Carillon Technologies Limited Designed Experiments

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Tools for Experimentation

• Analysis of Variance - ANOVA
  • Zero Factor
  • One Factor
  • Two Factor
• Experimental pattern
  • Factorial design
  • Fractional Factorial
  • Taguchi's Orthogonal Arrays
• Hypothesis Testing & Control Charts

What is ANOVA?

The term analysis of variance (ANOVA) is used in the field of study called designed experiments. In this field the goal is to try to maximize the amount of information that is collected when an experiment (production trial) is performed.

The technique was developed by Sir Ronald Fisher in the 1930's as a way to interpret the results from agricultural experiments.

The normal way in which things are usually done in experiments is to hold everything constant while only varying one item at a time. This is a most inefficient way to do things and not very representative of what happens in the real world.

In designed experimental approaches items are allowed to vary simultaneously and the respective data is gathered and analyzed. This analysis can not only detect differences in means, but effects of interactions.

As mentioned the area of ANOVA is a whole field of study in itself, and we will only look at one of the simpler types. One word of caution should be given before ever starting any data collection, the data gathering should be randomized allowing equal chance of occurrence. This is necessary to prevent any bias that might result in misinterpretation.

Steps and Zero Factor ANOVA - A good place to start to understand the concept

One Factor ANOVA

Two Factor ANOVA

Terminology & Definitions

• Response variable: A variable observed or measured in an experiment, sometimes called the dependent variable.
• Factor: Independent or causal variable, a variable that is deliberately varied or changed in a controlled manner in an experiment.
• Background variable: Noise variable or blocking variable, a variable that potentially can effect a response variable in an experiment, but is not of interest as a factor.
• Nuisance variable: an unknown variable that can effect a response variable in an experiment.
• Experimental units: The smallest division of material in an experiment.
• Blocks: Groups of experimental units treated similarly in an experimental design.
• Level: A given value or specific setting of a quantitative factor.
• Effect: the change in the response variable that occurs as a factor or background variable is changed.
• Replication: Repeating the experiment to gain insight into the possible interaction between variables.
• Randomization: Sequencing the order of the test in order to minimize the effects of nuisance variables.

Issues

• Planned Grouping -How do we address background variables (blocking)?
• Randomization
• Replication
  

Factorial Experiments

1. We can study the effects of several factors in the same set of experiments.
2. We can test for the effect of each factor at all levels of the other factors and can discover whether or not this effect changes as the other factors change.
3. We can test not only for the effects of the factors separately, but also for interactions.
4. Factorial experiments are more sensitive in the detection of small effects.

Example: Full Factorials

Purpose:

1. To pinpoint the most important variables - i.e. Red X, Pink X
2. To separate and quantify the main and interaction effects of the important variables.

Methodology:

• The power of the full factorial is that every one of the up to four chosen variables is tested with all levels of every other variable.
• Thus all the possible combinations of factors and levels are tested, allowing for the systematic separation and quantification of all main effects, as well as, interaction effects.

Procedure:

1. Select the factors to be investigated. (A,B,C,D)
2. Determine the levels for each factor (- and +)
3. Draw up the matrix or go to the worksheet.
4. Randomize the sequence of testing.
5. Run an experiment with each combination.
6. Repeat Steps 4 and 5 using another random order for the second test sequences.
7. Calculate the average of the two readings for each combination.
8. Determine the Effects by subtracting the (-) totals from the (+) totals, and then determining the average effects.
9. Perform analysis by use of End Counts and/or Normal Probability paper and by plotting main and interaction effects.

For example: we are interested in the effects of temperature, humidity, and accelerator concentration on the effectiveness of a reaction.

We have picked two levels for each factor

Temperature 70° & 80°

Humidities 20 & 60%

Concentration 2 & 5%

Two - level, three factor or (2)^3 = 8 runs

Hypothesis Testing

Link To Seven Steps of Hypothesis Testing