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Correlations

Correlations. Renan Levine POL 242 July 12, 2006. Association. :. Today: Correlations. Correlation is a measure of a relationship between variables. Measured with a coefficient [Pearson’s r ] that ranges from -1 to 1. Measure strength of relationship of interval or ratio variables

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Correlations

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  1. Correlations Renan Levine POL 242 July 12, 2006

  2. Association :

  3. Today: Correlations • Correlation is a measure of a relationship between variables. Measured with a coefficient [Pearson’s r] that ranges from -1 to 1. • Measure strength of relationship of interval or ratio variables • r = Σ(Zx * Zy)/n – 1 • Zx=Z 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.

  4. 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.

  5. Positive Relationship • If two variables are related positively or directly • r > 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.

  6. 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 77:1 (Mar. 1983), pp. 55-74 More fun examples: http://www.nationmaster.com/correlations/eco_gdp-economy-gdp-nominal

  7. Example II r = 0.84

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

  9. Example – Negative / Inverse r = -0.68 Time/SRBI: Oct 3-6, ‘08 red= Republicans, blue=Democrats, grey diamonds=Independents

  10. 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.

  11. Compare Lots more noise here. Typical of public opinion data. Most points close to a line.

  12. 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 Pearson’s r for public opinion data may be low for an association in aggregate data.

  13. Guidelines for Public Opinion Data

  14. Rough Guidelines for Aggregate Data

  15. 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

  16. 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.

  17. Use Yo’ Brain • Computer cannot tell you if it’s 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): Putnam’s 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?option=com_content&task=view&id=99&Itemid=140

  18. 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.

  19. 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!

  20. 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 (Pearson’s r)

  21. 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, don’t spend any time discussing significance.

  22. 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 Pearson’s r): • Spearman: ordinal x ordinal • Point-biserial: one interval/ratio, one dichotomous • Phi: two dichotomous variables • All interpreted the same way

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