Download Review

Lesson 4 - R
Chapter 4 Review
More about Relationships Between
Two Variables
Objectives
• Identify settings in which a transformation might be
necessary in order to achieve linearity.
• Use transformations involving powers and
logarithms to linearize curved relationships.
• Explain what is meant by a two-way table, and
describe its parts.
• Give an example of Simpson’s paradox.
• Explain what gives the best evidence for causation.
• Explain the criteria for establishing causation when
experimentation is not feasible.
Vocabulary
• None new
Residuals
● One difference between math and stat is that
statistics assumes that the measurements are not
exact, that there is an error or residual
● The formula for the residual is always
Residual = Observed – Predicted
● This relationship is not just for this chapter … it is
the general way of defining error in statistics
● The least squares regression line minimizes the sum
of the square of the residuals
Residual on the Scatter Diagram
The model line
The residual
The observed value y
The predicted value y
The x value of interest
Least-Squares Regression Model Scope
Response
Scope of the model
Areas outside the scope of the model
Explanatory
Linear relationship outside the scope of the model
is not guaranteed!
Things to Remember
• In the Least Square Regression model:
– R² gives us the % of the variation in the response
variable explained by the model
– the mean of the response variable depends on the
linearity of the explanatory variable
– the residuals must be normally distributed with
constant error variance
– the εi will be normally distributed (0,σ²)
– we are estimating two values (b0 b1) and therefore
lose 2 degrees of freedom in the t-statistic
– the prediction interval for an individual response will
be wider than the confidence interval for a mean
response
– procedures are robust
Things to Remember
• In Multiple Regression models:
– the adjusted R² gives us the % of the variation in the
response variable explained by the model
• R² is adjusted based on # of sample and number of
explanatory variables in the model
– multicollinearity may be a problem if a high linear
correlation exists between explanatory variables
• Rule: |correlation| > 0.7 then multicollinearity possible
– the procedure used in our book for multiple
regression modeling is called backwards step-wise
regression
• Rule: remove explanatory variable with highest p-value and
then rerun the model check adjusted R² values
Multiple Regression Model
yi = β0 + β1x1i + β2x2i + … + βkxki + εi
where
yi is the value of the response variable for the ith individual
β0, β1, β2, , βk ,are the parameters to be estimated based on the
sample data
x1i is the ith observation for the first explanatory variable,
x2i is the ith observation for the second explanatory variable
and so on
εi is am independent random error term that is normally
distributed with mean 0 and variance = σ²
i = 1, 2, 3, …, n, where n is the sample size
• The adjusted R² is used in multiple regression models
– The adjusted R² will decrease if a variable is added to the model
that does little to explain the variation in the response variable.
– The adjusted R² will increase if a variable is taken from the model
that does little to explain the variation in the response variable.
ANOVA
• We use the Analysis of Variance (ANOVA) method to
compare three or more means
• ANOVA analyzes the differences in the sample
means using sums of squares and an F test
• One-way ANOVA analyzes models where there is
only one factor that differentiates the groups
• Two-way ANOVA analyzes models where there are
two factors that differentiate the groups
Summary and Homework
• Summary
• Homework
– pg