**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

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