1. Example-1
Solve using Two-way ANOVA method
| Observation | A | B | C | | 1 | 10,8,7,9,6 | 7,4,3,2 | 11,9,10,9,11 | | 2 | 1,2,1,4,2 | 6,7,6,5 | 4,3,6,4,3 | | 3 | 3,2,3,3,4 | 2,1,2,3 | 5,6,4,5,5 |
Solution:
Given problem
| Observation | `A` | `B` | `C` | | `1` | 10, 8, 7, 9, 6 | 7, 4, 3, 2 | 11, 9, 10, 9, 11 | | `2` | 1, 2, 1, 4, 2 | 6, 7, 6, 5 | 4, 3, 6, 4, 3 | | `3` | 3, 2, 3, 3, 4 | 2, 1, 2, 3 | 5, 6, 4, 5, 5 |
Row and column sums
| `A` | `B` | `C` | Row total `(x_(a))` | | `1` | 40 | 16 | 50 | `106` | | `2` | 10 | 24 | 20 | `54` | | `3` | 15 | 8 | 25 | `48` | | Col total `(x_(b))` | `65` | `48` | `95` | `208` |
`sum x^2=10^2 + 8^2 + 7^2 + ... + 4^2 + 5^2 + 5^2=1362 ->(A)`
`sum (x_(b)^2)/(n_(b))=65^2/15+48^2/12+95^2/15`
`=4225/15+2304/12+9025/15`
`=281.6667+192+601.6667`
`=1075.3333 ->(B)`
`sum (x_(a)^2)/(n_(a))=106^2/14+54^2/14+48^2/14`
`=11236/14+2916/14+2304/14`
`=802.5714+208.2857+164.5714`
`=1175.4286 ->(C)`
`sum (sum x_(ab)^2)/n_(ab)=40^2/5+16^2/4+50^2/5+10^2/5+24^2/4+20^2/5+15^2/5+8^2/4+25^2/5`
`=1600/5+256/4+2500/5+100/5+576/4+400/5+225/5+64/4+625/5`
`=320+64+500+20+144+80+45+16+125`
`=1314 ->(C)`
`(sum x)^2/(n)=(208)^2/(42)`
`=43264/42`
`=1030.0952 ->(D)`
Sum of squares total
`SS_T=sum x^2 - (sum x)^2/(n)=(A)-(D)`
`=1362-1030.0952`
`=331.9048`
Sum of squares between rows
`SS_A=sum (x_(a)^2)/(n_(a)) - (sum x)^2/(n)=(C)-(D)`
`=1175.4286-1030.0952`
`=145.3333`
Sum of squares between columns
`SS_B=sum (x_(b)^2)/(n_(b)) - (sum x)^2/(n)=(B)-(D)`
`=1075.3333-1030.0952`
`=45.2381`
Sum of squares between columns
`SS_(AB)=sum (sum x_(ab)^2)/(n_(ab)) - (sum x)^2/(n) - SSA - SSB = (B)-(D)- SSA - SSB`
`=1314-1030.0952-145.3333-45.2381`
`=93.3333`
Sum of squares Error (residual)
`SS_E=SS_T - SS_A - SS_B - SS_(AB)`
`=331.9048-145.3333-45.2381-93.3333`
`=48`
ANOVA table
| Source of Variation | Sums of Squares
SS | Degrees of freedom
DF | Mean Squares
MS | F | | A | `SS_A = 145.3333` | `a-1 = 2` | `MS_R=145.3333/2=72.6667` | `72.6667/1.4545=49.9583` | | B | `SS_B = 45.2381` | `b-1 = 2` | `MS_C=45.2381/2=22.619` | `22.619/1.4545=15.5506` | | AB | `SS_(AB) = 93.3333` | `(a-1)(b-1) = 4` | `MS_(AB)=93.3333/4=23.3333` | `23.3333/1.4545=16.0417` | | Error (residual) | `SS_E = 48` | `n-ab = 33` | `MS_E=48/33=1.4545` | | | Total | `SS_T = 331.9048` | `n-1 = 41` | | |
This material is intended as a summary. Use your textbook for detail explanation.
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