Correlation @UWE_JT9 @dave_lush Scientific Practice Correlation @UWE_JT9 @dave_lush

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

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)

Correlation Correlation coefficient, r = -1

Correlation Correlation coefficient, r = 1

Correlation Correlation coefficient, r = 0.992

Correlation Correlation coefficient, r = 0.954

Correlation Correlation coefficient, r = 0

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

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

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

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!

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= -0.227

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!

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)

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…

Interpreting r Our 21 random data pairs  df 21-2 = 19 Critical r (at 0.05, two-tailed) is 0.433 Our r = 0.227 (ignore sign) So p > 0.05

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

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