STATS 15- correlation and regression

Question 1

Correlation research design

Answer 1

• Experiments may be impractical/Unethical for some research questions
• “Does cholesterol affect the probability of heart disease”
• “Does smoking shorten peoples life expectancy”
• But we can look for relationships between such variables

Question 2

Relationship between 2 variables

Answer 2

• Is there a relationship between IQ and Exam marks
• Bivariate data- each participant there are 2 different variables measure, we see for any relationships (C.f. within-subject design)

Question 3

Start with a scatter plot

Answer 3

• Height and intelligence
• Data suggests no relationship between height & intelligence

Question 4

Strong positive Correlation, R=1

Answer 4

• strong positive correlation

Question 5

Strong NEGATIVE correlation, R= -1

Answer 5

Question 6

No correlation- Height and intelligence

Answer 6

Question 7

Non-linear correlation

Answer 7

• NB: correlations are not about how steep any slope is but about the variation of values around the slope (how well values fit the slope)
  
  • 1= perfect fit
• Strong positive 0-8
• Strong negative 8-14

Question 8

Correlations: Hypotheses testing

Answer 8

• Null hypothesis
• No relationship between variables X and Y above that expected by chance alone
• NB: Correlations are not about how steep any slope is, but about the variation of values around the slope

Question 9

Measuring the degree of relationship

Answer 9

• Pearson product moment (r)- how well does a straight line fit the data
• r = -1 (perfect negative relationship)-X decreases as Y increases
• r = +1 (perfect positive relationship)-X increases as Y increases
• r = 0 (No linear relationship)
• Is affected by outliers and by number of pairs of data

Question 10

Pearsons

Answer 10

• Assumption
• linear relationship between X and Y
• Continuous random variables
• Both variables must be normally distributed
• X and Y must be independent of each other

Question 11

Measuring the degree of the relationship

Answer 11

• Pearson, r=
• • or - sign indicates = direction of relation
• Value indicates strength. Valye closer to either -1 or +1 reflect a strong correlation valyes close too 0 mean weak/no correlation
  
  • r= -0.65 quite strong negative correlation
  • r= +0.65 equally strong positive correlation
  • P value indicates significance

Question 12

Some real data

Answer 12

• Positive but not very strong relationship
• With quite a lot of variability

Question 13

And if we collect more data

Answer 13

*

Question 14

The null hypothesis (r=0)

Answer 14

• What is the chance that there really isn’t a correlation (r=0)
• OR
• That we got our value of r by chance p<0.05

Question 15

What it all means

Answer 15

• r= +0.701 ; p<0.001 (N=30)
• • 0.701 => How close are the points to a straight line
• p<0.001 =>How likely is it that the true correlation co-efficient is actually zero (no correlation) and we got this r value by chance
• N=30 => how may pairs of points there are

Question 16

Interpret correlations cautiously

Answer 16

• Correlation does NOT imply causality
• Correlations are affected by range restrictions
  
  • E.g. Height… you don’t get many people above 7ft
• Correlations are affected by outliers
• Correlation only measures the degree of LINEAR relationships
  
  • Plot the graph to see if linear

Question 17

Correlation and causation

Answer 17

• Ice cream sales and the number of shark attacks on swimmers are correlated
• The number of cavities in primary school children and vocabulary size has a strong positive correlation
  
  • Cant say vocab causes cavities (Probably due to age)
• The more tvs per citizen the longer the average life expectancy of a country
• Patients operated on by surgeons with cleaner hands live longer

Question 18

Non-parametric alternatives

Answer 18

• Spearman’s Rho
  
  • Spearmans rank correlation co-efficient
  • • Kendall’s tau
  • Cross tabulation, c.f. Chi squared
• Fewer assumptions, robust to outliers, but also less sensitive

Question 19

Advantages of Non-parametric correlation. ranking 1

Answer 19

• Can convert non-linear data to linear- allow us to perform linear statistical test on the data

Question 20

Advantages II

Answer 20

• Less sensitive to outliers
• ranking can distribute the data more evenly

Question 21

Spearman’s Rho (p, rs)

Answer 21

• Tests for a relationship between the ranks of 2 variables
  
  • So put paired variables in tables and rank each one
  • Compare differences in ranks
• E.g. Is there a relationship between age and shoe size

Question 22

Spearman’s Rho- Formula

Answer 22

• d= Difference in rank
• N = Number of participants
• E = sum of

Question 23

Example- Positive correlation

Answer 23

Question 24

Reporting the result

Answer 24

• The Spearman’s Rho test was applied to the data and a significant positive correlation was found between age and shoe size
• (r_s= 0.9, n=5, p<0.05)
• As shoe size increased, age increased
• No implication of causality

Question 25

Another rank correlation: Suitable for contingency table data

Answer 25

• contingency table is when you relate on variable to another
• e.g. age of students v classification of degree they got

Question 26

Why not use chi-squared

Answer 26

• Because three cells have less than 5
  
  • See nonparametric testing lectures
• Can use Kendall’s Tau
• A method for measuring the association between variables in cross tabulations

Question 27

How does Tau work

Answer 27

• If there was a positive correlation between age and classificaiton
• younger = 1st / oldest= 3rd
• we would expect most of the data to fall between the tram lines if it was positively correlated

Question 28

How does it work

Answer 28

• we would expect most of the data to fall between the tram lines if it was negatively correlated
• This doesn’t happen
• If there was a positive correlation between age and classificaiton

younger = 1st / oldest= 3rd

Question 29

How does it work

Answer 29

• Essentially, Kendall’s Tau is a statistic which for each cell compares the number of cases below and to the right of the cell with those above and to the right
  *

Question 30

How does it work

Answer 30

• There is not a similar balance in the data meaning it is not correlated

Question 31

Reporting the result

Answer 31

• Tau tells us both the size and direction (like Spearman’s Rho) of the correlation
  
  • Computers can also compute the significance value for the sample size used
• E.g. A Kendall Tau test for ordered contingency tables suggested no significant relationship between age and degree class (tau =-0.44, N=180, p>0.05

Question 32

Reporting correlations

Answer 32

• Describe data (Including scatterplot)
• Describe relationship in words
• Quote N- if we have a large N we are more likely to see real correlation (or lack of)
• Quote co-efficient (with value of p)- links with how significant our results are

Question 33

Summary of correlations

Answer 33

• Pearsons product moment: r
• Spearman’s Rho: rs- suitable for non-parametric data
• Kendall’s Tau: T- data represented in a contingency table
• When to use each
  
  • What research design
  • Normal data