Psychology 412

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Psychology 412

• Instructor Adam Kramer
• Week 2

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Last week

• Questions lead to measurement
• Measurement leads to variability
• Summaries can be made
• Variability quantifies the likelihood of the
  summary
• Standard deviations are useful
• The central limit theorem gives us p

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Some comments
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Sampling Distributions

• You have a sample size of n
• Variance and SD have are EXPECTED to vary as a
  function of n
• You only have ONE sample size, so how much
  variability does IT have?

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t distributions
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But the original distribution...

• The prior slide shows t-distributions for MEANS,
  not for POINTS...

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UNIFORM distribution
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SKEWED distribution
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Back to our data...

• Are we SATISFIED?
• Moreso than a KNOWN population z
• Moreso than an ESTIMATED pop (i.s.) t
• Moreso than a CONSTANT (o.s.) t
• Do our scales DIFFER?
• SWL measure 1 vs measure 2

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Dependent samples

• SWL1 and SWL2 are not independent
• Dependence means correlation
• Or, knowing one tells us about the other
• This is because they share raters
• Namely, us.
• So we cant compare us to ourselves the same way
  we compare us to the student body.

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Dependent samples

• Solution Remove the dependence.
• Are the measures different?

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Dependent samples

• Are they different?
• On average, do they differ
• On average, are the differences nonzero?
• Hypothesis µ1-µ20
• A one-sample t-test on the differences
  sd2.748, se.97, µ0.125, t(7)0.123, n.s.

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Not different...

• Is that because n is small?
• Are they related to each other at all?
• Relative, of course, to the variability

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Variance again.

• The average squared deviation from the mean.
• But now we have two variances two average
  squared deviations from each mean
• Do they go up and down together?

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Two variances

• When one observation is above the mean for the
  variable, is the other?
• How much higher?
• Or lower, or neithersimilarity of highness is
  what matters

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Comparing deviations

• Two variances
• Multiply them!
• Review -11-1, 111, -1-11
• Applied If x and y are on the same side of the
  mean, the product is positive
• If x and y are on opposite sides, negative

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Comparing deviations

• Result
• If a person is above average or below average
  on both measures, the product is high.
• If a person is above average on one and below
  on the other, the product is low
• So, if we add them

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Adding deviation products

• With the variance, we couldnt add
• Because the sum of deviations from average is
  zero
• With the deviation products, we can!
• Because each deviation from average is enhanced
  by another variables deviation!
• This value is called the cross product.

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Covariance

• Variance averaged squared deviations
• We average co-deviation in the same way
• and call it covariance.
• but theres no nifty greek symbol for it. (

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Our example
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Un-squaring covariance

• Variance was square rooted to get stdev because
  variance is the average squared deviation
• Covariance is not squared deviation, its the
  product of two unsquared deviations
• BUT WAIT! Stdev is an un-squared unit!

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Un-squaring covariance

• Divide the covariance by which stdev?

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Un-squaring covariance

• The result
• A product term over
• a square-rooted square
• product term, over sqrt(n-1), just like stdev
• The bottom is always positive
• The top determines the sign on average,
• are both variables on the same side of the
• mean, or opposite sides?

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Un-squaring covariance

• The deviation on top is removed on the bottom

• For both variables.
• The result is fully un-scaled the scale is
  gone
• Hence, it can only range between 0 and 1, with
  the sign determined by the top

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Correlation

• Its the correlation!
• You probably saw that coming.

• Correlation r is between -1 and 1
• Distance from zero indicates relatedness
• Positive Similar, Negative Opposite
• Zero Not related

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Correlation

• Because it is unscaled, the scales become
  irrelevant
• Its as if both variables were put on the same
  level it doesnt matter if we say x is
  correlated with y or y with x
• One variable may have had more variance, but that
  has been removed.

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Correlation and Prediction

• So, the degree to which variables are related
  should indicate the degree to which one predicts
  the other
• Predict Guess one from the other
• Degree to which How well we can do it
• Back to variabilityhow much can we account for?
• We seek to explain variance.

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Correlation and Prediction

• But that pesky -1 to 1 range
• How much does an overlap of -.5 mean?
• Squaring to the rescue!
• The square of the sum of products over the
  product of the sums of squares
• Or, covariance out of ALL variance!

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Correlation and Prediction

• Covariance out of ALL variance sounds like a
  proportion
• Indeed it is. R2 is the proportion (percent) of
  standardized variance that one variable
  explains of the other
• But is it meaningful?
• Er, significant

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Testing correlations

• You may have noted how t-formula like the
  correlation formula is.
• Indeed, r is distributed as
• n-2 because we computed TWO means
• This transformation gives r the infinite scope of
  t but undefined at r1.0.

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Our data

• We find r-.60large, negative, but insignificant
• Remember, n8 is tiny!
• How do we interpret this?

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Using correlations

• People know what a correlation is
• R2, too
• But they are weirdly distributed
• An actual correlation of 0.5 could be
  under-estimated by up to 1.5, but overestimated
  by only up to 0.5.

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Using correlations

• Transfrom the r to a z under the null hypothesis
• Gives you
• how many sd
• your r is above no relationship
• But your deviation and n are gone!
• You cant estimate how accurate your r is.

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Two steps back
This is where Tuesdays lecture ended.

• Two basic tools
• Differences, similarities
• But wait!
• The tools come after the data
• The data comes after the research question

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The story of a correlation

• Forming a research question
• Lots of people in this world
• Some of them are really bad at emotion regulation
• Man, thats unfortunate for them. I wonder why it
  is

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The story of a correlation

• Forming a research question
• Can I think of anything that would describe which
  people have bad regulation patterns? Or good
  ones?
• Hmm, types of people.

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The Big 5

• Personality trait sounds like a good way of
  typing people!
• Openness, Conscientiousness, Extraversion,
  Agreeableness, Neuroticism
• Closed-mindedness, Ignoring of others,
  Introversion, Disagreeableness, Emotional
  Stability
• Which ones go where?

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Measuring constructs

• How would you measure personality?
• How would you measure emotion regulation?

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Measuring constructs

• How would you measure personality?
• A bunch of questions, aggregated
• One question per trait
• How would you measure emotion regulation?
• A bunch of questions, aggregated
• One question per strategy

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Research design

• Bring someone in, ask them lots of questions!
• Time-tested, advisor-approved, tastes great, too!
• Like 2 pencil lead.
• Why might this not work?
• What questions do we want to ask?
• Depends on what answers we want to get

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Scales

• Continuous or categorical?
• Is this person an INTROVERT?
• Does this person have a LEVEL of Int/Ext?
• Similar for emotional regulation strategies?
• What is our question?

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Emoreg Scales

• Whats good emoreg?

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Emoreg Scales

• Whats good emoreg?
• SWL, Savoring

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Time-lapse sequence here

• Write the IRB proposal, reserve the room, run the
  study, collect the data
• Compute the scale values
• Designing scales is a statistical issue we might
  get to, but were talking about pre-validated
  scales

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EDA
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EDA
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EDA
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EDA

• Sweeeet. Looks good.
• The good thing about pre-designed scales
• We may proceed!

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Questions to Hypotheses

• We hypothesize that
• Higher E predicts higher SWL
• Higher N yields lower SWL
• Higher A predicts higher SWL, Savoring

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Hypotheses to strategies

• We would test that by
• Higher E predicts higher SWL
• Es SWL compared to I (independent sample t)
• Correlate SWL with E
• Es SWL above 50 (one-sample t)
• Higher N yields lower SWL
• Higher A predicts higher SWL, Savoring