Correlations Revisited

The IOC Coordination Commission were told that 80 per cent of the land had already been acquired. London Mayor Ken Livingstone added that he was hoping that, by the time the public enquiry starts at the end of next month, four-fifths of the land would have been acquired.

Radio Oxford news report 20 April 2006

Size of the correlation (Cohen, 1988)

Negative Positive

Small -.10 to -.29 .10 to .29

Medium -.30 to -.49 .30 to .49

Large -.50 to -1.00 .50 to 1.00

Calculate Pearson’s r using z scores.

• What is this formula telling us to do?

• The text uses N in the denominator.– This is related to using

n-1 when calculating variance (population vs. sample).

• If you want to get the same result as SPSS use n-1.

1

Z

n

Zr YX

Definitional formula for Pearson’s r

22 )()(

))((

YYXX

YYXXrXY

Computational formula for Pearson’s r.

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][)(

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))((

22

22

N

YY

N

XX

N

YXXY

rXY

Covariance

• An index to the degree that to variables share variance (i.e., vary together).– By itself has no meaning.

• Much like variance.

– Needs to be standardized.– Text shows the total of cross products (the numerator)

• Definitional formula below

1

))((

n

YYXXSxy

Computational Formula for Covariance

)1(2

nn

YXXYnSxy

Calculating Correlation from Covariance

YX

XYXY SS

Sr

2

Back to magnitude of effect

• Coefficient of determination – Also known as

• Shared variance• The proportion of variance

accounted for• Systematic variance• Percentage of variance

accounted for

• Coefficient of nondetermination – Proportion of variance not

accounted for

2r

21 r

Problems associated with Pearson’s r

• Lack of linear relationship– (e.g., anxiety and test performance)

• Restricted (truncated) range– Can reduce the magnitude of the correlation

• Sample size• Outliers• Two populations

– It appears there is not a correlation (or the correlations is low), but when you stratify there is a correlation.

• Extreme scores– Selection bias

• Causal arguments– Correlation does not equate causation.

Probability

I think you're begging the question, said Haydock, and I can see looming ahead one of those terrible exercises in probability where six men have white hats and six men have black hats and you have to work it out by mathematics how likely it is that the hats will get mixed up and in what proportion. If you start thinking about things like that, you would go round the bend. Let me assure you of that!

Agatha ChristieThe Mirror Crack's

Misunderstanding of probability may be the greatest of all impediments to scientific literacy.

Stephen Jay Gould

The Personal Probability Interpretation

Personal probability of an event = the degree

to which a given individual believes the event will happen.

Sometimes subjective probability used because the degree of belief may be different for each individual.

Restrictions on personal probabilities:• Must fall between 0 and 1 (or between 0 and 100%).• Must be coherent.

Probability Definitions and Relationships

Sample space: All the possible outcomes that can occur.

Simple event: one outcome in the sample space; a possible outcome of a random circumstance.

Event: a collection of one or more simple events in the sample space; often written as A, B, C, and so on.

Assigning Probabilities

• A probability is a value between 0 and 1 and is written either as a fraction or as a proportion.

• A probability simply is a number between 0 and 1 that is assigned to a possible outcome of a random circumstance.

• For the complete set of distinct possible outcomes of a random circumstance, the total of the assigned probabilities must equal 1.

Classical Approach

• A mathematical index of the relative frequency of likelihood of the occurrence of a specific event.– Based on games of chance– The specific conditions of the game are

known.

Determining the probability of an Outcome (Classical)

A Simple Lottery

Choose a three-digit number between 000 and 999.

Player wins if his or her three-digit number is chosen. Suppose the 1000 possible 3-digit numbers (000, 001, 002, 999) are equally likely.

In long run, a player should win about 1 out of 1000 times. Probability = 0.0001 of winning.This does not mean a player will win exactly once in every thousand plays.

Example: Probability of Simple EventsRandom Circumstance:

A three-digit winning lottery number is selected.

Sample Space: {000,001,002,003, . . . ,997,998,999}. There are 1000 simple events.

Probabilities for Simple Event: Probability any specific three-digit number is a winner is 1/1000. Assume all three-digit numbers are equally likely.

Event A = last digit is a 9 = {009,019, . . . ,999}. Since one out of ten numbers in set, P(A) = 1/10.

Event B = three digits are all the same = {000, 111, 222, 333, 444, 555, 666, 777, 888, 999}. Since event B contains 10 events, P(B) = 10/1000 = 1/100.

Estimating Probabilities from Observed Categorical Data - Empirical Approach

Assuming data are representative, the probability of a particular outcome is estimated to be the relative frequency (proportion) with which that outcome was observed.

Methods of sampling

• Simple random selection– Every member of the population has an equal

chance of being selected.

• Systematic– Every Xth person.

• Stratified– Random sampling by subgroup.

• Why?

Determining the probability of an Outcome – Empirical Approach

Observe the Relative Frequency of random circumstances

The Probability of Lost Luggage

“1 in 176 passengers on U.S. airline carriers will temporarily lose their luggage.”

This number is based on data collected over the long run. So the probability that a randomly selected passenger on a U.S. carrier will temporarily lose luggage is 1/176 or about 0.006.

Proportions and Percentages as Probabilities

• The proportion of passengers who lose their luggage is 1/176 or about 0.006 (6 out of 1000).

• About 0.6% of passengers lose their luggage.• The probability that a randomly selected

passenger will lose his/her luggage is about 0.006.• The probability that you will lose your luggage

is about 0.006.Last statement is not exactly correct – your probability depends on other factors (how late you arrive at the airport, etc.).

Example: Probability of Male versus Female Births Long-run relative frequency of males born in the United States is about 0.512 (512 boys born per 1000 births)

Table provides results of simulation: the proportion is far from .512 over the first few weeks but in the long run settles down around .512.

Nightlights and Myopia Assuming these data are representative of a larger population, what is the approximate probability that someone from that population who sleeps with a nightlight in early childhood will develop some degree of myopia?

Note: 72 + 7 = 79 of the 232 nightlight users developed some degree of myopia. So we estimate the probability to be 79/232 = 0.34.

Complementary Events

Note: P(A) + P(AC) = 1

One event is the complement of another event if the two events do not contain any of the same simple events and together they cover the entire sample space.

Notation: AC represents the complement of A.

Example:A Simple Lottery (cont)A = player buying single ticket winsAC = player does not winP(A) = 1/1000 so P(AC) = 999/1000

Mutually Exclusive Events

Two events are mutually exclusive if they do not contain any of the same simple events (outcomes).

Example; A Simple Lottery

A = all three digits are the same.B = the first and last digits are different

The events A and B are mutually exclusive.

Independent and Dependent Events• Two events are independent of each other

if knowing that one will occur (or has occurred) does not change the probability that the other occurs.

• Two events are dependent if knowing that one will occur (or has occurred) changes the probability that the other occurs.

Example Independent Events •Customers put business card in restaurant glass bowl. •Drawing held once a week for free lunch. •You and Vanessa put a card in two consecutive wks.

Event A = You win in week 1.Event B = Vanessa wins in week 2

• Events A and B refer to to different random circumstances and are independent.

Event A = Alicia is selected to answer Question 1.Event B = Alicia is selected to answer Question 2.

• P(A) = 1/50.• If event A occurs, her name is no longer in the bag; P(B) = 0.• If event A does not occur, there are 49 names in the

bag (including Alicia’s name), so P(B) = 1/49.

Events A and B refer to different random circumstances, but are A and B independent events?

Knowing whether A occurred changes P(B). Thus, the events A and B are not independent.

Example: Dependent Events