When trying to explain some of the patterns you have observed in your species and community data, it sometimes helps to have a look at relationships between variables – both physical and biological Correlation and linear regression Is it possible to quantify your observations? For Example……….. What patterns can see when you look at the above data?

Correlation and linear regression: not the same, but are related Correlation: quantifies how X and Y vary together Linear regression: line that best predicts Y from X Use correlation when both X and Y are measured Use linear regression when one of the variables is controlled

Direction C1 vs C2 C1 C C1 vs C2 C1 C2 Positive Large values of X = large values of Y, Small values of X = small values of Y. e.g., height and weight Large values of X = small values of Y Small values of X = large values of Y e.g., speed and accuracy Negative 3 characteristics of a relationship Direction Positive(+) Negative (-) Degree of association Between–1 and +1 Absolute values signify strength Form Linear Non-linear

Form Linear Non-linear Degree of association C1 vs C2 C1 C C1 vs C2 C1 C2 Strong (tight cloud) Weak (diffuse cloud)

Pearson’s r Absolute value indicates strength +/-indicates direction A value ranging from to 1.00 indicating the strength and direction of the linear relationship Correlation :a statistical technique that measures and describes the degree of linear relationship between two variables ObsXY A11 B13 C32 D45 E64 F75 Dataset X Y Scatterplot

Some Examples………….

(X–X)(Y–Y)   r (X–X)  2 (Y–Y) 2  √ Sum of Squares (Sample) Mean Sum of Squares (sample) (Variance) Σ N mean Remember…………… Standard Deviation s = 1.5 (Variance) √ Square units? How to Calculate Pearson’s r

(X–X)(Y–Y)   r (X–X)  2 (Y–Y) 2  √ The equation for r YX, of variationtotal YX, of covariation  r Means this in words……… NUMERATOR: For each set of X and Y values - you are looking at the deviation of X from its mean, and the deviation of Y from its mean – to get a feel for their joint deviation – or covariation. This is summed across all sets of X-Y values to provide an overall index of co- variation. DENOMINATOR: This is simply total variation of X and Y (see previous slide)

For Example…………. (X–X)(Y–Y)   r (X–X)  2 (Y–Y) 2  √ = 834 / √( x 1010) = 834 / √ = 834 / = 0.994

Some issues with r Outliers have strong effects Restriction of range can suppress or augment r Correlation is not causation No linear correlation does not mean no association Outliers Child 19 is lowering r Child 18 is increasing r

The restricted range problem The relationship you see between X and Y may depend on the range of X For example, the size of a child’s vocabulary has a strong positive association with the child’s age But if all of the children in your data set are in the same grade in school, you may not see much association Common causes, confounds Two variables might be associated because they share a common cause. There is a positive correlation between ice cream sales and the number of drowning incidents.. Also, in many cases, there is the question of reverse causality

Non-linearity Practice time Proficiency Some variables are not linearly related, though a relationship obviously exists

The correlation coefficient, r, is a statistic Its significance can be determined by checking it against the appropriate critical value [for a set level of probability, degree of freedom and alpha (1 or 2 tailed)] in a table of r values. When you check the table – ignore the sign of your value If your value is greater than the critical value, then it is considered significant. It summarises the co-variation or correlation between the two variables and varies (excluding negatives) from 0 to 1 Before checking it, however, you need to set up a null hypothesis (H 0 ) What would such an hypothesis be?

If r is the correlation coefficient, what is r 2 ? The amount ofcovariationcompared to the amount of total variation “The percent of total variance that is shared variance” E.g. “If r =.80, then X explains 64% of the variability in Y” (and vice versa) MSExcel can generate r 2 values…………….

A CAUTIONARY NOTE r = 0.93 r = r = BUT……..

THE END Image acknowledgements – Content acknowledgements – Dr Vanessa Couldridge, UWC