Correlation

Use of Correlation

to measure and describe the association between two scale variables

correlation research design

observing two measured variables as they occur naturally; no manipulations, both variables continuous and does not imply causation (measured in a single group of participants)

knowing a score on one variable allows us to

predict with some degree of precision the score on another variable

outlier

data points that stand apart from the data set

importance of outliers

the can affect the strength of the correlation

characteristics of the relationship between two variables

direction, strength and form

direction

sign of the correlation describes direction of the relationsips

positive correlation

x inc, y inc - line slops up to the right

negative correlation

x inc, y dec - line slops downward to the right

strength

the degree to which x and y are related, indicated by the correlation coefficient's absolute value.

form

linear relationship (assume we are working with straight lines)

Pearson Correlation - r

most common statistic used to measure the strength and the direction of the linear association between two variables

something that pearsons r indicats

how much two variables vary together compared to how much they vary separately

rho - p

correlation coefficient of a population

sum of products of deviations

measure the co variability between two variables

sum of squares

used to measure variability of a single product

significance of correlation coefficient

to test if the correlation large simply due to random chance factors? need to be rejected

how is the significance of a correlation tests

t-test or confidence interval

3 possible directions of causality

include direct causation, reverse causation, and bidirectional causation.

r is invariant under linear transformations

The property that the correlation coefficient remains unchanged when the data are transformed linearly, meaning it is unaffected by changes in scale or location.

Share variance R²

measures the proportion of variability in one variable that can be determined from the relationship with the other variable

issue of small samples (n< 30)

correlation is highly unstable - confidence interval will be too wide

benefit of large sample size

more precise and narrower confidence interval

why transform r to Z_r

to achieve a normal distribution of correlation coefficients for hypothesis testing because r’s sampling distribution is generally skewed

why is r’s distribution skewed

r = 0; sampling distribution is close to normal

r approaches ±1; becomes more skewed

Point Biserial correlation

correlation is done when one variable is continuous and the other is a dichotomous categorical variable

significant test of a point biserial correlation is identical to

an independent sample t-test