Confounding in DOE lecture - 34

 

Introduction to Confounding

Lecture - 34

Introduction

The principal objective of blocking is to reduced heterogeneity among the experimental units and the objective of block is vain when the block size is sufficiently large, the purpose of blocking is futile and the efficiency of the particular design is limited. In factorial experiment all possible treatment combinations are accommodate in a block and the number of replications is always equal the number of blocks. The factorial experiment is efficient technique for small number of factors and its level.

e.g. let we have 2^3 factorial experiment. The number of treatment combination is 8 and listed as: (1), a, b, ab, c, ac, bc, abc. The arrangement in a block is given by;

Block

(1)

a

b

ab

c

ac

bc

abc

But when the number of factors and its levels are increasing the treatment combination is also increases. It required a large block size to accommodate all treatment combination in a block. The large size of block is not able to maintain homogeneity in the experimental units and function of block is questionable as well as objectionable. In such situation two designs are suitable either to choose a connected design like BIBD or unconnected technique like confounding to address these problems. If all treatment combinations are important and desired to estimate then BIBD is suitable on the other hand if high order is not important and not desired to estimate, then confounding technique is suitable.

Confounding Factor  

A confounding factor is factor that influences the dependent and independent variable and in the presence of such factor the precise estimate is not possible. Furthermore, the interpretation is also not possible.

Example:

Let two factors (A: throat infection, B: Sevier fever) of Covid -19 and consider two levels of each factor i.e. 0 & 1 (0: No, 1: Yes). Then we have  possible symptoms combinations. If the two factors are at high levels, then there is risk. Several studies shown that a third factor C (C: Past Chronic diseases) in the presence of C develop risk. This factor is confounded in the study.

Confounding

A factorial experiment, in which the number of factors or the number of levels of factors in the experiment is large, then it would require a large block size to accommodate all treatment combinations. As the size of block is large, the inherent variability tends to increase and the efficiency of design is reduced. To overcome this difficulty; the concept of confounding was introduced to reduce the inherent variability by sacrificing information on some high order interactions by dividing the experiment into more than one block. The confounding technique is used, when n < t.

Where:

n: block size

t: Number of treatment combinations.

Thus, confounding is a design technique for arranging a complete factorial experiment in blocks, where block size is smaller than the number of treatment combination in one replicate and used to reduce block size by sacrificing information on the high order interaction. This high order interaction deemed less important and can be obtained from block difference.

To explain the concept of confounding: Let us consider 2^3 factorial experiment (i.e. 3 factors each with two levels). The  possible treatment combinations are: 

(1), a, b, ab, c, ac, bc, abc.

Suppose we divide the 8 treatment combinations into two blocks. The contrast which estimating ABC as:

Block – I: consist of that treatment combination for which contrast estimating ABC appears as at low level ( - ive sign) and block – II: consist of those treatment combinations for which the contrast estimating ABC appears at upper levels (+ive sign).

Thus, contrast ABC = Block II – Block I, i.e. the block difference or simply we can say that ABC is confound with blocks. That’s the contrast between treatment combinations in two blocks estimate the interaction ABC. Therefore, it is not possible to separate the true interaction from block differences and thus the interaction ABC is confounded with blocks.

Alternatively, the 2^3 treatment combinations  can be expressed as:

(1), a, b, ab, c, ac, bc, abc.

These treatment combinations can be represented by X1, X2, X3, where Xi take the values 0 and 1.

Block – I: consist treatment combinations for which

 Block – II: consist treatment combinations for which

The interaction ABC is estimated by the contrasts between two groups.

TYPES of CONFOUNDING

There are two types of confounding.

1.      Complete Confounding

2.      Partial Confounding

Complete Confounding

 In case of multiple replications, if the same interaction is confounded in all replications, then such confounding is called complete confounding.

Partial Confounding

In case of multiple replications, if the different interactions are confounded in different replications, then such confounding is called partial confounding.

Construction of Blocks of 2^k Factorial Experiment

Suppose in a 2^k factorial experiment and it is desired to confound a single effect. Confounding a single effect divided 2^k treatment combinations into 2 blocks of 2 ^(k-1) treatment combinations in which each block. Confounding the effect A, B, C,…, K. with blocks. The treatment combinations of two blocks can be obtained by solving the following two equations:

Let we have 2^3 factorial experiment and we want to confound ABC into 2 blocks, we solve the following equations:

The block which satisfying the equation X1 + X2 + X3 = 0 mod 2, is called principal block or key block. This block has the property that any elements except (0, 0, …, 0) may be generated by adding two other elements in this block mod 2.

For example: 110 +101 = 011 mod 2

Treatment combinations in the other block may be generated by adding mode 2, any element in the new block  to the element in the key block. The treatment combination in block II in the above example can be generated by adding 100 to every element in the key element.

Analysis

Let we have enough material to run 3 replications of 2^3 factorial experiment. The ABC interactions are confounded with blocks. We required 6 blocks to replicate each treatment combination of 2^3 factorial experiment. 

The experimental plan after complete randomization might be appear as:

The outline of the ANOVA:

SV	d. f
Block	6  - 1
A	1
B	1
AB	1
C	1
AC	1
BC	1
Error	12
Total	23

The treatment combination of 2^3 factorial experiment is carried out in 3 replications in 6 blocks of size with ABC is confounded.

Note that ABC is not included in the ANOVA table because it is the same as one of the block contrasts, i.e. ABC is completely confounded with blocks.

Method for Construction of Blocks of 2^k Factorial Experiment

 Consider 2^3 factorial experiment and it is desired to confound a single effect in 2 blocks. Divided 2^(k-1) treatment combinations in two blocks of treatment combinations in which each block.

That’s 

Confounding the effect A, B, C,…, K. with blocks. The treatment combinations of two blocks can be obtained by solving the following linear combination:

Where:

xi: the i^th level of factor i^th factor.

αi : The exponent appearing on the i^th factor in the effect is to be confounded.

e.g. in 2^3  factorial experiment ABC is confounded in blocks.

Block I consist the treatment combinations:

Block II consist the treatment combinations:

Example: If ABC is confounded in four replications of  factorial experiment, then make two columns of ANOVA.

Solution: 2^3 factorial experiment consists three factors (A, B, C). Confounded with ABC in 

The linear combination is:

The two equations:

ANOVA table:

SV	df
Block	8 – 1
A	1
B	1
AB	1
C	1
AC	1
BC	1
ABC	1
Error	18
T0tal	31

Example: An experiment as conducted to study the effect of 3 factors in 4 replications is given below:

Test the hypothesis about main effects and interaction effects.

Solution:

The main effects and interaction effects are obtained by using contrast method.

Where:

Now AB interaction is computed as contrast total from the columns in which AB is not confounded.

Similarly, interaction AC is computed as contrast total from the columns in which AC is not confounded.

BC interaction is computed as:

ABC interaction is computed as:

ANOVA Table:

Even – Odd Rules

The even – odd rules is used to determine which set of treatment combination is confounded. A treatment combination will be confounded, if the treatment effect must have even number or no number of the common factor with elements with the principle block.

Example: A 2^3 factorial experiment with factors A, B, and C is arranged in two blocks of four plots each as follow:

Block I	(1)	ab	ac	bc
Block II	a	b	c	abc

The treatment contrast that is confounded with block is:

(a). AB        (b). AC        (c). BC         (d). ABC

Solution: We need to check all the above contrast with the treatment combination of block I:

Example: 

A chemical product is produced in a pressure vessel. A factorial experiment is carried out in the pilot plant to study the factors thought to influence the filtration rate of this product. The four factors are temperature (A), pressure (B), concentration of formaldehyde (C), and stirring rate (D). each factor is present at two levels. the design matrix and the response data obtained from a single replicate of the 2^4 experiment are shown in the table. The 16 runs are made in a random order. The process engineer is interested in maximizing the filtration rate.

	A	B	C	D	Filtratin Rate
1	-1	-1	-1	-1	45
a	1	-1	-1	-1	71
b	-1	1	-1	-1	48
ab	1	1	-1	-1	65
c	-1	-1	1	-1	68
ac	1	-1	1	-1	60
bc	-1	1	1	-1	80
abc	1	1	1	-1	65
d	-1	-1	-1	1	43
ad	1	-1	-1	1	100
bd	-1	1	-1	1	45
abd	1	1	-1	1	104
cd	-1	-1	1	1	75
acd	1	-1	1	1	86
bcd	-1	1	1	1	70
abcd	1	1	1	1	96

In addition, he wants to introduce a block effect so that the utility of blocking can be demonstrated. Suppose that when he selects the two blocks batches of raw material required to run the experiment, one of them is much poorer quality and, as a result all responses will be 20 units lower of this material batch than in the other.

Solution:

The experiment is 2^4 = 16 treatment combination and cannot be run in a single replicate (one batch of raw material). We divide 2^(k-1) means divide into two blocks. 8 units per block. The experiment is ½ 2^k. The highest order of interaction (in this case is ABCD) will be confounded with blocks.

Divided into two blocks by confounding ABCD. 

Block – I consists + ive sign and Block – II consists – ive sign of ABCD.

Block I	Yield	Block II	Yield
1	45	a	71
ab	65	b	48
ac	60	c	68
bc	80	abc	65
ad	100	d	43
bd	45	abd	104
cd	75	acd	86
abcd	96	bcd	70
	566		555	1121

The block effect and ABCD is confounded, and the block I material is less effect by 20.

To develop SEE, combine all minor effects.

The effect of block and the interaction effect of ABCD is indistinguishable.

ANOVA Table

SV	df	SS	MS	F	Ftab
Block (ABCD)	1	1387.5625	1387.5625
A	1	1870.5625	1870.5625	95.09	5.12
C	1	390.0623	390.0623	19.83	5.12
D	1	855.5625	855.5625	43.49	5.12
AC	1	1314.0625	1314.0625	66.80	5.12
AD	1	1105.5625	1105.5625	56.20	5.12
Error	9	177	19.66
Total	15

The factors temperature (A), concentration of formaldehyde (C), stirring rate (D),  the interaction of temperature &  concentration of formaldehyde and the interaction of temperature & stirring rate has a significant effect on the production of a pressure vessel. 

• Read More: Confounding in more two blocks