Blocking and Confounding

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Introduction
• Experimentation is an essential part of the scientific method.
• Most problems in science require observation of the system at work and

experimentation to explicate information about how the system works.
• In many situations, experimenters are required to study effects two or more

factors, wherein in each complete trial or replicate of the experiment all possible
combinations of the levels of the factors are investigated.

• This situation is referred as crossed experiments or factorial experiments.
• Unfortunately, this sometimes limited by the available resources.
• This chapter covers the ways to handle situations when it is impossible to

perform all of the runs of the factorial experiments under homogenous
conditions.

• In this connection the concept of blocking and confounding is discussed in 2k

factorial.
• So, within 2k factorial design, how blocking is done when confounding arises and

how to deal such situations is described with examples.

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BLOCKING AND CONFOUNDING SYSTEM FOR TWO-LEVEL FACTORIALS

• Blocking
➢In the statistical theory of the design of experiments (DoE), blocking is nothing

but arranging experimental units in groups (blocks) that are similar to one
another. Usually.

➢A blocking factor is a source of variability which is not a primary interest of
the experimenter.

➢The objective of blocking is to reduce the impact of uncontrolled variables on
the experimental responses.

➢The examples of using blocking in experiments include experiments on
different machines, different operators or multiple times over the duration of
study.

➢Experimenter’s interest is to allocate treatments across these nuisance
factors.

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Why blocking?

• Usually it may not be possible for experimenter to directly compare
treatments because the differences cannot be separated from the
effect of the nuisance factors.

• Here blocking technique deals with controlling nuisance variables.
• Sometimes, it is impossible to perform all 2 factorial experiments

under homogeneous condition.
• In such cases blocking technique is used to make the treatments

equally effective across many situations.
• For example, when we do not have sufficient resources such as raw

materials or the operators or when we deliberately create different
heterogeneous situations, blocking is used to handle such situations.

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What is Blocking?
• A block is defined by a homogenous large unit, including, raw materials, areas,

places, plants animals, humans, etc.
• where samples or experimental units drawn are considered identical twins, but

independent.
• If there are n replicates in factorial design then in each set of homogeneous

conditions, first block is defined and then each replicate is run in one of the blocks.
• Likewise, in each set of non-homogeneous conditions, first block is defined and then

each replicate is run in one of the blocks.
• When there are n replicates of the design, then each replicate is a block
• Each replicate is run in one of the blocks (time periods, batches of raw material, etc.).
• The number of runs within the block is randomized.
• When experimenter is in doubt for any factor (variable), this factor is blocked. In

order to block any of the nuisance variables, randomize variables as much as possible
to help balance out unknown nuisance effects.

• The known nuisance factors which cannot be controlled are measured.

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• Even if there is not an obvious nuisance factor, a better way is to conduct an
experiment in blocks, just to protect against the loss of data or situations
where the complete experiment can not be finished.

• For example, consider a 2² factorial design wherein on complete settings all
the factorial runs require 4 experiments to be conducted.

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• Suppose if experimenter want to replicate
this experiment 3 times and have no
enough raw materials to go for 4 x 3 = 12
experimental runs under this situation.

• In this case, experimenter may take
batches of raw material as one batch with
4 experiments with all the complete
settings, second batch second replication,
third batch third replication etc.

• This means, if there are n replicates in 2
factorial experiment then each set of
homogeneous conditions defines a block
and each replicate is run in one of the
blocks.

• It is obvious to take randomized runs in
each block

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Example:
• Let us see an example to understand how the blocking is used in experiments.

• Consider an experiment to study effect of the concentration of the reactant and amount of
catalyst on the conversion (yield) in a chemical process.

• In this case there are two factors namely;

– concentration of reactant (A) and

– amount of catalyst (B).

• Suppose the experimenter is interested in knowing the effect of these two factors on the
conversion of reactants as response variable, the yield (Y).

• The objective of the experiment is to determine if adjustment to either of these two factors
would increase the yield. To do this, the experimenter has to adjust either A or B or both in such
a manner that the yield will increase.

• To increase the yield let the reactant concentration (factor A) has two levels 15% and 25%.

• Thus, the reactant concentration is a factor A and 15% and 25% are its two levels of interest.

• The amount of catalyst is factor B with the high level of 2 pounds and the low level of 1 pound.

• The experiment is replicated three times (n = 3), so there will be 12 runs in all.
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• The presentation in Table for above
mentioned chemical process where 12
runs are required in 2² factorial
designs.

• Suppose that only four experimental
trials can be made from single batch of
raw materials, then four independent
settings would be (1), a, b, ab.

• But in this case the experimenter has
raw material in such a quantity that he
can conduct only four experiments of a
single batch.

• If experimenter wish to go for 3
replications he would require at least 3
batches of raw material or it may
happen that he may not be or he will
sacrifice some of the a factorial
experiments per batch.

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• This procedure is adequate to
conduct 4 experimental runs for
every batches of raw material used.
The table shows three blocks
namely: block 1 = (1), a, b, ab, block
2 (1), a, b, ab and block 3 (1), a, b,
ab.

• Thus, in 2² factorial designs overall 4
treatment combinations are possible
using 3 batches of raw materials.
The raw material represents the
blocks. This is called blocking in
experiment with 2 factorial design.

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Graphical Presentation of Blocking in 2k Factorial Design
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• In many experiments when an investigator compares a set of treatments, often
there is the possibility of one or more sources of variability in the experimental
measurements.

• It is important to consider such variability, if any, during the design stage of the
experiment.

• For example, we might be investigating four different excipients for formulation
using say two different machines, which would be expected to display different
degrees of efficiency with the formulation. Or we might not be able to run all of
the experimental combinations in one session so we would want to take into
account systematic differences that are due to experiments in the various sessions.

• The simplest situation is where we would have a single factor that we want to
control for in the experiment.

• In a complete block design all treatments occurs the same number of times in
every block, usually one replicate of all treatments per block.

• There will be situations where the number of treatments is too large for all of
them to be included in every block of the design.

• In such situations an incomplete block design would be used for running an
experiment.