Linear Regression & Correlation
1/1/11
SP Notes
- used to compare the relationship between two variables where the relationship appears to
be continuous (ie. BP & blood loss)
- linear regression = drawing of a line that best describes the association between two
variables
- correlation = 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 by choosing a course which minimises the
sum of the squared vertical distances between the individual points (yi) and their imaginary
equivalents (y) on the line = the least squares fit
- the plot of y at x = the regression of y on x
- equation that describes the line & 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.
CORRELLATION
- 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
Jeremy Fernando (2011)
r
r
r
r
=
=
=
=
0.2
0.4
0.7
1.0
to
to
to
or
0.4 - mild association
0.7 - moderate association
1.0 - strong association
-1.0 - perfect correlation
Calculation
(1) assessment of amount of residual scatter around regression line (greater scatter > poorer
correlation)
(2) r = 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 & summed.
- the sum is used in Spearmans rank correlation equation to give rs which is interpreted in
the same way as r.
Jeremy Fernando (2011)