Correlation

1. Correlation • Chapter 9 of Howell • Relational research • Aim: Is there a relationship between 2 variables in the population? • Method: measure two variables per subject, compare them • We need analysis tools to investigate relational hypotheses

2. Looking for relationships • How do we decide if two variables are “related” • What is a relationship? • Think of things that are related • Rain and cold (always happen together) • learning and performance • What do these have in common?

3. Related variables • Rain and cold • the colder it gets, the more it rains OR • the lower the temperature, the more it rains • Learning and performance • the more you learn, the better you perform OR • the higher the learning, the higher the performance

4. Working with two variables • Imagining 2 vars at once can be confusing • Draw a picture (maybe show relationship) • Show both vars at once • Scatterplot shows this • x axis shows one variable (the IV) • y axis shows the other variable (the DV)

5. Drawing a scatterplot • We need a measurement for both vars for each subject • Example: hours spent studying and 206f mark • Subject Hours Mark • 1 3 55 • 2 4 57 • 3 3 60 • 4 2 75 • 5 6 65

6. Drawing a scatterplot • Step 1: create a set of axes • one var on each axis • Step 2:for each subject, draw a point which shows its values for x and y values. • Check: You will have drawn one dot per subject (some dots might overlap)

7. Drawing our example • Step 1: draw a set of axes (labelled & scaled) 75 60 M A R K 45 30 15 Hours 1 2 3 4 5 6 7

8. Drawing our example • Step 2: draw subject 1 75 60 Subject 1: Draw a dot where hours = 3 and mark = 55 M A R K 45 30 15 Hours 1 2 3 4 5 6 7

9. Drawing our example • All the dots are drawn 75 60 M A R K 45 30 15 Hours 1 2 3 4 5 6 7

10. A “real” scatterplot (n = 100) Can you see any trends in the data?

11. Looking for trends in the picture • We can examine the scatterplot to look for trends • We are looking for one of three trends: Downwards (negative) Even (zero) Upwards (positive)

12. Positive trends • Imagine a balloon around the data - is it vaguely pointing upwards? • Tells us: low values of x are associated low values of y AND high values of x are associated with high values of y • Low x -> Low y • High x -> High y

13. Negative trends • Is the balloon around the data vaguely pointing downwards? • Tells us: low values of x are associated high values of y AND high values of x are associated with low values of y • Low x -> High y • High x -> Low y

14. Zero trends • Is the balloon around the data vaguely horizontal? • Tells us: No pattern - high values associated with both high and low values • Low x -> Any y • High x -> Any y • (No trend!)

15. Identifying the type of trend • To decide - draw a balloon around the data, see which way it slopes: • Sloping upwards - postive • Sloping downwards - negative • Horizontal - no relation

16. Problems with sausages & balloons • Look at these graphs (and their balloons) Flat (no trend) These are obviously different relationships (or…. Are they?) Slopey upwards (positive trend)

17. Lying with scatterplots • The axis scale can hide/emphasise the slope of the data • small differences are hidden by large scales • small differences are emphasised by large scales • Because we focus on the picture rather than the data, it is easy to be fooled! • We need a “crook-proof” method for detecting relationships

18. Co-variance • Essence of a relship: 2 variables, each exhibiting variance • variation in both temperature and rainfall • BUT they tend to vary together (change together) • As one changes, the other changes also • This behaviour is known as covariance • two variables variances are “tied” to some degree • Expressed as a number (eg. cov = 245)

19. Direction of covariance • Relationships can be positive or negative • Positive: high x implies high y and vice-versa • Negative: high x implies low y and vice-versa • We express this “direction” as the sign of the covariance number • pos relationships have pos number (eg. cov = 200) • neg ones have neg numbers (eg. cov = -200)

20. Strength of relationships • Relationship between calorie intake and weight • For some people: positive relationship (less calories means less weight; more calories, more weight) • But for some, it doesn’t work (less calories, same weighr!!) • This is a weak relationship • only works some of the time • The stronger a relationship, the more people it occurs in

21. Magnitude of covariance • The sign of the covariance tells us about the direction of relationship • The magnitude of the number tells us about the strength of the relationship • ignore the minus sign • a higher cov value means a stronger relationship • eg. “cov = -350” is a stronger relationship than “cov = 220” • The actual value of cov means nothing • similar to variance values - funny units!

22. Pearson’s Product Moment • A different way to express covariation is to use Pearson’s product moment (“r”) • Uses nice units, can compare across variables • sometimes incorrectly referred to as “correlation” • Pearson’s product moment is a standard measure (easy to interpret units)

23. Understanding r • It is written as a single number • eg. r = 0.354 • But is has 2 parts!! • A sign (+ or -) • A magnitude (the number if you ignore the sign) • The sign of r is simply the direction of the relationship • a plus: positive relationsip • a minus: negative relationship

24. Understanding r • The magnitude of the sign gives a rough idea of the strength of the relationship • remember: ignore the sign! (look only at the number) • 0 means no relationship at all • 1 means a perfect, super strong relationship • Values in between mean varying strength • eg. 0.3 is a weaker relationship than -0.8 (ignore the sign!!) • Remember: “strength of relationship” simply means “how many people will it happen for”

25. Linking r & scatterplots r = 0.07 r = 0.3 r = 0.97 r = 0.76

26. Scatterplots & r • A low r means the dots are widely scattered • High r means the dots cluster close by, forming a line • The direction (sign of r) is simply the slope of the line (up or down) r = 0.97 r = -0.97

27. Direction of relationship • The direction of a relationship is not too important • Tells us more about the scales used than the data • Consider this: we can correlate cold to rain, or heat to rain • Cold to rain: positive relationship • Heat to rain: negative relationship • Because heat & cold are opposites • When you see a neg relationship, think about the scale used

28. Statistical Significance of r • r simply tells you about the sample • need to test its significance to tell if it applies to the population • We test the Ho that r=0 (no relationship in the population) • Use the usual hypothesis testing strategy

29. Statistical significance of r • If Ho is false, then the relationship we found also applies in the population • Computer will give you a p value, so it is simple to test • if p is less than your alpha, reject Ho - the relationship exists in the population

30. Strength of r: accurately • r allows us to look at relationships differently • How closely tied are x and y actually? • What proportion of the variance of y is actually because of x? • To what extent are the scores of y being contributed to by x?

31. Example • Think of somebody’s salary (R4000) • a part of that is due to their education • a part of it is due to the specific employer • a part of it is due to the person’s personality • We can ask: what proportion of that person’s salary is due to their education? • 10%? 50%? 90%?

32. Covariance again • It is the same with two co-varying variables • Some of the variable’s variance will be due to the other var • Some will be due to other factors • r allows us to accurately pin down this proportion

33. Working out the proportion: R2 • To find this proportion simply square your r value • just r2, multiply by 100 • It is actually written with a capital: R2 • Eg. if r = 0.6, then R2 is • 0.6 x 0.6 = 0.36 • 0.36 x 100 = 36 • 36% of the variance of y is due to x