Intro to JMP
There are two different experimental designs:
In the simplest design of experiments (DOE), there are to factors and two levels.
We will be using the same factors throughout: Xa abd Xb. The two levels considered will high level (+1) and low level (-1) with the mean being zero.
We also have the output, y, which is a function of Xa and Xb (y = Xa, Xb)
There are three effects on the results of the experiments, namely: Xa, Xb and Xab.
There are four experiments that could be conducted by taking different signs for the factors Xa and Xb
Therefore, we can deduce from the data above that the output, y, has four parameters associated with it. The equation for y can now be written as:
Y = P0 + PaXa + PbXb + PabXaXb
As an example, we will study the parameters of the output, y.
In this example, we will define y as the surface finish. Y is considered to be a measured value on a scale of 1-10. Xa is defined to be the speed of the machine and Xb as the depth of the surface.
Below is a sample of an outcome of an experiment:
Sign of Xa - +
Since Y = P0 + PaXa + PbXb + PabXaXb, we can use the results shown above to get the following four equations:
If the four equations above are added, we get 4P0 = 15.2 and thus, we can find P0 to be 3.8
Below is a sample of a graph that could be used to determine the effect of moving the parameters Xa and Xb up, down, to the left or to the right on the output, y.
It is also used to study the effect of parameters on how the reference value, P0, is going to change the process and outcome when values of factors Xa and Xb are changed.
JMP is used to perform a designed experiment rather than one chosen at random. It is a useful tool used by many companies to design their experiments. Experimentation is important since it is the fundamental of the scientific method. Experiments are run a few times and some factors are varied in order to get the response of interest. Some factors are varied while others are kept constant. The factors that do not participate in the experiment are kept constant. Throughout the design of the experiments, factors are changed to see how they affect the results of the experiment.
In order to learn about a system, the inductive method is used, which involves learning about a system by changing a few factors and observing the results. While designing experiments, trial and error is conducted; however, when experiments are not conducted properly, the results will not be accurate. Therefore, when the effects of factors on the response of experiments are studied, proper control of conditions is necessary. That is in order to ensure that the results of the experiments are a result of the change in variables and not as a result of uncontrolled conditions.
JMP is software used to describe factors and responses of an experiment as well as to create the design of the experiment. The catalog of JMP can be seen below:
In order to learn about a system, the inductive method is used, which involves learning about a system by changing a few factors and observing the results. While designing experiments, trial and error is conducted; however, when experiments are not conducted properly, the results will not be accurate. Therefore, when the effects of factors on the response of experiments are studied, proper control of conditions is necessary. That is in order to ensure that the results of the experiments are a result of the change in variables and not as a result of uncontrolled conditions.
JMP is software used to describe factors and responses of an experiment as well as to create the design of the experiment. The catalog of JMP can be seen below:
There are three types of design experiments:
1. Screening design: if the number of far tots affecting the response of the experiment is greater than 5, screening design is used to screen for the factors that affect the response more than others.
2. Response surface design: this design is used when the number of factors is less than five and the number of levels is greater than two.
3. Full factorial design: this type of design is used to get how a factor affects the response of an experiment.
A simple example showing how JMP is used for designing experiments is as follows:
Acme Piñata discovers that it's piñatas break easily and so it wants to know what factors affect the peeling strength of its flour paste. The strength of the past is basically how well the pieces of paper are glued to one another and this resisting peeling.
In the design of this experiment, nine factors were considered to check which factors affect the peeling strength of the paste and how these factors could be changed to optimize the peeling strength of the flour paste.
Batches of flour paste were prepared to determine the effect of the following nine factors on peeling strength:
To start designing the experiment, we must first decide on the type of design to be used. Since the number of factors is greater than 5 (9), screening design will be used. Next the response as well as factors must be specified on JMP as follows:
1. Screening design: if the number of far tots affecting the response of the experiment is greater than 5, screening design is used to screen for the factors that affect the response more than others.
2. Response surface design: this design is used when the number of factors is less than five and the number of levels is greater than two.
3. Full factorial design: this type of design is used to get how a factor affects the response of an experiment.
A simple example showing how JMP is used for designing experiments is as follows:
Acme Piñata discovers that it's piñatas break easily and so it wants to know what factors affect the peeling strength of its flour paste. The strength of the past is basically how well the pieces of paper are glued to one another and this resisting peeling.
In the design of this experiment, nine factors were considered to check which factors affect the peeling strength of the paste and how these factors could be changed to optimize the peeling strength of the flour paste.
Batches of flour paste were prepared to determine the effect of the following nine factors on peeling strength:
- Flour :1/8 cup of white unbleached flour or 1/8 cup of whole wheat flour
- Sifted : flour was sifted or not sifted
- Type : water-based paste or milk-based paste
- Temp : mixed when liquid was cool or when liquid was warm
- Salt : formula had a dash of salt or had no salt One of the constraints when designing experiments is the budget. In this example, we will restrict the number of times the experiment is run to 16.
- Liquid : 4 teaspoons of liquid or 5 teaspoons of liquid
- Clamp : pasted pieces were tightly clamped together or not clamped during drying
- Sugar : formula contained 1/4 teaspoon or no sugar
- Coat : whether the amount of paste applied was thin or thick
To start designing the experiment, we must first decide on the type of design to be used. Since the number of factors is greater than 5 (9), screening design will be used. Next the response as well as factors must be specified on JMP as follows:
Next the actor constraints are defined and since the number of runs is limited to 16 while the number of factors is large, only the main effects of the factors will be used. The number of runs is then specified to 16.
The " make design" button is then clicked and JMP finds the optimal design for the experiment by searching through a mathematical surface.
The " make design" button is then clicked and JMP finds the optimal design for the experiment by searching through a mathematical surface.
JMP uses balanced equations by using statistical data to design experiments and do the 9 factors are divided equally among all 16 runs.
Experimental design is a long process that requires lots of preparation. There are two types of experimental design; factorial and fractional factorial design with the difference being that fractional factorial design allows the number of experiments to be reduced as long as the higher order factors are of little importance.
Before proceeding, a few definitions must be laid out:
Factors: They are quantities that affect the outcome of experiments.
Levels: They are the values of factors for which data is gathered. Every value, from low to high is considered a level. An example of levels would be speed. The number of values in an experiment should be equal to the number of levels.
Replication: This process refers to the gathering of results repeatedly for a set of levels for all factors. Replication is used to get statistical models of what is being studied.
We shall focus mostly on factors as well as factorial design in this blog.
There are two types of factors:
Experimental design is a long process that requires lots of preparation. There are two types of experimental design; factorial and fractional factorial design with the difference being that fractional factorial design allows the number of experiments to be reduced as long as the higher order factors are of little importance.
Before proceeding, a few definitions must be laid out:
Factors: They are quantities that affect the outcome of experiments.
Levels: They are the values of factors for which data is gathered. Every value, from low to high is considered a level. An example of levels would be speed. The number of values in an experiment should be equal to the number of levels.
Replication: This process refers to the gathering of results repeatedly for a set of levels for all factors. Replication is used to get statistical models of what is being studied.
We shall focus mostly on factors as well as factorial design in this blog.
There are two types of factors:
- Independent factors may have independent effect on the outcome of experiment. Independent factors are considered to be the more important type of factors.
- Interacting factors are factors that interact to produce an effect on the outcome of the experiment. Interacting factors are the less important factors.
There are two different experimental designs:
- Simple design: In the simple design, one factor is varied at a time. Moreover, there are a large number of experiments needed. It is also assumed that the factors do not interact.
- Full Factorial design: Every combination of factor levels is tested and all interactions are captured. Like the simple design, a large number of experiments is needed.
In the simplest design of experiments (DOE), there are to factors and two levels.
We will be using the same factors throughout: Xa abd Xb. The two levels considered will high level (+1) and low level (-1) with the mean being zero.
We also have the output, y, which is a function of Xa and Xb (y = Xa, Xb)
There are three effects on the results of the experiments, namely: Xa, Xb and Xab.
There are four experiments that could be conducted by taking different signs for the factors Xa and Xb
| Experiment # | Sign of Xa | Sign of Xb |
| 1 | - (low) | - (low) |
| 2 | - (low) | + (high) |
| 3 | + (high) | - (low) |
| 4 | + (high) | + (high) |
The summations of Xa, Xb, XaXb, Xa^2 and Xb^2 between 1 & 4 are shown below
Σ (Xa)^2 = Σ (Xb)^2 = 4
Σ (Xa) = Σ (Xb) = Σ (XaXb) = 0
The different level combinations shown above for each of the four experiments will be used to find the product XaXb as follows:
Σ (Xa)^2 = Σ (Xb)^2 = 4
Σ (Xa) = Σ (Xb) = Σ (XaXb) = 0
The different level combinations shown above for each of the four experiments will be used to find the product XaXb as follows:
| Sign of Xa | Sign of Xb | Sign of XaXb |
| - (low) | - (low) | - (low) |
| - (low) | + (high) | - (low) |
| + (high) | - (low) | + (high) |
| + (high) | + (high) | + (high) |
Therefore, we can deduce from the data above that the output, y, has four parameters associated with it. The equation for y can now be written as:
Y = P0 + PaXa + PbXb + PabXaXb
As an example, we will study the parameters of the output, y.
In this example, we will define y as the surface finish. Y is considered to be a measured value on a scale of 1-10. Xa is defined to be the speed of the machine and Xb as the depth of the surface.
Below is a sample of an outcome of an experiment:
Sign of Xb
Sign of Xa - +
- 2 8.2
+ 1.5 3.5Since Y = P0 + PaXa + PbXb + PabXaXb, we can use the results shown above to get the following four equations:
- 2 = P0 - Pa - Pb + Pab
- 1.5 = P0 - Pa + Pb - Pab
- 8.2 = P0 + Pa - Pb- Pab
- 3.5 = P0 + Pa+ Pb + Pab
If the four equations above are added, we get 4P0 = 15.2 and thus, we can find P0 to be 3.8
Below is a sample of a graph that could be used to determine the effect of moving the parameters Xa and Xb up, down, to the left or to the right on the output, y.
It is also used to study the effect of parameters on how the reference value, P0, is going to change the process and outcome when values of factors Xa and Xb are changed.
Next, the above linear system is solved and the following values are obtained for the parameters:
P0 = 3.8
Pa = 2.05
Pb = -1.3
Pab = -1.05
The main question here is: “What is the importance of each factor?”
To find the answer to this question, a term known as variance is calculated.
Variance of y^2 = (Σ (yi – y)^2) /(2^2 -1)
To solve the above equation, we first solve for yi – y.
yi – y = P0 + Pa + Pb + Pab – P0
Since Y i= P0 + Pa+ Pb+ Pab and the mean y = 4P0/4 = P0
Substituting, variance of y^2 = (4(Pa+ Pb+ Pab)^2)/3 = (Pa)^2 + 2PaPb + 2PaPab + (Pb)^2 + 2PbPab + (Pab)^2
σ y^2= (Pa)^2 + (Pb)^2 + (Pab)^2 = 5.6 + 2.25 + 1.47 = 9.32
Now the percentage effect of each of the factors is calculates:
A = 5.6/9.32 * 100 = 60.1 %
B = 2.25/9.32 * 100 = 24.1 %
C = 1.47/9.32 * 100 = 15.8 %
Therefore, we conclude that A&B produce the main effect and AB produces the interacting effect.
P0 = 3.8
Pa = 2.05
Pb = -1.3
Pab = -1.05
The main question here is: “What is the importance of each factor?”
To find the answer to this question, a term known as variance is calculated.
Variance of y^2 = (Σ (yi – y)^2) /(2^2 -1)
To solve the above equation, we first solve for yi – y.
yi – y = P0 + Pa + Pb + Pab – P0
Since Y i= P0 + Pa+ Pb+ Pab and the mean y = 4P0/4 = P0
Substituting, variance of y^2 = (4(Pa+ Pb+ Pab)^2)/3 = (Pa)^2 + 2PaPb + 2PaPab + (Pb)^2 + 2PbPab + (Pab)^2
σ y^2= (Pa)^2 + (Pb)^2 + (Pab)^2 = 5.6 + 2.25 + 1.47 = 9.32
Now the percentage effect of each of the factors is calculates:
A = 5.6/9.32 * 100 = 60.1 %
B = 2.25/9.32 * 100 = 24.1 %
C = 1.47/9.32 * 100 = 15.8 %
Therefore, we conclude that A&B produce the main effect and AB produces the interacting effect.
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