Correlation MARE 250 Dr. Jason Turner

Correlation Coefficient Correlation Coefficient (r)(Pearson) – measures the extent of a linear relationship between two continuous variables (responses) Pearson correlation of cexa Ant and cexa post = 0.811 P-Value = 0.000 IF p < 0.05 THEN the linear correlation between the two variables is significantly different than 0 IF p > 0.05 THEN you cannot assume a linear relationship between the two variables

Correlation Coefficient Correlation test is used to determine the relationship between two responses – Specifically it gives you two pieces of information: 1) p-value is used to determine whether a linear relationship exists i.e. - is relationship significantly different than zero 2) Correlation value (R) – used to determine strength and direction of the relationship - value between 0 & -1 or 0 & 1. Closer to 1 or -1 – the stronger the linear relationship; positive number – positive direction of relationship, negative number – negative direction of relationship

Correlation Coefficient

Coefficient Relationships The coefficient of determination (r2) is the square of the linear correlation coefficient (r) We will use coefficient of determination in regression (next week)

Correlation vs. Regression Correlation coefficient (Pearson) – measures the extent of a linear relationship between two continuous variables (“Responses”) Linear regression investigates and models the linear relationship between a response (Y) and predictor(s) (X) Both the response and predictors are continuous variables (“Responses”)

When Correlation vs. Regression? Correlation coefficient (Pearson) – used to determine whether there is a relationship or not Linear regression - used to predict relationships, extrapolate data, quantify change in one versus other is weighted direction

When Correlation vs. Regression? IF Correlation – variables are equally weighted in both direction IF Regression – then it matters which variable is the Response (Y) and which is the predictor (X) Y – (Dependent variable) X – (Independent) X causes change in Y (Y outcome dependent upon X) Y Does Not cause change in X (X –Independent)

Effects of Outliers Outliers may be influential observations Length (cm) A data point whose removal causes the correlation equation (line) to change considerably Consider removal much like an outlier If no explanation – up to researcher r = -0.728 r = -0.852

Correlation vs. Causation Two variables may have a high correlation without being related/connected For example…You might find a strong correlation between depth and urchin density at Onekahakaha when possibly there is little true causation (cause-effect relationship) In actuality the relationship is probably driven by salinity being very low in shallow, nearshore waters and higher in deeper waters further from the freshwater outflow

Correlation vs. Causation THEREFORE… You must determine whether there is a scientific basis for the comparison before you test for it…

Correlation – How to? STAT – Basic Statistics - Correlation

Correlation – How to? Enter all response variables of interest into “Variables” box

Correlation – How to? Output is a matrix table with Pearson Correlation scores and p-values

Correlation – How to? GRAPH – Scatterplot – Simple Enter all response variables of interest into “Variables” box as X – Y combinations

Correlation – How to? Scatterplots are valuable graphic tools

Correlation – How to? For more than 2 variable – use a matrix plot