Hi! I’m back again. This week, I will be showing how to perform full factorial data analysis and fractional data analysis from a case study. Here is the case study I am tasked to analyse with.
Full factorial data analysis
I have done the analysis in an excel file and here is how I do it. Firstly, I organised the data into a factorial design table like this.
‘+’ indicates high level of factor
‘-’ indicates low level of factor
Secondly, I want to find out how changing the level of the factor affects the amount of pollutants discharged. Hence, I calculate the total value of y first when a certain factor is at a certain level and find the mean thereafter.
Next, I plotted a line graph to show the difference better.
Effect of factors and their rankings
For factor A:
When the concentration of coagulant added increases from 1% to 2%, the amount of pollutant discharged by the plant increases from 5.00 lb/day to 17.50 lb/day.
For factor B:
When the treatment temperature increases from 72oF to 100oF, the amount of pollutant discharged by the plant increases from 10.50 lb/day to 12.00 lb/day.
For factor C:
When the stirring speed increases from 200 rpm to 400 rpm, the amount of pollutant discharged by the plant increases from 4.00 lb/day to 18.50 lb/day.
Ranking of factors from largest effect to smallest effect on the amount of pollutant discharged:
Stirring speed (Factor C), Concentration of coagulant added (Factor A), Treatment temperature (Factor B)
Interaction effect (A x B)
At LOW B, average of LOW A = (5 + 4) / 2 = 4.5
At LOW B, average of HIGH A = (30 + 3) / 2 = 16.5
At LOW B, total effect of A = 16.5 - 4.5 = 12 (increase)
At HIGH B, average of LOW A = (6 + 5 ) / 2 = 5.5
At HIGH B, average of High A = (33 + 4) / 2 = 18.5
At HIGH B, total effect of A = 18.5 - 5.5 = 13 (increase)
There is a slight difference in the gradient of both lines, so there is very small interaction between A and B.
Interaction effect (A x C)
At LOW C, average of LOW A = (5 + 6) / 2 = 5.5
At LOW C, average of HIGH A = (30 + 33) / 2 = 31.5
At LOW C, total effect of A = 31.5 - 5.5 = 26 (increase)
At HIGH C, average of LOW A = (4 + 5) / 2 = 4.5
At HIGH C, average of HIGH A = (3 + 4) / 2 = 3.5
At HIGH C, total effect of A = 3.5 - 4.5 = -1 (decrease)
There is a notable difference in the gradient of both lines (one is + and the other is -), so there is significant interaction between A and C.
Interaction effect (B x C)
At LOW C, average of LOW B = (5 + 30) / 2 = 17.5
At LOW C, average of HIGH B = (6 + 33) / 2 = 19.5
At LOW C, total effect of B = 19.5 - 17.5 = 2 (increase)
At HIGH C, average of LOW B = (4 + 3) / 2 = 3.5
At HIGH C, average of HIGH C = (5 + 4) / 2 = 4.5
At HIGH C, total effect of C = 4.5 - 3.5 = 1 (increase)
There is a slight difference in the gradient of both lines, so there is very small interaction between B and C.
Conclusion
From this data analysis, we can conclude that factor B (treatment temperature) has very little impact on the amount of pollutants discharged due to its very little interaction with factors A and C and the small difference in the amount of pollutants discharged when the level of B changes. Factor A (concentration of coagulant) and C (stirring speed) have a significant impact on the amount of pollutants discharged due to the large interaction with each other and rather large difference in the amount of pollutants discharged when changing the levels of A and C respectively. So to reduce the amount of pollutant discharged into the environment as much as possible to reduce the harmful effects of global warming, all 3 factors must be at a low level.
Fractional factorial data analysis
For fractional factorial data analysis, I have to choose 4 experiments from the full factoria data that are orthogonal. So how do we know which 4 points are orthogonal?
From this diagram above, you can see that points 1,4,6,7 and 2,3,5,8 are orthogonal because each point is diagonal to each other on different faces of the cube. For this task, I will be choosing points 2,3,5,8 to perform fractional factorial data analysis. Again, I have done the task in an excel file.
Effects of factors and their rankings
For factor A:
When the concentration of coagulant added increases from 1% to 2%, the amount of pollutant discharged by the plant increases from 5.00 lb/day to 17.00 lb/day.
For factor B:
When the treatment temperature increases from 72oF to 100oF, the amount of pollutant discharged by the plant decreases from 17.00 lb/day to 5.00 lb/day.
For factor C:
When the stirring speed increases from 200 rpm to 400 rpm, the amount of pollutant discharged by the plant decreases from 18.00 lb/day to 4.00 lb/day.
Ranking of factors from largest effect to smallest effect on the amount of pollutant discharged:
Stirring speed (Factor C), Concentration of coagulant added (Factor A), Treatment temperature (Factor B)
*Factor A and C have the same effect on the amount of pollutant discharged*
Conclusion
From the data analysis, we can conclude that factor C has the most effect on the amount of pollutant discharged due to the large difference when changing its level from low to high. Factor A and B also play a part in the amount of pollutant discharged, but not as much as factor C. In order to reduce the amount of pollutant released into the environment as much as possible, factor A must be at a low level and factor B and C must be at a high level.
Link to excel file:
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