BA 275 Quantitative Business Methods Agenda Simple Linear Regression Inference for Regression Inference for Prediction

Regression Analysis A technique to examine the relationship between an outcome variable (dependent variable, Y) and a group of explanatory variables (independent variables, X1, X2, … Xk). The model allows us to understand (quantify) the effect of each X on Y. It also allows us to predict Y based on X1, X2, …. Xk.

Types of Relationship Linear Relationship Nonlinear Relationship Simple Linear Relationship Y = b0 + b1 X + e Multiple Linear Relationship Y = b0 + b1 X1 + b2 X2 + … + bk Xk + e Nonlinear Relationship Y = a0 exp(b1X+e) Y = b0 + b1 X1 + b2 X12 + e … etc. Will focus only on linear relationship.

Simple Linear Regression Model population True effect of X on Y Estimated effect of X on Y sample Key questions: 1. Does X have any effect on Y? 2. If yes, how large is the effect? 3. Given X, what is the estimated Y?

Least Squares Method Least squares line: It is a statistical procedure for finding the “best-fitting” straight line. It minimizes the sum of squares of the deviations of the observed values of Y from those predicted Sum of Squares is minimized. Bad fit.

Initial Analysis Summary statistics + Plots (e.g., histograms + scatter plots) + Correlations Things to look for Features of Data (e.g., data range, outliers) do not want to extrapolate outside data range because the relationship is unknown (or un-established). Summary statistics and graphs. Is the assumption of linearity appropriate?

Correlation r (rho): Population correlation (its value most likely is unknown.) r: Sample correlation (its value can be calculated from the sample.) Correlation is a measure of the strength of linear relationship. Correlation falls between –1 and 1. No linear relationship if correlation is close to 0. r = –1 –1 < r < 0 r = 0 0 < r < 1 r = 1 r = –1 –1 < r < 0 r = 0 0 < r < 1 r = 1

Correlation (r vs. r) Sample size P-value for H0: r = 0 Ha: r ≠ 0

Fitted Model: Least Squares Line b0 b1 Least squares line: estimated_Price = –15.1245 + 76.1745 Area.

Hypothesis Testing Key Q1: Does X have any effect on Y? b0 H0: b1 = 0 Ha: b1 ≠ 0 b1 SEb1 SEb0 Degrees of freedom = n – p – 1 p = # of independent variables used.

Interval Estimation Key Q2: If so, how large is the effect? b0 b1 SEb1 SEb0 Degrees of freedom = n – p – 1 p = # of independent variables used.

Prediction and Confidence Intervals Key Q3: Given X, what is the estimated Y? What is your estimated price of that 2000-sf house on the 9th street? Quick answer: estimated price = -15.1245 + 76.1745 (2) = 137.2245 What is the average price of a house that occupies 2000 sf? What is the difference?

Prediction and Confidence Intervals

Prediction and Confidence Intervals Prediction interval Confidence interval

Model Comparison: A Good Fit? SS = Sum of Squares = ???