Factorial Design (DOE)

To make a factory's product stronger, should you adjust the temperature first? What happens if you raise the pressure? And could raising temperature and pressure together have a bigger effect than changing each one on its own?

Factorial design is a way to combine several experimental conditions systematically and analyze the effect of each condition and how they influence one another. It is the most widely used design in the design of experiments (DOE).

Where one-way ANOVA looks at mean differences across a single factor, a factorial design handles two or more factors at once. What matters most is that it can detect interactions, where the effect of one factor depends on the level of another.

It is used in research where several conditions have to be tuned, such as manufacturing processes, food development, agricultural experiments, and service operations. In this article we'll walk through a single example covering the basic idea of factorial design, main effects and interactions, the run layout and results table, interpreting the ANOVA, how it differs from other analyses, and what to watch out for.

Factorial design is a way of designing and analyzing experiments that combines the levels of several experimental factors systematically to estimate the main effect of each factor and the interactions between factors. A full factorial design runs every combination, while a fractional factorial design runs only a selected subset.

If you vary only temperature while holding pressure and processing time fixed, you learn the effect of temperature only under those conditions. But if raising the temperature has a bigger effect at high pressure, that difference is a temperature × pressure interaction.

Because a factorial design observes combinations of conditions, it can analyze not only the average effect of each factor but also cases where performance improves or worsens under particular combinations. The catch is that the cost of the experiment grows quickly as the number of combinations increases.

① Choose the factors and levels

In this example we set two levels for each of three factors: temperature (160/180°C), pressure (20/30 bar), and processing time (20/30 min).

② Lay out every combination

With three two-level factors, there are 2³ = 8 experimental conditions. Replicating each combination with 3 independent production batches gives 24 runs.

③ Randomize the run order

Randomize the run order so it isn't confounded with equipment wear, operators, or dates, and record blocks (day, machine, etc.) separately where needed.

④ Estimate main effects and interactions

A model evaluates the mean difference for each factor, the two-factor interactions, and the three-factor interaction.

A hypothetical factory evaluated the tensile strength (MPa) of its product under different process conditions. Assume 24 independent batches were randomly assigned to the 8 conditions, with three measurements per condition. The values below are data constructed for learning purposes.

The example was generated with fixed rules to illustrate the effects between factors. It is not data that validates a real process or proves a causal effect.

Averaging over all levels of the other factors, the mean strength is 55 MPa at 160°C and 65 MPa at 180°C. So the main effect of temperature is +10 MPa.

A main effect here is a mean difference that depends on the levels included in the analysis and on how the other factors were laid out. Significance has to be tested separately, using the error from the replicate runs.

At 20 bar, raising the temperature from 160°C to 180°C increases strength from 54.5 MPa to 59.5 MPa, a gain of 5 MPa; at 30 bar it rises from 55.5 MPa to 70.5 MPa, a gain of 15 MPa.

If an interaction is large or statistically important, don't justify the process choice with a single main-effect number; it's better to check the simple effects within each condition.

In a balanced 2³ full factorial design you can estimate the main effects A, B, and C, the two-factor interactions AB, AC, and BC, and the three-factor interaction ABC. With 24 independent runs, the model effects have 7 degrees of freedom and the residual has 16.

The example focuses on understanding the effects by designing the data pattern. Real significance testing should be done only after reviewing model choice, residual assumptions, and the independence of the runs.

Experimental design isn't something you change after you see the results. The variables and levels, the number of replicates, and the randomization method must be set before the experiment.

① Check the residuals and variance

Review the residuals for normality, equal variance, outliers, and process drift. If needed, consider a transformation or a different model.

② Identify the independent experimental units

Measuring the same batch several times is not the same as producing several independent batches. Avoid pseudoreplication.

③ Don't ignore interactions

If interactions matter and you pick the best condition from main effects alone, you can make the wrong process decision.

④ Verify the best condition separately

The maximum in a factorial design is a result within the factor levels you chose this time. Verify the recommended condition with a separate confirmation run, and don't rush to generalize beyond the range of the experiment.

Effects in a factorial design can be tested with ANOVA or linear regression. Design of experiments is the framework for the design, and ANOVA is one of the tools for analyzing the resulting data.

In a paper or technical report, first present the factors and their levels, whether the design is full or fractional factorial, the number of replicates and the independent experimental units, the randomization and blocking setup, and the response variable. Then report the main-effect and interaction estimates, confidence intervals, F-tests, and p-values as appropriate.

In a real report, add the standard errors and confidence intervals of the estimates, appropriate test statistics, model diagnostics, and the results of confirmation runs.

If the response variable isn't continuous, you can estimate the factors and interactions with a suitable generalized linear model such as binomial or Poisson. The principles of experimental design stay the same, but the analysis model is chosen to fit the nature of the outcome data.

In the manufacturing example, we set temperature, pressure, and processing time at two levels each and compared all eight conditions. Temperature had the largest average main effect, but the gain in strength from raising the temperature differed between high and low pressure. So there are limits to choosing the best condition from the average effect of a single factor.

In real research you need to check whether the experiment had enough replication and randomization, and whether the residual and variance assumptions hold. Interpret main effects and interactions together, and even the condition that gave the best result in the experiment should not be declared the optimum for the whole process without a separate confirmation run.