Design of Experiments Midterm

Experiment

A structured set of coherent tests that are analyzed as a whole to gain an understanding of the process

Efficiency

A efficient experiment derives the required information with the least expenditure of resources

Examples of resources

Time, Material, Energy, Money

Stages of deigning an experiment

1. Gain knowledge of the process 

2. Develop clear goals and objectives

3. Choose a response variable

Precise (definition)

Repeatable but not necessarily accurate

Relevant (definition)

Related to the design objective

Statistical 

if it unlikely to have been caused by change

Meaningful

if it large enough to have a real world impact

Degrees of freedom

A way of counting the amount of data in an experiment. For every average of the data reduces the total DF by 1

Something is statistically different if

The data sets have a low likely hood that the data groups come from the same population

T test

T is the calculated difference represented in units of standard error. The greater the T the more likelihood they are from difference populations.

P Values do not tell you

1. The probability that Ho is true

2. The meaningfulness (or impact importance) of the observed effect

3. Note that P=0.05 is an arbitrary convention

Alpha Risk (type 1)

The probability of deciding the samples are different when they are the actually the same

Result: Supplier is scrapping good parts

Beta Risk (Type 2)

The probability of deciding the samples are the same when they are really different (More difficult to control)

Result: The company buys bad parts

F- Test

Compares the variance looking for a difference in variance that is larger than can be expected by random chance

F<=1 Variances are likely the same

F>>1 The variances are likely different

ANOVA (Analysis of Variance)

Variance between groups is the “signal” (Main effect)

Variance within groups “groups” (Error)

F stat for Anova

Signal/Noise

Requirements for ANOVA

1. Normally distributed

2. Independent

3. Homogenous 

Total Value for ANOVA (Sum of squares)

Calculated by summing the square of the difference between the values and the mean. It is also equal to the main effects and the error. 

What is the purpose of a 2 way ANOVA

Identify the factors that cause significant changes in the response variable

How an ANOVA avoids a large alpha risk

Variance is pooled

Error (definition)

Difference between data point and true mean

Residual (definition) 

Difference between data point and population mean

What do we if the null hypothesis is rejected in a 2 way ANOVA

If we have more than 2 groups we should run a post-hoc analysis to find which individual groups are actually different

Interpreting a 2 way ANOVA

1. Interaction: Always look at this first, if there is an interaction effect, we ignore the main effects and must move to step three

2. Main effect: If there is no significant interaction, we can look at the main effects. If nothing is significant, we accept the null hypothesis and stop

   1. Post Hoc. If there are significant main effects (or interaction) we want to explore, we can then run a post-hoc

Why should we be careful when we use a T-Test

Repeated tests run the risk of a type 1 error

What is factorial design

Factorial design is a way of setting up experiments so that all possible combinations of factors are tested. Full Factorial designs allow us to

1. Detect interaction

2. Asses main effects

3. Create a maximally efficient experimental setup

How Full Factorial Summary

Efficiency: The full factorial gives the same precision for main factors but with fewer total runs

Requires attention to detail: With fewer runs, the influence of each data point has a large effect on the results. Record keeping is critical! Label everything with the correct treatment combination not just the run number.

Assumptions for full factorials

1. Errors are homogenous

   1. Errors are Normally distributed

2. Errors are independent (Randomized from testing order)

ANOVA vs Factorial

Factorial: Full factorials describe a method of setting up or disning your experiment before you run it. it is especially helpful when you need to move beyond 2 factors and 2 levels.

ANOVA: When you want to analyze the results, use ANOVA to compare populations and find interactions

Summary of all tests discussed

T-Test: Allows us to directly compare2 groups. BUT must use judiciously, as repeated tests run the risk of a type 1 error

ANOVA: Allows us to compare multiple groups by po