Outline

1
Outline

• Correlation and Covariance
• Bivariate Correlation Coefficient
• Types of Correlation
• Correlation Coefficient Formula
• Correlation Coefficient Computation
• Short-cut Formula
• Linear Function (Intercept and Slope)

2
Correlation and Covariance

• It asks how two variables are related
• When x changes, how does y change?
• Underlying information is covariance
• Cov(x,y)E(x-xbar)(y-ybar)
• Cov(x,y)Cov(y,x)
• Cov(x,x)Var(x), variance is a special type of
  covariance (covariance of a variable and itself)

3
Bivariate Correlation Coefficient

• (Karl Pearson product moment) correlation
  coefficient
• Bivariate correlation coefficient (BCC) for two
  interval/ratio variables
• Differentiated from Spearmans rank correlation
  coefficient (nonparametric)
• Differentiated from partial correlation
  coefficient that controls the impact of other
  variables
• No causal relationship imposed. X?Y or Y?X
• BCC is used for prediction

4
Bivariate Correlation Coefficient

• BCC ranges from -1 to 1 (So does Gamma ?)
• Covariance component can be negative
• means positive relationship when x increases 1
  unit, y increases r unit
• 0 means no relationship.
• - means negative relationship when x increases 1
  unit, y decreases r unit.
• http//noppa5.pc.helsinki.fi/koe/corr/cor7.html

5
Positive relationship
6
Negative relationship
7
No relationship
8
Correlation Coefficient

• Ratio of the covariance component of x and y to
  the square root of variance components of x and y

9
Correlation Coefficient (short-cut)
Textbook suggests a short-cut formula below but
it is not recommended.
10
Illustration example 10-2, p.526
No x y (x-xbar) (y-ybar) (x-xbar)2 (y-ybar)2 (x-xbar)(y-ybar)
1 43 128 -14.5 -8.5 210.25 72.25 123.25
2 48 120 -9.5 -16.5 90.25 272.25 156.75
3 56 135 -1.5 -1.5 2.25 2.25 2.25
4 61 143 3.5 6.5 12.25 42.25 22.75
5 67 141 9.5 4.5 90.25 20.25 42.75
6 70 152 12.5 15.5 156.25 240.25 193.75

Sum 345 819     561.5 649.5 541.5
Mean 57.5 137          
SSxx SSyy SPxy

Correlation coefficient Correlation coefficient Correlation coefficient Correlation coefficient 0.8967      
11
Hypothesis Test

• How reliable is a correlation coefficient?
• r is a random variable drawn from the sample ?
  is its corresponding parameter
• H0 ? 0, Ha ? ? 0
• TS follows the t distribution with dfn-2
• If H0 is not rejected, r is not reliable
  regardless of its magnitude (? 0)

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Illustration Example 10-3, p.529

• Step 1. H0 ? 0, Ha ? ? 0
• Step 2. a.05, df4 (6-2), CV2.776
• Step 3. TS4.059, r.897
• Step 4. TSgtCV, reject H0 at the .05 level
• Step 5. ? ? 0

13
Linear function

• A function transforms input into output in its
  own way
• Ex ysquare_root(x). Whey you put x (input) into
  the funciton square_root(), you will get y
  (output).
• Linear function consists of a intercept and
  linear combinations of variables and their slops.
  Y a bX cX2
• Slopes are constant

14
Intercept and Slope of a function

• A linear model Y a b X
• Dependent variable Y to be explained
• Independent variable X that explains Y
• Y-Intercept a the coordinate of the point at
  which the line intersects Y axis.
• Slope b the change of dependent variable Y per
  unit change in independent variable X

15
Illustration