Multiple Regression

Question 1

What does a confounder affect?

Answer 1

The exposure-outcome association because it is a common cause the of exposure and outcome.

Controlling for the confounder gives the adjusted exposure-outcome association.

Question 2

What does a mediator affect?

Answer 2

The E-O associations because it is caused by the exposure which in turn affects the outcome.

Question 3

What do mediators address?

Answer 3

How or why the exposure (E) causes outcome (O).

Question 4

When is a value classified as a moderator?

Answer 4

If E-O association varies depending on the values/levels of Z.

Question 5

What do Moderators address?

Answer 5

When or for whom the exposure (E) causes outcome (O).

Question 6

What can bias in an estimated E-O association arise due to?

Answer 6

Many reasons, one common reason is not taking into account the effects of confounders in estimating the E-O association

Question 7

What is an unadjusted association?

Answer 7

E-O association is estimated ignoring the possible effect of C

Question 8

What is an adjusted association?

Answer 8

E-O association is estimated by taking account of the possible effect of C

Question 9

What are possible ways to adjust for confounders?

Answer 9

Effects of confounders can be dealt with:

at the design stage, or

at the analysis stage

Question 10

How can confounders be dealt with at the study design stage?

Answer 10

Matching - observational studies

Randomisation of exposure/treatment – clinical trials

Question 11

How can confounders be dealt with at the analysis stage?

Answer 11

Stratified analysis - works for one/few categorical confounders

Statistical modelling (multiple regression analysis)

Question 12

What does a multiple regression model investigate?

Answer 12

The relationship between an outcome variable and more than one predictor variables simultaneously.

Question 13

What must an outcome variable be for linear regression?

Answer 13

A continuous variable.

Question 14

What must a Predictor/exposure variable be?

Answer 14

Continuous or categorical.

Question 15

What is the multiple regression formula?

Answer 15

Y = B0 + B1x1+ ….. +BkXk+ E

Coefficients (β’s) - partial regression coefficients

βj represents the change in average y for one unit change in xj (holding all other x’s fixed)

In other words, βj is the effect of the predictor xj adjusted for the effects of all other predictors in the model

Question 16

How can you calculate the adjusted effect of an effect and outcome relationship?

Answer 16

2 options:

Option 1 (multiple regression) is more practical and almost universally used.

Option 2 (regression of residuals) is rarely used in practice, but is useful to understand the underlying mechanism of adjustment.

Question 17

How can we adjust the E-O association for C?

What is the resulting model?

Answer 17

Include C in the regression model of O on E as an additional predictor, which will automatically adjust the E-O association for C.

The resulting model is a multiple linear regression model (more than one predictors):
O = B0 + B1E + B2C + E

The coefficient of E (i.e., B1) in this case will represent the adjusted E-O association controlling for the effect of C .

Question 18

For multiple linear regression, the fitted model is what instead of a line?

Answer 18

Hyperplane

Question 19

What can a multiple regression deal with?

Answer 19

Any number of confounders.

Question 20

How can all confounders be adjusted for in a multiple regression model?

Answer 20

Including them simultaneously as additional predictors.

Question 21

What can restrict the number of variables that can be included in a regression model?

What is a way to resolve this?

Answer 21

Sample size

A rule of thumb is to ensure that there are more than 10 observations (data points) per predictor variable. A sample of size 100, for example, will allow us to consider up to 10 predictors.

Question 22

What does R-squared (R2) measure?

Answer 22

Amount of variance in outcome (y) that can be explained by differences in predictor variables.

Question 23

What is R2 an indicator of?

Answer 23

Goodness-of-fit or prediction power of a fitted regression model.

Question 24

What is R2 generally expressed as?

Answer 24

Percentage and ranges between 0 and 100 (perfect prediction).

Question 25

What are assumptions of a linear regression?

Answer 25

Normality: The error terms are normally distributed.

Variance Homogeneity: The error terms have the same variance irrespective of the values of X (i.e., variance does not depend on X),

Linearity (continuous X): There is a linear relationship between X and Y.

Independence: The observations Y are independent from each other (thus, conditional on X, the error terms are also independent).

Question 26

How can we check for assumptions of a linear regression?

Answer 26

There are a number of diagnostics that use the residuals (differences between observed and predicted values):

Normality: Histogram, Normal probability plot of residuals.

Homogeneity of variance: Plot residuals vs. predicted values (called Residual plot).

Linearity: Partial residual plot –

Shows dependent variable plotted against each predictor variable, both adjusted for all other predictor variables.

Question 27

What do partial residual plots show?

Answer 27

The net relationship where the influence of other variables is partialled out.

Plots suggest a linear relationship between each predictor variable and the outcome variable