Correlations

Renan Levine

POL 242

July 12, 2006

Association

: Crosstabulation

Category Specifics

Symmetry Specification

Measure of Association

Indication of Direction

Nominal X Nominal Only (2 X 2) Symmetrical Phi Yes

Nominal X Nominal Greater than (2X2) - Cramer's V No

Nominal X Ordinal At least (2 X 3) - Cramer's V No

Ordinal X Ordinal Square (e.g., 3 X 3) SymmetricalKendall's Tau-

bYes

Ordinal X Ordinal Rectangle  (e.g., 3 X 4) AsymmetricalKendall's Tau-

cYes

Interval X Interval (not taught yet)

- -Pearson's R (not yet

taught)Yes

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.

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.

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.

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

Example II0

1020

3040

5060

7080

9010

0F

eelin

g T

herm

omet

er P

alin

0 10 20 30 40 50 60 70 80 90 100Feeling Thermometer McCain

Opinion towards Palin & McCain

r = 0.84

Negative or Inverse Relationship Variables can be inversely or negatively

related High values of X are associated with low values

of Y.

Example – Negative / Inverse0

1020

3040

5060

7080

9010

0F

eelin

g T

herm

omet

er O

bam

a

0 10 20 30 40 50 60 70 80 90 100Feeling Thermometer McCain

Opinion towards Obama & McCain r = -0.68

red= Republicans, blue=Democrats, grey diamonds=Independents

Time/SRBI: Oct 3-6, ‘08

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.

Compare

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

010

2030

4050

6070

8090

100

Fee

ling

The

rmom

eter

Pal

in

0 10 20 30 40 50 60 70 80 90 100Feeling Thermometer McCain

Opinion towards Palin & McCain

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.

Guidelines for Public Opinion DataMAGNITUDE OF ASSOCIATION QUALIFICATION COMMENTS

0.00 No RelationshipKnowing the independent variable does not reduce the number of errors in

predicting the dependent variable at all.

.00 to .15 Not Useful Not Acceptable

.15 to .20 Very Weak  Minimally acceptable

.20 to .25 Moderately Strong  Acceptable

.25 to .30 Fairly Strong  Good Work

.30 to .40 Strong  Great Work

.40 to .70 Very Strong/Worrisomely StrongEITHER an excellent relationship OR the two variables are measuring the same

thing

.70 to .99 Redundant (?) Proceed with caution: are the two variables testing the same thing?

1.00 Perfect Relationship. If we the know the independent variable, we can predict the dependent variable with

absolute success. 

Rough Guidelines for Aggregate DataMAGNITUDE OF ASSOCIATION QUALIFICATION COMMENTS

0.00 No RelationshipKnowing the independent variable does not reduce the number of errors in

predicting the dependent variable at all.

.00 to .30 Not useful, very weak Not Acceptable

.30 to .50 Weak  Minimally acceptable

.50 to .70 Fairly Strong  Acceptable

.70 to .85 Strong  Good Work

.80 to .90 Very Strong/Worrisomely StrongEITHER an excellent relationship OR the two variables are measuring the same

thing

.90 to .99 Redundant (?) Proceed with caution: are the two variables testing the same thing?

1.00 Perfect Relationship. If we the know the independent variable, we can predict the dependent variable with

absolute success. 

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

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.

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

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.

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!

Webstats Output                      - -  Correlation Coefficients  - -

             Q375A1     Q305       Q375A3     Q1005

Q375A1       1.0000      .2916      .5320     -.3163            (  686)    (  666)    (  667)    (  672)            P= .       P= .000    P= .000    P= .000

Q305          .2916     1.0000      .2679     -.1272            (  666)    ( 2776)    (  660)    ( 2721)            P= .000    P= .       P= .000    P= .000

Q375A3        .5320      .2679     1.0000     -.2020            (  667)    (  660)    (  682)    (  666)            P= .000    P= .000    P= .       P= .000

Q1005        -.3163     -.1272     -.2020     1.0000            (  672)    ( 2721)    (  666)    ( 3181)            P= .000    P= .000    P= .000    P= .  

Coefficients (Pearson’s r)

N

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.

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