# Linear regression and Correlation

OVERVIEW

• Linear regression and correlation used to compare the relationship between two variables where the relationship appears to be continuous (e.g. tachycardia and blood loss)
• linear regression is the drawing of a line that best describes the association between two variables
• correlation is the closeness of association between two continuous variables.

ASSUMPTIONS

• relationship is linear
• observations are independent
• outcomes of interest are dependent on observations
• observations must be normally distributed

LINEAR REGRESSION

• data fed into computer
• independent variables (x) vs dependent variables (y)
• computer draws line of best fit through the points using least squares fit — it by chooses a course which minimises the sum of the squares of the vertical distances between the individual points (yi) and their imaginary equivalents (y) on the line
• the plot of y at x = the regression of y on x
• equation that describes the line and proposed relationship:

y = a + b.x

y = predicted points on regression line
b = slope of line (regression co-efficient), defines the proposed relationship
a = intercept of y axis when x = 0

b > 0 – positive relationship
b < 0 – negative relationship
b = 0 – line of no slope -> no relationship

• the larger the sample the closer b will be to the true effect in the population
• the precision can be gauged by reporting b with SE & CI.

CORRELATION

• Pearsons correlation co-efficient (r) is used to assess how likely the proposed relationship is
• it is based on quantifying the residual scatter around the regression line

r = 0 – no association at all
r = 0.2 to 0.4 – mild association
r = 0.4 to 0.7 – moderate association
r = 0.7 to 1.0 – strong association
r = 1.0 or -1.0 – perfect correlation

Calculation

• (1) assessment of amount of residual scatter around regression line (greater scatter > poorer correlation)
• (2) r is a ratio of variance

r = square root of (regression SS / total SS)

SS = regression line of sum of squares

• (3) complex equation!

Spearman’s rank correlation (rs)

• non-parametric test for small samples (<10 patients)
• variables are ranked separately
• differences between the pairs of ranks for each patient is calculated, squared and summed
• the sum is used in Spearman’s rank correlation coefficient equation to give rs which is interpreted in the same way as r

## Critical Care

Compendium

##### Chris Nickson

Chris is an Intensivist and ECMO specialist at the Alfred ICU in Melbourne. He is also the Innovation Lead for the Australian Centre for Health Innovation at Alfred Health and Clinical Adjunct Associate Professor at Monash University. He is a co-founder of the Australia and New Zealand Clinician Educator Network (ANZCEN) and is the Lead for the ANZCEN Clinician Educator Incubator programme. He is on the Board of Directors for the Intensive Care Foundation and is a First Part Examiner for the College of Intensive Care Medicine. He is an internationally recognised Clinician Educator with a passion for helping clinicians learn and for improving the clinical performance of individuals and collectives.

After finishing his medical degree at the University of Auckland, he continued post-graduate training in New Zealand as well as Australia’s Northern Territory, Perth and Melbourne. He has completed fellowship training in both intensive care medicine and emergency medicine, as well as post-graduate training in biochemistry, clinical toxicology, clinical epidemiology, and health professional education.

He is actively involved in in using translational simulation to improve patient care and the design of processes and systems at Alfred Health. He coordinates the Alfred ICU’s education and simulation programmes and runs the unit’s education website, INTENSIVE.  He created the ‘Critically Ill Airway’ course and teaches on numerous courses around the world. He is one of the founders of the FOAM movement (Free Open-Access Medical education) and is co-creator of litfl.com, the RAGE podcast, the Resuscitology course, and the SMACC conference.

His one great achievement is being the father of two amazing children.