Correlation• MEASURING ASSOCIATION• Establishing a degree of association

between two or more variables gets at the central objective of the scientific enterprise. Scientists spend most of their time figuring out how one thing relates to another and structuring these relationships into explanatory theories. The question of association comes up in normal discourse as well, as in "like father like son“.

Scatterplots

A. scatter diagram

A list of 1,078 pairs of heights would be impossible to grasp. [so we need some method that can examine this data and convert it into a more conceivable format]. One method is plotting the data for the two variables (father's height and son's height) in a graph called a scatter diagram.

B. The Correlation CoefficientThis scatter plot looks like a cloud of points which

visually can give us a nice representation and a gut feeling on the strength of the relationship, and is especially useful for examining outliners or data anomalies, but statistics isn't too fond of simply providing a gut feeling. Statistics is interested in the summary and interpretation of masses of numerical data - so we need to summarize this relationship numerically. How do we do that - yes, with a correlation coefficient.

The correlation coefficient ranges from +1 to -1

r = 1.0

r = .85

r = .42

R = .17

R = - .94

R = - .54

R = - .33

• Computing the Pearson's r correlation coefficient

• Definitional formula is:

Convert each variable to standard units (zscores). The average of the products give the correlation coefficient. But this formula requires you to calculate z-scores for each observation, which means you have to calculate the standard deviation of X and Y before you can get started. For example, look what you have to do for only 5 cases.

Dividing the Sum of ZxZy (2.50) by N (5) get you the correlation coefficient = .50

• The above formula can also be translated into the following – which is a little easier to decipher but is still tedious to use.

yxSSSS

SPr

))(( YYXXSP

2)( XXSSx

2)( YYSS y

• Or in other words …..

22YYXX

YYXXr

• Therefore through some algebraic magic we get the computational formula, which is a bit more manageable.

2222 YNYXNX

YXNXYr

Interpreting correlation coefficients• Strong Association versus Weak

Association: strong: knowing one helps a lot in predicting the other. Weak, information about one variables does not help much in guessing the other. 0 = none; .25 weak; .5 moderate; .75 < strong

• Index of Association• R-squared defined as the proportion of the

variance of one variable accounted for by another variable a.k.a PRE STATISTIC (Proportionate Reduction of Error))

Significance of the correlation

• Null hypothesis?

• Formula:

• Then look to Table C in Appendix B

• Or just look at Table F in Appendix B

21

2

r

Nrt

Limitations of Pearson's r

• 1) at best, one must speak of "strong" and "weak," "some" and "none"-- precisely the vagueness statistical work is meant to cure.

• 2) Assumes Interval level data: Variables measured at different levels require that different statistics be used to test for association.

• 3) Outliers and nonlinearity• The correlation coefficient does not always give a true

indication of the clustering. There are two main exceptional cases: Outliers and nonlinearity.

r = .457 r = .336

4. Assumes a linear relationship

0

10000

20000

30000

40000

50000

60000

0 5 10 15 20 25 30

Education

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4) Christopher Achen in 1977 argues (and shows empirically) that two correlations can differ because the variance in the samples differ, not because the underlying relationship has changed.

Solution?

Regression analysis

Three Types of Unobtrusive Research

1. Content analysis - examine written documents such as editorials.

2. Analyses of existing statistics.

3. Historical/comparative analysis - historical records.

What is Content Analysis?

• Study of recorded human communication

• Topic Appropriate for CA– “who says what, to whom, how, and with

what”– Effects of the Media

ExampleInvestigated the media’s role in framing the welfare

privatization debate with a content analysis of ABC, CBS & NBC evening news & special programs from 1/1/94 to 8/22/96. Specials include Nightline, 20/20 and This Week with David Brinkley on ABC; 60 Minutes, 48 Hours and Face the Nation on CBS.

Searched LexisNexis and the Vanderbilt Television Archives for all transcripts pertaining to the issue of how welfare should be administered, and found 191 stories.

At the time of the study NBC’s transcripts are not available on LexisNexis prior to 1997. Authors searched for stories using the Vanderbilt News Archives and then purchased pre-1997 transcripts from Burrell’s Transcripts.

Coding, Counting and Record Keeping

• Unit of Analysis

• Manifest vs. Latent Content coding

• Analysis:– Counting– Qualitative evaluation

Coding: Pro-Privatization Frames

CAUSE OF PROBLEM/PROBLEM/SOLUTION9. Delivery / dependency / faith-based10. Delivery / economic costs / faith-based11. Delivery / dependency / non-profits12. Delivery / econ. costs / non-profits13. Delivery / dependency / for-profits14. Delivery / econ. costs / for-profits16. Gen govt / dependency / faith-based17. Gen govt / econ. costs / faith-based18. Gen govt / dependency / non-profits

Coding: Anti-Privatization Frames

CAUSE OF PROBLEM/PROBLEM/SOLUTION3. Privatization / job loss / don’t privatize4. Privatization / job loss / don’t devolve5. Privatization / accountability / don’t privatize6. Privatization / accountability / don’t devolve11. Secular / job loss / don’t privatize12. Secular / job loss / don’t devolve13. Secular / accountability / don’t privatize

Hypothesis & Findings

• Authors hypothesized that mainstream (corporate owned) media would be biased toward privatization.

• Findings did not support such a hypothesis. Media coverage was remarkably balanced (with slight leaning against privatization)

Strengths of Content Analysis

• Economy of time and money.• Easy to repeat a portion of the study if

necessary.• Permits study of processes over time.• Researcher seldom has any effect on the

subject being studied.• Reliability.

Weaknesses of Content Analysis

• Limited to the examination of recorded communications.

• Problems of validity are likely.

Analyzing Existing Statistics

• Can be the main source of data or a supplemental source of data.

• Often existing data doesn't cover the exact question.

• Reliability is dependent on the quality of the statistics.

• Examples: Census data, Crime Stats

Analyzing Existing Statistics

• Can be the main source of data or a supplemental source of data.

• Often existing data doesn't cover the exact question.

• Reliability is dependent on the quality of the statistics.

• Examples: Census data, Crime Stats

Problems with Existing Statistics

• Problems with Validity– What’s available v. what is needed

• Problems with Reliability– Moreno Valley Example