BIVARIATE DATA: CORRELATION AND REGRESSION

1
BIVARIATE DATA CORRELATION AND REGRESSION

• Two variables of interest X, Y.
• GOAL Quantify association between X and Y
  correlation.
• Predict value of Y from the value of
  X regression.
• EXAMPLES (height, weight), (yrs. of education,
  salary), (hrs. of studying, exam score), (SAT
  score, GPA), (chemical reaction time,
  temperature), (rainfall, runoff volume), (demand,
  price), etc.
• BIVARIATE DATA PAIRS (X, Y) (x1, y1), (x2, y2),
  , (xn, yn).
• (xi, yi) ith observation, values of X and y on
  the ith subject.
• Correlation studies study type and amount of
  association between X and Y.
• Regression studies aim to predict Y from X by
  constructing a simple equation relating Y to X.

2
CORRELATION

• GRAPHICAL REPRESENTATION OF BIVARIATE DATA
  SCATTER
• PLOT plot observations (x1, y1), (x2, y2), ,
  (xn, yn) as points on the plane.
• Types of association/relationship between vars
  positive, negative, none.
• Positive association Two variables are
  positively associated if large values of one tend
  to be associated (occur) with large values of the
  other variable and small values of one tend to be
  associated with small values of the other
  variable.

Positive association
Example Height and weight are usually positively
associated
3
CORRELATION, contd.

• Negative association Two variables are
  negatively associated if large values of one tend
  to be associated (occur) with small values of the
  other. The variables tend to move in opposite
  directions.
• No association If there is no association, the
  points in the scatter plot show no pattern.

Negative association
No association
E.g. High demand often occurs with low price.
4
CORRELATION COEFFICIENT

• Measure of strength of association correlation
  coefficient rxy.
• Data (x1, y1), (x2, y2), , (xn, yn).
• Sample statistics sample means
• Sample standard deviations sx and sy.
• Sample correlation coefficient
• Correlation coefficient measures strength of
  LINEAR association.

5
PROPERTIES OF THE SAMPLE CORRELATION COEFFICIENT
rXY.

• rxygt 0 indicates positive association between X
  and Y.
• rxy lt 0 indicates negative association between X
  and Y.
• rxy 0 indicates no association between X and Y.
• -1 rxy 1, the closer rxy to 1, the
  stronger the relationship between X and Y.
• Computational formula for r given sample stats
• we can compute r as

6
CORRELATION COEFFICIENT AND ASSOCIATION
Moderate association
Strong association
r 0.75
r 0.9
r 0.5
Weak association
No association
Almost perfect association, but not linear, r
small.
r 0.28
r - 0.3
7
CORRELATION, CONTD.

• Correlation does not imply CAUSATION!
• Watch out for hidden (lurking) variables.
• Example. Study of fires. Xamount of damage, Y
  of firefighters.
• rXY0.85. The more firefighters, the more damage?
• Hidden variable Size of the fire.

8
EXAMPLE

• In a study of income an savings, data was
  collected from 10 households. Both savings and
  income are reported in thousands of in the
  following table.
• Find the
  correlation coefficient between income and
• savings.
• Solution
  Summary statistics Xincome, Ysavings
• Sxi 463
  , Sx2i 23533, Syi 27.4 , Sy2i 120.04 ,
• Sxi yi
  1564.4.
• 
  

income savings 25 0.5 28 0.0 35 0.8 39 1.6 44 1.8
48 3.1 52 4.3 65 4.6 55 3.5 72 7.2
There is a strong positive association between
family income and savings.
9
EXAMPLE, MINITAB
Correlations (Pearson) Correlation of income and
savings 0.963, P-Value 0.000