Lecture 9: Correlation

What is the correlational method

• what is the output

• statistical technique that estimates the strength and sign of a relationship between two variables (X, Y)

  • There are diff types

    • Difference cases in which they are used (Pearson r for linear relationships)

• Output: correlation coefficient

Correlation coefficient

• absolute ranges

• sign means what

• range from 0-1

  • Higher = more stronger relationships

• Sign reflects directionality

  • + = same direction

  • - = difference directions from each other

Interpreting correlation coefficients

• How to describe each

  • 1

  • 0.9

  • 0.5

  • 0

  • -0.5

  • -0.9

  • -1

• The try to find for each graph on the left

  • Which is be more common

• How is correlation affected by sampling

• perfect + correlation

• high + correlation

• low + correlation

• no correlation

• low - correlation

• high - correlation

• perfect - correlation

• will commonly see these

  • bottom right: weak negative

  • Top right: no correlation

• Won’t see these

  • bottom left: weak positive

  • top left: perfect positive

• Low correlations are more commonly to be found .2 or .3

• correlation affected, as less data points are used, and not representative of the population that you are testing

Regression towards the means

• extreme stores (far away from the mean) tend to be followed by less extreme score that will be more closer to the mean (regressing)

Purpose of using correlations (4 reasons)

• prediction (Y from X)

  • given one, you can predict another with a reasonable accuracy

• validation (construct validity)

  • e.g. using test scores to predict behaviour

• reliability (inter-rater reliability)

  • rater 1 and rater 2 should have similar results or else correlation does not work

• Theory verification

The correlational matrix

• looking at the correlation with several others

• e.g. neuroticism is associated with openness and agreableness

The calculation process for the Spearman correlation

• first thing to do first always before calculation

• What the first calculation required: what is it called

  • What is the alternative equation for it

  • Calculate it for the data on the left

• What is the second calculation

  • Calculate it with the data on the left

• What is the final correlation equation

  • Calculate with the data on the left

• graph the data so then see if the values match with the graph and allow you to identify cases where r values are misleading (and outliers you may have to control, outliers affect the correlation)

• Do a sum of products

  • 𝑆𝑃 = Σ (𝑋 − 𝑀𝑋) (𝑌 − 𝑀𝑌)

    • or 𝑆𝑃 = Σ𝑋𝑌 − (ΣXΣY)/n

    • In the example: SP = 2

• The the sum of squares for each variable

  • SS x = 5

  • SS y.= 10

• r = SP / sqrt(SSxSSy)

  • r = 0.28

Calculate the correlation for the data on the left

• ans : 0.875

Correlations are about patterns

• what does adding or subtracting a constant from all X scores affect the correlation

• how does multiplying all x scores (or y) affect the correlation

• How does multiplying all x scores by a negative constance affect the correlation

• No effect (all moves up, but eh placement of the score relative to each other doesn’t change)

• No effect (+, all moves up, and the placement of the scares relative to each other doesn’t change)

• The magnitude of the correlational doesn’t change, but the placement of the scores are opposite so the the sign of the correlation is the opposite - picture on the left

  • If both scores are multiplies by a negative constant (- x - = +, there is no change to the correlation)

What about r², what does it tell you compared to r?

• what are two things this is called

• R² is effect size, which explains the variability in X is accounted for by variability in Y

  • Which R² can be explain the other variable not accounted for that affects the relationship

• Called effect size and coefficient of determination

Hypothesis testing

• what are generally looking for in the correlation test

• Two-tailed

• one-tailed

• Alpha threshold

Looking for a result that is more extreme than is probable if there was no effect - null hypothesis

• non-directional two tailed

  • H0: p = 0

  • H1L p does not = 0

• Directional one-tailed - positive shown

  • H0 p <=0

  • H1: P>0

• a threshold: a = 0.05 or 0.01

Test statistic for correlation

p is 0 because of null hypothesis

Practice

• Sample of n = 30 scores, r = 0.35

• Two-tailed, alpha = 0.05

• df = 28, tcrit = 2.048

• what is the t-score and is it significant?

---

• Sample of n = 20 scores, r = 0.79

• One-tailed, alpha = 0.01

• df = 18, tcrit = 2.55

• What is the t-score and is it significant

t = (0.35 – 0) / √ ((1-0.352)/(30-2)) = 1.98

• not significant

---

t = (0.79 – 0) / √ ((1-0.792)/(20-2)) = 5.47

• t > tcrit, this one is significant

five issues with correaltion

• correlation is reserved for linear relationship

• restricted range

• correlation is sensitive to outlier

• correlation is not equal to causation

• correlations tells you nothing about the mechanism

Explain why correlational calculations are not useful for non-linear relationship

• what two characteristics of a non-linear relationships

• they are monotonic (consistent and one direction) but has an asymptote (no loner linear)

• The r value understates the the relationships of the two variables

  • Misrepresentation by correlations (r = 0.00 but still very meaningful)

Restricted range issue

• explain why there is a restricted range, and what for?

• Give an example

• Therefore, what should you avoid?

• the correlation can only be made by a certain range (low to medium scores), but not high scores (only prediction is consistent is a particular range)

• e.g. SAT score and GPA

  • Low scores tends to struggle

  • but higher scores average out (have an asymptote)

• You cannot predict any outcomes outside of the data scores you have (outside the range)

Outliers

• explain outliers to correlations

• e.g. out outlier can even change the direction of the correlation (becomes positive and super strong), especially for smaller samples

• Therefore need to be very careful when making final conclusions (and what to expect)

Correlational does not equal causation

• What is this called?

• What is the multiple possibilities of the correlation

  • three possible things

• post-hoc fallacyL If A preceded B and A+B is correlated, A must cause B

  • Explanation for superstitious and series concern in developmental fields

• Three possible reasons for correlations

  • Is causally related, but directionality and mechanism unclear (how? and which causes what?)

  • The relationship is spurious (meaningless, coincidental)

  • Two things could be related, but there could be a third factor that causes the causation or correlation

    • e.g. ice cream and criminal activity correlated, but both is correlated with temperature

partial correlations: what is it (explain)

• what is the equation

• Practice with the data on the left and R XY given as 0.923

• the correlation of two things, after controlling for another variable (e.g. ice cream and criminal activity and temp, after controlling for temp is obviously 0)

• Practice

  •  rXY = 0.923 (already done)

  • rYZ = 0.961

  • rXZ = 0.961

  • rXY.Z = 0

Spearman alternative correlations

• what data would you use this for?

• What is the equation + assumption need to do beforehand

• the relationship is monotonic but has a asymptote or the r value is not useful based on the original data, but converting the scores into ranks make the relationships much stronger

• Assuming there aren’t ties to the scores, the alternative formula will work (known as the spearman correlation or rs)

What happens if the scores are tied

• take the means of the same ranked scores, and then do calculations as normal

Point-biserial correlation

• what is it

• between one continuous variance and a two category variable dichotomous

• do binary numerical variable and then do the person correlation as normal

The Phi Coefficient: what is it?

• the correlations between two dichotomous variables

• record both into a binary numerical variable

• Do the Pearson correlation