16- Association and Correlation II

Question 1

What is Covariation?

Answer 1

means “Varying together” or “Varying jointly”

it is the cross product for each observation of deviations from means an

Question 2

Are signs important for Covariance?

Answer 2

Yes, they tell us if it is a high or low covariation as well at the direction (positive or negative).

1) Linear points with positive slope = High positive covariation
2) Lin

Question 3

Characteristics of Covariation

Answer 3

1) Magnitude of covariance is dependant on the units of x and y 2) Mutliple pairs of variables are not directly comparable 3) To be able to compare multiple pairs you would need a standard form of covariance

Question 4

What is Correlation?

Answer 4

A dimensionless measure that indicates the nature and degree of linearity between two variables

It is a standardized form of covariance, ranges from values between -1 t

Question 5

What are the Correlation ranges for interpretation?

Answer 5

1. High Perfect +ve = +1.00
2. Moderate +ve = +0.50
3. Weak +ve = +0.25
4. No Correlation = 0.00
5. Weak -ve = -0.25
6. Moderate -ve = -0.50
7. High Perfect -ve= -1.00

Question 6

What is the Pearson’s Correlation Coefficient?

Answer 6

The covariance of x and y divided by the product of the population standard deviations

Question 7

Purpose of Pearson’s Correlation Coefficient

Answer 7

to help quantify associations that we see on a scatter plot/chart or map

Question 8

Assumptions of Pearson’s Correlation Coefficient

Answer 8

1. Variables must be interval or ratio
2. Data pairs must be selected randomly from the population
3. Relationship between X and Y is linear
4. Constant Variation (homoscedasticity)
5. The variables X and Y must share a joint bivariate normal distribution

Question 9

Pearson’s Correlation Coefficient Values

Answer 9

1. Values -1 and +1 correspond to a strong negative and strong positive linear relationship between the variables X and Y
2. Value of 0 indicates no linear relationship exists between X and Y ( they are independent)

Question 10

Spearman’s Correlation Coefficient

Answer 10

Computes the linear correlation on the ranks of xy

Question 11

Properties of Spearmans Correlation Coefficient

Answer 11

1. Relaxes normality and linearity assumption
2. Data can be ordinal
3. Measures the difference between the ranks
4. Coefficient is interpreted the same way as Pearson’s (value of 0 = No association)

Question 12

Kendalls Correlation

Answer 12

measures the strength of the monotonic relationship between X and Y

It is resistant to the effect of a small number of outliers and is used

Question 13

When to use Spearmans?

Answer 13

When…
* Data Types: analyzing relationships between ordinal, interval and ratio variables
* Assumption of Monotonicity: robust against non-linear relationships however Spearmans is well-suited for detecting monotonic relationships, including both positive and negative monotonic trends
* Larger Sample Sizes: more powerful and efficient when you have larger sample sizes

Question 14

When to Use Kendall’s?

Answer 14

• Data Types: when data is strictly ordinal
• Ties in Data: effective in handling tied values (when two or more data points have the same rank)
• Smaller Sample Sizes: works well with limited data
• When you Want to Emphasize the Relative Order: assesses association based on the orfer or ranking of data points rather than the actual values

Question 15

Possible Issues with Correlations

Answer 15

• Non-Linear Associations
• Correlation does not equal causation
• Spatial Aggregation Impacting Analysis

Question 16

Problem: Non- Linear Association

Answer 16

Occurs when data is clearly associated but not in a linear fashion

solution: to use a transformation on one or both values and then calcula

Question 17

Problem: Correlation Does Not Equal Causation

Answer 17

Association may be the result of interdependency (autocorrelation); happens when there is a correlation of a random variable with itself

Question 18

Problem: Spatial Aggregation

Answer 18

At indiviual level, there may be a high correlation between two variables but at a provinical level this association becomes weak or non-existant

Question 19

Recap: Correlation

Answer 19

• Pearson’s R is most widely used and most useful for linear relationships
• Spearman’s can be used as a non-parametreic equivalent to Pearson’s
• Use Kendall’s when dealing with monotonic, non-linear trends
• Use chi-squared test and Cramer’s V for nominal data presented in a contingency table
• Always plot data before calculating correlations