correlation - university of michigan dearbornacfoos/courses/381/05... · 2018-01-30 · •pearson...

Correlation

Experimental Psychology

Arlo Clark-Foos

Preview

• Standardized Scores (z scores)

• Correlation

– Characteristics of

– Relationship to Cause-Effect

– Partial Correlations

• Psychometrics

• Reliability

• Validity

Las Cucarachas

• How heavy are these cockroaches?

– 8.0 drachms

– .25 pound

– 98 grams

– Need to put these on the same scale!

– Not always this easy (these were all measures of weight)

(14.17 grams)

(113.40 grams)

(98 grams)

Equal Footing

• Different variables, often different scales

• Need standardization

– z score (z is italicized)

• the number of standard deviations a particular score is

from the mean

• Why is this useful?

• z distribution

– Scores vs. Distributions

z Distributions

Mean = 0, SD = 1 (Always!)…why?

Raw Score z Score

• Hypothetical Test Scores

– Mean = 70, SD = 10, you scored an 85.

Raw Score z Score

• NEED: Population parameters for the

mean (μ) and standard deviation (σ)

Xz

Changing Gears:

Correlations

What is a correlation?

A general term to describe the association (or

relation) between two variables

– co-relation

Correlation Coefficient

• A statistic that quantifies the relation

between two variables

• Pearson (r) correlation coefficient

– Measures the strength of a linear

relationship

Correlation Coefficient

• Characteristics

1. Can be positive or negative

2. Ranges from -1.00 to 1.00

3. Strength (magnitude) of correlation is

indicated by size of resulting value, not

the sign.

Positive Correlation

r > 0

Positive Correlations

• Positive correlation between

smoking and lung cancer

• Positive correlation between SAT

scores and high school grades

• Positive correlation between the

amount of diet soda you drink and

your weight

Negative Correlation

r < 0

Negative Correlations

• Negative correlation between

education and years in jail

• Negative correlation between

amount of time a baby is held

and how much she cries

• Negative correlation between

amount of TV watched and

grades

No Correlation

r = 0

Strength of Correlations

Strong, positive Weak/moderate, positive

Cohen (1988)

• How strong is your correlation?

Correlation & Causation

• True: After WWII, there was a

strong positive correlation

between the birth rate and

the number of storks in

Copenhagen.

– Why?

– Increasing population

• More buildings– Increasing numbers of storks

• More baby makers– Increasing numbers of babies

Illusory Correlations

Capricorn: Some rather boring and mundane tasks, perhaps

involving paperwork, could take up much of your time today,

Capricorn. You could get easily distracted and be tempted to

set it aside and do something more interesting, but don't fall

into this trap. You'll want your recent pattern of success to

continue, and so it's best to get the boring stuff out of the

way and then move on to what's exciting. Hang in there!

Pisces: Some paperwork involving business enterprises could be

executed today, Pisces, possibly in your home. This might be

a new and unexpected development. It could have you feeling

a bit disoriented, but try to pull yourself together and make

the most of it. Whatever the opportunity, all signs say that it

might prove quite successful, so don't let it pass you by. Go

with the flow and see where it takes you. You could be

pleasantly surprised!

Correlation is NOT Causation

• Third Variables

Karl Pearson

• Pearson correlation coefficient

– r (sample) or ρ (population)

– Let’s use attendance and exam grade as

an example…

N

zzr

yx

X Y

X Y

X M Y Mr

SS SS

or

Correlation Example

Method 1: Correlation with Standard Scores

N

zzr

yx

Correlation with z

1. Plot the raw scores for each variable on a

scatter plot to see if there might be a

linear relationship

– If so, proceed with calculating the Pearson

correlation coefficient.

2. Create z scores corresponding to each value

– Do this for both variables of interest for each

person, keeping pairs of scores together

Xz

Scatterplot

Absences

0 2 4 6 8 10

Mean E

xam

Gra

de

40

50

60

70

80

90

100

110

Level the Field

1. Plot the raw scores for each variable on a

scatter plot to see if there might be a

linear relationship

– If so, proceed with calculating the Pearson

correlation coefficient.

2. Create z scores corresponding to each value

– Do this for both variables of interest for each

person, keeping pairs of scores together

Xz

Correlation Example

Xz

Correlation Example

Xz

Correlation Example

3. Pearson Correlation Coefficient

– Calculate the cross-products

• Multiple each pair of z scores together

4. Find the average of all of the cross-

products• This is your Pearson correlation coefficient

Cross-products

Correlation Example

3. Pearson Correlation Coefficient

– Calculate the cross-products

• Multiple each pair of z scores together

4. Find the average of all of the cross-

products• This is your Pearson correlation coefficient

Cross-products

∑(Cross Products) / Number of Pairs =

Pearson Correlation Coefficient

r = -.85

Method 2: Correlation with Deviations and Sums of Squares

X Y

X Y

X M Y Mr

SS SS

Please Use:

Same Data, New Method

Correlation with z

1. Plot the raw scores for each variable on a

scatter plot to see if there might be a

linear relationship

– If so, proceed with calculating the Pearson

correlation coefficient.

– Caveat: Not always necessary (e.g., SPSS, SAS)

2. Numerator: Create deviations (Score –

Mean) for each variable, keeping pairs

together

Scatterplot

Absences

0 2 4 6 8 10

Mean E

xam

Gra

de

40

50

60

70

80

90

100

110

Correlation

1. Plot the raw scores for each variable on a

scatter plot to see if there might be a

linear relationship

– If so, proceed with calculating the Pearson

correlation coefficient.

2. Numerator: Create deviations (Score –

Mean) for each variable, keeping pairs

together

Same Data, New Method

X Y

X Y

X M Y Mr

SS SS

Numerator: Deviations &

Cross Products

X Y

X Y

X M Y Mr

SS SS

Correlation Example

3. Denominator: Create Sums of Squares

(Sum of all Squared Deviations) for

each variable

4. Denominator: Multiply your Sums of

Squares

5. Take the Square Root of that product

6. Divide Numerator by Denominator to

get your Correlation Coefficient

X Y

X Y

X M Y Mr

SS SS

3. Denominator: Create

Sums of Squares

X Y

X Y

X M Y Mr

SS SS

Correlation Example

3. Denominator: Create Sums of Squares

(Sum of all Squared Deviations) for

each variable

4. Denominator: Multiply your Sums of

Squares

5. Take the Square Root of that product

6. Divide Numerator by Denominator to

get your Correlation Coefficient

X Y

X Y

X M Y Mr

SS SS

Correlation Example

4. Denominator: Multiply your Sums of Squares

SSX = ∑(X - Mx2) = 56.4

SSY = ∑(Y - MY2) = 2262

(SSX)(SSY) = (56.4)(2252) = 12576.8

5. Take the Square Root of that product

12576.8 = 357.18

X Y

X Y

X M Y Mr

SS SS

Correlation Example

3. Denominator: Create Sums of Squares

(Sum of all Squared Deviations) for

each variable

4. Denominator: Multiply your Sums of

Squares

5. Take the Square Root of that product

6. Divide Numerator by Denominator to

get your Correlation Coefficient

X Y

X Y

X M Y Mr

SS SS

6. Find Correlation

Coefficient (r)

• Divide Numerator by Denominator to get your

Correlation Coefficient

𝑟 =−304

357.18= −.851

• r = -.85

Absences

0 2 4 6 8 10

Mean

Exam

Gra

de

40

50

60

70

80

90

100

110

X Y

X Y

X M Y Mr

SS SS

Cautions and Issues Related to

Correlations

Correlation Example

• There appears to be a negative correlation

between attendance and exam grades

• What else could contribute to this?

Correlation is NOT Causation

• Restriction of Range

r = .05

Correlation is NOT Causation

• Restriction of Range

r = .56

Correlation is NOT Causation

• Outlier Effects

r = .39

Correlation is NOT Causation

• Outlier Effects

r = -.135

Applications of Correlational

Techniques

Some examples

of Reliability

• Test-Retest

– Determines whether scale being used

provides consistent information every time

it is given

• Split-Half Reliability

– Measures internal consistency of a test or

scale by typically correlating the odd-

numbered items with the even-numbered

items

– Coefficient/Cronbach’s alpha α)

• Calculated by taking the average of all

possible split-half correlations

Some examples of

Validity

• Criterion-related

– The scale of measurement of interest is

correlated with a criterion, which is some

external standard (e.g., an observable behavior)

– Postdictive/Predictive

• The scale or measurement of interest is

correlated with a criterion measured in the past

or future

– Concurrent

• The scale or measurement of interest is

correlated with a criterion measured at the same

time

Partial Correlation

• A technique that quantifies the

degree of association between two

variables, after statistically

removing the influence of a third

variable by holding it constant

Psychometrics

• Branch of statistics used in the development of

tests and measures and/or the evaluation of

statistical data on the behavior of animals

(typically humans)

– Examples

• Test for cultural biases in SAT/ACT/GRE

• Identify high-achieving employees

• Analyze online behavioral patterns, biostatistics, and much

much more

• “critical shortage” (NYT, Herszenhorn, 2006)

Review

• Correlation

– Characteristics of

– Relationship to Cause-Effect

– Partial Correlations

• Standardized Scores (z scores)

• Psychometrics

• Reliability

• Validity