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Correlation. Review and Extension. Questions to be asked…. Is there a linear relationship between x and y? What is the strength of this relationship? Pearson Product Moment Correlation Coefficient (r) Can we describe this relationship and use this to predict y from x? y=bx+a

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### Correlation

Review and Extension

Questions to be asked…

- Is there a linear relationship between x and y?
- What is the strength of this relationship?
- Pearson Product Moment Correlation Coefficient (r)

- Can we describe this relationship and use this to predict y from x?
- y=bx+a

- Is the relationship we have described statistically significant?
- Not a very interesting one if tested against a null of r = 0

Other stuff

- Check scatterplots to see whether a Pearson r makes sense
- Use both r and r2 to understand the situation
- If data is non-metric or non-normal, use “non-parametric” correlations
- Correlation does not prove causation
- True relationship may be in opposite direction, co-causal, or due to other variables

- However, correlation is the primary statistic used in making an assessment of causality
- ‘Potential’ Causation

Possible outcomes

- -1 to +1
- As one variable increases/decreases, the other variable increases/decreases
- Positive covariance

- As one variable increases/decreases, another decreases/increases
- Negative covariance

- As one variable increases/decreases, the other variable increases/decreases
- No relationship (independence)
- r = 0

- Non-linear relationship

Covariance

- The variance shared by two variables
- When X and Y move in the same direction (i.e. their deviations from the mean are similarly pos or neg)
- cov (x,y) = pos.

- When X and Y move in opposite directions
- cov (x,y) = neg.

- When no constant relationship
- cov (x,y) = 0

- Covariance is not very meaningful on its own and cannot be compared across different scales of measurement
- Solution: standardize this measure
- Pearson’s r:

Factors affecting Pearson r compared across different scales of measurement

- Linearity
- Heterogeneous subsamples
- Range restrictions
- Outliers

Linearity compared across different scales of measurement

- Nonlinear relationships will have an adverse effect on a measure designed to find a linear relationship

Heterogeneous subsamples compared across different scales of measurement

- Sub-samples may artificially increase or decrease overall r.
- Solution - calculate r separately for sub-samples & overall, look for differences

Range restriction compared across different scales of measurement

- Limiting the variability of your data can in turn limit the possibility for covariability between two variables, thus attenuating r.
- Common example occurs with Likert scales
- E.g. 1 - 4 vs. 1 - 9

- However it is also the case that restricting the range can actually increase r if by doing so, highly influential data points would be kept out

Effect of Outliers compared across different scales of measurement

- Outliers can artificially and dramatically increase or decrease r
- Options
- Compute r with and without outliers
- Conduct robustified R!
- For example, recode outliers as having more conservative scores (winsorize)

- Transform variables

What else? compared across different scales of measurement

- r is the starting point for any regression and related method
- Both the slope and magnitude of residuals are reflective of r
- R = 0 slope =0

- As such a lone r doesn’t really provide much more than a starting point for understanding the relationship between two variables

Robust Approaches to Correlation compared across different scales of measurement

- Rank approaches
- Winsorized
- Percentage Bend

Rank approaches: Spearman’s rho and Kendall’s tau compared across different scales of measurement

- Spearman’s rho is calculated using the same formula as Pearson’s r, but when variables are in the form of ranks
- Simply rank the data available
- X = 10 15 5 35 25 becomes
- X = 2 3 1 5 4
- Do this for X and Y and calculate r as normal

- Kendall’s tau is a another rank based approach but the details of its calculation are different
- For theoretical reasons it may be preferable to Spearman’s, but both should be consistent for the most part and perform better than Pearson’s r when dealing with non-normal data

Winsorized Correlation compared across different scales of measurement

- As mentioned before, Winsorizing data involves changing some decided upon percentage of extreme scores to the value of the most extreme score (high and low) which is not Winsorized
- X = 1 2 3 4 5 6 becomes
- X = 2 2 3 4 5 5

- Winsorize both X and Y values (without regard to each other) and compute Pearson’s r
- This has the advantage over rank-based approaches since the nature of the scales of measurement remain unchanged
- For theoretical reasons (recall some of our earlier discussion regarding the standard error for trimmed means) a Winsorized correlation would be preferable to trimming
- Though trimming is preferable for group comparisons

Methods Related to M-estimators compared across different scales of measurement

- The percentage bend correlation utilizes the median and a generalization of MAD
- A criticism of the Winsorized correlation is that the amount of Winsorizing is fixed in advance rather than determined by the data, and the rpb gets around that
- While the details can get a bit technical, you can get some sense of what is going on by relying on what you know regarding the robust approach in general
- With independent X and Y variables, the values of robust approaches to correlation will match the Pearson r
- With nonnormal data, the robust approaches described guard against outliers on the respective X and Y variables while Pearson’s r does not

Problem compared across different scales of measurement

- While these alternative methods help us in some sense, an issue remains
- When dealing with correlation, we are not considering the variables in isolation
- Outliers on one or the other variable, might not be a bivariate outlier
- Conversely what might be a bivariate outlier may not contain values that are outliers for X or Y themselves

Global measures of association compared across different scales of measurement

- Measures are available that take into account the bivariate nature of the situation
- Minimum Volume Ellipsoid Estimator (MVE)
- Minimum Covariance Determinant Estimator (MCD)

Minimum Volume Ellipsoid Estimator compared across different scales of measurement

- Robust elliptic plot (relplot)
- Relplots are like scatterplot boxplots for our data where the inner circle contains half the values and anything outside the dotted circle would be considered an outlier
- A strategy for robust estimation of correlation would be to find the ellipse with the smallest area that contains half the data points
- Those points are then used to calculate the correlation
- The MVE

Minimum Covariance Determinant Estimator compared across different scales of measurement

- The MCD is another alternative we might used and involves the notion of a generalized variance, which is a measure of the overall variability among a cloud of points
- For the more adventurous, see my /6810 page for info matrices and their determinants
- The determinant of a matrix is the generalized variance

- For the more adventurous, see my /6810 page for info matrices and their determinants
- For the two variable situation
- As we can see, as r is a measure of linear association, the more tightly the points are packed the larger it would be, and subsequently smaller the generalized variance would be
- The MCD picks that half of the data which produces the smallest generalized variance, and calculates r from that

Global measures of association compared across different scales of measurement

- Note that both the MVE and MCD can be extended to situations with more than two variables
- We’d just be dealing with a larger matrix

- Example using the Robust library in S-Plus
- OMG! Drop down menus even!

Remaining issues: Curvature compared across different scales of measurement

- The fact is that straight lines may not capture the true story
- We may often fail to find noticeable relationships because our r, whichever method of “Pearsonesque” one we choose, is trying to specify a linear relationship
- There may still be a relationship, and a strong one, just more complex

Summary compared across different scales of measurement

- Correlation, in terms of Pearson r, gives us a sense of the strength of a linear association between two variables
- One data point can render it a useless measure, as it is not robust to outliers
- Measures which are robust are available, and some take into account the bivariate nature of the data
- However, curvilinear relationships may exist, and we should examine the data to see if alternative explanations are viable

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