Characteristics of a relationship

1
Overview

• Correlation
• -Definition
• -Deviation Score Formula, Z score formula
• -Hypothesis Test
• Regression
• Intercept and Slope
• Unstandardized Regression Line
• Standardized Regression Line
• Hypothesis Tests

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Associations among Continuous Variables
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3 characteristics of a relationship
Direction Positive() Negative (-) Degree of
association Between 1 and 1 Absolute values
signify strength Form Linear Non-linear
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Direction
Positive
Negative

Large values of X large values of Y, Small
values of X small values of Y. - e.g. IQ and
SAT
Large values of X small values of Y Small
values of X large values of Y -e.g. SPEED and
ACCURACY
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Degree of association
Strong(tight cloud)
Weak(diffuse cloud)
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Form
Linear
Non- linear
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Regression Correlation
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What is the best fitting straight line?
Regression Equation Y a bX How closely are
the points clustered around the line? Pearsons R
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Correlation
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Correlation - Definition

• Correlation a statistical technique that
  measures and describes the degree of linear
  relationship between two variables

Scatterplot
Y
X
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Pearsons r

• A value ranging from -1.00 to 1.00 indicating the
  strength and direction of the linear
  relationship.
• Absolute value indicates strength
• /- indicates direction

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The Logic of Correlation
MEAN of X
MEAN of Y
For a strong positive association, the
cross-products will mostly be positive
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The Logic of Correlation
MEAN of X
MEAN of Y
For a strong negative association, the
cross-products will mostly be negative
Cross-Product
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The Logic of Correlation
MEAN of X
MEAN of Y
For a weak association, the cross-products will
be mixed
Cross-Product
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Pearsons r
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Deviation Score Formula
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Deviation Score Formula
.99
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Pearsons r
For a strong positive association, the SP will be
a big positive number
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Pearsons r
Deviation score formula
For a strong negative association, the SP will be
a big negative number
SP (sum of products)
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Pearsons r
Deviation score formula
For a weak association, the SP will be a small
number ( and will cancel each other out)
SP (sum of products)
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Pearsons r
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Z-score formula
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Z-score formula
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Z-score formula
r .99
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Formulas for R
Z score formula
Deviations formula
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Interpretation of R

• A measure of strength of association how
  closely do the points cluster around a line?
• A measure of the direction of association is it
  positive or negative?

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Interpretation of R

• r .10 very small association, not usually
  reliable
• r .20 small association
• r .30 typical size for personality and
  social studies
• r .40 moderate association
• r .60 you are a research rock star
• r .80 hmm, are you for real?

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Interpretation of R-squared

• The amount of covariation compared to the amount
  of total variation
• The percent of total variance that is shared
  variance
• E.g. If r .80, then X explains 64 of the
  variability in Y (and vice versa)

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Hypothesis testing with r

• Hypotheses
• H0 ? 0
• HA ? ? 0

Test statistic r
Or just use table E.2 to find critical values of r
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Practice
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Practice
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Properties of R

• A standardized statistic will not change if
  you change the units of X or Y. (bc based on
  z-scores)
• The same whether X is correlated with Y or vice
  versa
• Fairly unstable with small n
• Vulnerable to outliers
• Has a skewed distribution

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Linear Regression
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Linear Regression

• But how do we describe the line?
• If two variables are linearly related it is
  possible to develop a simple equation to predict
  one variable from the other
• The outcome variable is designated the Y
  variable, and the predictor variable is
  designated the X variable
• E.g. centigrade to Fahrenheit
• F 32 1.8C
• this formula gives a specific straight line

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The Linear Equation

• F 32 1.8(C)
• General form is Y a bX
• The prediction equation Y a bX
• Where
• a intercept
• b slope
• X the predictor
• Y the criterion

a and b are constants in a given line X and Y
change
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The Linear Equation

• F 32 1.8(C)
• General form is Y a bX
• The prediction equation Y a bX
• Where
• a intercept
• b slope
• X the predictor
• Y the criterion

Different bs
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The Linear Equation

• F 32 1.8(C)
• General form is Y a bX
• The prediction equation Y a bX
• Where
• a intercept
• b slope
• X the predictor
• Y the criterion

Different as
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The Linear Equation

• F 32 1.8(C)
• General form is Y a bX
• The prediction equation Y a bX
• Where
• a intercept
• b slope
• X the predictor
• Y the criterion

Different as and bs
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Slope and Intercept

• Equation of the line
• The slope b the amount of change in y with one
  unit change in x
• The intercept a the value of y when x is zero

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Slope and Intercept

• Equation of the line
• The slope
• The intercept

The slope is influenced by r, but is not the same
as r
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When there is no linear association (r 0), the
regression line is horizontal. b0.
and our best estimate of age is 29.5 at all
heights.
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When the correlation is perfect (r 1.00),
all the points fall along a straight line with a
slope
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When there is some linear association (0ltrlt1),
the regression line fits as close to the points
as possible and has a slope
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Where did this line come from?

• It is a straight line which is drawn through a
  scatterplot, to summarize the relationship
  between X and Y
• It is the line that minimizes the squared
  deviations (Y Y)2
• We call these vertical deviations residuals

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Regression lines
Minimizing the squared vertical distances, or
residuals
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Unstandardized Regression Line

• Equation of the line
• The slope
• The intercept

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Properties of b (slope)

• An unstandardized statistic will change if you
  change the units of X or Y.
• Depends on whether Y is regressed on X or vice
  versa

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Standardized Regression Line

• Equation of the line
• The slope
• The intercept

A person 1 stdev above the mean on height would
be how many stdevs above the mean on weight?
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Properties of ß (standardized slope)

• A standardized statistic will not change if
  you change the units of X or Y.
• Is equal to r, in simple linear regression

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Exercise

• Calculate
• r
• b
• a
• ß
• Write the regression equationWrite the
  standardized equation

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Exercise

• Calculate
• r .866
• b .375
• a 3.125
• ß .866
• Write the regression equationWrite the
  standardized equation

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Regression Coefficients Table
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Summary
Correlation Pearsons r
Unstandardized Regression Line
Standardized Regression Line
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Exercise in Excel

• Calculate
• r
• b
• a
• ß
• Write the regression equationWrite the
  standardized equation
• Sketch the scatterplot and regression line