Correlation

Descriptive technique: Describes the relationship between two variables Variables are observed or measured but rarely manipulated thus we CANNOT infer causation

You have a pair of observations -one for each variable Eg height and weight Anxiety and introversion You measure all people on both variables N = number of pairs

Characteristics of the relationship measured 1)Direction is it positive or negative- Indicated by the sign of the correlation 2) The strength of the relationship Indicated by the number of the correlation +/- 1 is a perfect correlation 0 indicates no linear relationship 3) The form – is it linear?

Mean heights of a group of children in Kalama, an Egyptian village that is the site of a study of nutrition in developing countries. The data were obtained by measuring the heights of all 161 children in the village each month over several years. Age in months Height in cm

Scatterplot of age vs height

Correlation does not mean causation

Before Jonas Salk found the polio vaccine researchers looked for relationship with polio and anything – any protection Found a correlation between soft-drink sales and polio. The more soft-drinks sold the more cases of polio…… Or factoid... The more churches per square mile the higher the crime.....

So why use correlation if we can’t infer causation ??? 1) prediction – if know the relationship then can predict the value of one variable if know the other. 2) validity – if design a new test and you want to see if it tests the same thing as the old one. … 3) reliability – will the test change over time or over different observers 4) test a theory

r = amount that X and Y vary together amount that X and Y vary separately r = covariability of X and Y variability of X and Y separately

r is unchanged if 1) interchange the two variables call height X and age Y or height Y and age X 2) Add a constant to all the values of one variable 3) Multiply all the values of one variable by a constant. (changes mean and sd but not the relative positions)

age ht Ht+2Ht X 2

+2X2

Coefficient of Determination r is correlation r 2 is coefficient of determination Tells you how much of the variability is explained by the correlation

Interpretation 1)No causation 2) Value of r is influenced by range 3) A zero correlation tells you that there is no linear relationship. 4) Outliers can have a huge impact web