what is correlation
consideration of whether there is any relationship or association between two variables
describe the correlation model
 both Y and X are random variables;
 sample observations are obtained by selecting a random sample of the units of association and taking on each a
measurement of X and a measurement of Y
 sample observations are obtained by selecting a random sample of the units of association and taking on each a
measurement of X and a measurement of Y
define correlation analysis
a statistical tool used to study the closeness of the relationship between two or more variables.
what is the correlation matrix
presents correlation coefficients among a group of variables
 used by investigators to portray all possivle bivariate combos of set variables in order to determin patterns of interesting associations in order to study them further
what is the correlation coefficient
the index which defines the strength of association between two variables
can be used to predict the value of one of the variables using another if a relationship exists
to determine relationship random samples must be taken from both sets of the two variables. this data is known as bivariate data
what is the basic rule for determining a relationship betw/ two variables
 the two sets of data are presented as ordered pairs
 dependant variable= y= the one who's value is being predicted
 indepentant II =x= the one used to make the prediction
 ordered pairs are plotted on a graph and a relationship is inferred before calculations are done
 ordered pairs are plotted on a graph and a relationship is inferred before calculations are done
what is a scatter diagram
a diagram thgat shows the relationship between two variables by plotting the x,y pairs
independant values (x) are plotted on x axis
dependant values (y) are plotted on y axis
the coordiate of the two points form a correlation on the graph
what is the pearson correlation coefficient (p)
A population parameter that measures the degree of association betw/ 2 varialbes
 natural parameter for bivariate nominal data
 requires interval or ratio measurements
 used to asses the straight line association between X&Y
 bivariate normal distrubution is a probablilty of distrubutions of X & Y aswell as the density of base pairs
 this allows for both positive and negative dependance betw/ X&Y
list the 5 correlation assumptions
 each value of X has a normally distributed subpopulation of Y values
 each value of Y has a normally distributed subpopulation of X values
 joint distribution of X&Y is a normal distrubution called 'bivariate normal distribution'
 subpopulations of Y values have the same variance
 subpopulations of X values have the same variance
what is bivariate normal distibution
the joint normal distribution of X&Y
inferencial values can only be taken from normal joint x,y distro(bivariate)
no inferences can be made from non normal distrubutions although descriptive means can be used
five parameters of BIVARIATE DISTRUBUTION
σx : σy: standard deviations of each data set
µx µy : means for each data set
p: correlation coefficient= measures strength of X&Y
what is the pearson coefficient
coeffecient used to asses the straight line assoc betw/ x & y and requires interval or ratio values
symbol for the sample correlation coefficient is r,
correlation varies from negative one to positive one (–1 r +1).
r1 is perfect negative x,y relationship
r+1 is perfect positive x,y relationship
r=1 is a straight line
what is pearson product moment correlation
numerical measure of the degree of association between two variables
 provides a quantitative measure of the extent to which the two variables are associated
 calculated from the bivariate data by a formula
using values of data points  value of correlation coefficient calculated from a sample is denoted by the letter r
 value of correlation coefficient calculated from a population is denoted by the Greek letter ρ
pearson product moment correlation continued
 correlation coeficients only show assoc not causeation
 if r=1 it doesn't mean p=1 ( an assoc in sample doesnt mean assoc in pop)
 however a large sample size(no of pairs) increases the size of r and therefore suggests a high correlatio w/in the pop
list the types of correlations

r = +1, the two variables have perfect positive correlation. This means that on a scatter diagram, the points all lie on a straight line that has a positive slope
 If r = –1, the two variables have perfect negative correlation. This means that on a scatter diagram, the points all lie on a straight line that has a negative slope
 if its betw/ 0 and 1 two variables are positively correlated, but not perfectly so, the
coefficient lies between
 if its between –1 and 0 the two variables are negatively correlated, but not perfectly so,

r is 0: two variables have no overall
upward or downward trend whatsoever,
the

curvilinear relationship: positive/negative relationship till a certain point then after theis the realtionship inverses
coefficient lies between
upward or downward trend whatsoever,
the
confidence interval for pearson's correlation
aka
Fisher's rtoz transformation
Fisher developed a transformation of r that tends to
become normal quickly as N increases.
used to conduct tests of r and calc CI
z=0.5ln (1+r/1r)
z=+/ (criterionz) x (standard deviation)
criterion z=1.96 in case of 95% ci
can be used to calc upper and lower limits
what method tests for the statistical significance of a correlation coeficient
based on a ttest that evalutes the H0 that p =0 in the population
 ε = error term/ noise term/ residual term = random unobserved component
 during hypothesis tests it has a normal distro w/ a mean of 0 and unknown cariance σ^{2} independadnt of x
what is the phi coefficient
its a product—moment coefficient of correlation variation of Pearson’s definition of r when the two states of each variable are given values of 0 and 1 respectively.
purpose of phi's coefficient
designed for the comparison of truly dichotomous distributions ( only have 2 points on their scale for an unmeasurable attribute) i.e nominal values
aka known as the YUTE (φ)
relates to 2x2 tables
often used in psychoological and educational testing d/2 freq of applying dichotomy onto a continuous variable and PASS/ FAIL categories are found based on a threshold score
what is YULE'S Q and what is it used for
: a nominal measure of association used to determine the association betw/ variables
or
the ratio of dx betw/ the products of diagonal cell freq and the sum of products of diagonal cell frequencies
 used to analyse the strength and direction of association between two dichotomous variables(e.g.Gender, yes/no, T/F, aggree/disagree)
 developed for variables w/ only 2 values
 uses a 2x2 table where each variable is a dichotomy
 distribution free statistic
6 benefits of YULES Q
 no corrections required
 computed from 2x2 table w/o computising chi squared
 best meaningfully applied test for dichoutomous data
 no stringent assumptions for its application
 quick and easy calc
 measures the porportional reduction in error assoc w/ predictong one variable from the order
4 cons of YULE'S Q
 can only use 2x2 tables
 if data fits into larger tables yule's q can't be used unless the data collapses to a 2x2 table
 collapsing data into smaller categoris causes info loss
 better to avoid collapsing data w/ YULE'S Q
what is spearman's rank order coefficient p_{sp}
 alternative measure of the degree of assoc betw/ 2 variables

non parametric version of pearson product moment using _{sp }to dx
 measures the association betw/ ranks of observations
conditions for spearman's rho (1904) and kendall's tau (1938)
 x & y can have other joint distributions other than the bivariate one
 the correlation betw/ x&y has the property of positve/ negative correlation
what does spearmans correlation coefficient determine
the strength and direction of the monotonic relation between 2 variables instead of the strength of the linear(pearson)
monotonic relationship
 either as one variable increases so does the other
 as the value of one variable increases the other decreases
assumptions made in the spearman's coefficient
 requires ordinal( scale of agreement)/ interval/ ratio(iq score) scale
 a monotonic relationship exists betw/ the two variables
spearman's rho
each measurement is seperatley ranked for Xs and Ys in increasing order
then the pair for each Xi/Yi is replaced with the no of their rank
formula is apllied and the result is SPEARMAN'S RHO
 rho is always betw/ 1 and 1
 rho is pearson correlation applied to ranks
 if y is monotonically increasing function of x then Yi matches Xi
extra info on spearman's
 can be used when two variables are not normally distributed
 not very outlier(values outside usual pattern) senseitive
 therefore vailid results can still be obtained w/ outliers in the data
 therefore vailid results can still be obtained w/ outliers in the data