Correlation Coefficients Pearson’s Product Moment Correlation Coefficient  interval or ratio data only What about ordinal data?

Spearman’s Rank Correlation Coefficient r s = 1 -  di2di2 i=1 i=n n 3 - n 6

Spearman’s Rank Correlation Coefficient: Example

A Significance Test for r s SE r s = 1 n -1 t test = rsrs SE r s = r s n -1 df = n - 1

Spearman’s Rank Correlation Coefficient: Example

Pearson’s r - Assumptions 1.Interval or ratio scale data 2.Selected randomly 3.Linear 4.Joint bivariate normal distribution  S-Plus (qqnorm)

Spearman’s Rank Correlation Coefficient Ordinal data already in a ranked form Interval or ratio data convert it to rankings

Spearman’s Rank Correlation Coefficient TVDI (x) Rank (x) Theta (y) Rank (y) Difference (d i )

A Significance Test for r s

 S-Plus

TVDI (x) Theta (y)

Correlation  Direction & Strength We might wish to go a little further Rate of change Predictability Correlation  Regression

Deterministic  perfect knowledge Probabilistic  estimate  not with absolute accuracy (or certainty) Two Sorts of Bivariate Relationships

Travel  at a constant speed Deterministic  time spent driving vs. distance traveled A Deterministic Relationship s = s 0 + vt s: distance traveled s 0 : initial distance v: speed t: time traveled time (t) distance (s) slope (v) intercept (s 0 ) Truly deterministic  rare

More often  probabilistic e.g., ages vs. heights (2 – 20 yrs) A Probabilistic Relationship age (years) height (meters) Good relationship Unpredictability or error

Sampling and Regression Our expectation (less than perfect) Collecting data  measurement errors  height Other factors (not accounted for in the model)  plant growth vs. T

Simple vs. Multiple Regression Simple linear regression  y  x Multiple linear regression  y  x 1, x 2, … x n

Model y = a + bx + e Simple Linear Regression x: independent variable y: dependent variable b: slope a: intercept e: error term x (independent) y (dependent) b a error: 

Scatterplot  fitting a line Fitting a Line to a Set of Points x (independent) y (dependent) Least squares method Minimize the error term e

Sampling and Regression Sampled data  model y = a + bx + e Attempt to estimate a “true” regression line y =  +  x +  Multiple samples  several similar regression lines  the population regression line

Minimize the error term e The line of best fit  ŷ = a + b Least Squares Method y ŷ = a + bx ŷ (y - ŷ)

Estimates and Residuals Errors e = y – ŷ Residuals  Underestimate  Overestimate

Errors (residuals)  e = (y - ŷ) Overall error  Simply sum these error terms  0  Square the differences and then sum them up to create a useful estimate Minimizing the Error Term SSE =   (y - ŷ) 2 i = 1 n 

Minimizing the SSE   (y - ŷ) 2 i = 1 n min a,b n   (y i - a - bx i ) 2 i = 1 min a,b =

Least squares method  Finding Regression Coefficients   (x i - x) (y i - y) i = 1 n b =   (x i - x) 2 i = 1 n a = y - bx

Interpreting Slope (b) Slope of the line (b  the change in y due to a unit change in x b > 0 b < 0

Regression Slope and Correlation  (x i - x)(y i - y) i=1 i=n (n - 1) s X s Y r =   (x i - x) (y i - y) i = 1 n b =   (x i - x) 2 i = 1 n b = r sysy sxsx