Psychology 412

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Psychology 412

• Instructor Adam Kramer
• Week 3

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Last week

• Sampling distributions lead to p-values
• Dependent data
• Deriving a correlation
• The standardized average directional distance
  from the mean
• Research overview Correlations

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Re-deriving correlation

• Standardized average directional difference
• You can also standardize first
• Then, r is the covariance of the standardized data

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Using correlations

• Correlations are standardized twice
• Once for each variable
• The result is directionless.
• Our task, however, is likely prediction
• If R2 tells us the explained variance, which
  variance do we care to explain?

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Predicting

• Back to Extraversion and SWL
• r .357, t(271) 6.29, p
• So, E explains SWL variance
• So, if we use E, we can do BETTER than guessing
  the mean for SWL

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Predicting Our task
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Predicting Mean SWL?
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Predicting Mean BFIE?
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Predicting Means?
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Predicting Some diagonal?
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Predicting Your choice?
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Predicting Why?
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Predicting Why?

• Minimization of squared residuals
• Residual is the unexplained part
• In this case, the distance from the line
• Were trying to predict SWL
• Squared because it differentiates

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Residuals
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Residuals
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Predicting Why?
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Predicting How?

• We use the correlation, but un-standardize it
• Put it back in y-units by multiplying by ?y
• Divide by ?y again, and youre in y units per x
  unit.

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Predicting SWL from E

• SWLy, Ex
• (by convention, y is the DV)
• y units per x unit
• We expect someones SWL to increase by 0.41 for
  every increase in BFIE
• But thats only MORE, where do we start?

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Predicting SWL from E

• Start at zero.
• The regression line hits (µY,µX)
• It has to, or it wouldnt minimize variance
• (variance is computed relative to the mean)
• So, go back µX slope points, and youre at 0 on
  the x axis, and 40.311 on the y axis

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Modeling

• SWL 40.311 Extraversion0.411
• We say, We have a MODEL of SWL, based on
  extraversion.
• Generally, Y ?0?1X
• The DV is a function of the IV, plus intercept
  estimates are represented with indexed betas
  (sometimes alpha is the intercept)

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Modeling

• One issue here is that were not really dealing
  with SWL and E
• These were just self-report scales
• There is likely some error involved
• Y ?0?1X?
• The regression model.

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Testing the model

• How good is our model?
• Can it fly like a REAL airplane?
• We have estimated two parameters
• So, we have two parameters to test
• Y ?0?1X
• Is ?0 significantly different from 0?
• Is ?1 significantly different from 0?
• These sound like t-test questions

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t-tests

• In general, weve been looking at mean
  differences over standard errors to compute t
• Really, t-tests are more general if you can
  estimate a standard error for ANY value, t is
  the ratio.

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Standard errors

• The standard error of the model has to do with
  the deviation from the line
• Y given X
• Numerator is deviations from the line
• Denominator n-2 because we estimated the mean for
  X and Y

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Standard errors

• Standard error of the intercept and slope are not
  the same.
• wtf.

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Testing our parameters

• Is ?0 significantly different from 0?
• When BFIE is zero, is SWL significantly different
  from zero?

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Testing our parameters

• Is ?0 significantly different from 0?
• When BFIE is zero, is SWL significantly different
  from zero?
• t(271) 10.44, p
• YES. Even total introverts have some modicum of
  satisfaction with their life.

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Testing our parameters

• Is ?1 significantly different from 0?
• Does BFIE help us predict SWL better than the
  mean of SWL?

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Testing our parameters

• Is ?1 significantly different from 0?
• Does BFIE help us predict SWL better than the
  mean of SWL?
• t(271) 6.32, p
• YES. BFIE helps us predict SWL every point of
  extraversion predicts 0.411 SWL.
• Note that the t is the same as for rwhy?

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Reporting our regression

• Coefficients
• Estimate Std. Error t value Pr(t)
  
• (Intercept) 40.34084 3.97257 10.155
• bfie 0.40994 0.06516 6.291 1.26e-09
  
• Computers do the math for us!
• An appropriately-reported regression
• A line for each estimated parameter
• List estimate, SE, t, p

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Intercepts

• Even total introverts have some modicum of
  satisfaction with their life.
• Uhhduh. So what?
• The intercept tests Y when X is zero
• If we change Y (SWL), we change the intercept
• What is a MEANINGFUL level to test Y against?

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Intercepts

• If we first center our dataset, the units remain
  the same, but 0 becomes synonymous with the
  mean.
• So the intercept would test Is SWL significantly
  above or below the mean when BFIE is zero?
• And we could center BFIE
• Is SWL significantly above or below the mean for
  the AVERAGE BFIE?

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Intercepts
This is where lecture ended on Tuesday.

• Estimate Std. Error t value
  Pr(t)
• (Intercept) -0.03148 1.20244 -0.026
  0.98
• center(kbfie) 0.40994 0.06516 6.291
  1.26e-09
• The intercept is no longer significant at mean
  extraversion level, SWL is no different from its
  mean
• Note that the estimate, SE, t are the same for
  BFIE
• Why ask? Because we can.

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Intercepts

• Centering is not the only way
• Test whether when X0 is Y50? 100?
• Add 100 to the Y variable

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Intercepts

• Centering is not the only way
• Test whether when X0 is Y50? 100?
• Add 100 to the Y variable
• Estimate Std. Error t value
  Pr(t)
• (Intercept) 40.34084 3.97257 10.155
• bfie 0.40994 0.06516 6.291 1.26e-09
  
• versus
• (Intercept) 140.34084 3.97257 35.327
• bfie 0.40994 0.06516 6.291 1.26e-09
  
• The estimate went up by 100 t by 15!

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Slopes

• every point of extraversion predicts 0.411
  SWL.
• Once again, so what does 0.411 mean?
• The slope converts X units to Y units, but X and
  Y may not have meaningful units

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Slopes

• The meaning of a unit
• A one-point increase is 1/7 of the scale
• But if everyones a 4 5 or 6, thats a lot
• Meaningful units
• 0.411 contented sighs per day
• Percentage points on a test
• Standard deviations
• Addresses the variability within the variable
  scale doesnt matter as much

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Slopes

• Standardization can help
• If we standardized SWL, we would see how many
  standard deviations in SWL are predicted by
  unit-changes in E
• If we standardized E, we would see how much a
  standard deviation worth of difference in E
  predicted an increase in SWL units
• If we standardize both, we would see how many SD
  in SWL are predicted by a 1-SD increase in E

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Slopes

• Estimate Std. Error t value
  Pr(t)
• (Intercept) -0.001483 0.056640 -0.026
  0.98
• scale(kbfie) 0.356068 0.056598 6.291
  1.26e-09
• As BFIE goes up by one sd, we expect SWL to go up
  by 0.356 SD
• SE, t, p the same
• Same centered intercept
• The standardized slope is the correlation!

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Residuals

• Remember those residuals? You can plot them.
• EPDAA doesnt have the ring of EDA
• Plot a residual for a data point against the
  points predicted value

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Residuals

• Looks like a mess!
• Thats good.
• Its as if we took the regression line and
  rotated our plot so that its flat.
• Note the scale differences
• Range is

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Residuals

• New data set Airplane load factor and stall
  speed
• LF The amount of weight placed on a maneuvering
  aircraft relative to its ground weight
• Stall speed The speed at which the wing loses
  its lift

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Residuals

• OK, do some EDA.
• Hmm, its definitely line-likemaybe kinda curvy
• May as well try a regression

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Residuals

• OK, do some EDA.
• Hmm, its definitely line-likemaybe kinda curvy
• May as well try a regression
• Ooh, nice! Good fit!
• Not perfect, but good.

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Residuals

• Estimate Std. Error t value Pr(t)
  
• (Intercept) 35.2289 1.5102 23.33
• LF 17.9096 0.4609 38.85
• Looks good!
• Each additional point in load factor predicts
  almost 18MPH higher stall speed
• So, clearly, if youre gonna maneuver, ygotta
  fly faster.
• And my God, R2 .9655! Almost perfect!
• Well, it is an engineering issue, not psych.

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Residuals

• That does NOT look messy.
• The deviation from the line depends on what we
  predict
• The issue is LINES
• We shot a line through a curve theres still a
  curve.

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Residuals

• Our line explains variance
• But our errors depend on one of our variables
• We are clearly missing something.

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Residuals

• When we plot residuals, were looking for
  patterns. A pattern indicates that our regression
  is better for predicting SOME data points than
  OTHERS.
• Were still explaining variance, were still
  making better estimates than nothing, but there
  is something we are missing.
• If we can articulate how the error relates to
  where on the line we are

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Residuals

• What to do?
• Best option Explain it away. WHY is that curve
  there?
• Probably because a line is insufficient
• Another option Accept it.
• We have a significant LINEAR relationship it
  explains 95 of the variance. There is something
  left, and we dont know what
• Is our research question answered?

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Explain it away.

• In many cases, we can transform the data so that
  it no longer irks us
• Experience teaches us what graphs look likelog,
  sqrt, square, sine

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Explain it away.

• XLF on left, Xsqrt(LF) on right
• Estimate Std. Error t value
  Pr(t)
• (Intercept) 6.748e-04 1.156e-03 0.583
  0.562
• sqrt(lfLF) 5.400e01 6.851e-04 78823.615
  
• R2 1 (physics, not psychology)

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Assumption 1 Linearity

• assumes a linear relationship between X and Y
• Why would we assume this?

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Assumption 1 Linearity

• Predicting beyond the data
• In our example, the further from the line we get,
  the worse our line performs
• Prediction accuracy is not consistant
• But do we need to predict there?
• A clearly flawed model
• A pattern in the residuals means were missing
  something
• But do we care?

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Assumption 1 Linearity

• R2, ?, t are inaccurate
• They do NOT estimate parameters
• They are too small when curving makes them err
• to the extent that our data does not represent
  the population.
• They still describe the data set, and they still
  test the linear component.

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Assumption 2 Normality

• Regression assumes that errors are normally
  distributed along the regression line
• NOT that X or Y is normal!

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Assumption 2 Normality
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Assumption 2 Normality

• The distance here from the line is constant, not
  normal
• In this case, there are two effects.
• Once again, the model is incomplete
• What is the grouping?

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Assumption 2 Normality

• However, the regression line still represents the
  data
• Explanation is mediocre
• Prediction is bad
• But there is on average a linear effect.

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Assumption 2 Normality

• What to do about it?
• If the errors are NOT normally distributed, what
  does that mean?

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Assumption 2 Normality

• What to do about it?
• If the errors are NOT normally distributed, what
  does that mean?
• Variation in X or Y is due to something else
• Neither X nor Y is measured precisely there may
  be multiple causes of a certain score
• Random causes are normally distributed in
  relevance, so nonrandom effects indicate another
  cause of the report
• So, seek the other cause

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Assumption 3 Homoscedasticity

• A word too big to fit on one title line
• Scadesticity refers to the extent to which
  variance is consistent.

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Assumption 3 Homoscedasticity

• Heteroscedastic plots make us worry about our
  slope
• Wide-varianced points have more influence

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Assumption 3 Homoscedasticity

• Solutions
• Transform the data, or give the outliers less
  influencedont end up like this graph

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Assumption 3 Homoscedasticity

• But if you do, same issues
• Does this answer your question?
• The line still summarizes the data linearly

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Assumption 4 Independence

• Every X has a corresponding Y
• But the pairs should be independent.
• Each x and y value contribute equally to the x
  and y mean each value contributes equally to the
  x and y sd
• So if you can predict one x from another, those
  two xs are closer together than the others
• So the variance is underestimated.

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Assumption 4 Independence

• Example Parent/child height
• Hypothesis You can predict a childs height from
  a biological parents
• Method Get height for mother, father, children
• One data point for each parent/child pair
• How does this violate independence?

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Assumption 4 Independence
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Assumption 4 Independence
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Assumption 4 Independence

• But how do you compute the means and covariance?
• How many observations do you have?
• How many degrees of freedom?
• There are ways to handle data like these, but
  simple regression is not the way.

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Assumption 5 Independence?

• of errors.
• How far off from the line we are for one (x,y)
  data point should not predict how far from the
  line we are for any other data point.

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Assumption 5 Independence?

• Example Happiness over the month.

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Assumption 5 Independence?

• Example Happiness over the month.

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Assumption 5 Independence?

• Oh, right, weekends.

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Assumption 5 Independence?

• t(29)0.06, n.s. What does this tell us?

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Assumption 5 Independence?

• What does this tell us?
• Theres more to the story than just the
  regression
• Overall, happiness does not change over the
  course of the month
• The correlation is negative but insignificant.

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Assumptions

• underlie conclusions.
• You make assumptions in your interpretation
• The math is deterministic
• The regression line is always the best fit to the
  data
• But whether thats good enough fit is your
  decision

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Assumptions

• Regression estimates a line, its the best line
• If a line is not the way, find another way
• If theres more than a line in your data, that is
  a good thing. Your data answers questions you
  didnt ask!
• Oh, right, questions!
• Does fitting a line to the data answer your
  question?