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STAT151 (74)
Chapter 3


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University of Alberta
Paul Cartledge

Ch. 3 – Intro to Correlation and Regression Ch. 2 deals with univariate data. This chapter, however, considers bivariate data and how two numerical variables are related. Methods of description are introduced here and formalized in Ch. 11. Terminology: x y Explanatory variable Response variable Independent variable Dependent variable Predictor variable Predicted variable Notation: - bivariate sample of size n: { (x1, 1 ), (2 ,2y ), …, nx n y ) } - sample means: x , y - sample std dev.: sx, sy Displaying relationships: Def’n: An association exists between two variables if a particular value for one variable is more likely to occur with certain values of the other variable. A scatterplot is a graphical display of two quantitative variables. - x-variable goes on the x-axis, y-variable on the y-axis - origin (0,0) may be included Look for: - form of relationship (i.e. any obvious pattern) - strength of relationship (i.e. closeness of fitting to a line) - direction of relationship (i.e. positive or negative association) - any unusual observations or outliers x y 1 1 2 2 4 1 3 2 (graph of above data used to discuss scatterplot traits further) Correlation: Def’n: Pearson’s Sample Correlation Coefficient r is given by n ⎛ ⎞⎛ ⎞ r = 1 ⎜ xi − x ⎟⎜y i− y ⎟ = 1 z z n 1 ∑i=⎝ sx ⎠⎜ sy ⎟ n − ∑ i i ⎝ ⎠ where z ixithe “standardized” observation for x andiz is thi “standardized” observation for y ior i = 1, …, n (example graphs of correlation drawn in class: 1. strong positive linear; 2. weak positive linear; 3. strong negative linear; 4. no pattern; 5. parabola; 6. exponential) Properties of r: • Can only be calculated for numerical data. • A measure of the LINEAR relationship between two variables. • -1 ≤ r ≤ 1 • The magnitude of r (or absolute value) measures the strength of the relationship: o If r = ± 1, then the points follow a straight line. o If r = 0, then the pattern of scatter suggest no linear relationship. • The sign of r indicates the nature of the relationship: o Positive association if r > 0, o Negative association if r < 0. • The two variables x and y play symmetric roles. • Location and scale invariance (unitless). • We can have r = 0, even when the data reveal a strong nonlinear relationship. o e.g. y = x 2 • Correlation does not imply causation (or vice versa). • Since r depends on the mean and std. dev., it is sensitive to outliers. 3.3 Intro to Simple Linear Regression Ex3.1) Suppose you had 4 variables for the Oilers roster: height, weight, jersey, age - which relationships might be valid? - how can we describe the relationship between any pair? - how do we use the description to make predictions? - how do we quantify errors in estimates and predictions? Def’n: The regression line predicts the value for the response variable y as a straight-line function of the value x of the explanatory variable. Equation for the regression line: ŷ = a + bx - a is the intercept: the height of the line at x = 0. - b is the slope: the amount by which y increases when x increases by 1 unit. - ŷ (“y-hat”) denotes the predicted value of y (or mean y for a given value of x). What about a new student who gets a mark of 80.1%? No observation so can we estimate the final mark based on the pattern of the other observations? Try and fit a line through the data and use it as a model for final percentage given midterm percentage; then, use the line to estimate (or, interpolate) the final percentage for a student that gets 8
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