Design of Experiment

What does that mean?

DOE is a branch of applied statistics that deals with planning, conducting, analysing, and interpreting controlled tests to evaluate the factors that control the value of a parameter or group of parameters. DOE is a powerful data collection and analysis tool that can be used in a variety of experimental situations.

Full Factorial V.S. Fractional Factorial

- Both are design analysis

Full factorial consists of all possible runs, while Fractional factorial selects few possible runs only. Full factorial can provide comprehensive data to identify the effect on each variable. Nonetheless, fractional factorial have also sufficient data to do so, though it is more effective, information may be slightly less accurate/ reliable due to the lack of several runs.

Interaction Effect

An interaction effect happens when one explanatory variable interacts with another explanatory variable on a response variable

Practical!!

During our practical, we were tasked to learn more about how the 3 factors namely, arm length, start angle and stop angle would affect the flying distance of a ball using a catapult at 2 different levels + and - using what we learned during tutorial, DOE!! The objective of this is to find out which is most significant effect on the distance of the ball. In order to do this we had to use Full and Fractional Factorial to determine it. 

Full Factorial Design

Darius, Zhi Yao and I were tasked to collect data for the full factorial design (64 runs). I was the one recording the data, Zhi Yao setting the catapult, while Darius was measuring the distance the ball travelled.

From the graph shown, When arm length increases from 26.8cm to 32cm, the flying distance of projectile decreases from 134.04cm to 109.89cm. 

When start angle increases from 7 degrees to 18 degrees, the flying distance of projectile decreases from 133.10cm to 110.83cm. 

When stop angle increases from 55 degrees to 90 degrees, the flying distance of projectile decreases from 144.30cm to 99.63cm. 

According to the gradients of the 3 lines, the ranking of the significance of the factor on the flying distance of the projectile can be identified. 

Most significant: Stop Angle Significant: Arm Length 

Least significant: Start Angle

Interaction effect:

Interaction between A and B:

For the interaction of A x B, since both gradients are both negative and only differ by a small margin, we can say that the interaction between them is low and not very significant.

Interaction between A and C:

The gradients of both lines are different (one is positive one is negative). Therefore, there is a significant interaction effect between A and C.

Interaction between B and C:

There is a significant interaction between B and C as their gradients differ by quite a lot. However, both gradients are still negative, so the effect of interaction is not as strong as A x C.

Overall, for full fractional analysis, the interaction with the strongest effect on the inflight distance is A x C followed by B x C and lastly with A x B with the least effect.

Effect	Interaction
Most	A x C
Moderate	B x C
Least	A x B

Fractional Factorial:

Interaction between A and B:

The gradient of both lines is different (one is positive one is negative). Therefore, there is a significant interaction between A and B.

Interaction between A and C:

There is a significant interaction between A and C as their gradients are different (one is positive one is negative). However, their difference in gradient is not as big of a difference as the interaction between A and B.

Interaction between B and C:

The gradients of both lines are similar (both negative) but are not parallel to each other. Thus, they still have an interaction with each other, but it is not significant.

Overall, the interaction with the most significant effect is A x B, followed by A x C and B x C.

Effect	Interaction
Most	A x B
Moderate	A x C
Least	B X C

Conclusion:

Interaction between A & B

Full factorial showed negative gradients for both high and low levels, showing that the interaction between A & B is low. However, for fractional factorial there is 1 positive and negative gradient which shows a significant interaction. In conclusion, for A & B, full factorial and fractional factorial showed distinct differences. Thus, not being able to justify a concrete interact effect for A x B.

Interaction between A & C

Full factorial showed 2 different gradient 1 positive and negative each, showing a significant effect on C. Likewise, for fractional factorial, there is also a significant effect as there are both positive and negative gradient shown. This means that full factorial and fractional factorial have the same outcome and it is justified that A and C show significant interaction effect.

Interaction between B & C

There is an interaction between B & C since the 2 gradients differ by quite a lot, however both lines showed a negative gradient which meant it is less impactful compared to A & C. Likewise, for fractional factorial, both gradients are negative, just different in steepness.

From the above comparison, since interaction between A & C, and B & C have the same outcome for both full and fractional factorial, it can be concluded that A & C and B & C have more accurate data. A & C would be more significant compared to B & C and A & B, since A & B’s fractional factorial showed only difference in gradient.

Most to Least: A & C, B & C, A & B

Links to Excel Files

Full Factorial Design

Fractional Factorial Design

Individual Case Study

Full Factorial:

Factor A= diameter
Factor B= microwaving time
Factor C= power

When diameter of the bowl increases, the bullets formed increased from 0.575g to 1.475g.

When the microwaving time increases, the bullets formed decreased from 2g to 0.9g.

When the power increases, the bullets formed decreased from 2.35g to 0.55g.

Power being the steepest means that the effect is the most prominent. Microwaving time being the 2nd steepest, hence the 2nd most prominent effect.

Hence:

in order of effect on bullets, 

Power > Microwaving Time > Diameter of the bowl

Interaction between the factors (Full factorial design data)

Link: Full Factorial

A x B

At LOW B, Average of low A = (0.7+3.1)/2= 1.9 (-)

At LOW B, Average of high A = (3.5+0.7)/2 = 2.1 (+)

At LOW B, total effect of A = (2.1 - 1.9)= 0.2 (increase) 

At HIGH B, Average of low A = (1.6 + 0.5)/2 = 1.05 (-)

At HIGH B, Average of high A = (1.2 + 0.3)/2 = 0.75 (+)

At HIGH B, total effect of A = (0.75 - 1.05) = -0.3 (decrease) 

The gradient of both lines are different, positive and negative gradient. Therefore, there’s a significant interaction between A and B.

A x C

At LOW C, Average of low A = (1.6+3.1)/2= 2.35 (-)

At LOW C, Average of high A = (3.5+1.2)/2 = 2.35 (-)

At LOW C, total effect of A = (2.35 - 2.35)= 0 (no effect)

At HIGH C, Average of low A = (0.7+0.5)/2= 0.6 (-)

At HIGH C, Average of high A = (0.7+0.3)/2 = 0.5 (+)

At HIGH C, total effect of A = (0.5 - 0.6) = -0.1 (decrease) 

The gradient of both lines is different, positive and negative gradient. Therefore, there’s an interaction between A and C, but the interaction is small. 

B x C

At LOW C, Average of low B = (3.5 + 3.1)/2 = 3.3 (-)

At LOW C, Average of high B = (1.6 + 1.2)/2 = 1.4 (+)

At LOW C, total effect of B = (1.4 - 3.3) = -1.9 (decrease) 

At HIGH C, Average of low B = (0.7 + 0.7)/2 = 0.7 (-)

At HIGH C, Average of high B = (0.3 + 0.5)/2 = 0.4 (+)

At HIGH C, total effect of B = (0.4 - 0.7) = -0.3 (decrease) 

The gradient of both lines are different, positive and negative gradient. Therefore, there’s a significant interaction between B and C. 

Link: Interaction

Frictional factorial

From the graph above, power of the microwave has the steepest gradient which means that it is the most significant factor. Followed by the microwaving time and finally diameter.

Hence:

in order of effect on bullets:

Power > Microwaving Time > Diameter of bowl

Link: Fractional Factorial