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Basic Quantitative Methods in the Social

Sciences(AKA Intro Stats)

- 02-250-01
- Lecture 9

Assignment Due and Course Evaluations

- All four modules of the assignment are due in the

first 5 minutes of class. NO assignment will be

accepted after 405 PM. - Course evaluations will be completed during the

first 10 minutes of class.

Correlation

- We are often interested in knowing about the

relationship between two variables. - Consider the following research questions
- Does the incidence of crime (X) vary with the

outdoor temperature (Y) in Detroit? - Does pizza consumption (X) have anything to do

with how much time one spends surfing the web

(Y)? - Does severity of depression (X) vary as a

function of Ecstacy use (Y)? - Do the occurrence of pimples (X) increase as air

pollution increases (Y) in Windsor?

Correlation

- These are all examples of relationships.
- In each case, we are asking whether one variable

(X) is related to another variable (Y). Stated

differently Are X and Y correlated? - More specifically Are changes in one variable

reliably accompanied by changes in the other? - Correlation coefficients can be calculated so

that we can measure the degree to which two

variables are related to each other.

Scatter Plot Used to Describe Correlation

- We can plot the X and Y points on a Scatter plot.
- We plot the Y scores on the vertical axis and the

X scores on the horizontal axis. - We then can draw a straight line to try to

represent or describe the points on our scatter

plot.

Graphing Relationships

- When our height and weight scores are plotted, we

see some irregularity. - We can draw a straight line through these points

to summarize the relationship. - The line provides an average statement about

change in one variable associated with changes in

the other variable.

r .770

Correlation

AGE

WEIGHT

Imagine if.

- All of the dots fell exactly on the line? What

would that mean? - All of the dots clustered close to the line, but

few fell on the line What would that mean? - The dots were widely dispersed around the line,

such that the line is only a vague representation

of how the scatterplot looks. What would that

mean?

Correlation Positive R

- Lets look at some different scatter plots.
- A positive relationship.

Various degrees of linear correlation

Correlation Negative R

- Lets look at some different scatter plots.
- A negative relationship.

Various degrees of linear correlation

Correlation No Relationship

- Lets look at some different scatter plots.
- No Relationship

What Direction Relationship Is Described in This

Scatter Plot?

Logic Dictates

- We can measure the distance between each dot and

the line. - If a perfect correlation (1.000) is represented

by all of the dots falling on the line, while a

line whose dots vary around it indicates a weaker

correlation - The degree to which the two variables are

correlated can be thought of as the mean distance

between the dots and the line. This is calculated

algebraically.

Covariance

- Conceptually, the correlation between X and Y is

based on covariance a statistic representing

the degree to which two variables vary together. - Like variance, covariance is based on deviations

from the mean. - r is calculated as
- But wait! Just like calculating variance, there

is an easier formula

The Pearson Product-Moment Correlation

Coefficient (r)

- r is a quantitative expression of the degree to

which two variables are correlated in a linear

relationship. - Linear relationship This means that the

scatterplot points are clustered more or less

symmetrically about a straight line, such that

the line is an adequate representation of the

relationship. - Non-linear or curvillinear relationship The

scatterplot points do not cluster around a

straight line. Example? Arousal/performance

Characteristics of r

- r has two components
- The degree of relationship
- The direction of relationship
- r ranges from 1.000 to 1.000

Are X Y Correlated?

The Pearson r

(SC) (SU)

SCU

N

r

Note This formula really is the same as the one

in the book, just slightly rearranged.

We Need

- Sum of the Xs SC
- Sum of the Ys SU
- Sum of the Xs squared (SC)2
- Sum of the Ys squared (SU)2
- Sum of the squared Xs SC2
- Sum of the squared Ys SU2
- Sum of Xs times the Ys SCU
- Number of Subjects (N)

Correlation Arithmetic

The Pearson r

(15) (17)

57

5

r

The Pearson r

255

57

5

r

The Pearson r

57

51

r

The Pearson r

6

r

The Pearson r

6

r

The Pearson r

6

r

57.8

63

45

55

The Pearson r

6

r

5.2

10

The Pearson r

6

r

52

The Pearson r

6

r

7.2111

The Pearson r

.832

r

Hypothesis Testing with Correlations

- H0 ? 0 (? rho population correlation

coefficient) - Ha ? ? 0 (there is a significant relationship

between X and Y) - Technically, you could do a one-tailed test for

correlations (? lt0 or ? gt0), but for our purposes

we will always test whether there simply is a

relationship therefore, we will always do a

two-tailed test for correlations. - Find the critical value for .05 with dfn-2

(where N is the number of paired observations) in

Table E.2 p. 440

The Pearson r

.832

r

Is an r of .832 significant?

See Table E.2 (p.440) for n - 2 df ( 5 - 2 3

df) and an alpha (a) of .05

The Pearson r

.832

r

Is an r of .832 significant?

The Critical r .878 r .832 Therefore, the

correlation is NOT significant

Popcorn Consumption

- Researcher X hypothesizes that popcorn

consumption varies as a function of stress. He

gives a random sample of 5 people a self-report

measure of stress that produces scores ranging

from 1 (little or no stress) to 10 (very

stressed), and then has them watch a movie. He

measures how many kernels of popcorn each of them

eat. Is popcorn consumption correlated with

stress?

Are X Y Correlated?

Stress

Ratings of Kernals

The Pearson r

(SC) (SU)

SCU

N

r

We Need

- Sum of the Xs SC
- Sum of the Ys SU
- Sum of the Xs squared (SC)2
- Sum of the Ys squared (SU)2
- Sum of the squared Xs SC2
- Sum of the squared Ys SU2
- Sum of Xs times the Ys SCU
- Number of Subjects (N)

Correlation Arithmetic

The Pearson r

(SC) (SU)

SCU

N

r

The Pearson r

(29) (40)

256

5

r

The Pearson r

1160

256

5

r

The Pearson r

256

232

r

The Pearson r

24

r

The Pearson r

24

r

The Pearson r

24

r

189

320

370

168.2

The Pearson r

24

r

50

20.8

The Pearson r

24

r

1040

The Pearson r

24

r

32.2490

The Pearson r

.744

r

The Pearson r

.744

r

Is an r of .744 significant?

See Table E.2 (p.440) for n - 2 df ( 5 - 2 3

df) and an alpha (a) of .05

The Pearson r

.744

r

Is an r of .744 significant?

The Critical r .878 r .744 Therefore, the

correlation is NOT significant

A Useful Means of Interpretation Variance

- r is not the most useful interpretation of a

correlation. - r2 is more useful. r2 is the proportion of the

variance of the Y scores that is accounted for by

X. - You need so much information in order to make an

error free prediction of Y. r2 is roughly equal

to the percentage of that information that you

possess just by knowing X.

Why Do Some People Have High-Self-Esteem While

Others Have Low Self-Esteem?

- Say 100 people are given a self-esteem inventory

(e.g., I think I am a person of worth, from

1strongly disagree to 5 strongly agree) - They are also asked to fill out measures of

body-satisfaction (I think I have a good body),

social-esteem (I think I am a good friend), and

academic-esteem (I am a good student). - Correlations are calculated between overall

self-esteem and the other variables (3

correlations).

Explaining Self-Esteem

- The entire pie Overall self-esteem
- The different pieces represent different

variables that explain the variability (or

variance) in self-esteem scores (in other words,

these variables explain why some people have high

self-esteem, low self-esteem, very low, etc. etc.)

So

- Body-esteem accounts for (or explains) 16 of the

variance in overall self-esteem. - Social-esteem explains? (.540)(.540) .290, so

it explains 29 of the variance in overall

self-esteem.

Correlation Errors in Interpreting r

- Common errors in interpreting a correlation

coefficient - Interpreting r in direct proportion to its size
- Not a percentage
- Not proportionate across the range (.2 not half

of .4) - The correlation coefficient is an ordinal

statistic. So r0.750 represents a stronger

relationship than r0.520 - Interpreting in terms of arbitrary descriptive

labels - Small - medium large

More Errors Interpreting Correlation

- Correlation does NOT imply Causation!
- X causes Y to change ? Examples?
- Y causes X to change ? Examples?
- W causes changes in X and Y! ? Examples?
- SO body-esteem might account for 16 of the

variance in self-esteem, but this does not mean

that body-esteem causes self-esteem. - For the trip to Hawaii and the Samsonite Luggage

Psychologists used to think that having been

sexually abused causes bulimia. How could

researchers demonstrate that this is true?

Factors that affect the size of a correlation

- Nature of the relationship between X and Y.
- Heterogeneous subsamples if the sample could be

subdivided into 2 distinct sets based on another

variable (e.g, males vs. females) - Truncated range.
- Range restricted in size.
- May cause correlation to appear lower than it

really is (or higher than it is for non-linear

relationships) - Without the full range of scores it is not

possible to calculate the correlation accurately.

Lets look at why

Underlying Assumptions for r

- X and Y need to be adequately represented by a

straight line function. Stated differently, the

relationship must be linear. - If r is to be used inferentially
- Homoscedasticity The variabilities of X at

different values of Y are equal. E.g.,

variability in weight for 65 people is equal to

variability in weight for 55. - Normality X is normally distributed at all

values of Y (e.g., weight is normally distributed

for 65 people and for 55 people. - Vice-versa as well (Y at values of X)

Work on it

- Say were interested in knowing whether exam

grades are related to number of hour spent

studying. Ten students report how many hours they

studied for an exam. Here are the data

Work on it!

- State the Ho and Ha.
- Test the hypothesis.

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