1. Scientific Practice
Correlation

2. Where We Are/Where We Are Going
We have looked at experiments in which we actively do something to see if we can influence something else
eg the effect of a drug (or drugs) on BP
a manipulative experiment
these are analysed using tests such as t-tests or ANOVA
But what if we are interested in the influence of height on BP?
we can’t go around ‘manipulating’ people’s heights!
instead, we use natural variation to explore the relationship
called correlational

3. Correlation An apparent association between two variables
as one variable changes, so does the other
could be in the same direction, eg height and weight
could be opposite, eg BP and life expectancy
Relationships can be demonstrated graphically
is there a linear ‘line of best fit’ I could draw?
The degree of association can be quantified in terms of its correlation coefficient, r
aka Pearson Product Moment Correlation Coefficient
0  1 (-1  +1 when direction taken into account)
strength of ‘link’ between the two (elastic  rigid)
r is independent of the units used (eg feet/inches)

4. Correlation
Correlation coefficient, r = -1

5. Correlation
Correlation coefficient, r = 1

6. Correlation
Correlation coefficient, r = 0.992

7. Correlation
Correlation coefficient, r = 0.954

8. Correlation
Correlation coefficient, r = 0

9. Correlation & Its Pitfalls
Correlation between height and weight in male children
By convention, x-axis used for independent variable, y-axis for dependent variable
ie can’t help thinking of causation…
weight  height?
if anything…
height  weight
reverse causation

10. Correlation & Its Pitfalls
This is a bit better, as the idea that increased height causes increased weight makes more sense (generally, taller bodies need more cells!)
BUT…
correlation does not imply causation
third factors

11. Correlation & Its Pitfalls
Third Factors
Eg there is a (positive) correlation between the number of fire engines at the scene of a fire and the damage done
that does not mean that the fire engines cause the damage!
the fire is the third factor; the size of the fire determines…
the amount of damage done and
the number of fire engines called out

12. Correlation & Its Pitfalls
Third Factors can be used to nefarious ends…
Eg there is a correlation between smoking an ill health
tobacco lobby often cites that association does not imply causation
could be that other factors cause both ill health and the increased tendency to smoke
eg stress
tobacco industry excels at exploiting the idea that there are no certainties in science!

13. Interpreting r
Even if we generate random data, we will always get a non-zero value for r
21 random x/y pairs generated in Excel r=

14. Interpreting r
So how do we tell ‘random’ associations from ‘real’ ones?
how do we determine the significance of r?
The usual 4 steps apply…
1) Generate the Null Hypothesis
there is no association between the two variables
ie r is zero so the value we get is just ‘bad luck’
2) Generate the test statistic
easy!
r is the test statistic!

15. Interpreting r 3) Work out the probability, p
this is the probability of getting an r value as big as we did even though the underlying population of all readings has an r of zero (ie the Null Hypo)

16. Interpreting r 3) Work out the probability, p
there is a critical value of r that needs to be exceeded
depends on df, which is N-2
the greater the N, the lower is the critical r
depends on level of probability required (eg p=0.05)
the smaller the p, the higher is the critical r
use a table…

17. Interpreting r Our 21 random data pairs  df 21-2 = 19
Critical r (at 0.05, two-tailed) is 0.433
Our r = (ignore sign)
So p > 0.05

18. Interpreting r 4) Interpret the probability
there is a greater than 5% chance that the Null Hypothesis is true
so we can’t reject the Null Hypo
so we accept that there is no significant association between the two variables

19. Summary
Correlation is a units-independent estimation of the degree of association between two variables
it estimates their tendency to change together
Correlation coefficient range is -1  +1
ignoring sign, it is 0  1
no correlation  maximum correlation
‘out of step’  ‘marching together’
Significance of r can be tested
is r significantly different to zero?
Correlation does not imply causation
reverse causation and third factors