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 (yijŸ)2 Total sum of squares

SST = b S (YiŸ)2 Between treatment sum of squares

SSB = t S (bjŸ)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