In full factorial experiments, all the combinations of all the levels of all factors are tested. For example, if the experiment studies the effect of 4 factors A, B, C, D, at `a`, `b`, `c`, d levels respectively, then the number of treatments is `abcd`. If each treatment is realized with `n` runs, then the number of runs will be `abcdn`.

To investigate further, let's consider an experiment with 2 factors A and B. We would like to study the effect of `a` levels of factors A and `b` levels of factor B on response `Y`. Each level of factor A combines with all the `b` levels of factor B and vice versa. There are `ab` treatments. If each treatment is realized with `n` runs, then the total runs of experiment are `abn`.

After experiment, the results of response Y are shown on Table 1.

Table 1 Response `Y` in full factorial experiment with 2 factors

		Factor B						Mean
		Level 1	Level 2	. . .	Level `j`	. . .	Level `b`
Factor A	Level 1	`y_(111)`	`y_(121)`	. . .	`y_(1j1)`	. . .	`y_(1b1)`	`bar y_(a1)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(11k)`	`y_(12k)`	. . .	`y_(1jk)`	. . .	`y_(1bk)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(11n)`	`y_(12n)`	. . .	`y_(1jn)`	. . .	`y_(1bn)`
	Level 2	`y_(211)`	`y_(221)`	. . .	`y_(2j1)`	. . .	`y_(2b1)`	`bar y_(a2)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(21k)`	`y_(22k)`	. . .	`y_(2jk)`	. . .	`y_(2bk)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(21n)`	`y_(22n)`	. . .	`y_(2jn)`	. . .	`y_(2bn)`
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
	Level `i`	`y_(i11)`	`y_(i21)`	. . .	`y_(ij1)`	. . .	`y_(ib1)`	`bar y_(ai)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(i1k)`	`y_(i2k)`	. . .	`y_(ijk)`	. . .	`y_(ibk)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(i1n)`	`y_(i2n)`	. . .	`y_(ijn)`	. . .	`y_(ibn)`
	. . .	. . .	. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
		. . .	. . .	. . .	. . .	. . .	. . .
	Level `a`	`y_(a11)`	`y_(a21)`	. . .	`y_(aj1)`	. . .	`y_(ab1)`	`bar y_(aa)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(a1k)`	`y_(a2k)`	. . .	`y_(ajk)`	. . .	`y_(abk)`
		. . .	. . .	. . .	. . .	. . .	. . .
		`y_(a1n)`	`y_(a2n)`	. . .	`y_(ajn)`	. . .	`y_(abn)`
Mean		`bar y_(b1)`	`bar y_(b2)`	. . .	`bar y_(bj)`	. . .	`bar y_(b b)`

In Table 1.

`i` is the index of factor A, `j` is the index of factor B, `k` is the index run in `a` treatment,

`y_(ijk)` is the value of response `Y` in `k^(th)` run with level `i` of factor A and level `j` of factor B.

`bar y_(ai)` is the mean of level `i` of factor A (consists of `bn` values),

`bar y_(bj)` is the mean of level `j` of factor B (consists of `an` values),

`bar y_(ij)` is the mean of treatment `ij` (consists of `n` values),

`bar y` is the mean of all abn values of `Y` (also known as grand mean).

 

In principle, analysis of variance (ANOVA) for multiple factor experiments is realized similar to that for single factor experiments. It consists of:

• calculate sums of squares `SS`,
• calculate means of squares `MS` by dividing `SS` to suitable degree of freedom `df`,
• calculate values `F_(oi)=(MS_i)/(MS_E)`,
• compare `F_(oi)` to critical value `F_i`*,
• then, conclude.

Now examine this procedure further in detail for experiments with 2 factors.

Calculate sums of squares `SS`

Sums of squares for experiments 2 factors A and B consist of:

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

`SS_A=bnsum_(i=1)^a (bar y_(ai)-bar y)^2`(2)

`SS_B=ansum_(j=1)^b (bar y_(bj)-bar y)^2`(3)

`SS_(AB)=nsum_(i=1)^a sum_(j=1)^b (bar y_(ij)-bar y_(ai)-bar y_(bj)+bar y)^2`(4)

`SS_E=sum_(i=1)^a sum_(j=1)^b sum_(k=1)^n (y_(ijk)-bar y_(ij))^2`(5)

in which `SS_T` is total sum of squares, `SS_A` is sum of squares of factor A, `SS_B` is sum of squares of factor B, `SS_(AB)` is sum of squares of interaction between A and B, `SS_E` is sum of squares of errors.

We can prove that :

`SS_T=SS_A+SS_B+SS_(AB)+SS_E`(6)

Degrees of freedom of these components are  `df_T=abn-1` ; `df_A=a-1` ; `df_B=b-1` ; `df_(AB)=(a-1)(b-1)` ; `df_E=ab(n-1)`

Calculate means of squares

Means of squares for experiments 2 factors A and B consist of:

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

(7)

`MS_B=(SS_B)/(b-1)`

(8)

`MS_(AB)=(SS_(AB))/((a-1)(b-1))`

(9)

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

(10)

Calculate values `F`

`F_A=(MS_A)/(MS_E)`

(11)

`F_B=(MS_B)/(MS_E)`

(12)

`F_(AB)=(MS_(AB))/(MS_E)`

(13)

When we use softwares to analyze the data of this type of experiment, the result is shown in the form of Table 2.

Table 2 Result of analysis of variance for full factorial experiment with 2 factors

Source of variation	Degree of freedom	`SS`	`MS`	`F_o`	`F`*	`p` value
Factor A	`a-1`	`SS_A`	`MS_A`	`(MS_A)/(MS_E)`	`F_A`*
Factor B	`b-1`	`SS_B`	`MS_B`	`(MS_B)/(MS_E)`	`F_B`*
Interaction AB	`(a-1)(b-1)`	`SS_(AB)`	`MS_(AB)`	`(MS_(AB))/(MS_E)`	`F_(AB)`*
Error	`ab(n-1)`	`SS_E`	`MS_E`
Total	`abn-1`	`SS_T`

 

This is a special case of full factorial experiments in which each factor is studied at two levels only, designated at high level and low level (and coded as + and −). So if `k` factors are studied, the number of treatments will be `2^k`.

For example, if we would like to study the effect of 3 factors `X_1`, `X_2`, `X_3`, the number of treatments is 8 and the matrix of coded factors of this experiment (without replication) is shown on Table 3.

Table 3 Matrix of coded factors for experiment `2^3`

Run	`X_1`	`X_2`	`X_3`
1	−	−	−
2	−	−	+
3	−	+	−
4	−	+	+
5	+	−	−
6	+	−	+
7	+	+	−
8	+	+	+

The numbers of treatments and runs of this type of experiment are low. But it cannot investigate the factors in depth. So `2^k` experiments are mainly used in screening experiments, in which we would like to separate important factors and unimportant ones.

 

An experiment was carried out to compare anti-settling properties of 3 additives A, B, and C for nectar. Each additive was tested with two levels of concentration: 0,5% and 1%. There were three runs for each treatment. The result is given in Table 4.

Table 4 Settling time of runs in experiment (unit: day)

	0,5%			1%
	Run 1	Run 2	Run 3	Run 1	Run 2	Run 3
A	61	67	65	95	88	97
B	84	77	81	103	114	116
C	57	55	61	94	92	88

This experiment studied the effect of 2 factors: type of additive (with 3 levels A, B and C) and additive concentration (with 2 levels 0,5% and 1%). Hence, there were 6 treatments. Each treatment was realized with 3 runs, so this experiment consists of 18 runs. The response is settling time.

The result of analysis of variance is shown in Table 5.

Table 5 Result of analysis of variance

Source of variation	Degree of freedom	`SS`	`MS`	`F_o`	`F`*
Additive	2	1525,8	761,9	41,0	1,56
Concentration	1	4324,5	4324,5	252,4	1,46
Additive*Concentration	2	17,3	8,7	0,4657	1,56
Error	12	223,3	18.5
Total	17	6090,9

From this result, we conclude that:

• Type of additive affects the settling time.
• Additive content affects the settling time.
• There is not interaction between these factors on settling time.