Further Techniques in Multiple Linear Regression

Question 1

What does linear regression assume?

Answer 1

That an independent predictor variables and the outcome dependent variable are related linearly.

Question 2

How do we fix a non linear regression model?

Answer 2

• By transforming the variables in the model
• By fitting polynomial relationship instead of a straight line

Question 3

How does polynomial fitting work?

Answer 3

• The principle of fitting a straight line applies, but we are just turning one predictor variable into two or more
• Instead of predicting y on x, we predict y on x, and x^2 (and possibly x^3, x^4, etc)

Question 4

List an advantage of polynomial fitting.

Answer 4

Allows us to deal with obvious non-linear relationship without having to specify in advance what the appropriate transformation would be

Question 5

How do we fit a quadratic model?

Answer 5

• We transform x to x^2
• y = β(0)+ β(1)x+ β(2)x^2 + e

Question 6

How do we interpret quadratic terms?

Answer 6

• Where x > 0 and x^2 < 0, y is increasing in x at first, but will eventually turn around and be decreasing
• Where x < 0 and x2 > 0, y is decreasing in x at first, but will eventually turn around and being increasing

Question 7

What is an advantage of transformed data?

Answer 7

• Often less skewed
• Outliers are less extreme

Question 8

List three examples of transformed data.

Answer 8

• Logarithms
• Inverse
• Square root

Question 9

What is an interaction effect?

Answer 9

When the effect of an explanatory variable depends on the level of another explanatory variable.

Question 10

Give an example of an interaction effect.

Answer 10

• Assume male happiness increases with years of marriage, whereas females happiness decreases with years of marriage
• The relationship between happiness and years of marriage may be linear, but it would not be independent of sex.

Question 11

Write the regression equation of a model with interaction terms.

Answer 11

y = β0+ β(1)x^1+ β(2)x^2+ β(3) (x(1)x(2)) + e
where β3 (x1x2) is a multiplicative term of the two main effects

Question 12

What is the most common centring method?

Answer 12

Mean centring

Question 13

How can we ensure B(0) is interpretable?

Answer 13

By changing x by centring the age variable.

Question 14

What changes when you carry out mean centring?

Answer 14

The intercept which now corresponds to the average age, e.g. x(centred), not x=0