Correlations Revisited

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Correlations Revisited
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• 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

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Size of the correlation (Cohen, 1988)
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Calculate Pearsons 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.

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Definitional formula for Pearsons r
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Computational formula for Pearsons r.
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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

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Computational Formula for Covariance
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Calculating Correlation from Covariance
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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

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Problems associated with Pearsons 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.

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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

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• Misunderstanding of probability may be the
  greatest of all impediments to scientific
  literacy.
• Stephen Jay Gould

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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.

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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.
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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.

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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.

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Determining the probability of an Outcome
(Classical)
A Simple LotteryChoose 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.
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Example Probability of Simple Events
Random 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.
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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.
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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?

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Determining the probability of an Outcome
Empirical Approach
Observe the Relative Frequency of random
circumstances
The Probability of Lost Luggage1 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.
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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.).
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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.
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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.
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Complementary Events
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.
Note P(A) P(AC) 1
ExampleA Simple Lottery (cont) A player
buying single ticket wins AC player does not
win P(A) 1/1000 so P(AC) 999/1000
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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.
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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.

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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.

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Example Dependent Events
Event A Alicia is selected to answer Question
1. Event B Alicia is selected to answer
Question 2.
Events A and B refer to different random
circumstances, but are A and B independent
events?

• 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 Alicias name), so P(B)
  1/49.

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