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 is an Intensivist and ECMO specialist at The Alfred ICU, where he is Deputy Director (Education). He is a Clinical Adjunct Associate Professor at Monash University, the Lead for the Clinician Educator Incubator programme, and a CICM First Part Examiner.
He is an internationally recognised Clinician Educator with a passion for helping clinicians learn and for improving the clinical performance of individuals and collectives. He was one of the founders of the FOAM movement (Free Open-Access Medical education) has been recognised for his contributions to education with awards from ANZICS, ANZAHPE, and ACEM.
His one great achievement is being the father of three amazing children.
On Bluesky, he is @precordialthump.bsky.social and on the site that Elon has screwed up, he is @precordialthump.
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