Factorial Experiments





 We want to test how does the regularity of feeding with a liquid fertilizer affects the growth of tomato plants?” Thus, 20 plants were selected be fed at 5 different rates. The total 100 plants should be assigned at random to the 5 feeding rates.


Experimental Design???


If we introduce another tomato plant, which has, been genetically modified from our original strain. We repeat the same set-up that we used for the first plant with the second.


We are varying two factors: the level of feeding and type of tomato.


Now it is a two-factor design (two-way design).





If we have set up the experiment identically for the two strains, we have a fully-crossed factored (fully crossed) design.


We have all possible combinations of the two factors. We have individuals of both strains at all feeding rates.


If we could only apply three of the feeding rates to the new strain, but still used all five for our original strain, then our design would be incomplete design.


The statistical analysis of fully crossed design is easy to do, whereas analyzing incomplete designs is significantly more difficult.


Avoid incomplete designs whenever possible


An n-factor or n-way design varies n different independent factors and measures response to these manipulations.





If we want to examine the effects of both diet and exercise regime on dog’s health, then it would be a two-factor design. If we consider three different diets and four different exercise regimes then there are three levels to factor “diet” and four levels to factor “exercise”.


It would be a 3X4 two-factor (way) design


For the tomato example:

It is a fully cross-factored, two factor design with five levels for factor One and two for factor two.


5X2 fully crossed two way design


Thus we have 10 treatment groups, and as long as we have the same number of plants in each treatment and more than one plant in each treatment then we have a fully replicated and balanced design.




Give example of three factor or four factor design.


If we want to investigate whether factor A is influenced by independent factors B and C. if the value of A is affected by the value of B when the value of C is kept constant, then we can say that there is a main effect due to factor B. Similarly if A is affected by C when B is kept constant, then there is a main effect due to C. If the effect of B on A is affected by the value of C, or equivalently if the effect of C on A is affected by the value of B, then there is an interaction between the factors B and C.


Why not do two separate one-way design for the two different plants?

The answer

1)     Two-way design allows us to investigate the main effects in a single experiment (whether growth rate increases with feeding rate in both species and whether one species grows faster then the other when both are given the same feeding rate).

2)     It allows us to examine the interaction effect (whether the two species differ in the effect that feeding rate has on their growth).