Week 13: DOE - Inter-Factor Interactions

Welcome back to the blog!

For this week, we explored deeper into DOE and observed the interactions between each factor.  You may ask, "Why would we need this?" Using these interactions, we can find which two factors' particular level can maximise our response variable. 

We were tasked to do DOE on a case study regarding a wastewater experiment. Our response variable is the amount of pollutant released (lb/day). The 3 factors are:

Factor A - Concentration of coagulant added. high (+) = 2%wt, low (-) = 1%wt

Factor B - Treatment Temperature. high (+) = 100 deg Fahrenheit, low (-) = 72 deg Fahrenheit

Factor C - Stirring Speed. high (+) = 400rpm, low (-) = 200rpm

We first did full factorial method. The data is listed as below:

Using this data, we calculated the averages for each factor's level, as shown below:

Now, with those averages, we plot a graph for the high and low level averages of each factor. This is how it looks:

As seen from the graphs as well as the total difference calculation, factor C has the highest influence with factor A close behind. Factor B has the least influence on pollutant discharge amount. Tl;dr: C > A > B

We also see the effect of each factor through its gradient. For a higher pollutant discharge amount, one must increase factor A and B, while decreasing factor C.

We then move on to doing the factor interactions. This means that for each factor level, one must find the average for another factor's high and low level. For example, to compare factors A and B, one must find the averages of the runs containing only high A and high B and also find the averages of runs containing only high A and low B. This would create one graph. Another graph must be made using the average of runs containing only low A and high B and runs containing only low A and low B.

...This is quite confusing, so here's what the table looks on Excel:

Using these data averages, we then create graphs (again, yay) for the interactions. This is how it looks for interactions between A and B, A and C, and lastly B and C:

The significance of each interaction is seen if they intersect each other on the graph. As seen from the graphs, the interaction between A and C is the most significant. Although the two other interactions don't seem to intersect from the data, they would intersect soon. However, both their influences on the amount of pollutant discharged would be comparatively much lower than if A and C are the only ones changed. Tl;dr: AxC > BxC = AxC

With this, we can conclude that factor A and C are exceptionally important in affecting the pollutant discharged amount. This is also supported by the individual influence of factor A and C where they are both quite significant as compared to factor B.

After that, we move to fractional factorial method. The data points that I used are these:

This is statisically good as all 3 factors appear equally (i.e at most twice) and there is no run bias.

As usual, we find the average of each factor's level, shown below:

Then, we graph these averages to find their significance:

From the graph and calculations, we can see that factor C is the most influential. Factor A and B have about equal amount of influence to the response variable. Tl;dr: C > A = B

Moving on to interactions, we did the same thing and placed the data points to something that can be visualised better:

We then graph the same interactions, A to B, A to C, and B to C:

Through these graphs, we find that the graphs of all 3 interactions intersect. However, I would think that the interaction between A and B are much more significant compared to the other two as the interaction is "further away" from the data points used to plot. In that sense, the two other interactions seem to be similar. Tl;dr: AxB > BxC = AxC

To conclude, through individual comparison, C would be the most influential. Although the interaction with A and B is more significant, the  interaction with B and C further makes the change more significant. Hence, factor C and B are the most important to look further into than factor A.

Through this case study, I found out the use of two factors' interactions to establish which factors to investigate more on. I also saw the drastic conclusion difference between full and fractional factorial methods. In future, I hope to use the same method to create more efficient testing and delve into more important factors as compared to other factors.

To end off, here's the Excel file for all your reading pleasure: https://drive.google.com/uc?export=download&id=1D9C1ksNTsEBtlGirEehw5tDwDOCOAmPt

Thank you for reading!