Consider a general case of single factor experiments. The experiment is carried out to study the effect of factor A with `a` levels. To simplify the experiment, we realize `n` runs for each level. So experiment consists of `an` runs. In the method of Completely Randomized Design, `an` runs are realized in a random order.

For each run, we obtain value `y_(ij)` of response `Y` corresponding to run `j` of level `i`. With each level `i` of factor A, there are `n` values of `y_i` for `n` runs. So there are `an` values of `Y` as on Table 1.

Table 1 Value of `Y` for `an` runs of experiment

		Level
		1	2	. . .	`i`	. . .	`a`
Runs	1	`y_(11)`	`y_(21)`	. . .	`y_(i1)`	. . .	`y_(a1)`
	2	`y_(12)`	`y_(22)`	. . .	`y_(i2)`	. . .	`y_(a2)`
	. . .	. . .	. . .	. . .	. . .	. . .	. . .
	`j`	`y_(1j)`	`y_(2j)`	. . .	`y_(ij)`	. . .	`y_(aj)`
	. . .	. . .	. . .	. . .	. . .	. . .	. . .
	`n`	`y_(1n)`	`y_(2n)`	. . .	`y_(i n)`	. . .	`y_(an)`
Mean		`bar y_1`	`bar y_2`	. . .	`bar y_i`	. . .	`bar y_n`

The example below will illustrate these notions.

Example 4

A manufacturer of fish ball would like to improve the firmness of his product by soybean protein. To study the effect of this ingredient on firmness and sensory quality, an experiment was realized with 5 levels of soybean protein content: 10%, 14%, 18%, 22%, 26% (5 treatments). With each level of soybean protein content, there are 4 runs. So this experiment consists of 20 runs. These runs are realized in random order, for example 7, 15, 2, 11, 19, 4, 5, 18, 6, ...

The measurement of firmness of these runs are presented in Table 2.

Table 2 Firmness of 20 runs of experiment

	Protein content (%)
	10	14	18	22	26
Run 1	181	194	204	210	216
Run 2	179	197	208	214	214
Run 3	182	201	205	210	212
Run 4	183	195	206	209	215
Mean	181,3	196,8	205,8	211,5	214,3

 

The variation of response `Y` is due to:

• effect of factor A
• random errors

The variation of `Y` due to factor A is characterized by:

`SS_A=nsum_(i=1)^a (bar y_i-bar y)^2`

(2)

(`SS` is the acronym of sum of squares)

The variation of `Y` due to random errors is characterized by:

`SS_E=sum_(i=1)^a sum_(j=1)^n (y_(ij)-bar y_i)^2`

(3)

Total sum of squares `SS_T` is defined by:

`SS_T=sum_(i=1)^a sum_(j=1)^n (y_(ij)-bar y)^2`

(4)

We can prove that:

`SS_T=SS_A+SS_E`(5)

We also define:

`MS_A=(SS_A)/(a-1)`

(6)

`MS_E=(SS_E)/(a(n-1))`

(7)

(`MS` is the acronym of mean of squares)

In formulae (6) and (7), the denominators of `MS_A` and `MS_E` are degrees of freedom `df_A` and `df_E` respectively.

We also can prove that the ratio:

`F=(MS_A)/(MS_E)`

(8)

conforms to Fisher distribution with degrees of freedom `df_A` and `df_E`.

Therefore, `F` represents the difference between variation due to factor A and variation due to random errors.

Because ANOVA is a form of hypothesis testing, so to conclude about the differences between these variations, we have to compare `F_o` (calculated from data) to critical value `F"*"`.

Because this case is a one-sided test with rejection region being in the right of `F`* and significant level `alpha` so:

  `F`*`=F_(alpha,df_A,df_E)=F_(alpha,a-1,a(n-1)`

If :

• `F_o>F`* : we conclude that factor A does affect on `Y`.
• `F_o< F`* : we conclude that effect of factor A on `Y` is insignificant statistically.

When we use softwares to realize ANOVA, the results are shown in the form of Table 3.

Table 3 Result of analysis of variance

Source of variation	Degree of freedom	`SS`	`MS`	`F_o`	`F`*	`p` value
Treatment	`a-1`	`SS_A`	`MS_A`	`(MS_A)/(MS_E)`	`F_(alpha,a-1,a(n-1)`
Error	`a(n-1)`	`SS_E`	`MS_E`
Total	`an-1`	`SS_T`

 

Manual analysis of variance by procedure above is tedious and the errors can propagate if we use formulae (2), (3) and (4). To facilitate the analysis and increase the accuracy, we use another way to calculate `SS_T` and `SS_A`.

To simplify the formulae, put:

`y_(i*)=sum_(j=1)^n y_(ij)`(9)

and   `y_(* *)=sum_(i=1)^a sum_(j=1)^n y_(ij)`(10)

Then we calculate `SS_T` and `SS_A` by formulae:

`SS_T=sum_(i=1)^a sum_(j=1)^n y_(ij)^2\ -\ (y_(* *)^2)/(an)`(11)

`SS_A=1/nsum_(i=1)^a y_(i*)^2\ -\ (y_(* *)^2)/(an)`(12)

with :

`CF=y_(* *)^2/(an)`

(13)

`CF` is denoted as correction factor.

The example below will illustrate these notions.

 

To study the effect of preferred colour on intelligence of human being, an experiment was conducted with 3 colours A, B and C. For each colour, 10 persons are tested and their `IQ` are measured. The result is shown in Table 4.

Table 4 `IQ` and preferred colour.

	Colour A	Colour B	Colour C
	102	89	51
	88	100	76
	106	92	90
	93	76	117
	98	64	103
	104	104	64
	90	66	64
	103	98	50
	99	90	89
	92	82	67

This is a single factor experiment used to study the effect of factor "preferred colour" on response `IQ`. Factor is studied with 3 levels (`a=3`) and 10 runs for each level (`n=10`). So the experiment consists of 30 runs.

For a manual analysis, we develop Table 4 to obtain Table 5.

Table 5 Preliminary calculation for manual analysis of variance

	Colour A	Colour B	Colour C	`sum`
	102	89	51
	88	100	76
	106	92	90
	93	76	117
	98	64	103
	104	104	64
	90	66	64
	103	98	50
	99	90	89
	92	82	67
`y_(i*)`	975	861	771	2607
`y_(i*)^2`	950.625	741.321	594.441	2.286.387
`sum y_(ij)^2`	95.427	75.857	63.877	235.161
Mean	97,5	86,1	77,1

From Table 5, we obtain:

  `y_(* *)=sum_(i=1)^a sum_(j=1)^n y_(ij)=2607`   `sum_(i=1)^a sum_(j=1)^n y_(ij)^2=235.161`

  `sum_(i=1)^a y_(i*)^2=2.286.387`   `CF=y_(* *)^2/(an)=2607^2/(3xx10)=226.548,3`

  `SS_T=sum_(i=1)^a sum_(j=1)^n y_(ij)^2\ -\ (y_(* *)^2)/(an)=235.161-226.548,3=8612,7`

  `SS_A=1/nsum_(i=1)^a y_(i*)^2\ -\ (y_(* *)^2)/(an)=2.286.387/10-226.548.3=2090,4`

  `SS_E=SS_T-SS_A=8612,7-2090,4=6522,3`

  `MS_A=(SS_A)/(a-1)=(2090,4)/(3-1)=1045,2`

  `MS_E=(SS_E)/(a(n-1))=(6522,3)/(3xx(10-1))=241,567`

  `F_o=(MS_A)/(MS_E)=(1045,2)/(241,567)=4,327`

With confidence level of 95%, critical value of `F` is:
 `F`*`=F_(alpha,a-1,a(n-1))=F_(0,05,2,27)=3,354`.

Because `F_o>F`* ; we conclude preferred colour affects `IQ`.