Correlation

1
Correlation

• K300 Class 26
• April 16, 2009

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Overview

• Types of analyses
• Correlation and regression
• Scattergrams
• Correlation
• Correlation and causation

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Types of analyses

• One-way analysis of variance (ANOVA)
• Whether mean (of quantitative, interval or ratio
  variable) varied with category (nominal or
  ordinal variable)
• Chi-square test of independence
• Whether distribution across one set of categories
  dependent upon distribution across another set of
  categories (two nominal or ordinal variables)

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Types of analyses

• So we have quantitative with categorical
  variables (ANOVA)
• And categorical with categorical variables
  (chi-square test for independence)
• What about quantitative with quantitative
  variables?
• Thats correlation and regression

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Correlation and regression

• Correlation addresses whether one variable varies
  as another varies
• Regression addresses ability of one variable to
  predict another variable
• All part of the relationship of one quantitative
  variable to another quantitative variable

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Scattergrams

• Start with plots of value of one variable versus
  value of other variable
• See if there is a pattern
• Does one variable tend to increase (or decrease)
  as the other variable increases?

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Scattergrams

• Looking at large metropolitan areas
• Is the median rent people pay for rental housing
• related to the median family income in the
  metropolitan area?

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Rent versus income
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Correlation

• Measure of the extent to which one variable
  varies with the other in a linear (straight-line)
  relationship

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Simple correlation example

• Is there relationship between number of radio ads
  aired per week and amount of sales (in thousands)?

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Scattergram
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Correlation coefficient r
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Calculating correlation coefficient
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Calculating correlation coefficient
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Hypothesis test about the correlation coefficient

• Is there a correlation, a relationship, in the
  population?
• Or is there no correlation, no relationship, in
  the population? (null hypothesis)

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Step 1 state hypotheses

• H0 ? 0 (population correlation 0)
• H1 ? ltgt 0 (population correlation ltgt 0)

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Step 2 critical value

• a 0.05
• d.f. n 2 6 2 4
• C.V. t 2.776

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Step 3 test value
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Step 4 make decision

• t value for test value greater than critical
  value
• Reject null hypothesis

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Step 5 summarize the results

• There is a significant linear relationship
  between sales and number of radio ads aired per
  week

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Alternative hypothesis test for correlation

• Since t test statistic for correlation
  coefficient depends only on correlation
  coefficient r and number of cases n, which
  determines the number of degrees of freedom
• Can develop table for critical values of the
  correlation coefficient r itselfno need to
  compute a separate test statistics
• Use Table I, Critical Values for PPMC (Pearson
  Product-Moment Correlation Coefficient)

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Step 1 state hypotheses

• H0 ? 0 (population correlation 0)
• H1 ? ltgt 0 (population correlation ltgt 0)

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Step 2 critical value

• a 0.05
• d.f. n 2 6 2 4
• C.V. r 0.811

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Step 3 test value

• r 0.988

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Step 4 make decision

• r value of 0.988 greater than critical value of
  0.811
• Reject null hypothesis

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Step 5 summarize the results

• There is a significant linear relationship
  between sales and number of radio ads aired per
  week

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Some other issues

• Direction of relationship
• Assumption of linear relationship
• Assumption of bivariate normal distribution for
  doing hypothesis test

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Direction of relationship

• Increase of one variable with increase of another
  gives positive correlation
• 0 lt r lt 1
• Decrease of one variable with increase of another
  gives negative correlation
• -1 lt r lt 0

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Assumption of linear relationship

• Correlation assumes relationship between one
  variable and the other is linear
• Straight line on scatterplot
• Proportional change in one variable always equal
  to proportional change in other
• Perfect nonlinear relationships could produce
  zero correlation

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Assumption of bivariate normal distribution

• Doing the hypothesis test for the correlation
  coefficient requires assumption of bivariate
  normal distribution
• Both variables normally distributed
• Combined distribution (in three dimensions) is
  also normal (normal distribution hat)
• Hypothesis test is robust with respect to the
  assumption, however
• Means that it will still generally give
  reasonable results even when assumption is
  violated

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Correlation and causation

• Correlation does not necessarily imply causation
• Because one variable increases does not
  necessarily mean that this causes other variable
  to increase

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Correlation and causation

• Correlation between two variables can be caused
  by relationships to third variable
• Number of births in towns in England may be
  correlated to number of storks, but