1. 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. 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. 3 Related variables Rain and cold the colder it gets, the more it rainsOR 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. 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. 5 Drawing a scatterplot We need a measurement for both vars for each subject Example: hours spent studying and 206f mark SubjectHoursMark 1355 2457 3360 4275 5665

6. 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. 7 Drawing our example Step 1: draw a set of axes (labelled & scaled) 15 30 45 60 75 1234567 MARKMARK Hours

8. 8 Drawing our example Step 2: draw subject 1 15 30 45 60 75 1234567 MARKMARK Hours Subject 1: Draw a dot where hours = 3 and mark = 55

9. 9 Drawing our example All the dots are drawn 15 30 45 60 75 1234567 MARKMARK Hours

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

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

12. 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. 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. 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. 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. 16 Problems with sausages & balloons Look at these graphs (and their balloons) Flat (no trend) Slopey upwards (positive trend) These are obviously different relationships (or…. Are they?)

17. 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. 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. 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. 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. 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. 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. 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. 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. 25 Linking r & scatterplots r = 0.07 r = 0.76 r = 0.97 r = 0.3

26. 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 = 0.97r = -0.97

27. 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. 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. 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. 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. 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. 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. 33 Working out the proportion: R 2 To find this proportion simply square your r value just r 2, multiply by 100 It is actually written with a capital: R 2 Eg. if r = 0.6, then R 2 is 0.6 x 0.6 = 0.36 0.36 x 100 = 36 36% of the variance of y is due to x