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Solution
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Solution provided by AtoZmath.com
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Two-way ANOVA calculator
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1. | 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 |
2. | Observation | A | B | C | | 1 | 4.1,3.9,4.3 | 3.1,2.8,3.3 | 3.5,3.2,3.6 | | 2 | 2.7,3.1,2.6 | 1.9,2.2,2.3 | 2.7,2.3,2.5 |
3. | Observation | A | B | C | D | | 1 | 421 | 325 | 320 | 362 | | 2 | 118 | 102 | 122 | 56 | | 3 | 591 | 518 | 552 | 509 | | 4 | 14 | 26 | 20 | 2 | | 5 | 116 | 14 | 26 | 46 |
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Example1. 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` | | |
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