Chapter 5 Linear Models
Linear Models are studied intensively because:
•Easiest to understand and analyze
•Relationships are often linear
•Variables with non-linear relationship can often be transformed
into linear relationship through an appropriate transformation
•Even when a relationship is non-linear, a linear model may
provide an accurate approximation for a limited range of values.
5 Linear Models
• Least Square Line:
• Let X and Y be two quantitative variables.
• Let (x1, y1), (x2, y2), …, (xn, yn) be data
points collected on n individuals
• Goal: Find a linear model in the form
Y=a+bX
that represents the given data set
5 Linear Models
5 Linear Models
5 Linear Models
5 Linear Models
Formulas for computing b and a
• b = Sxy/Sxx
• a = average(y) – b average(x)
• Sxy = Sum(xy) –Sum(x)Sum(y)/n
• Sxx = Sum(x^2) –(Sum(x))^2/n
5 Linear Models
• Remarks:
• 1. Use linear regression if the scatter plot looks
reasonably straight
• 2. Check for outliers which could drastically influence
regression lines
• 3. If data seem to cluster in scatter plot, look data more
carefully
• 4. Large magnitude of slope indicates steeper line
• 5. Negative slope shows negative association
• 6. Positive slope shows positive association