Correlation and Linear Regression

Question 1

In what situation is linear regression correlation a useful method of comparison

Answer 1

With continuous data

Question 2

What is correlation?

Answer 2

Correlation tells us whether there is any association between two continuous variables and what is the strength of their association

Question 3

What is linear regression?

Answer 3

Quantifies the relationship between two variables when one of them depends on another. This allows the mean of one variable to be estimated for a given value of the other

Question 4

What sort of hypothesis testing can be done with linear regression?

Answer 4

Can carry out a t test and be easily extended to incorporate adjustments for baseline imbalances with a continuous outcome

Question 5

What is the standard method of calculating correlation?

Answer 5

Pearson’s correlation coefficient / product moment correlation coefficient

Question 6

Describe the notation of Pearson’s correlation coefficient

Answer 6

p (rho) for the population value and the estimated value is r

Question 7

What is Pearson’s Correlation Coefficient?

Answer 7

Measures the scatter of the points around an underlying linear trend and can take a value from -1 to +1

Question 8

How do we interpret r

Answer 8

Positive if higher values of one variable are associated with higher values of the other variable

The correlation is negative if values of one variable decrease as the values of the other increases.

The closer the points are to an underlying trend the higher the correlation. Conversely the greater the spread of points around an underlying linear trend the lower the correlation

Question 9

How does r change with restriction of the range of a variable

Answer 9

When the range of one of the variables is limited r is weaker. This means that comparison of r in different studies may be misleading is ranges of variables are not comparable.

Question 10

How do you calculate a valid confidence interval for r?

Answer 10

Must have bivariate normal distribution

Question 11

How does bivariate data look on a scatter graph?

Answer 11

Elliptical

Question 12

What condition must be satisfied to produce a valid hypothesis test for correlated data?

Answer 12

At least one variable should be normally distributed

Question 13

How do we calculate the correlation coefficient for non-parametric data?

Answer 13

Spearman’s rank-order correlation coefficient (denoted as ps for the population value and rs for the estimate)

Question 14

How does Spearman’s rank-order correlation coefficient calculate correlation?

Answer 14

Rank correlation does not specifically assess linear association but assesses more generally whether there is a monotonic relationship between the variables.

Question 15

What are the two variables called in linear regression?

Answer 15

Variable x is the explanatory variable (dependent or outcome)
Variable y is the response variable (independent or predictor)

Question 16

How do we calculate a line of regression?

Answer 16

Least squares method to calculate the smallest possible residuals

Question 17

What assumptions are made that underlie the method of linear regression?

Answer 17

1. The response variable has a normal distribution at each value of x (i.e. the residuals should be normally distributed)
2. The variability of y must be the same across x (equal variance)
3. The relationship between x and y is linear
4. Observations are independent

Question 18

How do we guarantee observations are independent?

Answer 18

There is no test. Usually as long as and y are taken by individuals once then it is independent.

Question 19

What design of clinical trial would linear regression be inappropriate?

Answer 19

Cluster randomised trials

Question 20

What is the most important variable in the calculation of lines of best fit?

Answer 20

b (gradient)

Question 21

How do we calculate the confidence interval for the slope b?

Answer 21

Use the t-distribution

Question 22

What will t0.05 tend close to with increasing n?

Answer 22

1.96 - because the t-distribution approximates to the normal distribution as n increases

Question 23

How do we calculate the test statistic for a regression line hypothesis test?

Answer 23

Divide the estimate of b by its standard error and compare the result to the t-distribution with (n-2) degrees of freedom.

Question 24

P-values are not the most important measure of the appropriateness of lines of best fit. What other variable is important to comment on?

Answer 24

r^2 (the ratio of the model sum of squares and the total sum of squares) - this describes how well the model fits the data. Equivalent to the square of Pearson’s correlation coefficient

Question 25

What method may be used to analyse the variance of a regression model?

Answer 25

ANOVA testing (gives the model sum of squares)

Question 26

How do we check the assumption that the response variable (y) is normally distributed?

Answer 26

Plot the distribution of the residuals (ei)

Question 27

How do we check the assumption that the variability of y (variance) is the same as x changes?

Answer 27

Plot the residuals against x (the plot should not have a conical shape)

Question 28

What assumption, other than variance, can be checked when plotting the index of the residuals against x?

Answer 28

Linearity

Question 29

If one or more of the assumptions made when performing linear regression are not met, what common transformations may be possible to make it fit?

Answer 29

Natrual log, square, root and reciprocals

Question 30

How can we construct a hypothesis test using linear regression where one variable (x) is discrete (i.e. number of arms in a trial e.g. active and placebo)?

Answer 30

x = 0 = placebo and x = 1 = active
therefore b = difference in treatment between two groups
This is equivalent to two-sample t-test

Question 31

What are the benefits of using multiple linear regression to adjust for confounders in trial data?

Answer 31

1. Can adjust for multiple explanatory variables

2. Can be used to adjust for continuous data