Correlation And Mulitple Regression

Question 1

3 types of multiple regression

Answer 1

–simultaneous
–stepwise
–hierarchical

Question 2

What are correlation and regression for?

Answer 2

study of the relationship between two or more variables

Question 3

Regression

Answer 3

allows prediction of Y on the basis of knowledge of X

Question 4

Correlation

Answer 4

measures strength of relationship between X and Y

Question 5

Scatter plot

Answer 5

–2-D diagram
–1 point for each participant
–coordinates are scores on variables: e.g. (X1,Y) or (X2,Y)

Question 6

Correlation and scatter plot

Answer 6

–linked to degree to which points cluster around regression line
–value between -1 and +1

Question 7

Venn diagram

Answer 7

size of circles represent variance of variable

overlapping circles denote correlated variables

Question 8

what is the relationship between 2 variables once the effect of the other variables has been removed?

Answer 8

measures the strength of dependence between 2 variables that is not accounted for by the way in which they both change in response to variations in a selected subset of the other variables

Question 9

What is multiple regression for

Answer 9

learn about relationship between several independent variables (predictors) and one dependent variable (criterion)
 predictive tool

•examples
–estate agent analyzes selling price: for each house, he records size, number of bedrooms, average income in neighbourhood, subjective appeal, etc.
 how do these relate to the selling price?

–psychologist studies depression: for each participant, he records age, gender, stress, measure of neuroticism, etc.
 how do these relate to depression?

Question 10

Assessing goodness to fit

Answer 10

–multiple correlation coefficient
correlation between the criterion Y and the best linear combination of the predictors, Ŷ

–coefficient of determination (R2)
•proportion of variability in data set accounted for by statistical model
•square of multiple correlation coefficient

–F-ratio
improvement in prediction of criterion compared to inaccuracy of model

Question 11

Multiplied regression

Simulatanous (standard)

Answer 11

–no a priori model

–enter all IVs at once

Question 12

Multiple regression

Step wise

Answer 12

–no a priori model
–computer chooses, on statistical ground, an a posteriori model (best sub-set of IVs)
–capitalises on chance effects

Question 13

Multiple regression

Hierarchical (sequential)

Answer 13

–theoretically sound

–a-priori sequence of entry

Question 14

Factors affecting regression

Answer 14

• outliers & influential points
• homo/hetero-scedasticity
• singularity & multi-collinearity
• number of cases vs number of predictors
• range
• distribution

Question 15

Outliers and influential points

Answer 15

• points which deviate markedly from others in sample

* Cook’s distance of 1 or greater

Question 16

homoscedasticity vs heteroscedasticity

Answer 16

•homoscedasticity
variability of scores (errors) in one continuous variable same in second variable
 uniform scatter or dispersion of data points about the regression line.
(homogeneity of variance)

•heteroscedasticity
one variable is skewed or the relationship is non-linear

Question 17

Singularity and multi collinearity

Answer 17

•singularity
redundancy, one variable combination of 2 or more other variables
•multi-collinearity
variables are highly correlated ( > 0.90)

Question 18

Singularity and multi collinearity problems and solutions

Answer 18

•problems
–logical: don’t want to measure the same thing twice.
–statistical: singularity prevents matrix inversion (division) as determinants = zero

•screening & solutions
–high bivariate correlations (> 0.9)
 compute correlations amongst IVs, remove appropriate IV
–high multivariate correlations
 examine SMC (squared multiple correlation) of each IV w.r.t others
(tolerance = 1 – SMC)

Question 19

A small range…

Answer 19

Restricts power of tests

Question 20

Anacombes quartet:

Answer 20

same mean, variance, correlation, regression line

Question 21

What technique is used for two sets of independent variables?

Answer 21

What is common between the sets?

Canonical correlation

Question 22

What technique is used for many independent variables, when asking the question, what is relationship between 2 variables once effect of others removed?

Answer 22

Partial correlation

Question 23

One dependant variable, technique used

Predicting IV from DV’s

Answer 23

Multiple regression

Question 24

Relationship between IV and DV, technique used:

Answer 24

Multiple correlation