LAB 8: Correlation

In the lab, we will examine,

• What are relationships between variables
• How to interpret scatterplots
• How to use correlation coefficients: Pearson's r and Spearman's r_s
• How to test for significant correlation using r and r_s
• Cautions with the use of correlation
• Critical values for r and r_s are listed at the end of the lab

Recommended Web Links:
Interpreting Correlation Results: Background information on correlation from GraphPad
Correlation in Finland: Move your cursor over the sliding scale and see how the scatter plot changes.
 
Two correlation applets from the University of Illinois:

• Put Points: Click your mouse inside the empty scatter plot to add points and see what correlation you get. This is a good place to see the effect of outliers and other problems.
• Guess Correlation: Click New plots and four scatter plots appear. Test your skill by matching the plot to the Pearson's r value.
   
  

Introduction to Relationships

In this lab, we will examine the relationship or 'association' between two variables in a sample. Our questions are: what kind of relationship exists between the variables? How strong is this relationship?
 

Describing Relationships

There are two ways to summarize the relationship between two variables:

1. Graphically using scatterplots (introduced in Lab 1)
2. Numerically using a correlation coefficient.
    
   

1. Scatterplots

These graphs are an excellent way to get a sense of the relationship that exists between two variables. Scatterplots are usually based on the Cartesian coordinate system, where one variable is represented on the vertical or Y-axis and the other is represented on the horizontal or X-axis. The resulting pattern of points in the scatterplot shows the relationship between the variables, which can be described in terms of three criteria:

• strength: weak, moderate, strong
• direction: negative, positive, neutral
• shape: linear, curved, sinusoidal (s-shaped)

Example A: The table below contains 6 observations collected for a botany project. The diagram on the right presents a scatterplot of the data. What kind of relationship exists between tree height and soil moisture?
 

Tree ID Soil moisture (cm/m) Tree height (m) A 9 17 B 11 16 C 13 20 D 16 21 E 17 25 F 19 24

The scatterplot shows that soil moisture and tree height have a positive relationship: as soil moisture increases, tree height also increases. We can confirm this relationship by observing trees in a forest. Trees growing in dry or rocky areas tend to be smaller than trees growing in wetter areas or near creeks.

Example B: In a study on fuel efficiency and vehicle weight, data were collected from 6 vehicles. The data are presented as a scatterplot (below). What kind of relationship exists between these variables?
 

This scatterplot shows that fuel efficiency and vehicle weight have a negative relationship: as vehicle weight increases, fuel efficiency decreases. However, this relationship may be weak as other factors - driving style, engine size, cargo weight - also affect fuel efficiency. Notice that the points are fairly spread out and do not form a definite line.

What kind of relationship would you expect between the following variables:

1. number of cars per capita and carbon monoxide emissions?
2. stream velocity and bedload particle size?
3. elevation and average air temperature?
    
   

2. Correlation Coefficients

The descriptive terms associated with the scatterplots are useful. However, statisticians often want a more quantitative description of the relationship between two variables. Correlation coefficients and correlation analysis provides a numerical measure of the direction and strength of the relationship. The correlation between two variables ranges from -1 to +1 (see diagram below).

Some examples of correlation coefficients (and the scatterplots) are shown below:

Perfect, negative

Moderate, negative

Weak, positive

Strong, positive

There are two widely used correlation coefficients:

a. Pearson's product-moment correlation coefficient - also called Pearson's r
b. Spearman's rank correlation coefficient
 

A. Pearson's r

Pearson's r is the most commonly used correlation coefficient that measures the linear association between two variables. This coefficient is based on:

1. The variances of X and Y (S²_x and S²_y) which measure the amount of variation in each variable. The variance was introduced in Lab 1. The variances can also be written as S_x and S_y to reduce clutter.
    
   
2. The covariance of XY (S_xy), which measures how X and Y vary or 'change' together.

The variances and co-variance are influenced by the units of the variables (meters, kilograms, mm, etc.) and are difficult to interpret. Dividing the covariance of XY by the variances of X and Y removes the influence of units (this process is called 'standardizing'). The result, r, is a ratio that ranges between -1 and +1.

Formula:

where: Sx and Sy = the variances of X and Y Sxy = the covariance of XY

Note: We will not calculate this statistic by hand in this lab.

Pearson's r requires ratio or interval data that are normally distributed (it is a parametric statistic). For ordinal data or ratio/interval data that is skewed or biomodal, we use Spearman's rank correlation coefficient.
 

B. Spearman's rank ()

Formula:

where: d2 = the squared difference between the ranks for each observation N = number of observations

Example: The table below presents population and area information for 8 regions in a country. The data are presented as ranks from 1 to 8 (1 = smallest ,8 = highest)

Step 1: For each observation, calculate the difference between the ranks
Step 2: Square each difference and sum the squared differences.

Region	Population Rank	Area Rank	d	d2
A	1	1	0	0
B	2	2	0	0
C	3	8	-5	25
D	4	5	-1	1
E	5	3	2	4
F	6	6	0	0
G	7	4	3	9
H	8	7	1	1
Sum of squared differences (d2)				40

Step 3: Fill in the formula using the sum of squared differences = 40 and N = 8:




Note: if you have ratio or interval data that are skewed or bimodal, you can still use Spearman's . However, the ratio data much be converted into ranks before calculating the coefficient.

Spearman's and Pearson's r have the same properties:

• r and range from -1 to +1
• When X and Y both increase, r and are positive
• When X increases and Y decreases (or vice versa), r and are negative.
• When there is no relationship between X and Y, r and are close to 0. In other words, X and Y are uncorrelated.

Correlation coefficients are often applied in exploratory analysis. However, they can also be used to make inferences about population parameters, where (called 'rho') is the population correlation coefficient. The hypothesis tests for r and are summarized below.
 

 

Testing for Significant Correlation

Pearson's r

• two tailed, : 0 (significant correlation; may be negative or positive)
• one tailed (upper), : > 0 (significant positive correlation)
• one tailed (lower), : < 0 (significant negative correlation)

Generally, if we do not have a strong theory for the direction of the relationship, we use non-directional hypotheses. But if we know enough about the variables and can expect a certain direction of relationship, we can use directional hypotheses.

Assumptions

• The data are interval or ratio
• The variables have a linear relationship
• The data are a random sample of paired observations
• The variables are normally distributed (this assumption makes Pearson's r a parametric statistic)

Probability distribution and Test statistic

1. calculate a test statistic (t*) using r and compare t* to a critical value on the student's t distribution
2. compare r to a critical value based on a probability distribution of r values. The r probability distribution has a similar shape to the Z distribution.

Critical values

Decision rule

(upper) reject if r* is greater than
(lower) reject if r* is less than -

Multiple correlations

Spearman's r_s

Assumptions

• The data are ordinal or ratio/interval that have been 'downgraded' to the ordinal scale
• There must be at least five pairs of observations
• The data are a random sample of paired observations.
   
  Note: Both correlation tests assume that the sample is random. In a non-random sample, there may be some other factor(s) influencing the relationship between the two variables. However, often our data are collect in a specific geographic region (all municipalities within a city, all cities within a province, etc.) and our sample is not truly random. You can still use correlation, but you must acknowledge that this assumption may be violated.
   
  If several cities were selected from all possible cities in a province, then the sample is considered random and this assumption is not violated.

Probability distribution and Test statistic

1. calculate a test statistic (Z*) using and compare Z* to a critical value on the normal Z distribution
2. compare to a critical value based on a probability distribution of values.

Critical values

Decision rule

Example

2. Check assumptions

• The population and area data are ordinal ranks
• There are more than 5 observations
• The sample is random

3. State hypotheses
If you have some background information about the relationship, you can establish directional hypotheses. For example, for the relationship between population and area, we may expect a positive relationship (small population in a small region, larger population in a large region). Therefore, we can test for a significant positive correlation. If you are unsure of the type of relationship, you would use a non-directional hypothesis which will test for a significant correlation (it may be positive or negative).

In our example, we expect a positive relationship so we will establish directional (upper tail) hypotheses.

: there is no significant positive correlation between the variables ( = 0)
: there is a significant positive correlation between the variables ( > 0)

5. Select significance level
We will use the standard = 0.05 (95% confidence level)

5. Select probability distribution
Because is our test statistic, we use the probability distribution for our critical values

6. Establish critical value
At = 0.05 and n=8, = 0.643

7. Calculate test statistic
From above, the * = 0.524

8. Compare using decision rule
Rule (for upper tailed hypothesis): reject if * >
* (0.524) is less than (0.643), so we cannot reject

The p-value of 0.21 obtained from SPSS confirms the decision not to reject .

9. State inference
We infer with 95% confidence that there is no significant correlation between population and area in the 8 regions.
 

Cautions with Correlation

Causality and Spurious Correlation

Causality is defined as "the relation between a cause and its effect" (Webster's dictionary). Often, researchers are tempted to believe that a strong correlation indicates a causal relationship between two variables (i.e. variable A causes an effect in variable B).

Example:, a researcher collects housing and age data for various Victoria neighbourhoods. Correlation analysis shows a significant positive correlation between the number of apartment units per km² and the mean age of the population (older people tend to be concentrated in areas with many apartments). Should this researcher infer that high density neighbourhoods cause people to age? Or that older people cause higher population density?

No. There are many factors that influence the age structure of a neighbourhood. Seniors often move into apartments when maintaining a house becomes too much work. Therefore, neighborhoods with many apartments tend to have a larger proportion of seniors. The two variables are related but not linked in a 'cause and effect' way. The relationship may also be invalid if the study included more areas (i.e. the downtown area where the average age is lower).

Important Note: Because correlation analysis (like all other statistical analysis) depends on the data values, you may find significant correlations between any two variables. Example: The scatter plot on the right shows a moderate negative correlation (r = -0.51) between two variables. The catch: the data for these variables were generated using a random number table. The relationship between the two variables is purely numeric.

Aggregation

In geographic research, data are often summarized by aggregating individual observations (people, households, etc.) into groups. Aggregation gives researchers an overall view of patterns in the data. However, because aggregation averages individual observations, the true relationship may be hidden or a false relationship may develop (based purely on numerical association). Different aggregation strategies will often produce different results.
 

Other problems

Five sets of variables all have a correlation of 0.80. However, when you view the scatterplots (below), five very different pictures emerge.

A.	B.	C.
D.	E.

1. This scatterplot is what most people expect with a correlation of 0.80. It clearly shows a strong positive relationship.
    
   
2. An outlier (extreme data value) may influence the correlation analysis. In scatterplot B, the two variables have a correlation of 0.80. However, when the outlier is removed, the correlation is 0.17. In a small dataset, the effects of an outlier may be more severe than in a large dataset.
    
   
   Solution: Check your data to see if the outlier is a data entry error (a decimal place or digit may be wrong). If the outlier is a real measurement (it is not a mistake),you could remove this point and calculate the correlation on the rest of the data. You must be very cautious in this situation. It may be misleading to calculate a correlation using an extreme outlier, but it would also be misleading to report results after you have altered the data. If you re-calculate the correlation, you must report the outlier in your results.
    
   
3. These two variables have a perfect non-linear relationship because the points all line on a smooth line. Often non-linear relationships produce low correlation coefficients because the data 'seem' widely distributed. In scatter plot C, the correlation is 0.80 because the curve is not symmetrical (i.e. the right side of the curve makes the relationship seem strong and linear).
    
   
   Solution: As correlation analysis focuses only on linear relationships, it is not an appropriate way to summarize these non-linear data. If you detect a slight to moderate non-linear trend in your scatterplot, you can use correlation, but you must report the pattern in your research. Statisticians often used advanced techniques (transformations) to remove the non-linear relationship between the variables.
    
   
4. This scatterplot shows two clusters of points. The overall correlation of +0.80 is due to the differences between the two groups, but the correlation within each group is negative (lower group = -0.12 and upper group = -0.44)!
    
   
   Solution: Try and find out why there are two groups and report the within-group correlations. In the sampling design or data collection, one part of the population may have been excluded, produced a 'gap' in the data. Another reason may be that the population contained two distinct groups.
    
   Compare these scatter plots below. Until you conduct further research, you will not know if the 'true' relationship is negative (shown by the within-group correlations [A]) or positive (shown by the overall correlation [B]).

   A.

   B.

   
    
   
5. Two lines in the scatterplot may result from plotting the combined data from two distinctive groups. For example, one line may correspond to men, the other to women.
   Solution: Find out why there are two groups and calculate the correlation for each group. Even if you find no explanation, reporting a single correlation in your research without commenting on this pattern would be misleading.
    
   

 


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Critical values of Pearson's r
	Level of significance () for one tailed test
df	0.05	0.025	0.01	0.005	0.0005
(n-2)	Level of significance () for two tailed test
	0.1	0.05	0.025	0.01	0.001
1	0.988	0.997	1.000	1.000	1.000
2	0.900	0.950	0.980	0.990	0.999
3	0.805	0.878	0.934	0.959	0.991
4	0.729	0.811	0.882	0.917	0.974
5	0.669	0.755	0.833	0.875	0.951
6	0.622	0.707	0.789	0.834	0.925
7	0.582	0.666	0.750	0.798	0.898
8	0.549	0.632	0.716	0.765	0.872
9	0.521	0.602	0.685	0.735	0.847
10	0.497	0.576	0.658	0.708	0.823
11	0.476	0.553	0.634	0.684	0.801
12	0.458	0.532	0.612	0.661	0.780
13	0.441	0.514	0.592	0.641	0.760
14	0.426	0.497	0.574	0.623	0.742
15	0.412	0.482	0.558	0.606	0.725
16	0.400	0.468	0.543	0.590	0.708
17	0.389	0.456	0.529	0.575	0.693
18	0.378	0.444	0.516	0.561	0.679
19	0.369	0.433	0.503	0.549	0.665
20	0.360	0.423	0.492	0.537	0.652
25	0.323	0.381	0.445	0.487	0.597
30	0.296	0.349	0.409	0.449	0.554
35	0.275	0.325	0.381	0.418	0.519
40	0.257	0.304	0.358	0.393	0.490
45	0.243	0.288	0.338	0.372	0.465
50	0.231	0.273	0.322	0.354	0.442
60	0.211	0.250	0.295	0.325	0.408
70	0.195	0.232	0.274	0.302	0.380
80	0.183	0.217	0.257	0.283	0.357
90	0.173	0.205	0.242	0.267	0.338
100	0.164	0.195	0.230	0.254	0.321

Critical values of Spearman's
	Significance level () for one tailed test
df	0.05	0.025	0.01
N	Significance () for two tailed test
	0.1	0.05	0.025
5	0.900
6	0.829	0.886	0.943
7	0.714	0.786	0.893
8	0.643	0.738	0.833
9	0.600	0.683	0.783
10	0.564	0.648	0.746
12	0.506	0.591	0.712
14	0.456	0.545	0.645
16	0.425	0.507	0.601
18	0.399	0.476	0.564
20	0.377	0.450	0.534
22	0.359	0.428	0.508
24	0.343	0.409	0.485
26	0.329	0.392	0.465
28	0.317	0.377	0.448
30	0.306	0.364	0.432




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© University of Victoria 2000-2001     Geography 226 - Lab 8 Developed by S. Adams and M. Flaherty     Updated: September 29, 2001