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Correlation & Regression
Chapter 15
Correlation

statistical technique that is used to measure
and describe a relationship between two
variables (X and Y).
3 Characteristics

1. The direction of the relationship
– Positive correlation ( +)
– Negative Correlation (-)
2. The Form of the
Relationship

Relationships tend to have a linear
relationship. A line can be drawn through
the middle of the data points in each figure.
 The most common use of regression is to
measure straight-line relationships.
 Not always the case
Scatterplot

Visual representation of scores.
 Each individual score is represented by a
single point on the graph.
 Allows you to see any patterns of trends
that exist in the data.
Psychology 295
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HOMEWORK
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0
-2
12
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EXAM2
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30
3. The Degree of the
Relationship

Measures how well the data fit the specific
form being considered.
 The degree of relationship is measured by
the numerical value of the correlation (0 to
1.00)
– A perfect correlation is always identified by a
correlation of 1.00 and indicates a perfect fit.
– A correlation value of 0 indicates no fit or
relationship at all.
Example Correlations
Pearson Product-Moment
Correlation

Measures the degree and the direction of the linear
relationship between two variables
 Identified by r
degree to which X and Y vary together
r= degree to which X and Y vary separately
=
___covariability of X and Y____
variability of X and Y separately
How do we calculate the
Pearson Correlation?

Sum of products of deviations: provides a
parallel procedure for measuring the amount of
covariability between two variables.
Definitional formula SP =  (X-X) (Y-Y)
Computational
formula
SP = XY -
XY
n
Computational Formula
r =
SP
SSx SSy
Standardized Formula
r=
z
z
 xy
N 1
Using and Interpreting r
 Prediction
 Validity
 Reliability
 Theory
Verification
*“CORRELATION DOES NOT MEAN CAUSATION”
Restriction of Range

Occurs whenever a correlation is computed
from scores that do not represent the full
range of possible values.
 ie:IQ tests among college students.
 Correlations should not be generalized
beyond the range of data represented in the
sample.
Other Correlation Coefficients

Spearman r
– Two ranked (ordinal) variables

Point-biserial r
– Pearson r between dichotomous and continuous
variable

Phi Coefficient
– Pearson r between two dichotomous variables
Outliers

An individual with X and/or Y values that
are substantially different (larger or smaller)
from the values obtained for the other
individuals in the data set.
 An outlier can dramatically influence the
value obtained for the correlation.
 Always look at scatter plots to determine if
there are outliers.
Coefficient of Determination
 r2
measures the proportion of variability in
one variable that can be determined from
the relationship with the other variable.
 A correlation of r = .80 means that r2 = .64
or 64% of the variability in Y scores can be
predicted from the relationship with X.
Hypothesis Testing with r
Standard hypotheses:
 H0: r = 0 (There is no population correlation)
 H1: r  0 (There is a real correlation)
 Other hypotheses are possible, e.g.,
one-sided hypotheses or hypotheses
with r  0.
 If the alternative hypothesis prevails,
one can state that a correlation is
significant in the sample


There will always be some error between a sample
correlation (r) and the population correlation (r)
it represents.
 Goal of the hypothesis test is to decide between
the following two alternatives:
– The nonzero sample correlation is simply due to
chance.
– The nonzero sample correlation accurately represents a
real, nonzero correlation in the population.
– USE TABLE B 6.
CAPA Example

Questions 5-10
 Step 1) Calculate the SS for X
 Step 2) Calculate the SS for Y
 Formula for SS:
( X)2
 SS =  = (X-X2) OR SS=  X2 n
Calculation

Calculate XY to obtain
SP =  (X-X) (Y-Y) or
XY
Computational SP = XY n
Definitional formula
formula
cont’d

Calculate r:
SP
r= SSxSSy
 Compare to Table B6 and find the critical
value
 What can we determine??????
The errors in prediction are
the distances between
actual Y values and
prediction line
Best Fitting Line

The line that gives the best prediction of Y
 We must find the specific values for a and b
SP
b = SSx
a = Y –bX
Y = bX + a
Caution Be Aware

The predicted value is not perfect unless r =
1.00 or –1.00
 The regression equation should not be used
to make predictions for X values that fall
outside the range of values covered by the
original data (restriction of range).
The Spearman Correlation

Used for non-linear relationships
 Ordinal (ranked) Data
 Can be used as an alternative to the Pearson
 Measure of consistency
Ranks

Consistent relationships among scores
produces a linear relationship when the
scores are converted to ranks.
When is the Spearman
correlation used?

When the original data are ordinal, when
the X and Y values are ranks.
 When a researcher wants to measure the
consistency of a relationship between X and
Y, independent of the specific form of the
relationship.
– monotonic
Calculating a Spearman
Correlation
Step 1) Rank X and Y scores (separately)
Step 2) Use the Pearson correlation formula
for the ranks of the X and Y scores.
rs
Tied Scores

1.
2.
3.
When converting scores into ranks for the
Spearman correlation, there may be two or more
identical scores. If this occurs:
List the scores in order from smallest to largest
(include tied values)
Assign a rank (1st, 2nd ) to each position in the
list.
When two or more scores are tied, compute the
mean of their ranked positions, and assign this
mean value as the final rank for each score.
Special Formula for the
Spearman Correlation

X = (n+1)/2

SS= n(n2 –1)
12
6D2
rs = 1– n(n2-1) *D is the difference between the X
rank and the Y rank for each individual.
*N = number of pairs
Regression
Is the statistical technique for finding the
best-fitting straight line for a set of data.
 To find the line that best describes the
relationship for a set of X and Y data.

Regression Analysis

Question asked: Given one variable, can we
predict values of another variable?


Examples: Given the weight of a person, can we
predict how tall he/she is; given the IQ of a
person, can we predict their performance in
statistics; given the basketball team’s wins, can we
predict the extent of a riot. ...

Using regression analysis one can make this type
of prediction:

Predictor and Criterion

Regression analysis allows one to
  predict values of the criterion: point prediction
 estimate strength of predictability (significance
testing)
Regression line

makes the relationship between variables
easier to see.
 identifies the center, or central tendency, of
the relationship, just as the mean describes
central tendency for a set of scores.
 can be used for a prediction.
The Equation for a Line
Y = bX + a
– b = the slope
– a = y-intercept
– Y= predicted value
Example

Local tennis club charges $5 per hour plus an annual
membership fee of $25.
 Compute the total cost of playing tennis for 10 hours per
month.
(predicted cost) Y = (constant) bX + (constant) a
When X = 10
Y= $5(10 hrs) + $25
Y = 75
When X = 30
Y= $5(30 hrs) + $25
Y = $175
Least Squares Solution

Minimize the square root of the squared
differences between data points and the line
 The best fit line has the smallest total
squared error
 We seek to minimize
(Y - Y)2
•When estimating the parameters for slope and intercept,
one minimizes the sum of the squared residuals, that is,
prediction errors:
• least squares estimation.
The errors in prediction are
the distances between
actual Y values and
prediction line
Equations

The line that gives the best prediction of Y
 We must find the specific values for a and b
SP
b = SSx
a = Y –bX
Y = bX + a
Caution Be Aware

The predicted value is not perfect unless r =
1.00 or –1.00
 The regression equation should not be used
to make predictions for X values that fall
outside the range of values covered by the
original data (restriction of range).
Conclusion

Using methods of statistical inference in
regression analysis we ask whether the
regression line explains a significant
portion of the variance of Y.