Lecture 8

Question 1

What is correlation?

Answer 1

A measure of the strength of the relationship or association between two variables.

Question 2

How do we know if a statistical association between X and Y exists?

Answer 2

Variability in one variable leads to (affects, causes, or overlaps with) variability in the second variable.
Therefore, one variable can be used to explain SOME portion of the variability of the other variable.
In other words, having information on one variable decreases the variability of the other variables.

Question 3

What is a more inefficient way of predicting scores?

Answer 3

The predicted score for the ith person is the mean.

There is a lot of uncertainty with this estimation.

Question 4

If a relationship between X and Y exists…

Answer 4

we can use the information about X to decrease the uncertainty in our prediction of Yhati.

Question 5

How do we use the information about X to decrease the uncertainty in our prediction of Ycarroti?

Answer 5

First, we group the Y scores according to the X values.

Next, we could use the mean of the raw scores (Y) each treatment group (X) to establish Yia.

i.e. Use Ybara to establish an estimate of the raw score of a person in that group.

Question 6

Example
If the average range (variability) in Y given X equals 6, how many points of Y’s variability is NOT attributable to x? Why?

Answer 6

6!

i. e. it’s residual
- if 6 of the 16 points of variability in Y is not due to X, then 10 points in the variabilty in Y MUST be due to X.

i.e. 16-6=10
i.e. variability in Y that is attributable to X = total variability (total variabilty in Y not attributable to X).
Thus, when trying to predice Yi (an individual raw score), if we use our knowledge of X we reduce variability by 10 points.
i.e. we reduce our uncertainty

Question 7

Give the correlation ratio

Relate this to the example

Answer 7

η² (this is the greek letter eta) = variability in Y common to X / total variability in Y
Variability in Y (sums of squares between) and “common to” means attributable to

In our example:
10/16 = .63 –> Had to use variance instead of range as our measure of variability, η² = .74

Question 8

Actually, η² = SSbetween/SStotal

Describe!

Answer 8

• Is measure of the strength of the relationship between X & Y.
• Often used after F tests to determine practical significance –> i.e. is a measure of effect size

Question 9

Give the limitations of η² as a measure of effect size:

Answer 9

1. Relaibilty of variables restricts the magnitude of η²;
2. The more homogenous the population, the smaller η² –> restriction of range;
3. The magnitude of η² is affected by the number of levels of X;
4. Does not indicate the form of the relationship;
5. It is UNSTABLE –> it varies a lot from sample to sample –> therefore descriptive stat only, not inferential.

Question 10

Due to the limitations of η², what do we focus on?

Answer 10

The Pearson Product-Moment Correlation Coefficient (r).

Question 11

What is the Pearson Product-Moment Correlation Coefficient (r)?

Answer 11

Measures the degree direction of a linear relationship or association of two variables.

Question 12

Give the conceptual formula for r

Answer 12

r = degree to which X & Y vary together/degree to which X & Y vary seperately
i.e.
r = Covariability of X & Y/Variability of X & Y seperately

Question 13

what does it mean if r (correlation) is big?

Answer 13

It means that most of the variabilty of X & Y is due to how X & Y co-vary.

• If we have a perfect linear relationship, every cange in X is accomplished by a corresponding change in Y (and vice versa)
• If r = 0 (i.e. no relationship), a change in X does not correspond to a predictable change in Y.
  i. e. they don’t co-vary

Again: rxy = COVxy/σhatxσhaty

Question 14

Close up of the numerator of (COVxy(σhatxy):

give equation in words

Answer 14

Sum of the cross-products (SP - Sum of products of deviations) of the deviation scores divided by n-1

In other words, COVxy = average sum of the cross-products of the deviation scores of the 2 variables.

Question 15

Give the conceptual formula for Sum of products

Answer 15

Sum(Xi-Xbar)(Yi-Ybar) = SumXY - SumXSumY/N

Question 16

Note: Conceptual:
SS = (Y - Ybar)(Y - Ybar)
SP = Sum(X - Xbar)(Y - Ybar)
Computational:
SS = SumYY - SumYSumY/N
SP = SumXY - SumXSumY/N

Sum of squares (SS) uses squares and sum of products usees cross products

Answer 16

Note: Conceptual:
SS = (Y - Ybar)(Y - Ybar)
SP = Sum(X - Xbar)(Y - Ybar)
Computational:
SS = SumYY - SumYSumY/N
SP = SumXY - SumXSumY/N

Sum of squares (SS) uses squares and sum of products usees cross products

Question 17

When will we get large positive values for SP?

Answer 17

• Largest positive (Xi - Xbar) paired with largest positive (Yi - Ybar)
• Second largest positive (Xi - Xbar) <– deviation on Y
• Largest negative (Xi - Xbar) is paired with largest negative (Yi - Ybar)
• Second largest negative (Xi - Xbar) is paired wit second largest (Yi - Ybar)
  etc. ..

Question 18

Seeing from the largest possible values for SP, what does Covariance of X and Y (COVxy) measure?

Answer 18

The strength and direction of the relationship, independent of the number of scores (because we divide by N-1 which is our degrees of freedom).

Question 19

Why can’t we compare covariances from different scales?

Answer 19

Covariance of X and Y depends on the variability of X and Y, which depends on the measurement scales, therefore can’t compare covarianes from different scales. Also, differences in Covariance of X and Y may be due to differences in σcarrot. Therefore, X and Y must be correccted for differences in σcarrot.

Question 20

How do we correct for differences in σcarrot?

Answer 20

Accomplished by dividing COVxy (covariance of X and Y) by the product of σcarrotX and σcarrotY
i.e. σcarrotXσcarrotY (the denominator)

Question 21

Why does dividing COVxy by σcarrotXσcarrot Y correct for differences in σcarrot for r?

Answer 21

Because we divide by σcarrotX and σcarrotY, r is independent of the dispersions of the two variables and is a dimensionless index of a linear relationship –> i.e. is not dependent on the unit of measurement or variability of either X or Y.

Question 22

Z = X - Xbar/σ oh! r = Σ|(X-Xbar)/σcarrotX|(Y - Ybar)/σcarrotY|(numerator divided by n-1).
i.e. Σ2X2Y/n-1

Answer 22

Z = X - Xbar/σ oh! r = Σ|(X-Xbar)/σcarrotX|(Y - Ybar)/σcarrotY|(numerator divided by n-1).
i.e. Σ2X2Y/n-1

Question 23

What does r equal if X and Y are z scores?

Answer 23

r = σXY

Question 24

What does the equation r = Σ|(X-Xbar)/σcarrotX|(Y - Ybar)/σcarrotY|(numerator divided by n-1) mean?

Answer 24

That r is independent of the number of scores and variability.

Question 25

Describe how transformations affect r

Answer 25

r is unaffected by transformations.
Thus, r is identical if computed between:
- raw scores
- z scores
- if added or subtracted a constant to each raw score
- If multiplied each raw score by a constant
- if dividing each raw score by a constant
(These are all linear transformations

Question 26

what happens when we square r?

Answer 26

It results in a ratio of variance
i.e. r²xy = shared variance/total variance aka common variance/total variance and sooo many others (review briefly in notes)

Question 27

What do we get when we multiply r²xy by 100?

Answer 27

The percentage of σ² in Y accounted for b (knowing) X.

Question 28

What do we get when we minus 1 from r²xy?

Answer 28

1 - r²xy = proportion of σ² in Y NOT accounted for by X.

Question 29

What is r² another measure of?

Answer 29

effect size.

Question 30

Why is r² another measure of effect size?

Answer 30

It is the proportion of variance in Y attributable to X.
E.g. say r = .2, so r² = .04. Who cares? This is too small. Say Y is memory, and X is coffee consumption. So we understand 4% of memory by knowing someone’s coffee consumption.

Question 31

What can we do if there is a statistical association between two variables?

Answer 31

We can use this information to make predictions about one variable, or test hypotheses about group difference.
This brings us to Regression.

Question 32

Assumption: Yia = Y.. + alphaa + eia is true.

Answer 32

i.e. this linear model is correct.

Question 33

Is it an assumption of ANOVA that X and Y are linearly related?

Answer 33

“NO!” underliiiiined (Two points on final).