class notes Anova

ANALYSIS OF VARIANCE
The analysis of variance is a powerful statistical tool for tests of significance.

We consider the following types of ANOVA

I) One- way classification (One factor ANOVA) –Completely Randomized Design.
II) Two-Way classification (Two factors)-Randomized Block Design.
III) Latin Square Design-Three Way ANOVA




ONE - WAY ANOVA
One- way classification observations are classified according to one factor

WORKING PROCEDURE:
Null Hypothesis 𝐻0 : 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 … … …

Alternative Hypothesis 𝐻1 : 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 … … … ..

STEP1: To find the following
𝑁 = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

𝑇 = ∑ 𝑋1 + ∑ 𝑋2 + ∑ 𝑋3 + ⋯
𝑇2
Correction factor: 𝐶. 𝐹 = 𝑁


STEP2: To Calculate the following
𝑇𝑆𝑆 = ∑𝑋12 + ∑𝑋22 + ∑𝑋32 + ⋯ − 𝐶. 𝐹
(∑ 𝑋1 )2 (∑ 𝑋2 )2
𝑆𝑆𝐶 = 𝑛1
+ 𝑛2
+ ⋯ − 𝐶. 𝐹

𝑆𝑆𝐸 = 𝑇𝑆𝑆 − 𝑆𝑆𝐶

STEP3: O NE- WAY ANOVA TABLE

SOURCE OF SUM OF SQUARES DEGREES OF MEAN SUM OF VARIATION RATIO
VARIATION FREEDOM SQUARES

BETWEEN SSC = C–1= 𝑆𝑆𝐶 𝑀𝑆𝐸
𝑀𝑆𝐶 = 𝐹𝑐 =
COLUMNS 𝐶−1 𝑀𝑆𝐶

ERROR SSE = N–C= 𝑆𝑆𝐸
𝑀𝑆𝐸 =
𝑁−𝐶
TOTAL

, NOTE: IF CALCULATED VALUE OF F < TABULATED VALUE OF F, THEN 𝐻0 IS ACCEPTED.

IF CALCULATED VALUE OF F > TABULAED VALUE OF F, THEN 𝐻0 𝐼𝑆 𝑅𝐸𝐽𝐸𝐶𝑇𝐸𝐷.



PROBLEM 1:

The following figures relate to production in kgs. of three variables A, B, C of wheat sown on 12 plots



A 14 16 18

B 14 13 15 22

C 18 16 19 19 22



Is there any significant difference in the production of the Varieties.

SOLUTION:

Null Hypothesis:

𝐻0 ∶ 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑣𝑎𝑟𝑖𝑒𝑡𝑖𝑒𝑠.
Alternative Hypothesis:

𝑯 𝟏 : 𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑠𝑖𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑛𝑐𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑒𝑡𝑖𝑒𝑠.



𝑋1 𝑋2 𝑋3 𝑋12 𝑋22 𝑋32

14 14 18 196 196 324

16 13 1 256 169 256

18 15 19 324 225 341

22 19 484 361

20 400

TOTAL ∑𝑋2 = 64 ∑𝑋3 = 92 ∑𝑋12 = 776 ∑𝑋22 = 1074 ∑𝑋32 = 1702

∑𝑋1 = 48