Bivariate Linear Correlation

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Bivariate Linear Correlation
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Linear Function

• Y a bX

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Fixed and Random Variables

• A FIXED variable is one for which you have every
  possible value of interest in your sample.
• Example Subject sex, female or male.
• A RANDOM variable is one where the sample values
  are randomly obtained from the population of
  values.
• Example Height of subject.

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Correlation Regression

• If Y is random and X is fixed, the model is a
  regression model.
• If both Y and X are random, the model is a
  correlation model.
• Psychologists generally do not know this
• They think
• Correlation compute the corr coeff, r
• Regression find an equation to predict Y from X

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Scatter Plot
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Burgers (X) and Beer (Y)
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Burger (X)-Beer (Y) Correlation
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Burger (X)-Beer (Y) Correlation
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Hø ? 0

• df n 2 3
• Now construct a confidence interval

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• http//glass.ed.asu.edu/stats/analysis/rci.html
• r .8, n 5.
• This program has problems when limit is near or
  above 1.

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• Try r .8, n 10
• Clearly our estimate of ? is consistent, since
  the CI narrowed when N increased. The CI now
  excludes 0, that is, the correlation is
  significantly different from 0.

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Get Exact p Value

• COMPUTE p2CDF.T(t,df).

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Presenting the Results

• The correlation between my friends burger
  consumption and their beer consumption fell short
  of statistical significance, r(n 5) .8, p
  .10,95 CI -.28, .99.
• Among my friends, beer consumption was
  positively, significantly related to burger
  consumption, r(n 10) .8, p .006,95 CI
  .35, .96.

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Assumptions

• Homoscedasticity across YX
• Normality of YX
• Normality of Y ignoring X
• Homoscedasticity across XY
• Normality of XY
• Normality of X ignoring Y
• The first three should look familiar, we made
  them with the pooled variances t.

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Bivariate Normal
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When Do Assumptions Apply?

• Only when employing t or F.
• That is, obtaining a p value
• or constructing a confidence interval.

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Summary Statement

• The correlation between my friends burger
  consumption and their beer consumption fell short
  of statistical significance, r(n 5) .8, p
  .10, 95 CI -.28, .99.
• Among my friends, burger consumption was
  significantly positively related to beer
  consumption, ..........

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Shrunken r2

• This reduces the bias in estimation of ?
• As sample size increases (n-1)/(n-2) approaches
  1, and the amount of correction is reduced.

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Spearman Rho
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Pearson vs. Spearman
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Uses of Correlation Analysis

• Measure the degree of linear association
• Correlation does imply causation
• Necessary but not sufficient
• Third variable problems
• Reliability
• Validity
• Independent Samples t point biserial r
• Y a b? Group (Group is 0 or 1)

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Uses of Correlation Analysis

• Contingency tables -- ?
• Rows a b?Columns
• Multiple correlation/regression

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Uses of Correlation Analysis

• Analysis of variance (ANOVA)
• PolitConserv a b1 Republican? b2 Democrat?
• k 3, the third group is all others
• Canonical correlation/regression

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Uses of Correlation Analysis

• Canonical correlation/regression
• (homophobia, homo-aggression) (psychopathic
  deviance, masculinity, hypomania, clinical
  defensiveness)
• High homonegativity hypomanic, unusually frank,
  stereotypically masculine, psychopathically
  deviant (antisocial)

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Factors Affecting Size of r

• Range restrictions
• Without variance there cant be covariance
• Extraneous variance
• The more things affecting Y (other then X), the
  smaller the r.
• Interactions the relationship between X and Y
  is modified by Z
• If not included in the model, reduces the r.

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Power Analysis
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Cohens Guidelines

• .10 small but not trivial
• .30 medium
• .50 large

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PSYC 6430 Addendum

• The remaining slides cover material I do not
  typically cover in the undergraduate course.

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Correcting for Measurement Error

• If reliability is not 1, the r will underestimate
  the correlation between the latent variables.
• We can estimate the correlation between the true
  scores this way
• rxx and rYY are reliabilities

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Example

• r between misanthropy and support for animal
  rights .36 among persons with an idealistic
  ethical ideology

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H? ?1 ?2

• Is the correlation between X and Y the same in
  one population as in another?
• The correlation between misanthropy and support
  for animal rights was significantly greater in
  nonidealists (r .36) than in idealists (r .02)

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H? ?WX ?WY

• We have data on three variables. Does the
  correlation between X and W differ from that
  between Y and W.
• W is GPA, X is SATverbal, Y is SATmath.
• See Williams procedure in our text.
• See other procedures referenced in my handout.

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H? ?WX ?YZ

• Raghunathan, T. E, Rosenthal, R, and Rubin, D.
  B. (1996). Comparing correlated but
  nonoverlapping correlations, Psychological
  Methods, 1, 178-183.
• Example is the correlation between verbal
  aptitiude and math aptitude the same at 10 years
  of age as at twenty years of age (longitudional
  data)

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H? ? nonzero value

• A meta-analysis shows that the correlation
  between X and Y averages .39.
• You suspect it is not .39 in the population in
  which you are interested.
• H? ? .39.