Examples%20of%20Correlation

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G89.2229 Multiple Regression in Psychology

• Examples of Correlation
• Random variables and manipulated variables
• Thinking about joint distributions
• Thinking about marginal distributions
  Expectations
• Covariance as a statistical concept and tool

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Three examples of correlation

• All from bar exam study discussed last week
• Anxiety and Depression from POMS on day 29 (two
  days before bar exam)
• Anger and Vigor from POMS on day 29 (two days
  before bar exam)
• Anxiety and day to exam during week prior to
  start of exam.

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Anxious and Depressed Mood 2 Days Before Exam
r 0.64

• What do you notice about joint distribution?
• What is correlation?

4
Anger and Vigor 2 Days Before Exam
r -.19

• What do you notice about joint distribution?
• What is correlation?

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Anxious Mood in Days Before the Exam
r .25

• What do you notice about joint distribution?
• What is correlation?

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Random Variables vs. Manipulated Variables

• A random variable is a quantity that is not known
  exactly prior to data collection.
• E.g. anxiety and depression on any given day for
  a randomly selected subject
• A manipulated variable is a quantity that is
  determined by a sampling plan or an experimental
  design.
• E.g. Day to exam, level of exposure, gender
• This distinction will have implications on
  statistical analysis of bivariate association.

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Thinking about bivariate (Joint) distributions

• Suppose we sample persons and measure two
  behaviors.
• Both are random
• The variables might be related or independent
• The joint distribution contains information about
  each variable and the relation among them.
• When we ignore one of the two variables, and
  study the other, we say we are studying the
  Marginal distribution
• This term simply reminds us that another variable
  is in the background

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Expectations and Moments for Marginal
Distributions

• Suppose we measure X, and Y, but choose to study
  only X (ignoring Y).
• We can describe the marginal distribution of X
  using the mean, the variance, and other moments
  such as coefficient of skewness and kurtosis.
• The population moments of the variable are
  described with Expectation Operators.
• Expectation operators can be used to study means
  and variances.

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Expectation operators defined

• The population mean, m E(X), is the average of
  all elements in the population.
• It can be derived knowing only the form of the
  population distribution.
• Let f(X) be the density function describing the
  likelihood of different values of X in the
  population.
• The population mean is the average of all values
  of X weighted by the likelihood of each value.
• If X has finite discrete values, each with
  probability f(X)P(X), E(X)S P(xi)xi
• If X has continuous values, we write E(X) ò x
  f(x) dx

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Rules for Expectation operators

• E(X)mx is the first moment, the mean
• Let k represent some constant number (not random)
• E(kX) kE(X) kmx
• E(Xk) E(X)k mxk
• Let Y represent another random variable (perhaps
  related to X)
• E(XY) E(X)E(Y) mx my
• E(X-Y) E(X)-E(Y) mx - my
• Putting these together
• E( ) E(X1X2)/2 (m1 m2)/2 mThe
  expected value of the average of two random
  variables is the average of their means.

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Variance Operators

• Analogous to E(Y)m, is V(Y)E(Y-m)2 ò (y -
  m)2f(y) dy
• E(X-mx)2 V(X) sx2
• Let k represent some constant
• V(kX) k2V(X) k2sx2
• V(Xk) V(X) sx2
• Let Y represent another random variable that is
  independent of X
• V(XY) V(X)V(Y) sx2 sy2
• V(X-Y) V(X)V(Y) sx2 sy2
• A more general form of these formulas requires
  the concept of covariance

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Covariance A Bivariate Moment

• E(X-?x)(Y-?y) Cov(X,Y) ?XY is called the
  population covariance.
• It is the average product of deviations from
  means
• It is zero when the variables are linearly
  independent
• Formally it depends on the joint bivariate
  density of X and Y, f(X,Y).
• f(X,Y) says how likely are any pair of values of
  X and Y
• Cov(X,Y) ò ò(X-?x)(Y-?y)f(X,Y)dXdY

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Cov (X,Y) as an expectation operator

• For k1 and k2 as constants, there are facts
  closely parallel to facts for variances
• Cov(k1X, k2Y) Cov(X,Y) ?XY
• Cov(k1X, k2Y) k1k2Cov(X,Y) k1k2 ?XY
• Important special case
• Let Y (1/?Y)Y and X (1/?X)X V(X)
  V(Y) 1.0
• Cov(X,Y) (1/?Y) (1/?X) ?XY ?XY
• Cov (X,Y) is the population correlation for the
  variables X and Y, ?XY
• Since ?XY (1/?Y) (1/?X) ?XY, ?XY (?Y) (?X)
  ?XY

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An important use of correlation and covariance

• We are often interested in linear functions of
  two random variables aXbY
• a1, b1 gives sum
• a.5, b.5 gives average
• a1, b-1 gives difference
• What is the expected variance of WaXbY in
  general?
• Var(W) V(aXbY) a2 V(X)b2 V(Y) 2ab
  Cov(X,Y) a2 sx2 b2 sy2 2ab sx sy rxy
• This can be used to compute expected standard
  error of contrasts of sample statistics.

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Example

• Suppose we want to average the POMS anxious and
  depressed moods. What is the expected variance?
• In the sample on day 29,
• Var(Anx)1.129, Var(Dep)0.420Corr(A,D)
  0.64Cov(A,D).64(1.129.420)1/2 0.441
• Var(.5A.5D) .(25)(1.129)(.25)(.420)
  (2)(.25)(.441) 0.648

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Final Comment

• Standard deviations and variances are
  particularly useful when variables are normally
  distributed
• Expectation operators assume that f(X), f(Y) and
  f(X,Y) can be known, but they do not assume that
  these describe bell shape or normal distributions
• Covariances and correlations can be estimated
  with non-normal variables, but be careful about
  statistical tests.