + SSC + SSE

 

Shortcut formulas

 

TSS = S yijk2 – G2/n

 

SST = S Ti2/t – G2/n

 

SSR = S Rj2/t – G2/n

 

SSC = S Ck2/t – G2/n 

 

SSE = TSS – SST – SSR – SSC

 

 

Source

SS

df

MS

F

Treatment

SST

t-1

SST/t-1

MST/MSE

Rows

SSR

t-1

SSR/t-1

MSR/MSE

Columns

SSC

t-1

SSC/t-1

MSC/MSE

Error

SSE

t2-3t+2

SSE/t2-3t+2

 

Total

TSS

t2-1

 

 

 

 

Reject Ho for treatment if F > Fa, t-1, t2-3t+2

 

Reject Ho for rows if F > Fa, t-1, t2-3t+2

 

Reject Ho for columns if F > Fa, t-1, t2-3t+2