Welcome to the Fun Part!

• Very well done for all your hard work so far!

  • The concepts we have covered are complex and difficult

  • Mastery takes time and practice

  • You have all made an excellent start!!!!

• We will now begin putting these ideas into practice

  • Much less new information!

  • Applying the same ideas to different research questions/scenarios

• Finally at the confluence of your stats knowledge and R skill

• Let's get started!

Looking Ahead

• This week: Correlation

• Week 5: Chi-Square

• Week 6: t-test

• Week 7: The Linear Model

• Week 8: The Linear Model

What About the Lab Report?

• We will not start the lab report for a couple more weeks

  • Don't think about starting now - you can't!

• We will talk about the lab report in the lectures and work on it in the practicals

  • Make sure you come to your registered sessions

Objectives

After this lecture you will understand:

• The concepts behind statistical correlation

• How to interpret the values of the correlation coefficient r

• How to read a correlation matrix

• How to interpret and report significance tests of r

• The relationship between correlation and causation

Distributions, Test Statistics, and NHST

• Everything from the past few weeks we will now put into action!

• For each statistical analysis, we will have the same ingredients:

  • Data, from which we calculate...

  • A test statistic that represents the relationship of interest, which we compare to...

  • The distribution of that test statistic under the null hypothesis to get...

  • The probability p of getting a test statistic as large as the one we have (or larger) if the null hypothesis is true

Overall Reminder

• We want to believe true things about the world, and disbelieve false things

  • More accurately: we should believe things that are well-founded in reliable evidence, and disbelieve things that are not

• Statistics is a system to help us make decisions about whether, and to what degree, we believe something is supported by evidence

Correlation

• Essential question: how do two variables change in relation to each other?

• When one variable changes, does the other...

  • Change in a similar way?

  • Change in the opposite way?

  • Not change very much at all?

• In other words: to what degree do two variables behave the same way?

Correlation

• Quantifies the degree and direction of a relationship

• Typically used with two (or more) continuous variables

  • Can be used when one is categorical!

• Today's example: Gender and Sexuality data from the questionnaire

Correlation: Visualisation

Correlation

• People who gave high ratings for femininity tended to give low ratings for masculinity, and vice versa

• We might like to know:

  • How strong is this relationship?

  • Should we believe that it's real (ie representative of people/first-year psychology students in general?)

Correlation: Interpretation

• We can quantify the strength and direction of the relationship between femininity and masculinity with Pearson's correlation coefficient r

• Values range from -1 (perfect negative) through 0 (no relationship) to 1 (perfect positive)

Strength

• Absolute value of r between 0 and 1

  • 0: no relationship at all
  • 1: perfect relationship

Direction

• Whether the value of r is positive or negative

  • Positive: as one variable increases, the other tends to increase
  • Negative: as one variable increases, the other tends to decrease

Correlation: Let's Try It!

gensex %>% 
  select(Gender_fem_1, Gender_masc_1) %>% 
  cor(method = "pearson")

##               Gender_fem_1 Gender_masc_1
## Gender_fem_1     1.0000000    -0.7563823
## Gender_masc_1   -0.7563823     1.0000000

• So, our correlation coefficient r is -.76

• POP QUIZ: How can we interpret this?

Correlation: Interpretation

• The negative sign (-) means as femininity increases, masculinity tends to decrease (and vice versa)

• The absolute value of .76 is very strong - quite close to 1!

Correlation: Significance

• We now have our data, from which we calculated...

• Our test statistic r (-.76)

• We also know the distribution of r with different degrees of freedom

  • Or, rather...of t, for Reasons (TM)

• We can now ask how likely we are to get a value of -.76 (or larger) if in fact femininity and masculinity have a true r of 0

  • i.e. the null hypothesis is in fact true

  • We will use the standard significance level of .05 in this case

Correlation: Significance

## 
##     Pearson's product-moment correlation
## 
## data:  Gender_fem_1 and Gender_masc_1
## t = -20.128, df = 303, p-value < 0.00000000000000022
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
##  -0.8006746 -0.7038665
## sample estimates:
##        cor 
## -0.7563823

We can report this as: "There was a significant negative correlation between femininity and masculinity, r(303) = -.76, p < .001."

Correlation Matrices

• Correlations are often presented in matrices

• Each cell contains the correlation coefficient r for the variables in the corresponding row and column

• POP QUIZ: Why is there a diagonal line of 1s?

##             comfortable  masc   fem stability
## comfortable        1.00 -0.31  0.17      0.61
## masc              -0.31  1.00 -0.76     -0.28
## fem                0.17 -0.76  1.00      0.18
## stability          0.61 -0.28  0.18      1.00

Correlation Matrices

• More useful version with GGally::ggscatmat()

• Scatterplots, distributions, and r values

Correlation = Causation?

• Our analysis showed that higher ratings of femininity tended to correspond to lower ratings of masculinity, and vice versa

• Can we conclude from this that being more feminine causes you to be more masculine?

Correlation ≠ Causation!

• Why not? :(

• No distinction between cause and effect

  • Which is the chicken and which is the egg?

  • Which came first: femininity or masculinity?

• No experimental manipulation (randomisation)

• The problem of tertium quid - a third variable that influences both the variables you're actually measuring

Consider This...

• Consider the number of hours per day you and a friend on this course spend studying

  • Both tend to study less on similar days (e.g. the weekend)

  • Both tend to study more on similar days (e.g. right before an assessment is due)

  • So, you and your friend's hours studying are likely to be highly correlated

  • Does this mean that you studying more (or less) causes your friend to study more (or less)?

Consider This...

• Of course not! Which of you "causes" the other to study more/less?

• Tertium quid: An unmeasured third factor that influences both of you

  • In this case: being on the same course

• Some sources of variation:

  • Differences in experience or interest

  • Which electives you're each taking

  • Friends and family obligations

  • Part time work

More Examples

Say It With Me

CORRELATION DOES NOT
IMPLY CAUSATION

Correlation: VOCAB ALERT!

• In common language, "correlated" means "related to in some way, usually causally"

• In statistics-ese, it means "the (standardised) degree to which two or more variables covary", ie change in relation to each other

• "Correlation" is a technical term!

  • In your reports, do not say two things are "correlated" unless you report r as evidence!

  • Instead: variables "have a relationship"/"are related to each other"

Consider This

Correlation: Summary

• The correlation coefficient r quantifies the strength and direction of relationships between variables

• The p-value associated with r is the probability of encountering a value of r as large as the one we have, or larger, if in fact the true value of r in the population is 0

• Correlation DOES NOT IMPLY CAUSATION!!!!!!!

• More practice with interpreting r with this fun little game

A Few Reminders

• Recognise people who have helped your or others by nominating them for a SavioR award

• Give us feedback, ideas, or suggestions in the Suggestion Box

• Don't try to go it alone!

  • Ask to study with practical teams, friends on the course

  • Set up Zoom calls to work on the tutorials together

  • Be the change you wish to see in the world 😄

Looking Ahead

For the Quiz

• Revise all new definitions/concepts (see previous slide)

• Revise how to read the output of GGally::ggscatmat() and cor.test()

  • More practice in the tutorial!

• Do NOT need to memorise function names or syntax

Next Time

• Comparing frequencies with Chi-Square (χ²)