Elementary Ideas About Factorial Design

Engineering experiments often involve several factors and the job of an experimenter is to determine the influence of these factors on the output response of the system. Factorial experiment is an experimental strategy where factors are varied together instead of one at a time. In this article, I will just try to focus on some ideas about factorial design.

Let’s begin with some useful terms which are used in factorial design:

   
     2^2 factorial design: Two factors, each at two levels;  2^2 or 4 runs (here runs indicate total observations).
     2^3 factorial design: Three factors, each at two levels; 2^3 or 8 runs.

     Main effects: Individual effects of each factor.
     
     Interaction effects: Effects when the factors interact with each other.

     Replication: Repetition of the basic experiment. Suppose, four batteries are tested at each of three levels of temperature; so four replicates.  (We can say it also sample size; in this example, sample size is 4).

     Blocking: Blocking is used to reduce the variability from factors that may influence response but in which we are not directly interested. Suppose, batches of raw materials could be different due to supplier to supplier variability and we are not interested in this effect. So each batch of raw materials (considered as nuisance factors) would form a block.

     Analysis of Covariance: Blocking is used for controllable nuisance factors, but if the factors are uncontrollable then a procedure called analysis of covariance is used. If humidity cannot be controlled in any environment, it can be treated as covariate.

    Operating characteristic curve: This is used for selecting an appropriate sample size.

    Tukey’s test: This is used to test if two means are significantly different and thus all pair wise means are compared.

    Fisher LSD method: This is also used to compare means.

    Analysis of variance (ANOVA) table: Here sum of squares, degrees of freedom, mean square, test statistic for hypothesis are shown.

    Treatments: Different levels of a single factor.

    Contrast: Total effect of any factor.

    Levels: Levels can be said as just like different stages of any factor. An example is given below in this article.

   After determining which factors affect the response, the next step is to optimize the response that is to maximize or minimize the response. Response surface methodology(RSM) can be used for this optimization. 

 Two factor factorial design:

Suppose, there are two factors A and B. Factor A has 'a' levels and factor B has 'b' levels.There are 'n' replicates. These information can be represented like this:

Here, total runs or observations are abn.

To clarify what is levels of any factor, if we consider factor A is 'temperature' and data are collected at 15 degree, 20 degree and 50 degree to analyze the response; then these three stages of temperature are called three levels of this factor. So here, factor A has three levels

2^2 factorial design:

Suppose, we have two factors A and B; each at two levels. The levels are called ‘low’ and ‘high’ and are denoted as ‘-’ and ‘+’ respectively.

In this table, we can see the responses at different combinations for the factors. One replicate is used here. If there were more replicates, then cell total would be used for calculations.  

The interaction effect is denoted as AB. Effect of factor A is A and effect of factor B is B.

The four treatment combinations are represented by lowercase letters. The conventions are:

• When treatment combination of A at high level and B at low level, this is represented by a
• When treatment combination of A at low level and B at high level, this is represented by b
• Both factors are at high level, ab represents this
•  Both factors are at low level, (1) represents this

So we can see that, high level of any factor in the combination is denoted by corresponding lowercase letter and it also denotes that the other factor is at low level. 

Don't be confused with these lowercase letters and when we say factor A has 'a' levels. If it is said, factor A has 'a' levels; this does not mean that A is at high level. Here 'a' is just number of levels; like three levels of factor, a=3; five levels, a=5 etc. But in treatment combinations, this is the convention to use a to indicate that A is at high level and B is at low level. If we use the responses from above table, a will be 40; b will be 30; ab will be 12; (1) will be 20.

Algebraic signs for calculating effects in the 2^2 design are:

For example, to estimate A, the contrast is: – (1) +a – b + ab

If we use the responses given above, the main effect of factor A will be: 
                                                 
(here, n is replicate, k is factor)

2^3 factorial design:

Fractional factorial designs:

2^6 design requires 64 runs which make the calculations so difficult. So only a fraction of complete factorial experiment is done here, this is fractional factorial designs.

One-half fraction of  2^k design:

This design will contain runs of,
                                        

One-quarter fraction of  2^k design:

This design will contain runs of ,
                                                    

3^k factorial design:

This is k factors each at 3 levels. Three levels could be low, intermediate and high and are denoted as – 1, 0, + 1.

When factors are random:

In the above discussion, the factors are assumed to be fixed. Random factors are considered when there are large number of population levels and the factor levels are chosen at random. Suppose, there are several ovens in a working shop and three ovens are chosen at random for experiment. Here ovens will be treated as random factor.

When factors are both fixed and random:

Situations may arise when some factors would be fixed and others would be random.

Factorial design is a vast concept. Only some theoretical concepts are discussed here. Calculations can be done both by manually and with computer. Statistical software Minitab can be used to solve problems related with factorial design.

Reference: Design and Analysis of Experiments by Douglas C. Montgomery