The Factorial Experiment advantages and disadvantages over One-Factor-at-a-time Experiment.

### Factorial Experiment Advantages

**Required Less Number of Observations**

Let $A$ and $B$ be two factors. The information on a factor can be obtained by varying that factor and keeping the other factor fixed.

Effect of changing factor $A = A_2 B_1 – A_1B_1$

Effect of changing factor $B = A_1B_2 – A _1 B_1$

Three treatment combinations are used for two effects for error estimation we need two replicates so six observations are needed.

In the case of factorial experiments, one more combination $A_2B_2$ is utilized and we get:

Two estimates of $A$ are: $=A_2B_1 – A_1B_1 \qquad \text{ and } \qquad =A_2B_2 – A_1B_2$

Two estimates of $B$ are: $=A_1B_2 – A_1B_1 \qquad \text{ and } \qquad =A_2B_2 – A_2B_1$

Thus, by using four observations we can get the estimates of the same precision under a factorial experiment.

**More Relative Efficiency**

In the case of two factors the relative efficiency of factorial design to one-factor-at-a-time experimental design is $\frac{6}{4}=1.5$

This relative efficiency increases with the increase of the number of factors.**Necessary When Interaction is Present**

When using a one-factor-at-a-time design and the response of $A_1B_2$ and $A_2B_1$ is better than $A_1B_1$, an experimenter logically concludes that the response of $A_2B_2$ would be even better than $A_1B_1$. Whereas, $A_2B_2$ is available in factorial design.**Versatility**

Factorial designs are more versatile in computing effects. These designs provide a chance to estimate the effect of a factor at several levels of the other factor.

**Factorial Experiment Advantages in simple words**

The factorial Experiment Advantages without any statistical formula or symbol are:

- A factorial experiment is usually economical.
- All the experimental units are used in computing the main effects and interactions.
- The use of all treatment combinations makes the experiment more efficient and comprehensive.
- The interaction effects are easily estimated and tested through the usual analysis of variance.
- The experiment yields unbiased estimates of effects, which are of wider applicability.

### Factorial Experiments Disadvantages

- A factorial experiment requires an excessive amount of experimentation when there are several factors at several levels. For example, for 8 factors, each factor at 2 levels, there will be 256 combinations. Similarly, for 7 factors each at 3 levels, there will be 2187 combinations.
- A large number of combinations when used decrease the efficiency of the experiment. The experiment may be reduced to a manageable size by confounding some effects considered of little practical consequence.
- The experiment setup and the resulting statistical analysis are more complex.

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