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What is correlation?
correlation
measures the relationship between two variables
there are more than one type of correlation (depending on level of measurement)

Pearson product moment correlation
correlation
between interval/ratio variables
most widely used measure of correlation
has the most assumptions

Speraman rank correlation
correlation
between ordinal variables

Phi coefficient
correlation
between dichotomous variables

pointbiserial correlation
correlation
between a dichotomous & interval/ration variables
ex: right or wrong of an exam item and total score

biserial correlation
correlation
between a dichotomous (w/ underlying continuum) & interval/ratio variables
ex:anxiety (low, high) & depression

polyserial correlation
correlation
between a polytomous (3 or more levels) (w/ underlying continuum) & interval/ratio variables
ex: anxiety (low, medium, high) & depression

tetrachoric correlation
correlation
betwee dichotomous variables (w/ underlying continuum) and depression
ex: anxiety (low, high) & depression (low, high)

polychoric correlation
correlation
between polytomous variable (w/ underlying continuum)

Pearson Product Moment Correlation Assumptions
correlation
No Outliers
Linearity
Normal Distribution

Pearson Assumptions: No Outliers
correlation
no extreme scores
 two types:
 univariate
 multivariate

Pearson Assumption: No Outliers  Univariate
correlation
an extreme score that is far away from the distribution of a variable
only one variable

Pearson Assumption: No Outliers  Multivariate
correlation
an extreme score that is far away from the joint distribution of variables
it is possible for a case to be a multivariate outlier without bieng a univaritate outlier

Pearson Assumption: Linearity
correlation
variables should NOT show any nonlinear pattern

Pearson Assumption: Normal Distribution
correlation
distribution of a variable should be univariate normal
distribution of variables should be multivariate normal

Range of Correlation
correlation
 all correlation range from 1 to +1
 a measure of correlation is unitless (or standardized).
 r = 1 perfect negative relationship
 r = +1 perfect positive relationship
 r = 0 no relationship

Scatterplot
correlation
when the correlation is r= +1, the scatterplot appears in a straight line
as the correlation approaches 0, the variability of scores increases

Covariance
correlation
measures of relationship between two variables in raw units (unstandardized correlation)
possible range is infinity to +infinity

Types of Research Questions
correlation
1. Is there a significant relationship between x and y?
2. Is there a significant difference in the relationship of x and y between group 1 and group 2?
3. What is the (1alpha)%CI of the correlation?

Sampling Distribution Correlation
correlation
 in not normal except when r=0 due to restricted range of the correlation
 bound between 1 & +1

Onesample ttest for correlation
correlation
For testing H_{0}: p = 0, Greek letter rho  population correltaion
df = N  2
affected not only by N but also by r. Depends on the # of subjects & relationship

When to us rcritical
correlation
when testing more than one correlation becomes tedious
easier to compute rcritical value as long as there are same N

Fisher's rtoz transformation
when to use
correlation
For testing a H_{0}: p = p_{0}_{ }
For difference in a correlation between groups
For confidence interval of a correlation
Convert r to zscore

