Correlation and Regression

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Correlation and Regression

• Chapter 15

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A new sort of test

• One qualitative and one quantitative variable
  t-test
• Two qualitative variables chi square
• What if you have two quantitative variables, and
  you want to know how much they go together?
• E.g., study time and grades
• ? correlation

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Information in a correlation

• 1. direction of association (positive or
  negative)
• Captured by sign of correlation
• 2. strength of association (how much does one
  variable tell you about the other?)
• Captured by absolute value of correlation can
  vary between 0 and 1

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Watch wording

• Tell about, rather than dictate
• Correlation does not equal causation
• Any time neither variable is manipulated, you
  just know that two variables are related or two
  groups are different you dont know why

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Primary use of correlation

• Examining two quantitative variables
• There is a linear relationship between them

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Graphing correlations

• Scatterplot one variable on X axis, one variable
  on Y axis
• One dot per participant
• If most people above the mean on both variables
  or below the mean on both variables, positive
  correlation
• If most people below the mean on one variable,
  but above the mean on the other, negative
  correlation

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Calculating correlation

• 1. determine whether people are above the mean or
  below the mean, and by how much
• For each person, for each variable, subtract mean
  from score
• 2. combine this information together
• For each person, multiply result from 1 for each
  variable together cross-products
• Add this up across all participants
• ? sum of cross-products

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Taking into account variability of variables

• If just used sum of cross products, would end up
  with large value if there were many participants,
  or if the variable had a wide possible range of
  values
• ? need to correct for this
• Divide by square root of (sum of squares for one
  variable, times sum of squares for the other
  variable)
• ? value that can vary between -1 and 1

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Is it significant?

• Null hypothesis for correlation?
• How big is big enough that its not due to
  chance?
• Critical value of correlation
• Based on df n-2

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Fun with correlation

• Predicting one variable based on another
• Who will do best in college?
• Measuring whether a measure is valid
• Is it related to other measures of the same
  construct?
• Measuring whether a measure is reliable
• Is there a relationship between peoples answers
  at time 1 and peoples answers at time 2?

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Some points of caution

• Outliers
• Restricted range

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Telling the world

• r (df) correlation value, p information
• Or
• r correlation value, n sample size, p
  information

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What about effect size?

• Measured by r2
• Symbol for correlation r
• ? to calculate r2?
• Referred to as coefficient of determination

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What if the association isnt linear?

• Correlation, so far, only works with linear
  associations between variables
• Correlation weve talked about so far Pearson
  correlation
• Most common form of correlation
• If people just say correlation, they mean
  Pearson correlation

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Curvilinear associations

• Can still use a correlation to examine, as long
  as its not a U-shaped or inverted U-shaped
  correlation
• Can use Spearman correlation
• 1. replace raw scores with rank order
• 2. calculate Pearson correlation, using rank
  order data

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Independent samples t-tests are a special case of
correlation

• Point-biserial correlation used with one
  qualitative (with just 2 levels) variable and one
  quantitative variable i.e., the same situation
  when you can use an independent samples t-test
• 1. assign 0 to one group, and 1 to the other
• 2. compute a Pearson correlation

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Chi square is a special case of correlation

• Phi coefficient use when both variables are
  qualitative, with just two levels
• 1. assign 0 and 1 to the two levels of one
  variable
• 2. assign 0 and 1 to the two levels of the other
  variable
• 3. compute a Pearson correlation

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Correlation summary

• Most common form (Pearson) used with two
  continuous variables, in a linear association
• Spearman used with curvilinear associations
• Point-biserial used whenever an independent
  samples t-test can be used
• Phi used when a chi square for goodness of fit
  (with just 2 levels/variable) can be used
• Can vary between -1 and 1
• Does not tell anything about causation

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Back to the idea of prediction

• With correlation, you can predict the value of
  one variable based on the value of another
  variable
• If you know someones marital problems, you can
  predict that persons level of satisfaction
• But, if you knew more about that person you could
  do an even better job predicting satisfaction
• ? regression used to predict one quantitative
  variable from a whole mess of quantitative
  variables

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Building up to regression

• First, the equation for a line?
• Y bX a
• AKA Y mX b
• In both, have intercept and slope
• Intercept predicted value of Y when X is zero
• Slope how much Y is predicted to change as X
  changes
• Goal of regression line Minimize the discrepancy
  between predicted and actual values of Y

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Linking this to correlation

• Correlation slope of the regression line, if
  the scores are in z-scores
• ? predicted z score for Y variable correlation
  value z-score for X variable

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Difference between regression and correlation

• Correlation is a special case of regression, with
  just one predictor variable
• Regression lets you add in more predictor
  variables to
• Figure out how much of the Y variable is
  explained by a whole mess of predictor variables
• Figure out how much each predictor variable
  uniquely tells about the Y variable
• ? two tests for significance for whole model,
  and for each individual variable

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Be sure to review

• What correlation and regression are
• When to use each type
• How to calculate all types of correlations
• How to interpret results