Regression Lecture 8

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Regression Lecture 8
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Aims for Today - Regression

• Drawing lines on scatterplots
• The regression line Predicting values
• Correlation
• Ranked based correlation
• Break/Handout
• Examples by Dan
• Chile and maybe being hit by a car
• How tos

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Scatter Plot

• Plotting 2 continuous-ish variables
• Exploring their association
• One of the most used and most useful techniques
  in science.

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Several ways to make in SPSS.
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Default shows what appears to be a negative
relationship, but the graphs can be improved.
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Graphing 3 Variables (London et al., 2007)

• 4- to 9-year olds
• 2 week recall
• 10 month recall

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Can you see the 8s?
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Is this the right approach?

• Fitting a straight line (a linear relationship)

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Finding the Regression Line

• Very general procedure (easily expanded)
• Simple linear regression
• Easiest way is just to draw a straight line
  yourself
• A more formal method has some value
• and finding the ß0 and ß1 which minimize Sei2
  
• Least Squares is also used in t test and mean
• (least absolute value is used for the median)

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• minimizing the squared residuals min Sei2
• Is least squares regression
• better than eyeballing it?
• Are there better formal methods?

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Do you need to know the equations for ß0 and
ß1? Not reallyWould they be worth seeing
once? Probably
just look at, don't write
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Regressions sometimes used to predict values
(data based on Tytherleigh, 2002)
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Running a regression in R
lm is for Linear Model
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r2 or adjusted r2and r or R
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Assessing the Fit The Correlation
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Equation bit

• Top part determines whether positive or negative.
  If xi and yi are same side as their means,
  positive, otherwise negative.
• If as one goes up, the other goes up, positive.

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Correlation Strength of the linear relationship

• Can get to it in several ways.
• The correlation squared in the proportion of
  shared variance.
• The correlation can range only from -1 to 1.
• Does a correlation between x and y mean x caused
  y?
• Does a correlation between x and y mean that
  there is some causal relationship in the network
  of hypotheses that include x and y?
• Are the most parsimonious ones x -gt y and y -gt x?

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Significance Testing

• H0 ? (rho) 0
• Almost always use two tailed tests
• You must know the sample size
• r 0.1 is significant with n500 at 5
• r 0.4 is not significant with n20 at 5
• (Cohen sizes .1 small, .3 medium, .5 large)

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Significance Testing and Confidence IntervalsThe
equations
with df n - 2, and df 1, n - 2
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Making Confidence Intervals

• Several programs on web. http//glass.ed.asu.edu/
  stats/analysis/rci.html

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Notice the Normal and Basic Bootstraps give
impossible upper bounds.
BCa very similar to asymptotic methods
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r .64, w/o outlier r .92, w/o influential
point r .38
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Assumptions for Significance

• Random sampling
• It must make sense to talk about the response
  variable (the DV) as being continuous.
• No weird patterns (or non-linear in general) in
  residuals. Variance of residual homoscedastic
  (ie., not varying by other variables -
  heteroscedastic)
• Examination of outliers

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What to do if assumptions not meet(to get data,
install and load mrt. data(crime) and attach)
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Ranked based Correlation

• Spearman's rho
• Rank the data and use Pearson's stuff for ties.
• r .94 and Spearman's rS .78.

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In SPSS and R just tick a box or change the method
Doesn't print confidence interval
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• Same correlation estimate.
• But the CI really does meet appropriate
  assumptions.

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Break Time

• Short break, we have a lot to get through
  afterwards.
• In 4 groups
• Look at the handout that I am about to give you.
  Discuss how you would report your findings in a
  scientific journal versus People magazine. Are
  there any other statistics you would want to do?
• Talk about what you wrote for Suppose an
  undergraduate said "Since it is for looking at
  differences among means, why is it called an
  Analysis of Variance?"

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Some Examples

• Chile Heat To discuss re-expression and what to
  do with outliers.
• Automobile Accidents To discuss using theory to
  guide your statistics.

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Are smaller chiles hotter?

• How to measure length and heat.
• Length skewed

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Testing Normality
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par(mfrowc(1,2)) qqnorm(LENGTH)
qqline(LENGTH)qqnorm(log(LENGTH2.54))qqline(log
(LENGTH2.54)) par(mfrowc(1,1))
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Measuring Heat Scoville units or the number of
chiles?
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Command Summary

• r1 lt- lm(HEATLENGTH)
• r2 lt- lm(HEATLENGTHlt30LENGTHLENGTHlt30)
• r3 lt- lm(HEAT log(LENGTH 2.54))

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plot(r1)Nu Mex is hotter than predicted for
its length
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What to do with

• Genetically
• engineered.
• Depends on the population and purpose.

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What is a "linear model"
Y ßX e
Don't worry if you dislike matrix notation
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Vehicle-Pedestrian Accidents

• What is the relationship between the impact
  velocity of a vehicle and the throw of a
  pedestrian?
• A lot is known about how a body should move when
  hit by a car at a certain velocity.
• Good reason to suggest throwi k vi2 ei
• Dan will glance around to see if anyone
  looks interested in "why" this equation makes
  sense, and may skip the next two slides.

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Why Theoretical Sense?

• Body takes on impact
• horizontal velocity of
• the car, v, at an angle
• above the horizontal.
• Vertical velocity vy v sin ?
• Horizontal velocity vx v cos ?
• Time in air, t, is related only to vy. t 2
  vy / g, where g is the constant for gravity on
  Earth, about 10m/s2..

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• Without friction, vx is constant and thus throw
  should be
• and if ? is the same for all cars throw v2 k,
  where k is a constant.
• Thus, throwi k vi2 ei
• Simpler Only 1 unknown (k) to solve for AND it
  has some empirical meaning

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Wood, Simms Walsh (2005)
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Otte's work with crash test dummies
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• reglin lt- lm(distance speed)
• regpoly lt- lm(distance speed spsq)
• regmodel lt- lm(distance spsq - 1)

This can done in SPSS too. Tick
no intercept/constant.
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Summary
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This week's journal

• Try help(par)
• Write an equation in Word
• Access these data from web fishstock in .dat
  (use read.table) or .sav (use SPSS or read.spss)
• Variables are ocean (how much winter low
  temperature is above freezing in Celsius) and
  fishstock (gt 2cm in
• thousands per cubic kilometer).
• What are the correlation and the regression
  equation?
• Write a sentence about the results.

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