Chapters 36 and 37-Beyond Simple Linear Models

Question 1

What does nonlinear regression not mean?

Answer 1

Nonlinear regression does not simply mean that the graph of X vs Y will be curved.

Question 2

What does nonlinear regression mean?

Answer 2

Nonlinear regression means that Y is not linear with respect to the parameters.

Question 3

What is an example of nonlinear regression?

Answer 3

The Michaelis-Menten equation for enzyme activity

Question 4

For what should nonlinear models should usually be chosen for?

Answer 4

Nonlinear models should usually be chosen for theoretical reasons, not just by trying to fit a curved line equation to the data.

Question 5

What kind of equation is rarely appropriate for nonlinear regression?

Answer 5

Polynomial equations are rarely appropriate

Question 6

How would polynomial equations possibly be useful?

Answer 6

They might be useful for limited interpolation. These equations also make programs like MS Excel easier to fit data than truly nonlinear models.

Question 7

Why are polynomial equations still considered linear?

Answer 7

Because Y is linear with respect to the parameters.

Question 8

Finding the best fit parameters for a nonlinear equation is usually done how?

Answer 8

By many computer programs

Question 9

Like linear regression, what does nonlinear regression find?

Answer 9

Finds parameter values that minimize the sum of squares of the difference between actual and predicted Y values

Question 10

What does multiple regression do?

Answer 10

It extends linear regression to allow multiple independent (X) variables.

Question 11

What is an example of multiple regression?

Answer 11

Distinguish between effects of lead exposure vs age on kidney function

Question 12

Usually, multiple regression is considered distinct from _______.

Answer 12

multivariate analyses

Question 13

Multivariate usually means…

Answer 13

that there are multiple Y variables

Question 14

Example of multivariate?

Answer 14

How does soil texture affect quantities of all plant species in a community?

Question 15

What is a Regression Coefficient?

Answer 15

A parameter explaining the relationship of an independent variable to the dependent variable

Question 16

In simple linear regression, _______ is a regression coefficient.

Answer 16

slope

Question 17

In multiple linear regression, how many regression coefficients are there?

Answer 17

Two or more

Question 18

What do regression coefficients quantify?

Answer 18

Quantifies the relative impact of each independent variable on the dependent variable: ie the amount of change in Y for each unit of change in X

Question 19

Why is the regression coefficients chosen?

Answer 19

To minimize the sum of squares

Question 20

Dummy variable

Answer 20

Allows a discrete variable to be used. Presence value=1

Absence value=0

Question 21

What does the P value mean for regression coefficient?

Answer 21

P values test the null hypothesis that the coefficient is 0.

Question 22

What is a problem of finding the P value for a regression coefficient?

Answer 22

R2 goes up automatically when the number of independent variables increase.

Question 23

Adjusted R2

Answer 23

The adjusted R2
is always smaller and depends
on the # of X variables (decrease with) and #
of samples (increases with).

Question 24

Assumptions of multiple linear regression

Answer 24

Independent observations from one population
• The random component is Gaussian
• Linear relationships only (doesn’t guarantee a linear
relationship, but doesn’t detect other relationships)
• No interaction beyond that specified in the model
(e.g. assuming that the effect of lead is not greater
when age is greater), but interaction terms can be
created

Question 25

What if you don’t know which factors are important for

causing your dependent variable observations?

Answer 25

Could use an automatic selection process to choose
independent variables (from many for which you have
observations) that significantly improved your ability to
predict the Y values.

Question 26

e.g. forward-stepwise selection:

Answer 26

start with a very
simple model and use a computer to choose the X
variable that most improves the prediction.
• If significant keep it and choose another X
variable that best improves the prediction further
– until you reach a point where the best
additional variable cannot provide significant
improvement.
• Note: you just performed multiple comparisons

Question 27

How is automatic selection useful?

Answer 27

It may detect relationships that were not already obvious

Question 28

How should you approach using information from automatic selection?

Answer 28

because this involves multiple comparisons, the
R
2
, CIs, and P values cannot be trusted (if reported).

Question 29

Therefore, automatic selection must be…

Answer 29

considered an exploratory
analysis to generate the model (inductive process).
• If the model that you generate results in X variables
that make sense scientifically, then you could do a
completely independent analysis to test if those
variables predict Y significantly.

Question 30

Adding X variables will automatically

Answer 30

increase the R2
• Sample size needs to be much larger than the
number of X variables (10-40 times greater).

Question 31

Multicollinearity:

Answer 31

many X variables are strongly
correlated; adding correlated X variables will not
help predict Y better; avoid entering correlated Xs.

Question 32

Interaction:

Answer 32

may need to account for instances
when the effect of a particular X depends on the
value of another X. Use an interaction term.