Correlations

1
Correlations

• Renan Levine
• POL 242
• July 12, 2006

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Association

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Today Correlations

• Correlation is a measure of a relationship
  between variables. Measured with a coefficient
  Pearsons r that ranges from -1 to 1.
• Measure strength of relationship of interval or
  ratio variables
• r S(Zx Zy)/n 1
• ZxZ scores for X variable and Z scores for Y
  variable. Sum the products and divide by number
  of paired cases minus one.
• How to calculate Z scores can be found on-line.

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Correlation r

• Absolute values closer to 0 indicate that there
  is little or no linear relationship.
• Generally, 0.2-0.4 is weak, 0.4-0.6 is okay, 0.6
  or higher is strong.
• If correlation is very high, then its probably
  something related that you might considering
  indexing or choosing just one variable.
• The closer the coefficient is to the absolute
  value of 1 the stronger the relationship between
  the variables being correlated.

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Positive Relationship

• If two variables are related positively or
  directly
• r gt 0
• Variables track together high values on
  Variable X are associated with high values on
  Variable Y.
• Low values on X associated with low values.

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Example
Robert D. Putnam Robert Leonardi Raffaella Y.
Nanetti Franco Pavoncello. Explaining
Institutional Success The Case of Italian
Regional Government. The American Political
Science Review 771 (Mar. 1983), pp. 55-74
More fun examples http//www.nationmaster.com/cor
relations/eco_gdp-economy-gdp-nominal
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Example II
r 0.84
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Negative or Inverse Relationship

• Variables can be inversely or negatively related
• High values of X are associated with low values
  of Y.

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Example Negative / Inverse
r -0.68
Time/SRBI Oct 3-6, 08
red Republicans, blueDemocrats, grey
diamondsIndependents
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Data

• You need interval-level data.
• You will find many interval-level variables in
• Countries / World
• Provinces
• Election studies (feeling thermometers, odds of
  party entering government, etc)
• You can often use the index you created as an
  interval-level variable.

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Compare
Lots more noise here. Typical of public opinion
data.
Most points close to a line.
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Differences between Public Opinion and Aggregate
Data

• Although it is not uncommon to have one/some
  outliers in aggregate data, public opinion data
  tends to be noisy.
• Feeling thermometer example
• Many respondents gave both candidates a 50
• Quite a few respondents liked both candidates
• Even though most who liked McCain disliked Obama
• A high Pearsons r for public opinion data may be
  low for an association in aggregate data.

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Guidelines for Public Opinion Data
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Rough Guidelines for Aggregate Data
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Very Strong or Worrisome??

• Public Opinion above 0.40
• Aggregate above 0.80
• But these are just guidelines. It depends on how
  good the data is
• Lots of variation in data
• Large scale (10, 20, 100 pts like prediction
  odds, physicians per 100,000 people, feeling
  thermometer scales)
• Number of observations (N)
• Provinces dataset is small

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Outstanding or the same?

• You either have an outstanding relationship OR
  the variables may be measuring the same idea.
• Ex. unemployment and GDP both measure economic
  health
• Ex. Feeling thermometer Barack Obama and feeling
  thermometer for Joe Biden both measure attitudes
  towards the Democratic ticket
• Also inverse relationship
• Example above Obama and McCain feeling
  thermometers different sides of the same coin,
  as both seem to measure partisanship.

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Use Yo Brain

• Computer cannot tell you if its a good, strong
  relationship or two measures looking at the same
  thing.
• Need to understand what each variable is
  measuring
• Same thought process about the index creation.
• Use your knowledge of world and theory to decide
  whether two variables measure the same thing or
  two different things.
• Example (above) Putnams relationship between
  civic culture and government performance.
• Failed states survey - appears that the higher
  an indicator value, the worse off the country in
  that particular field.
• http//www.fundforpeace.org/web/index.php?optionc
  om_contenttaskviewid99Itemid140

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Flip side

• Relationship you expect is strong is surprisingly
  not ?!?!?
• Make certain both variables are interval
• Double check that you cleaned up data
• Missing values are missing
• Next week there may be the need to qualify the
  relationship as some sub-group of the data is not
  like the others and those need to be identified.
• Think about relationship maybe its not linear,
  so that relationship is only present for part of
  range.

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Usefulness

• Quick, easy way to look at several variables to
  see if they are related.
• With strong association, you can begin to think
  about predicting values of Y based on a value of
  X.
• Ex. Positive correlation you know a high value
  of X is associated with a high value of Y!

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Webstats Output
                      - -  Correlation
Coefficients  - -             Q375A1     Q305 
     Q375A3     Q1005Q375A1       1.0000     
.2916      .5320     -.3163            ( 
686)    (  666)    (  667)    (  672)           
P .       P .000    P .000    P .000Q305 
        .2916     1.0000      .2679    
-.1272            (  666)    ( 2776)    ( 
660)    ( 2721)            P .000    P .      
P .000    P .000Q375A3        .5320     
.2679     1.0000     -.2020            ( 
667)    (  660)    (  682)    (  666)           
P .000    P .000    P .       P .000Q1005
       -.3163     -.1272     -.2020   
 1.0000            (  672)    ( 2721)    ( 
666)    ( 3181)            P .000    P .000   
P .000    P .  
N
Coefficients (Pearsons r)
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Significance?

• Webstats will tell you whether or not the
  correlation coefficient is significant.
• Remember that this is just telling you whether
  the relationship may be due to chance.
• Not the strength of the relationship
• Almost unheard of to have a strong relationship
  that is insignificant when using survey data.
• So, dont spend any time discussing significance.

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What if non-interval/non-ratio?

• Usually more appropriate to use the other
  measures of association.
• Webstats will perform a correlation. Be ready for
  results to be less strong
• Program may report (instead of Pearsons r)
• Spearman ordinal x ordinal
• Point-biserial one interval/ratio, one
  dichotomous
• Phi two dichotomous variables
• All interpreted the same way