Two Factor-Factorial Design of Experiment - R Program

Factorial design is an important experimental design whose design consist of two or more factors. This design is most used to examined treatment variations and can combined a series of into one, for efficiency. The simplest type of factorial design is The Two Factor Factorial Design. In these two levels of factorial design have level of factor A, and level of factor B, then each replication contains all ab treatment combinations.

Written by: Pawani Liyanaarachchi ,BSc (Hons) in Applied science (statistics), University of Sri Jayewardenepura

Consider y_ijk be the observed response when factor A is at the i^th level (i=1,2,...,a) and factor B is at the j^th level (j=1,2,...,b) for the k^th replicate (k=1,2,...,n)

In this tutorial, we will study followings.

What is Replication?
Design Block for the Two Factor Factorial Design
Statistical model for Two Factor Factorial Design
Hypothesis Testing for Two Factor Factorial Design
Example problem for Randomized Complete Block Design

What is Replication?

The mean of replication is an independent repeat of each combination. Replication has two important properties. They are,

Replication allows the experimenter to estimate of experimental error.
If the sample mean used to estimate the true mean response for one of the factor levels, replication permits the experimenter to obtain a more precise estimate of this parameter.

Design Block for the Two Factor Factorial Design

Design Block for the Two Factor Design of Experiment

Statistical model for Two Factor Factorial Design

Y_ijk = µ + τ_i + β_j + (τβ)_ij + ε_ijk

where,

i=1,2,...,a
j=1,2,...,b
k=1,2,...,n

Y_ijk: observation for i_th level of factor A, j_th level of factor B for the k_th replicate
µ: overall mean
τ_i: effect of i^th level of factor A
β_j: effect oh j^th level of factor B
(τβ)_ij: effect of interaction between τ_i and β_j
ε_ijk: random error

Notations for Anova Table

Partitioning the SS

The total corrected sum of squares can be written as:

SS_T = SS_A +SS_B + SS_AB + SS_E

Hypothesis Testing for Two Factor Factorial Design

When we are doing hypothesis testing in this design we have to consider about row, column, and interaction effect also. There we consider 3 hypotheses in this two-factorial design.

H₀ : τ₁ = τ₂ = ... = τ_a = 0 vs H₁ : at least one T_i ≠ 0
H₀ : β₁ = β₂ = ... = β_b = 0 vs H1 : at least one β_j ≠ 0
H₀ : (τβ)_ij = 0 for all i,j vs H₁ : at least one (τβ)_ij ≠ 0

ANOVA table

ANOVA - Computing Formulas

This sum of squares also contains SS_A and SS_B. Therefore, the second step is to compute SS_AB as,

SS_AB = SS_Subtotal - SS_A - SS_B

The Total sum of squares

Total error sum of squares

SS_E = SS_T - SS_A - SS_B - SS_AB or SS_E = SS_T - SS_Subtotal

Example problem for Two Factor Factorial Design Experiment

A university student who studies an engineering faculty tests 3 types of fertilizers (A,B,C) for a new plant at 3 soil types (1,2,3). Four plants are tested at each combination of fertilizer type and soil type, and all 36 tests are run in random order. He wants to find out what effects do fertilizer type and soil type have on the growth of a plant.

Two Factor Factorial Design of Experiment

According to the above theoretical model and also using R code (in R studio platform) we can solve this problem.

R Program for Two Factor Factorial Design Experiment

Download R Program for Two Factor Factorial Design of Experiment

## setup framework for 2 factors and response variable

fertilizer<- as.factor(rep(1:3, each=12))

soil_type <- as.factor(rep(c(1,2,3), each=4, times = 3))

growth <- c(130, 155, 74, 180, 34, 40, 80, 75, 20, 70, 82, 58, 150, 188, 159, 126, 136, 122, 106, 115, 25, 70, 58, 45, 138, 110, 168, 160, 174, 120, 150, 139, 96, 104, 82, 60)

y<- data.frame(fertilizer,soil_type,growth)

## apply a linear model

yfit <- lm(growth ~ fertilizer+soil_type+fertilizer*soil_type, y)

## compute the analysis of variance table

anova(yfit)

Results of after running two factor factorial design experiment anova table

R program for Two Factor Factorial Design Experiment Anova Table

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