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ANOVA continued and Intro to Regression

ANOVA continued and Intro to Regression . I231B Quantitative Methods. Agenda. Exploration and Inference revisited More ANOVA (anova_2factor.do) Basics of Regression (regress.do).

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ANOVA continued and Intro to Regression

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  1. ANOVA continued and Intro to Regression I231B Quantitative Methods

  2. Agenda • Exploration and Inference revisited • More ANOVA (anova_2factor.do) • Basics of Regression (regress.do)

  3. It is "well known" to be "logically unsound and practically misleading" to make inference as if a model is known to be true when it has, in fact, been selected from the same data to be used for estimation purposes. - Chris Chatfield in "Model Uncertainty, Data Mining and Statistical Inference", Journal of the Royal Statistical Society, Series A, 158 (1995), 419-486 (p 421)

  4. Never mix exploratory analysis with inferential modeling of the same variables in the same dataset. • Exploratory model building is when you hand-pick some variables of interest and keep adding/removing them until you find something that ‘works’. • Inferential models are specified in advance: there is an assumed model and you are testing whether it actually works with the current data.

  5. Basic Linear Regression (one iv and one dv)

  6. Regression versus Correlation • Correlation makes no assumption about one whether one variable is dependent on the other– only a measure of general association • Regression attempts to describe a dependent nature of one or more explanatory variables on a single dependent variable. Assumes one-way causal link between X and Y. • Thus, correlation is a measure of the strength of a relationship -1 to 1, while regression measures the exact nature of that relationship (e.g., the specific slope which is the change in Y given a change in X)

  7. Basic Linear Model • Yi = b0 + b1xi + ei. • X (and X-axis) is our independent variable(s) • Y (and Y-axis) is our dependent variable • b0 is a constant (y-intercept) • b1 is the slope (change in Y given a one-unit change in X) • e is the error term (residuals)

  8. Basic Linear Function

  9. Slope But...what happens if B is negative?

  10. Statistical Inference Using Least Squares • We obtain a sample statistic, b, which estimates the population parameter. • We also have the standard error for b • Uses standard t-distribution with n-2 degrees of freedom for hypothesis testing. Yi = b0 + b1xi + ei.

  11. Why Least Squares? • For any Y and X, there is one and only one line of best fit. The least squares regression equation minimizes the possible error between our observed values of Y and our predicted values of Y (often called y-hat).

  12. Data points and Regression • http://www.math.csusb.edu/faculty/stanton/m262/regress/regress.html

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