In this post I will be documenting about how I performed full and fractional factorial design data analysis on the Case study 2 that was assigned to me.
Contents of this post:
1. About Case study 2
2. Full factorial design data analysis
- Determining the effect of single factors and their rankings
- Interaction effects of the single factors
- Conclusion on the full factorial design data analysis
3. Fractional factorial design data analysis
- Determining the effect of single factors and their rankings
- Conclusion on the full factorial design data analysis
4. Link to Google Drive containing the Excel files used
About Case study 2
In a waste water treatment facility, a combination of coagulant chemicals, treatment temperature
and stirring speed were identified as a critical factor to treat the waste-water to produce clean
water. The clean water produced is recycled back into the main proses and at the same time reduce
the amount of pollutant discharged by the plant.
8 runs were performed and the data are shown below.
The response variable (y) is the amount of pollutant discharged (lb/day)
A = concentration of coagulant added, 1% and 2% by weight
B = treatment temperature,72°F and 100°F
C = Stirring speed, 200 rpm and 400 rpm
Factor | Level |
A: Concentration of coagulant added (% weight) | 1 | 2 |
B: Treatment temperature (°F) | 72 | 100 |
C: Stirring speed (rpm) | 200 | 400 |
Translating the above data into "+" and "-" for easy reference:
Run | A | B | C | Y (lb/day) |
1 | - | - | - | 5 |
2 | + | - | - | 30 |
3 | - | + | - | 6 |
4 | + | + | - | 33 |
5 | - | - | + | 4 |
6 | + | - | + | 3 |
7 | - | + | + | 5 |
8 | + | + | + | 4 |
Full factorial design data analysis
To determine the effect of single factors, their rankings and the interaction effects of the factors, graphs will be plotted using the values of the average of the different runs. Firstly, the amount of pollutant discharged (Y) is added into MS Excel as seen from the image below. Since only the average of the amount of pollutant discharged (Y) was given, only the "Ave." column is filled up.
Once all the values have been added in, the average between the different runs when the factors are at different levels can be found as seen from the image below.
Determining the effect of single factors and their rankings:With the average known, graphs can be plotted to determine the effect of single factors and their rankings. The values used to plot the graph:
In the graph above, amount of pollutant discharged is plotted against the factor levels. The line with a steeper slope or higher gradient values has the most significant effect amongst the other factors.
Conclusion:
- Ranking of factors (most significant to least significant):
1. C (stirring speed)
2. A (concentration of coagulation added)
3. B (treatment temperature)
- Factor C with the steepest slope has the most significant effect on the amount of pollution discharged. When the stirring speed decreases from 400 rpm to 200 rpm, the average amount of pollutant discharged increases from an average of 4 lb/day to 18.5 lb/day. Therefore, the slower the stirring speed, the larger the amount of pollutant discharged.
- Factor A with the less steep slope has the second most significant effect. When the concentration of coagulation added increases from 1 % weight to 2 % weight, the average amount of pollutant discharged increases from an average of 5 lb/day to 17.5 lb/day. Therefore, the higher the concentration of coagulation added, the larger the amount of pollutant discharged.
- Factor B with the least steep slope has the least significant effect. When the treatment temperature increases from 72°F to 100°F, the average amount of pollutant discharged increases from an average of 10.5 lb/day to 12 lb/day. Therefore, the higher the treatment temperature, the larger the amount of pollutant discharged.
Interaction effects of the single factors:
Additionally, with the average known, the interaction effects between factors can also be found through some math and graph plotting on MS Excel.
The interaction of factors that will be analysed are:
1. A x B (Concentration of coagulation added x Treatment temperature)
2. A x C (Concentration of coagulation added x Stirring speed)
3. B x C (Treatment temperature x Stirring speed)
Interaction of A x B :
At low B,
Average of low A= (5 + 4) /2 = 4.5
Average of high A= (30 + 3) /2 = 16.5
Total effect of A= 16.5 - 4.5 = 12 (increase)
At high B,
Average of low A= (6 + 5) /2 = 5.5
Average of high A= (33 + 4) /2 = 18.5
Total effect of A= 18.5 - 5.5 = 13 (increase)
Conclusion:
The gradients of the two lines are slightly different so there is a small interaction between both factors.
Interaction of A x C :
At low C,
Average of low A= (5 + 6) /2 = 5.5
Average of high A= (30 + 33) /2 = 31.5
Total effect of A= 31.5 - 5.5 = 26 (increase)
At high C,
Average of low A= (4 + 5) /2 = 4.5
Average of high A= (3 + 4) /2 = 3.5
Total effect of A= 3.5 - 4.5= -1 (decrease)
Values used to plot the graph:
Conclusion:
The gradients of both lines are different whereby one has a negative gradient while another has positive gradient. Therefore there is a significant interaction between both A and C.
Interaction of B x C :
At low C,
Average of low B= (5 + 30) /2 = 17.5
Average of high B= (6 + 33) /2 = 19.5
Total effect of B= 19.5 - 17.5 = 2 (increase)
At high C,
Average of low B= (4 + 3) /2 = 3.5
Average of high B= (5 + 4) /2 = 4.5
Total effect of B= 4.5 - 3.5 = 1 (increase)
Values used to plot the graph:
Conclusion:Although it looks like it is parallel, after adding a trendline to check the exact gradient from the equation, both lines are not parallel. Since they are not parallel, there is interaction between B and C. However, the interaction is very small.
Final conclusion on Full Factorial Data Analysis:
With the help of plotting graphs through the means found and a little bit of math, these are the results that was observed through all the data analysis.
Rankings of the single factors on their influence over the amount of pollutant discharged:
Ranking | Factor | Significance (influential) |
1 | Stirring speed, C | Most influential |
2 | Concentration of coagulation added, A | Less influential |
3 | Treatment temperature, B | Least influential |
These rankings were obtained by plotting the graph of amount of pollutant discharged against the factors levels of the single factors. On the graph plotted, the steeper slope seen is the factor that has the most influence over the amount of pollutant discharged. In this case, factor C had the steepest slope amongst the other factors, hence it being the most significant. Factor C is followed by A and B, with B being the least significant of them all.
Interactions between factors and how significant their interactions are:
Factors analyzed | Interaction |
A x B | Small interaction Gradient at low B: 13 Gradient at high B: 12 |
A x C | Significant interaction Gradient at low C: 26 Gradient at high C: -1 |
B x C | Small interaction Gradient at low C: 2 Gradient at high C: 1 |
These interactions were obtained from the graphs that were plotted with the different factors at different factor levels against each other. From the graphs that were produced, the gradients were considered (whether the lines are parallel or whether the gradients were positive or negative). With that, [A x C] was seen to have the most significance amongst the rest as there is a very big difference between the gradients. After [A x C], [A x B] and [B x C] both have small interactions due to their small difference between gradients .
Fractional factorial design data analysis
Fractional factorial is different from full factorial in a sense that there are less runs being conducted to collect data but it is still able to produce sufficient information on determining the factor effects. It is more efficient and requires less amount of resources. However, there is a risk of losing some information when using fractional factorial design.
To perform fractional factorial design data analysis, runs has to be selected. For the selection of runs, there should be a balance between the amount of low levels (-) and high levels (+) whereby they occur the same number of times and is orthogonal. Therefore, in this case, I have selected 4 runs that are balanced and are statistically orthogonal.
The chosen runs:
- Run 3
- Run 4
- Run 5
- Run 6
With the runs chosen, the values are added into MS Excel for analysis to determine the effect of single factors and their rankings.
Once the values are in, the average will be calculated. However this time, the average is found through dividing the total by 2 instead 4 like in the full factorial design data analysis.
Determing the effects of single factors and their rankings: With the average known, graphs can be plotted to determine the effect of single factors and their rankings. The values used to plot the graph:
In the graph above, amount of pollutant discharged is plotted against the factor levels. The line with a steeper slope or higher gradient values has the most significant effect amongst the other factors.
Conclusion:
- Ranking of factors (most significant to least significant):
1. C (stirring speed)
1. B (treatment temperature)
2. A (concentration of coagulation added)
Note: Factors C and B are tied for first place so there is no third place for the ranking this time
-Factor A has the least significant effect on the amount of pollutant discharged. When the concentration of coagulation added increases from 1 % weight to 2% weight, the amount of pollutant discharged increases from an average of 5 lb/day to 18 lb/day. Therefore, the higher the concentration of coagulation added, the larger the amount of pollutant discharged.
-Both factor B and C have a more significant effect on the amount of pollutant discharged. Factor B and C both are equally influential on the amount of pollutant discharged as they have the same gradient with the difference being that B has a positive gradient while C has a negative gradient.
- For factor B, when the treatment temperature increases from 72°F to 100°F, the average amount of pollutant discharged increases from an average of 3.5 lb/day to 19.5 lb/day. Therefore, the higher the treatment temperature, the larger the amount of pollutant discharged.
-For factor C, when the stirring speed increases from 200 rpm to 400 rpm, the average amount of pollutant discharged decreases from an average of 19.5 lb/day to 3.5 lb/day. Therefore, the faster the stirring speed, the smaller the amount of pollutant discharged.
Final conclusion on Fractional Factorial Design Data Analysis:
These are the results observed from analysis the data given and the graphs plotted.
Rankings of the single factors on their influence over the amount of pollutant discharged:
Ranking | Factor | Significance (influential) |
1 | Stirring speed, C | Most influential |
1 | Treatment temperature, B | Most influential |
2 | Concentration of coagulation added, A | Least influential |
These are the rankings that were found through the graph plotted on the amount of pollutant discharged against the factors levels of the single factors. In this case, factors B and C has the steeper slope as compared to factor A, hence factor B and C having a higher ranking than factor C. However, factors B and C are tied for first place due to their gradients being the same. Therefore, factors B and C have equal influence over the amount of pollutant discharged.
Comparison between Full factorial and Fractional factorial:
The rankings of the effects of single factors are surprisingly different for the full and fractional factorial. For full factorial, factor C is the most influential followed by factor A then factor B. However for fractional factorial, factors C and B are more influential than factor A. These could have occured due to loss of information through external conditions that were uncounted for.
Link to Google Drive containing the Excel files used
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