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Lecture 2

Lecture 2. Last: Sections 1.1-1.3 (Read these) Today: Quick Review of sections 1.4, 1.6, 1.7 and 1.9 with examples Will not cover section 1.5 Next Day: 2.1-2.3 Review of Regression and Analysis of Variance (ANOVA)… Saturday at 12:00 in Frieze B166. Experiment.

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Lecture 2

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  1. Lecture 2 • Last: Sections 1.1-1.3 (Read these) • Today: Quick Review of sections 1.4, 1.6, 1.7 and 1.9 with examples • Will not cover section 1.5 • Next Day: 2.1-2.3 • Review of Regression and Analysis of Variance (ANOVA)… Saturday at 12:00 in Frieze B166

  2. Experiment • In an experiment, the experimenter adjusts the settings of input factors to observe the impact on the system • Better understanding of how the factors impact the system allows the experimenter predict future values or optimize the process • Should select experiment treatments so that data is easy (as possible) to analyze

  3. Linear Regression Model • Have N observations (y1 , y2,…, yN) • Have p covariates (regressors or explanatory variables) • Model: • Where the error terms are independent, identically distributed (iid) normal random variables

  4. Linear Regression Model • In matrix notation, • Where,

  5. The experiment is run in order to estimate the model • Least squares estimator of the regression coefficients: • Variance of least squares estimators:

  6. Notice, estimator and variance are functions of X • Does this indicate a strategy for choosing the design points or treatments?

  7. How would one test to see if a particular explanatory variable is statistically significant? • What does this imply about our choice of treatments?

  8. Residuals • Can verify some model assumptions by looking at residuals from model fit • Do this using plots

  9. One-Way ANOVA • Example - Comparing battery lifetimes • Is there a difference in battery life by brand? • Four brands of batteries when used in a one of those 'radio controlled' cars for kids (Schwarz, 1995). • A selection of brands was bought, and used in random order. The total time the car was able to be used was recorded to the nearest 1/2 hour. • Here is the raw data: • Which battery brand would you buy? Why?

  10. Model: • In matrix form: • Interpretation:

  11. Estimation of model parameters:

  12. Constraints:

  13. Assumptions: • This experiment is an example of a completely randomized experiment

  14. ANOVA Table

  15. Hypothesis • Want to test: • Test statistic:

  16. Multiple Comparisons • Which treatments are different? • Will use Tukey method:

  17. General Procedure: • Design experiment (collect data) • Plot data to gain intuition and check assumptions • Fit model • Residuals • Test hypothesis • Multiple comparisons (if necessary)

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