regression and correlation

Correlation and Regression
The test you choose depends on level of measurement:
Independent Dependent Test
Dichotomous Interval-Ratio Independent Samples t-test
Dichotomous
Nominal Interval-Ratio ANOVA
Dichotomous Dichotomous
Nominal Nominal Cross Tabs
Dichotomous Dichotomous
Interval-Ratio Interval-Ratio Bivariate Regression/Correlation
Dichotomous

 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”

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

 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).
Family Income
Self-esteem
Dependent Variables
Number of Children
Criminal Behavior

 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.
Family Income
Self-esteem
Dependent Variables
Number of Children
Criminal Behavior

 Let’s 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.

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
What do you think
the relationship is?
012345678910

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
Is it positive?
Negative?
No change?
012345678910

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
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.
012345678910

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
The slope of a line is the change in the corresponding
Y value for each unit increase in X (rise over run).
Slope = 0, No relationship!
Slope = 0.2, Positive Relationship!
1
Slope = -0.2, Negative Relationship!
1
0.5
012345678910

 The mathematical equation for a line:
Y = mx + b
Where: Y = the line’s position on the
vertical axis at any point
X = the line’s 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

 The statistics equation for a line:
Y = a + bx
Where: Y = the line’s position on the
vertical axis at any point (value of
dependent variable)
X = the line’s 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
^
^

 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.

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
How do we minimize the
distance between a line and all
the data points?
012345678910

• 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. Σ(Y - Y-bar)2

• 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 X’s).
∀ Σ(Y - Y)2
Y = line
^ ^

 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
 Σ (Y – Y) = 0; therefore, we use Σ (Y - Y)2
…and minimize that!
^
^
^ ^

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
012345678910
Illustration of Y – Y
= Yi, actual Y value corresponding w/ actual X
= Yi, line level on Y corresponding w/ actual X
∧
∧
5
-4
Y = 10, Y = 5
Y = 0, Y = 4
∧
∧

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
012345678910
Illustration of (Y – Y)2
∧
∧
5
-4
(Yi – Y)2
= deviation2
Y = 10, Y = 5 . . . 25
Y = 0, Y = 4 . . . 16
∧
∧
∧

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
012345678910
Illustration of (Y – Y)2
The goal: Find the line that minimizes
sum of deviations squared.
∧
∧
?
The best line will have the lowest value of sum of deviations squared
(adding squared deviations for each case in the sample.

Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
012345678910
∧
Y = a + bX
∧
∧
e

 We use Σ (Y - Y)2
…and minimize that!
 There is a simple, elegant formula for
“discovering” the line that minimizes the sum of
squared errors
Σ((X - X)(Y - Y))
b = Σ(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
^
^

Y
X0 1
12345678910
Considering that a regression line minimizes Σ (Y - Y)2
,
where would the regression line cross for an interval-ratio
variable regressed on a dichotomous independent variable?
^
For example:
0=Men: Mean = 6
1=Women: Mean = 4

Y
X0 1
12345678910
The difference of means will be the slope.
This is the same number that is tested for
significance in an independent samples t-test.
^
0=Men: Mean = 6
1=Women: Mean = 4
Slope = -2 ; Y = 6 – 2X

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

Correlation
Y,
crime
s
X,
self-esteem
10 15 20 25 30 35 40
So our equation is:
Y = 6 - .2X
The slope tells us direction of
association… How strong is
that?
012345678910
^

Correlation
12345678910
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).
Y
X

Correlation
12345678910
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).
Y
X

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.

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.
Σ(X - X)2
 Recall that s.d. of x, Sx = n - 1
Σ(Y - Y)2
 and the s.d. of y, Sy = n - 1
Sx
 Pearson correlation, r = Sy b

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

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!

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 line’s equation tells you how the mean on your dependent
variable changes as your independent variable goes up.
^
Y
^
X
Y

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 Σ ( Y - Y)2
 σ = n - 2 = n - 2
Y
^
X
Y
^
^

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
X
Y Conditional
standard
deviation
Original
standard
deviation

Error Analysis
 So guess what?
 We have a way to determine how much our
understanding of Y is improved when taking X
into account—it is based on the fact that
conditional standard deviations should be
smaller than Y’s original standard deviation.

Error Analysis
 Proportional Reduction in Error
 Let’s call the variation around the mean in Y “Error 1.”
 Let’s 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, let’s just stop at the basic measure, Sum of
Squared Deviations.
 Error 1 (E1) = Σ (Y – Y)2 also called “Sum of Squares”
 Error 2 (E2) = Σ (Y – Y)2 also called “Sum of Squared Errors”
Y
X
Y Error 2Error 1
∧

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) = Σ (Y – Y)2
 Error 2 (E2) = Σ (Y – Y)2
Y
X
Y
Error 2
Error 1
∧

R-squared
r2
= E1 - E2
E1
= TSS - SSE
TSS
= Σ (Y – Y)2 -
Σ (Y – Y)2
Σ (Y – Y)2
∧
r2
is called the “coefficient of
determination”…
It is also the square of the
Pearson correlation
Y
X
Y
Error 2
Error 1

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
everybody’s 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 finding—a small R2
only indicates that
you haven’t told the whole (or much of the) story with your variable.

Error Analysis, SPSS
Some SPSS output (Anti- Gay Marriage regressed on Age):
r2
Σ (Y – Y)2 - Σ (Y – Y)2
Σ (Y – Y)2
∧
196.886 ÷ 2853.286 = .069
Line to the Mean
Data points to the line
Data points to
the mean
Original SS
for Anti- Gay
Marriage

Error Analysis
Some SPSS output (Anti- Gay Marriage regressed on Age):
r2
Σ (Y – Y)2 - Σ (Y – Y)2
Σ (Y – Y)2
∧
196.886 ÷ 2853.286 = .069
Line to the Mean
Data points to the line
Data points to
the mean
0 18 45 89 Age
Strong Oppose 5
Oppose 4
Neutral 3
Support 2
Strong Support 1
Anti- Gay
Marriage
M = 2.98
Colored lines are examples of:
Distance from each person’s data point
to the line or model—new, still
unexplained error.
Distance from line or model to Mean for
each person—reduction in error.
Distance from each person’s data point
to the Mean—original variable’s error.

ANOVA Table
X
Y
Mean
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.
BSS
WSS
TSS
Line Intersects
Group Means

Dichotomous Variables
Y
X
0 1
12345678910 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).
^
0=Men: Mean = 6
1=Women: Mean = 4
Slope = -2 ; Y = 6 – 2X
Mean = 5 BSS
WSS
TSS

Regression, 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.
Recall that statistics are divided between descriptive
and 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).
Most extreme
5% of F’s
F = Regression SS / Residual SS

 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 3z
β

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
population’s slope, Ho, one determines whether a sample could have
been drawn from a population where the slope is equal Ho.
1. Two-tailed significance test for α-level = .05
2. Critical t = +/- 1.96
3. To find if there is a significant slope in the population,
Ho: β = 0
Ha: β ≠ 0 Σ ( Y – Y )2
n Collect Data n - 2
n Calculate t (z): t = b – βo s.e. =
s.e. Σ ( X – X )2
n Make decision about the null hypothesis
n Find P-value
∧

Back to the SPSS output:
The standard error and t
appears on SPSS output
…and the p-value too!

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.

regression and correlation

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regression and correlation