Chapters 2 and 10: Least Squares Regression
Readings: Chapters 2 & 10
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Introduction
•
Linear Regression
: examine association between two quantitative variables.
–
number of beers that you drink and your blood alcohol level
–
homework score and test score
•
Response variable Y
:
–
Dependent variable
–
Measures an outcome of a study
•
Explanatory variable X
–
Independent variable
–
Explains or is related to changes in the response variable (p. 105)
Example 1
: Which is the explanatory variable and which is the response variable?
a. The amount of time spent studying for an exam and the grade on the exam
b. The weight in kilograms and the height in centimeters of a person
c. Yield of corn in bushels per acre and the inches of rain in the growing season
General Procedure for Analyzing Two Quantitative Variables
1. Make a scatter plot of the data.
Describe the form, direction, and strength.
Look for
outliers.
2. Look at the
correlation
to get a numerical value for the direction and strength.
3. If the data is reasonably linear, get an equation of the line using least squares regression.
4. Look at the residual plot to see if there are any outliers or the possibility of lurking
variables. (Patterns bad, randomness good.)
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5. Look at the normal probability plot to determine whether the residuals are normally
distributed. (The dots sticking close to the 45 degree line is good.)
6. Look at hypothesis tests for the correlation, slope, and intercept.
Look at confidence
intervals for the slope, intercept, and mean response, and at the prediction intervals.
7. If you had an outlier, you should rework the data without the outlier and comment on
the differences in your results.
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Association
Scatter Plot
•
Show the relationship between 2 quantitative variables measured on the same individuals
•
Dots onlydon’t connect them with a line or a curve
•
Form
: Linear? Nonlinear? No obvious pattern?
•
Direction
: Positive or negative association? No association?
– Positive association
: When aboveaverage values of one variable tend to accom
pany aboveaverage values of the other, and belowaverage values also tend to occur
together.
– Negative association
: When aboveaverage values of one variable tend to accom
pany below average values of the other and visa versa.
– No association
: Hard to find a pattern in the dots
•
Strength
: how closely do the points follow a clear form? Strong or weak or moderate?
•
Look for OUTLIERS!
Form and direction of an association
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NonLinear
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No Relationship
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 Spring '08
 Staff
 Least Squares, Linear Regression, Regression Analysis, editor, Errors and residuals in statistics, Scatter plot

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