Correlation and Regression

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Correlation and Regression
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Correlation and Regression
The test you choose depends on level of
measurement Independent Dependent Test Dichoto
mous Interval-Ratio Independent Samples
t-test Dichotomous Nominal Interval-Ratio A
NOVA Dichotomous Dichotomous Nominal Nominal
Cross Tabs Dichotomous Dichotomous Interval-R
atio Interval-Ratio Bivariate Regression/Correlat
ion Dichotomous
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Correlation and Regression

• Bivariate regression is a technique that fits a
  straight line as close as possible between all
  the coordinates of two continuous variables
  plotted on a two-dimensional graph--to summarize
  the relationship between the variables
• Correlation is a statistic that assesses the
  strength and direction of association of two
  continuous variables . . . It is created through
  a technique called regression

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Bivariate Regression

• For example
• A criminologist may be interested in the
  relationship between Income and Number of
  Children in a family or self-esteem and criminal
  behavior.

Independent Variables Family Income Self-esteem
Dependent Variables Number of Children Criminal
Behavior
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Bivariate Regression

• For example
• Research Hypotheses
• As family income increases, the number of
  children in families declines (negative
  relationship).
• As self-esteem increases, reports of criminal
  behavior increase (positive relationship).

Independent Variables Family Income Self-esteem
Dependent Variables Number of Children Criminal
Behavior
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Bivariate Regression

• For example
• Null Hypotheses
• There is no relationship between family income
  and the number of children in families. The
  relationship statistic b 0.
• There is no relationship between self-esteem and
  criminal behavior. The relationship statistic b
  0.

Independent Variables Family Income Self-esteem
Dependent Variables Number of Children Criminal
Behavior
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Bivariate Regression

• Lets look at the relationship between
  self-esteem and criminal behavior.
• Regression starts with plots of coordinates of
  variables in a hypothesis (although you will
  hardly ever plot your data in reality).
• The data
• Each respondent has filled out a self-esteem
  assessment and reported number of crimes
  committed.

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Bivariate Regression
Y, crimes
What do you think the relationship is?
0 1 2 3 4 5 6 7 8 9 10
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression
Y, crimes
Is it positive? Negative? No change?
0 1 2 3 4 5 6 7 8 9 10
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression
Regression is a procedure that fits a line to the
data. The slope of that line acts as a model for
the relationship between the plotted variables.
Y, crimes
0 1 2 3 4 5 6 7 8 9 10
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression
The slope of a line is the change in the
corresponding Y value for each unit increase in X
(rise over run).
Y, crimes
Slope 0.2, Positive Relationship!
0 1 2 3 4 5 6 7 8 9 10
0.5
Slope 0, No relationship!
1
1
X, self-esteem
10 15 20 25
30 35 40
Slope -0.2, Negative Relationship!
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Bivariate Regression

• The mathematical equation for a line
• Y mx b
• Where Y the lines position on the
  vertical axis at any point
• X the lines position on the horizontal
  axis at any point
• m the slope of the line
• b the intercept with the Y axis, where
  X equals zero

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Bivariate Regression

• The statistics equation for a line
• Y a bx
• Where Y the lines position on the
  vertical axis at any point (value of
  dependent variable)
• X the lines position on the horizontal
  axis at any point (value of the independent
  variable)
• b the slope of the line (called the
  coefficient)
• a the intercept with the Y axis, where
  X equals zero

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Bivariate Regression

• The next question
• How do we draw the line???
• Our goal for the line
• Fit the line as close as possible to all the
  data points for all values of X.

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Bivariate Regression
How do we minimize the distance between a line
and all the data points?
Y, crimes
0 1 2 3 4 5 6 7 8 9 10
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression

• How do we minimize the distance between a line
  and all the data points?
• You already know of a statistic that minimizes
  the distance between itself and all data values
  for a variable--the mean!
• The mean minimizes the sum of squared
  deviations--it is where deviations sum to zero
  and where the squared deviations are at their
  lowest value. ?(Y - Y-bar)2

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Bivariate Regression

• The mean minimizes the sum of squared
  deviations--it is where deviations sum to zero
  and where the squared deviations are at their
  lowest value.
• Take this principle and fit the line to the
  place where squared deviations (on Y) from the
  line are at their lowest value (across all Xs).
• ?(Y - Y)2 Y line

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Bivariate Regression

• There are several lines that you could draw where
  the deviations would sum to zero...
• Minimizing the sum of squared errors gives you
  the unique, best fitting line for all the data
  points. It is the line that is closest to all
  points.
• Y or Y-hat Y value for line at any X
• Y case value on variable Y
• Y - Y residual
• ? (Y Y) 0 therefore, we use ? (Y - Y)2 and
  minimize that!

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Bivariate Regression
?
Illustration of Y Y Yi, actual Y value
corresponding w/ actual X Yi, line level
on Y corresponding w/ actual X

Y, crimes
?
5
0 1 2 3 4 5 6 7 8 9 10
?
Y 10, Y 5 Y 0, Y 4
?
-4
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression
?
Illustration of (Y Y)2 Yi, actual Y
value corresponding w/ actual X Yi, line
level on Y corresponding w/ actual X

Y, crimes
?
5
0 1 2 3 4 5 6 7 8 9 10
?
(Yi Y)2 deviation2 Y 10, Y 5 . . . 25 Y
0, Y 4 . . . 16
?
?
-4
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression
?
Illustration of (Y Y)2 Yi, actual Y
value corresponding w/ actual X Yi, line
level on Y corresponding w/ actual X The goal
Find the line that minimizes sum of deviations
squared.
Y, crimes
?
0 1 2 3 4 5 6 7 8 9 10
?
X, self-esteem
10 15 20 25
30 35 40
The best line will have the lowest value of sum
of deviations squared (adding squared deviations
for each case in the sample.
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Bivariate Regression
Y, crimes
?
?
Y a bX
e
0 1 2 3 4 5 6 7 8 9 10
?
X, self-esteem
10 15 20 25
30 35 40
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Bivariate Regression

• We use ? (Y - Y)2 and minimize that!
• There is a simple, elegant formula for
  discovering the line that minimizes the sum of
  squared errors
• ?((X - X)(Y - Y))
• b ?(X - X)2 a Y - bX Y
  a bX
• This is the method of least squares, it gives our
  least squares estimate and indicates why we call
  this technique ordinary least squares or OLS
  regression

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Bivariate Regression
Y

Considering that a regression line minimizes ? (Y
- Y)2, where would the regression line cross for
an interval-ratio variable regressed on a
dichotomous independent variable?
For example 0Men Mean 6 1Women Mean
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1 2 3 4 5 6 7 8 9
10
X
0 1
25
Bivariate Regression
Y
The difference of means will be the slope. This
is the same number that is tested for
significance in an independent samples t-test.
1 2 3 4 5 6 7 8 9
10

Slope -2 Y 6 2X
X
0 1
0Men Mean 6 1Women Mean 4
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Correlation

• This lecture has covered how to model the
  relationship between two variables with
  regression.
• Another concept is strength of association.
• Correlation provides that.

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Correlation
So our equation is Y 6 - .2X The slope tells
us direction of association How strong is that?
Y, crimes

0 1 2 3 4 5 6 7 8 9 10
X, self-esteem
10 15 20 25
30 35 40
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Correlation
Y
1 2 3 4 5 6 7 8 9
10
Example of Low Negative Correlation When there is
a lot of difference on the dependent variable
across subjects at particular values of X, there
is NOT as much association (weaker).
X
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Correlation
Y
1 2 3 4 5 6 7 8 9
10
Example of High Negative Correlation When there
is little difference on the dependent variable
across subjects at particular values of X, there
is MORE association (Stronger).
X
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Correlation

• To find the strength of the relationship between
  two variables, we need correlation.
• The correlation is the standardized slope it
  refers to the standard deviation change in Y when
  you go up a standard deviation in X.

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Correlation

• The correlation is the standardized slope it
  refers to the standard deviation change in Y when
  you go up a standard deviation in X.
• ?(X - X)2
• Recall that s.d. of x, Sx n - 1
• ?(Y - Y)2
• and the s.d. of y, Sy n - 1
• Sx
• Pearson correlation, r Sy b

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Correlation

• The Pearson Correlation, r
• tells the direction and strength of the
  relationship between continuous variables
• ranges from -1 to 1
• is when the relationship is positive and - when
  the relationship is negative
• the higher the absolute value of r, the stronger
  the association
• a standard deviation change in x corresponds with
  r standard deviation change in Y

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Correlation

• The Pearson Correlation, r
• The pearson correlation is a statistic that is an
  inferential statistic too.
• r - (null 0)
• tn-2 (1-r2) (n-2)
• When it is significant, there is a relationship
  in the population that is not equal to zero!

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Error Analysis

• Y a bX This equation gives the conditional
  mean of Y at any given value of X.
• So In reality, our line gives us the expected
  mean of Y given each value of X
• The lines equation tells you how the mean on
  your dependent variable changes as your
  independent variable goes up.

Y

Y
X
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Error Analysis

• As you know, every mean has a distribution around
  it--so there is a standard deviation. This is
  true for conditional means as well. So, you also
  have a conditional standard deviation.
• Conditional Standard Deviation or Root Mean
  Square Error equals approximate average
  deviation from the line.
• SSE ? ( Y - Y)2
• ? n - 2 n - 2

Y

Y
X


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Error Analysis

• The Assumption of Homoskedasticity
• The variation around the line is the same no
  matter the X.
• The conditional standard deviation is for any
  given value of X.
• If there is a relationship between X and Y, the
  conditional standard deviation is going to be
  less than the standard deviation of Y--if this is
  so, you have improved prediction of the mean
  value of Y by looking at each level of X.
• If there were no relationship, the conditional
  standard deviation would be the same as the
  original, and the regression line would be flat
  at the mean of Y.

Y
Conditional standard deviation
Original standard deviation
Y
X
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Error Analysis

• So guess what?
• We have a way to determine how much our
  understanding of Y is improved when taking X into
  accountit is based on the fact that conditional
  standard deviations should be smaller than Ys
  original standard deviation.

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Error Analysis

• Proportional Reduction in Error
• Lets call the variation around the mean in Y
  Error 1.
• Lets call the variation around the line when X
  is considered Error 2.
• But rather than going all the way to standard
  deviation to determine error, lets just stop at
  the basic measure, Sum of Squared Deviations.
• Error 1 (E1) ? (Y Y)2 also called Sum of
  Squares
• Error 2 (E2) ? (Y Y)2 also called Sum of
  Squared Errors

Y
Error 2
Error 1
Y
X
?
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R-Squared

• Proportional Reduction in Error
• To determine how much taking X into consideration
  reduces the variation in Y (at each level of X)
  we can use a simple formula
• E1 E2 Which tells us the proportion or
• E1 percentage of original error that
  is Explained by X.
• Error 1 (E1) ? (Y Y)2
• Error 2 (E2) ? (Y Y)2

Error 2
Y
Error 1
Y
X
?
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R-squared
r2 E1 - E2 E1 TSS - SSE
TSS ? (Y Y)2 - ? (Y Y)2 ? (Y
Y)2
r2 is called the coefficient of
determination It is also the square of the
Pearson correlation
Error 1
Y
?
Error 2
Y
X
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R-Squared

• R2
• Is the improvement obtained by using X (and
  drawing a line through the conditional means) in
  getting as near as possible to everybodys value
  for Y over just using the mean for Y alone.
• Falls between 0 and 1
• Of 1 means an exact fit (and there is no
  variation of scores around the regression line)
• Of 0 means no relationship (and as much scatter
  as in the original Y variable and a flat
  regression line through the mean of Y)
• Would be the same for X regressed on Y as for Y
  regressed on X
• Can be interpreted as the percentage of
  variability in Y that is explained by X.
• Some people get hung up on maximizing R2, but
  this is too bad because any effect is still a
  findinga small R2 only indicates that you
  havent told the whole (or much of the) story
  with your variable.

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Error Analysis, SPSS
Some SPSS output (Anti- Gay Marriage regressed on
Age)
r2
196.886 2853.286 .069
?
? (Y Y)2 - ? (Y Y)2 ? (Y Y)2
Line to the Mean
Data points to the line
Data points to the mean Original SS for Anti- Gay
Marriage
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Error Analysis
Some SPSS output (Anti- Gay Marriage regressed on
Age)
r2
196.886 2853.286 .069
?
? (Y Y)2 - ? (Y Y)2 ? (Y Y)2
Line to the Mean
Data points to the line
Data points to the mean
Colored lines are examples of Distance from each
persons data point to the line or modelnew,
still unexplained error. Distance from line or
model to Mean for each personreduction in
error. Distance from each persons data point to
the Meanoriginal variables error.
Strong Oppose 5 Oppose 4 Neutral
3 Support 2 Strong
Support 1
Anti- Gay Marriage M 2.98
0 18 45 89
Age
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ANOVA Table
Q Why do I see an ANOVA Table?
A We bust up variance to get R2. Each case has
a value for distance from the line (Y-barcond.
Mean) to Y-barbig, and a value for distance from
its Y value and the line (Y-barcond. Mean).
Squared distance from the line to the mean
(Regression SS) is equivalent to BSS, df 1. In
ANOVA, all in a group share Y-bargroup The
squared distance from the line to the data values
on Y (Residual SS) is equivalent to WSS, df
n-2. In ANOVA, all in a group share
Y-bargroup The ratio, Regression to Residual SS,
forms an F distribution in repeated sampling. If
F is significant, X explains some variation in Y.
Line Intersects Group Means
Y
BSS WSS TSS
Mean
X
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Dichotomous Variables
Using a dichotomous independent variable, the
ANOVA table in bivariate regression will have the
same numbers and ANOVA results as a one-way ANOVA
table would (and compare this with an independent
samples t-test).
Y
1 2 3 4 5 6 7 8 9
10
BSS WSS TSS
Mean 5

Slope -2 Y 6 2X
0 1
X
0Men Mean 6 1Women Mean 4
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Regression, Inferential Statistics
Recall that statistics are divided between
descriptive and inferential statistics.

• Descriptive
• The equation for your line is a descriptive
  statistic. It tells you the real, best-fitted
  line that minimizes squared errors.

• Inferential
• But what about the population? What can we say
  about the relationship between your variables in
  the population???
• The inferential statistics are estimates based on
  the best-fitted line.

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Regression, Inferential Statistics

• The significance of F, you already understand.
• The ratio of Regression (line to the mean of Y)
  to Residual (line to data point) Sums of Squares
  forms an F ratio in repeated sampling.
• Null r2 0 in the population. If F exceeds
  critical F, then your variables have a
  relationship in the population (X explains some
  of the variation in Y).

F Regression SS / Residual SS
Most extreme 5 of Fs
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Regression, Inferential Statistics

• What about the Slope or Coefficient?
• From sample to sample, different slopes would be
  obtained.
• The slope has a sampling distribution that is
  normally distributed.
• So we can do a significance test.

-3 -2 -1 0 1 2 3
z
?
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Regression, Inferential Statistics

• Conducting a Test of Significance for the slope
  of the Regression Line
• By slapping the sampling distribution for the
  slope over a guess of the populations slope, Ho,
  one determines whether a sample could have been
  drawn from a population where the slope is equal
  Ho.
• Two-tailed significance test for ?-level .05
• Critical t /- 1.96
• To find if there is a significant slope in the
  population,
• Ho ? 0
• Ha ? ? 0
  ? ( Y Y )2
• Collect Data
  n - 2
• Calculate t (z) t b ?o s.e.
• s.e.
  ? ( X X )2
• Make decision about the null hypothesis
• Find P-value

?
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Correlation and Regression
Back to the SPSS output
The standard error and t appears on SPSS output
and the p-value too!
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Correlation and Regression
Back to the SPSS output
?
Y 1.88 .023X So the GSS example, the slope is
significant. There is evidence of a positive
relationship in the population between Age and
Anti- Gay Marriage sentiment. 6.9 of the
variation in Marriage attitude is explained by
age. The older Americans get, the more likely
they are to oppose gay marriage.
A one year increase in age elevates anti
attitudes by .023 scale units. There is a weak
positive correlation. A s.d, increase in age
produces a .023 s.d. increase in anti scale
units.