Working with relationships between two variables

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Correlation Regression

• Working with relationships between two variables
• Size of Teaching Tip Stats Test Score

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Correlation Regression

• Univariate Bivariate Statistics
• U frequency distribution, mean, mode, range,
  standard deviation
• B correlation two variables
• Correlation
• linear pattern of relationship between one
  variable (x) and another variable (y) an
  association between two variables
• relative position of one variable correlates with
  relative distribution of another variable
• graphical representation of the relationship
  between two variables
• Warning
• No proof of causality
• Cannot assume x causes y

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Scatterplot!

• No Correlation
• Random or circular assortment of dots
• Positive Correlation
• ellipse leaning to right
• GPA and SAT
• Smoking and Lung Damage
• Negative Correlation
• ellipse learning to left
• Depression Self-esteem
• Studying test errors

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Pearsons Correlation Coefficient

• r indicates
• strength of relationship (strong, weak, or none)
• direction of relationship
• positive (direct) variables move in same
  direction
• negative (inverse) variables move in opposite
  directions
• r ranges in value from 1.0 to 1.0

-1.0 0.0
1.0
Strong Negative No Rel.
Strong Positive

• Go to website!
• playing with scatterplots

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Practice with Scatterplots
r .__ __
r .__ __
r .__ __
r .__ __
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Correlation Guestimation
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Samples vs. Populations

• Sample statistics estimate Population parameters
• M tries to estimate µ
• r tries to estimate ? (rho greek symbol ---
  not p)
• r correlation for a sample
• based on a the limited observations we have
• ? actual correlation in population
• the true correlation
• Beware Sampling Error!!
• even if ?0 (theres no actual correlation), you
  might get r .08 or r -.26 just by chance.
• We look at r, but we want to know about ?

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Hypothesis testing with Correlations

• Two possibilities
• Ho ? 0 (no actual correlation The Null
  Hypothesis)
• Ha ? ? 0 (there is some correlation The
  Alternative Hyp.)
• Case 1 (see correlation worksheet)
• Correlation between distance and points r -.904
• Sample small (n6), but r is very large
• We guess ? lt 0 (we guess there is some
  correlation in the pop.)
• Case 2
• Correlation between aiming and points, r .628
• Sample small (n6), and r is only moderate in
  size
• We guess ? 0 (we guess there is NO correlation
  in pop.)
• Bottom-line
• We can only guess about ?
• We can be wrong in two ways

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Reading Correlation Matrix
r -.904
p .013 -- Probability of getting a
correlation this size by sheer chance. Reject Ho
if p .05.
sample size
r (4) -.904, p?.05
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Predictive Potential

• Coefficient of Determination
• r²
• Amount of variance accounted for in y by x
• Percentage increase in accuracy you gain by using
  the regression line to make predictions
• Without correlation, you can only guess the mean
  of y
• Used with regression

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0
80
100
60
40
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Limitations of Correlation

• linearity
• cant describe non-linear relationships
• e.g., relation between anxiety performance
• truncation of range
• underestimate stength of relationship if you
  cant see full range of x value
• no proof of causation
• third variable problem
• could be 3rd variable causing change in both
  variables
• directionality cant be sure which way
  causality flows

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Regression

• Regression Correlation Prediction
• predicting y based on x
• e.g., predicting.
• throwing points (y)
• based on distance from target (x)
• Regression equation
• formula that specifies a line
• y bx a
• plug in a x value (distance from target) and
  predict y (points)
• note
• y actual value of a score
• y predict value

• Go to website!
• Regression Playground

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Regression Graphic Regression Line
See correlation regression worksheet
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Regression Equation

• y bx a
• y predicted value of y
• b slope of the line
• x value of x that you plug-in
• a y-intercept (where line crosses y access)
• In this case.
• y -4.263(x) 125.401
• So if the distance is 20 feet
• y -4.263(20) 125.401
• y -85.26 125.401
• y 40.141

See correlation regression worksheet
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SPSS Regression Set-up

• Criterion,
• y-axis variable,
• what youre trying to predict

• Predictor,
• x-axis variable,
• what youre basing the prediction on

Note Never refer to the IV or DV when doing
regression
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Getting Regression Info from SPSS
See correlation regression worksheet
y b (x) a y -4.263(20)
125.401
a
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Predictive Ability

• Mantra!!
• As variability decreases, prediction accuracy ___
• if we can account for variance, we can make
  better predictions
• As r increases
• r² increases
• variance accounted for increases
• the prediction accuracy increases
• prediction error decreases (distance between y
  and y)
• Sy decreases
• the standard error of the residual/predictor
• measures overall amount of prediction error
• We like big rs!!!

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Drawing a Regression Line by Hand

• Three steps
• Plug zero in for x to get a y value, and then
  plot this value
• Note It will be the y-intercept
• Plug in a large value for x (just so it falls on
  the right end of the graph), plug it in for x,
  then plot the resulting point
• Connect the two points with a straight line!