# BSCI 1511L Statistics Manual: 3.3 Two factor ANOVA

Introduction to Biological Sciences lab, second semester

## 3.3 Two-factor ANOVA

Another type of complex experimental design that can be analyzed using ANOVA is an experiment where there are more than one factors.  For example, students wanted to investigate the effects of soap and an antibacterial agent (triclosan) on Escherichia coli bacteria.  They could have tested these factors in two separate experiments.  But testing both factors simultaneously is better.

Table 17. Design of bacterial exposure experiment

 factors no soap soap no triclosan 5 replicates 5 replicates triclosan 5 replicates 5 replicates

The students set up four treatments: one containing only buffer, one containing buffer with triclosan, one containing liquid soap diluted with buffer, and one containing buffer with both triclosan and soap.  They exposed 5 E. coli cultures to each of the four treatments, plated the surviving bacteria, and recorded plate counts (click to see data) for a two-factor ANOVA.  In this example, there are two null hypotheses, one for each factor:

There is no difference in the mean number of colonies for the soap treatments.

There is no difference in the mean number of colonies for the Triclosan treatments.

The results are shown in Table 18.

Table 18. Analysis of variance of effect of soap and triclosan on Escherichia coli bacteria

 Source Degrees of freedom Sum of squares Mean square F ratio P Soap 1 4704500 4704500 7.1 0.0164 Triclosan 1 264500 264500 0.40 0.536 Error 17 11280500 663559 Total 19 16249500

The interpretation of the results is similar to before.  The variability in plate counts caused by the soap was much larger than the variability caused by the triclosan, as reflected in the much larger mean square.  The value of P reflects this as well, with soap significant at P=0.0164 and triclosan not significant.  We would reject the first null hypothesis (regarding the effect of soap) and would fail to reject the second null hypothesis (regarding the effect of Triclosan).

The example above is the simplest case of a two factor ANOVA.  It is possible to have more than two levels for each factor.  For example, the students could have tested three kinds of antibacterial agents in addition to buffer and could have used two different brands of soap in addition to buffer.  However, it is necessary to have a treatment group for every combination of the two factors.  With four levels of antibacterial agent and three levels of soap, the students would need to have 12 treatment groups.  With 5 replicates per group, that would require 60 replicates, which would be very labor intensive.  Thus the more complicated your design becomes, the larger your experiment needs to be in order to have enough replication to conduct statistical tests that have enough power to detect differences.  For this reason, it is best to keep your design simple and test only levels and factors that you believe have a reasonable chance of being important.