Lecture 2 zLast: Sections 1.1-1.3 (Read these) zToday: Quick Review of sections 1.4, 1.6, 1.7 and 1.9 with examples zWill not cover section 1.5 zNext Day:

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Presentation transcript:

Lecture 2 zLast: Sections (Read these) zToday: Quick Review of sections 1.4, 1.6, 1.7 and 1.9 with examples zWill not cover section 1.5 zNext Day: zReview of Regression and Analysis of Variance (ANOVA)… Saturday at 12:00 in Frieze B166

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

Linear Regression Model zHave N observations (y 1, y 2,…, y N ) zHave p covariates (regressors or explanatory variables) zModel: zWhere the error terms are independent, identically distributed (iid) normal random variables

Linear Regression Model zIn matrix notation, zWhere, z

zThe experiment is run in order to estimate the model zLeast squares estimator of the regression coefficients: zVariance of least squares estimators:

zNotice, estimator and variance are functions of X zDoes this indicate a strategy for choosing the design points or treatments?

zHow would one test to see if a particular explanatory variable is statistically significant? zWhat does this imply about our choice of treatments?

Residuals zCan verify some model assumptions by looking at residuals from model fit zDo this using plots

One-Way ANOVA zExample - Comparing battery lifetimes zIs there a difference in battery life by brand? zFour brands of batteries when used in a one of those 'radio controlled' cars for kids (Schwarz, 1995). zA 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. zHere is the raw data: zWhich battery brand would you buy? Why?

zModel: zIn matrix form: zInterpretation:

zEstimation of model parameters:

zConstraints:

zAssumptions: zThis experiment is an example of a completely randomized experiment

ANOVA Table

Hypothesis zWant to test: zTest statistic:

Multiple Comparisons zWhich treatments are different? zWill use Tukey method:

General Procedure: zDesign experiment (collect data) zPlot data to gain intuition and check assumptions zFit model zResiduals zTest hypothesis zMultiple comparisons (if necessary)