Week 4 - Correlations and linear regression

Question 1

If a research question has the words “relationship between X and Y” or “after controlling for Z, is there an association between X and Y?”
What kind of statistical analysis would be appropriate for these questions?

Answer 1

Correlations and regression

Question 2

What is a correlation?

Answer 2

A statistical technique for measuring the extent to which two variables are associated/ related.

Measures the pattern of responses across variables.

Assumes linear association between variables

Changes in one leads to predictable changes in another variable

Usually a bivariate association

Question 3

How is an association/ relationship determined?

Answer 3

When changes in one variable can show persistent and predictable changes in other variables

Question 4

Does correlation always mean causation?

Answer 4

No, because a third variable might be causing the observed associations

Question 5

What is the range of values for a correlation

Answer 5

-1 (perfect negative) to +1 (perfect positive)

0 = no association therefore represents the null hypothesis

Question 6

How s the significance of a correlation determined?

Answer 6

• sample size (n)
• alpha value (one (0.05) vs two-tailed (0.05)
• –> e.g. predicting one direction (positive or negative) or two direction

Question 7

Which type of alpha has greater statistical power? one-tailed or two tailed?

Answer 7

A one-tailed test because it only tests in one direction (very confident hypothesis is in that direction - back up with theory)

Question 8

What does variance measure?

Answer 8

How much the scores deviate from the mean of the distribution (one-variable)

variance = average squared distance from the mean

Question 9

What does covariance measure?

Answer 9

How much TWO variables differ from the mean

instead of sum of squares, sum of cross products are observed

Question 10

What are the problems with covariance?

How are they fixed?

Answer 10

UNIT OF MEASUREMENT - e.g. covariance of two variables might be measured in miles = 4.25 but then if converted to km the covariance is 11
–> standardise it (divide by the SD of both variables)

Question 11

The standardised version of a covariance is known as the ____

Answer 11

correlation coefficient

• unaffected by units of measurement
• makes the variances equal

Question 12

covariance = standardised/unstandardised

whereas Pearson correlation = standardised/unstandardised

Answer 12

covariance = unstandardised

Pearson correlation= standardised

Question 13

What does Pearson Correlation tell us?

Answer 13

Direction + strength of linear relationship between two interval/ratio variables (continuous data)

Question 14

What symbol denotes Pearson Correlation

Answer 14

r

r = strength and direction

Question 15

What does the size/magnitude of ‘r’ denote?

Answer 15

degree to which points fit on a straight line. Closer to 1 = more straight line indicating a linear relationship

+1 positive relationship
-1 negative relationship
0 = no relationship/ two variables are independent of one another

Question 16

What is a correlation matrix?

Answer 16

Represents each correlation between pairwise combination of variables.

Can be used for descriptive statistics/ exploratory analysis

Question 17

In a correlation matrix, each correlation is a separate test. What is the issue with multiple testing?

How can we fix this?

Answer 17

more tests (without seperate justifiable hypotheses) Increases the risk of a false positive

–> post hoc analysis = reporting associated found after data collection

Question 18

What are the assumptions of a Pearson Correlation?

Answer 18

• Parametric test therefore is assumes variables are normally distributed
• linear association
• variables measured on an interval or ratio scale

Question 19

How to deal with violation to the assumption of normality in a Pearson Correlation?

Answer 19

• if N >30, use CLT to justify preceding despite violation

- Spearman correlation

Question 20

How to deal with violation to the assumption of linearity in a Pearson Correlation?

Answer 20

• If relationship is monotonic, use Spearman correlation

- Otherwise, transformation to achieve linearity

Question 21

What are the two situations where you can use Spearman Correlation (r s or rho)

Answer 21

1. to find the association between two ordinal variables (X & Y consist of ranks)
2. to measure the consistency of direction of the association between two interval/ratio variables
   - -> variables converted to ranks first before Spearman is used

• Measures the degree of monotonic relationship between variables

Question 22

Do the Spearman Correlation Coefficient and Pearson Correlation Coefficient use the same formula?

Does this make the analysos more or less powerful?

Answer 22

yes, only the calculations are performed on ranked data instead in Spearman

Less powerful because data is lost during it’s conversion into ordinal data

Question 23

What is a monotonic relatonship?

Answer 23

Assumption that even tho the data doesn’t fit on straight line, data points are generally going in the same direction.

As Pearson correlation assumes linearity, use can use Spearman if data is non-linear but monotonic (increasing in the same direction)

Question 24

What do you use to find:

The proportion of variability in Y variable that can be attributed to variability in X

Answer 24

Coefficient of determination (r2/ r squared)

Shows how accurately one variable predicts the other

Question 25

For Spearman’s Correlation, what does the coefficient of determination tell us?

Answer 25

The proportion of variance in the RANKS that the two variables

Question 26

r = .411
r2 = 16.9

“16.9% of the variability in exam performance can be explained (overlaps) with variability in revision time.

What is the X variable/ Predictor?
Y variable/ Outcome ?
What type of variance does this represent?

Answer 26

X = revision time
Y = exam performance
Shared variance

Question 27

What is a partial correlation?

Answer 27

Measures association between two variables, controlling for the effect a third variable has on both

Question 28

What is a semi-partial/ part correlation?

Answer 28

Measures the relationship between two variables, controlling for the effect that a third variable has on ONE of the other variables

Question 29

What is a zero order correlation?

Answer 29

correlation between two variables when you do not control for anything

Question 30

What is a first order correlation?

Answer 30

partial correlation that controls for 1st variable

Question 31

What is a 2nd order correlation?

Answer 31

partial correlation that controls for two variables

Question 32

What is the directionality problem for correlations?

Answer 32

correlation =/= causation
third variable

it is not possible to determine which variable is the cause and which is the effect

Question 33

Can you compare correlation coefficients?

Answer 33

The best way is to use multiple regression to test whether the association between two variables differs by group

Question 34

Can partial correlations be performed on non-parametric data?

Answer 34

Yes, you can do Spearman partial correlations.

Spearman’s partial rank order correlation

Question 35

What is linear regression?

Answer 35

A model used to predict the value of one variable from another; describe the relationship using the equation of a straight line

Question 36

Straight line equation

Answer 36

Yi = b0 + b1X1 + Ei

Yi = the ith person’s score on the outcome variable

B0 = Y-intercept (value of Y when X=0) Point at which the regression line crosses the y-axis

B1 = regression coefficient for the predictor

• gradient (slope) of the regression line
• direction/ strength of relationship

Ei = the difference bewteen the actual and predicted value of Y for the ith person