Figure 15-3 (p. 512) Examples of positive and negative relationships. (a) Beer sales are positively related to temperature. (b) Coffee sales are negatively related to temperature.

1
Figure 15-3 (p. 512)Examples of positive and
negative relationships. (a) Beer sales are
positively related to temperature. (b) Coffee
sales are negatively related to temperature.
2
Figure 15-4 (p. 513) Examples of different
values for linear correlations (a) shows a
strong positive relationship, approximately
0.90 (b) shows a relatively weak negative
correlation, approximately 0.40 (c) shows a
perfect negative correlation, 1.00 (d) shows no
linear trend, 0.00.
3
The Pearson Correlation

• The Pearson correlation r measures the
  direction and degree of linear (straight line)
  relationship between two variables.
• The magnitude of the Pearson correlation ranges
  from 0 (indicating no linear relationship between
  X and Y) to 1.00 (indicating a perfect
  straight-line relationship between X and Y).
• The correlation can be either positive or
  negative depending on the direction of the
  relationship.

4
The Pearson Correlation

• r degree to which X and Y vary together divided
  by degree to which X and Y vary separately
• The Pearson correlation compares the amount of
• Covariability variation from the relationship
  between X and Y
• to the amount X and Y vary separately
• If there is a perfect linear relationship
• every change in X is matched by a change in the
  Y variable
• see fig 15.4a which illustrates a perfect
  negative correlation
• When X goes up one unit Y goes down one unit
• When X goes up two units Y goes down two units
• So X and Y covary

5
The Pearson Correlation

• To compute the Pearson correlation
• calculate the variability of X and Y scores
  separately by computing SS for the scores of each
  variable SSX and SSY
• Calculate Covariability which is the sum of
  products of deviation scores SP S (X-Mx)(Y-My)
• The Pearson correlation is found by computing the
  ratio of SP compared to square root of the
  SSxSSy
• r SP/?(SSX)(SSY) .

6
Excel file for generating a perfect correlation
7
The Pearson Correlation Calculations Example
15.2
Calculation of Pearson correlation r SP / v
(ssx)(ssy) r 6 / v (ssx)(ssy) r 6/ v
(10)(10) r 0.60 Note SS columns are not in
the textbook

• Calculating SP from definitional formula
• SP S (X-Mx)(Y-My)
• Using squared deviation table p 515

X Y X-Mx Y-My Products (X-Mx)2 (Y-My)2
1 3 -2 -2 4 4 4
2 6 -1 1 -1 1 1
4 4 1 -1 -1 1 1
5 7 2 2 4 4 4
 M3  M5     SP 6 SSx 10  SSy 10

8
The Pearson Correlation Calculations Example
15.2
SP Using Computational formula SP SXY (SXSY /
n) SP 66 - 12(20) /4 6 SS Using
Computational formula SSx SX2 (SX)2 /n SSx
46 (12) 2 / 4 10 SSy SY2 (SY)2 /n
SSy 110 (20)2 /4 10 Calculation of
Pearson correlation r SP / v (ssx)(ssy) r 6/
v (10)(10) r 0.60
X Y X2 Y2 XY
1 3 1 9 3
2 6 4 36 12
4 4 16 16 16
5 7 25 49 35
12 20 46 110 66
9
Calculating Sum of Products (SP) Example 15.3

• Using definitional formula table 15.1 page 518

10
Figure 15.5 (p. 517)Scatter plot of data from
Example 15.3
r SP/?(SSX)(SSY) r 28/ ?(64)(16) r
28/32 0.875
X Y
0 2
10 6
4 2
8 4
8 6
11

• Time For More Fun With SPSS

12
Using and Interpreting The Pearson Correlation

• Predictions
• knowing the relationship between SAT and GPA
• makes it possible to use SAT to predict GPA
• Validity
• comparing two tests of the same construct such as
  anxiety
• if they have high correlation their is construct
  validity
• Reliability
• Test Retest reliability
• Theory Verification
• When a theory makes a prediction about the
  relationship between two variables they can be
  tested with correlation
• Amount of sleep is positively related to GPA

13
Interpreting Correlations

• Correlations describe relationships
• but do not explain why they exist
• can not draw cause and effect conclusions
• However causation is not ruled out either
• Cigarette smoking is positively correlated with
  cancer
• Correlations are sensitive to the range of scores
• Correlations are sensitive to outliers
• Correlations are not proportions
• size of the r value is not directly related to
  strength of the relationship
• use r2 to interpret strength of the relationship

14
Figure 15-6 (p. 522)Hypothetical data showing
the logical relationship between the number of
churches and the number of serious crimes for a
sample of U.S. cities.

• Correlations describe relationships
• but do not explain why they exist
• can not draw cause and effect conclusions

15
Problem of Restricted RangeFigure 15-7 (p.
523)In this example, the green ellipse, when the
full range of X and Y values are used there is a
strong, positive correlation.However, the brown
circle, when the X values have a restricted
range of scores the correlation is near zero.

• Correlations are sensitive to the range of scores

16
Problem of OutliersFigure 15-8 (p. 524)A
demonstration of how one extreme data point (an
outlier) can influence the value of a correlation.

• Correlations are sensitive to outliers

17
Correlation and Strength of the Relationship

• Coefficient of Determination r2
• Using correlation for prediction
• Using SAT to predict GPA
• Based on degree of the relationship
• r value is not a good measure for predictions
• r2 measures the proportion of variability in one
  variable that can be determined by the other
  variable
• Small, Medium, Large see table 9.3
• Used as a measure of effect size for t test
• Amount of variance in the dependent explained by
  the independent

18
Figure 15.9 (p. 525) Three sets of data showing
three different degrees of linear relationships.
19
Calculations for Pearson Correlation Coefficient

• Definitional Formula
• r SP / v (ssx)(ssy)
• SP S (X-Mx)(Y-My)
• Computational Formula
• r SP / v (ssx)(ssy)
• SP SXY - SXSY / n
• z score formula (for samples)
• r Szxzy / n-1

20
The Spearman Correlation

• The Spearman correlation is used in two general
  situations
• (1) X and Y both consist of ranks
• Because it measures the relationship between two
  ordinal variables
• (2)When the relationship is non linear
• the two variables must be converted to ranks
  before the Spearman correlation is computed
• Because it measures the consistency of direction
  of the relationship between two variables.

21
Examples of relationships that are not linear
(a) relationship between reaction time and
age(b) relationship between mood and drug dose.
22
Relationship between practice and performance.
There is a consistent positive
relationship. Fig. 15-14, p. 536
23
The Spearman Correlation (cont.)

• The calculation of the Spearman correlation
  requires
• 1. Two variables are observed for each
  individual.
• 2. The observations for each variable are rank
  ordered. Note that the X values and the Y values
  are ranked separately.
• 3. After the variables have been ranked, the
  Spearman correlation is computed by either
• a. Using the Pearson formula with the ranked
  data.
• b. Using the special Spearman formula
  assuming there are few, if any, tied ranks

24
15.3
15.3
15.9
15.9
Performance
Performance
Practice
Practice
25
The Spearman Correlation Formulas and Calculations
Original Data 
X Y
3 12
4 10
10 11
11 9
12 2

• Example 15.10 Use the ranks for calculations
• SP SXY (SXSY / n) using computational
  formula
• SP 36 - 15(15) /5 -9
• SSx SX2 (SX)2 /n using computational
  formula
• SSx 55 (15) 2 / 5 10
• SSy SY2 (SY)2 /n using computational
  formula
• SSy 55 (15)2 /5 10
• rs SP / v (SSx)(SSy)
• rs -9 / v (10)(10) -0.90

26
Scatter plots of original scores and ranks for
Example 15.10
27
The Spearman Correlation Formulas and Calculations

• After the variables have been ranked
• Spearman correlation is computed by either
• a. Using the Pearson formula with the ranked
  data
• b. Using the special Spearman formula
• assuming there are few, if any, tied ranks
• Example 15.10 Always do the calculations on the
  ranks
• rs 1 - 6SD2 /n(n2-1) using special formula
• rs 1 - 6(38) / 5(25-1) -0.90
• But not if there are tied scores
• You are not responsible for this formula on the
  exam

Ranks
X Y D D 2  
1 5 4 16  
2 3 1 1  
3 4 1 1  
4 2 -2 4  
5 1 -4 16  

Sum 38
Original Data 
X Y
3 12
4 10
10 11
11 9
12 2
28
Ranking Tied ScoresExample from page 545
Use the Pearson correlation equation on the
ranked scores
29
point-biserial correlation as an alternative to
the Pearson Correlation

• The Pearson correlation formula can also be used
  to measure the relationship between two variables
  when one or both of the variables is dichotomous.
  
• A dichotomous variable is one for which there are
  exactly two categories for example, men/women or
  succeed/fail.
• The point-biserial correlation is used in
  situations where one variable is dichotomous and
  the other consists of regular numerical scores
  interval or ratio scale

30
point-biserial correlation as an alternative to
the Pearson Correlation

• The calculation of the point-biserial correlation
  proceeds as follows
• Assign numerical values to the two categories of
  the dichotomous variable(s). Traditionally, one
  category is assigned a value of 0 and the other
  is assigned a value of 1.
• Use the regular Pearson correlation formula to
  calculate the correlation.

31
point-biserial correlation as an alternative to
the Pearson Correlation

• The point-biserial correlation is closely related
  to the independent-measures t test introduced in
  Chapter 10.
• When the data consists of one dichotomous
  variable and one numerical variable, the
  dichotomous variable can also be used to separate
  the individuals into two groups.
• Then, it is possible to compute a sample mean for
  the numerical scores in each group.

32
point-biserial correlation as an alternative to
the Pearson Correlation

• In this case, the independent-measures t test can
  be used to evaluate the mean difference between
  groups.
• If the effect size for the mean difference is
  measured by computing r2 (the percentage of
  variance explained), the value of r2 will be
  equal to the value obtained by squaring the
  point-biserial correlation.

33
phi-coefficient as an alternatives to the
Pearson Correlation

• The phi-coefficient is used when both variables
  are dichotomous.
• The calculation proceeds as follows
• Convert each of the dichotomous variables to
  numerical values by assigning a 0 to one
  category and a 1 to the other category for each
  of the variables.
• Use the regular Pearson formula with the
  converted scores.

34
phi-coefficient as an alternatives to the
Pearson Correlation
35
phi-coefficient as an alternatives to the
Pearson Correlation