Pearson Product Moment Coefficient of Correlation:

1
Correlation Analysis

• Pearson Product Moment Coefficient of
  Correlation
• The variances and covariances are given by
• In general, when a sample of n individuals or
  experimental units is selected and two variables
  are measured on each individual or unit so that
  both variables are random, the correlation
  coef-ficient r is the appropriate measure of
  linearity for use in this situation.

2
Example The heights and weights of n 10
offensive backfield football players are randomly
selected from a countys football all-stars.
Calculate the correlation coefficient for the
heights (in inches) and weights (in pounds) given
in Table below.

• Table Heights and weights of n 10 backfield
  all-stars
• Player Height x
  Weight y
• 1 73 185
• 2 71 175
• 3 75 200
• 4 72 210
• 5 72 190
• 6 75 195
• 7 67 150
• 8 69 170
• 9 71 180
• 10 69 175

3

• Solution
• You should use the appropriate data entry method
  of your scientific calculator to verify the
  calculations for the sums of squares and
  cross-products
• using the calculational formulas given earlier
  in this chapter. Then
• or r .83. This value of r is fairly close to
  1, the largest possible value of r , which
  indicates a fairly strong positive linear
  relationship between height and weight.

4

• There is a direct relationship between the
  calculation formulas for the correlation
  coefficient r and the slope of the regression
  line b.
• Since the numerator of both quantities is Sxy,
  both r and b have the same sign.
• Therefore, the correlation coefficient has these
  general properties
• - When r 0, the slope is 0, and there is no
  linear relationship between x and y.
• - When r is positive, so is b, and there is a
  positive relationship between x and y.
• - When r is negative, so is b, and there is a
  negative relationship between x and y.

5
The relationship between r (correlation
coefficient) and the regression model
6
Figure Some typical scatter plots
7

• The population correlation coefficient r is
  calculated and interpreted as it is in the
  sample.
• The experimenter can test the hypothesis that
  there is no correlation between the variables x
  and y using a test statistic that is exactly
  equivalent to the test of the slope b in previous
  Section.

8

• Test of Hypothesis Concerning the correlation
  Coefficient r
• 1. Null hypothesis H 0 r 0
• 2. Alternative hypothesis
• One-Tailed Test Two-Tailed Test
• H a r gt 0 H a r ¹ 0
• (or H a r lt 0)
• 3. Test statistic
• When the assumptions are satisfied, the test
  statistic will have a Students t distribution
  with (n - 2) degrees of freedom.

9
When comparing to non-zero constant

• 1. Null hypothesis H 0 r r0
• 2. Alternative hypothesis
• One-Tailed Test Two-Tailed Test
• H a r gt r0 H a r ¹ r0
• (or H a r lt r0)
• 3. Test statistic
• When the assumptions are satisfied, the test
  statistic will have a Students t distribution
  with (n - 2) degrees of freedom.

10

• 4. Rejection region Reject H 0 when
• One-Tailed Test Two-Tailed Test
• t gt ta,n-2
  t gt ta/2, n-2 or t lt - ta/2,
  n-2 (or t lt -ta, n-2 when the alternative
  hypothesis is H a r lt 0 or H a r lt r0)
• or p-value lt a

11

• Example Refer to the height and weight data in
  the previous Example The correlation of height
  and weight was calculated to be r .8261. Is this
  correlation significantly different from 0?

Solution To test the hypotheses the value of
the test statistic is which for n 10 has a
t distribution with 8 degrees of freedom. Since
this value is greater than t.005 3.355, the
two-tailed p-value is less than 2(.005) .01,
and the correlation is declared significant at
the 1 level (P lt .01). The value r 2 .82612
.6824 means that about 68 of the variation in
one of the variables is explained by the other.
The Minitab printout n Figure 12.17 displays the
correlation r and the exact p-value for testing
its significance.
12

• r is a measure of linear correlation and x and y
  could be perfectly related by some curvilinear
  function when the observed value of r is equal to
  0.

13
Testing for Goodness of Fit

• In general, we do not know the underlying
  distribution of the population, and we wish to
  test the hypothesis that a particular
  distribution will be satisfactory as a population
  model.
• Probability Plotting can only be used for
  examining whether a population is normal
  distributed.
• Histogram Plotting and others can only be used to
  guess the possible underlying distribution type.

14
Goodness-of-Fit Test (I)

• A random sample of size n from a population whose
  probability distribution is unknown.
• These n observations are arranged in a frequency
  histogram, having k bins or class intervals.
• Let Oi be the observed frequency in the ith class
  interval, and Ei be the expected frequency in the
  ith class interval from the hypothesized
  probability distribution, the test statistics is

15
Goodness-of-Fit Test (II)

• If the population follows the hypothesized
  distribution, X02 has approximately a chi-square
  distribution with k-p-1 d.f., where p represents
  the number of parameters of the hypothesized
  distribution estimated by sample statistics.
• That is,
• Reject the hypothesis if

16
Goodness-of-Fit Test (III)

• Class intervals are not required to be equal
  width.
• The minimum value of expected frequency can not
  be to small. 3, 4, and 5 are ideal minimum
  values.
• When the minimum value of expected frequency is
  too small, we can combine this class interval
  with its neighborhood class intervals. In this
  case, k would be reduced by one.

17
Example 8-18 The number of defects in printed
circuit boards is hypothesized to follow a
Poisson distribution. A random sample of size 60
printed boards has been collected, and the number
of defects observed as the table below

• The only parameter in Poisson distribution is l,
  can be estimated by the sample mean 0(32)
  1(15) 2(19) 3(4)/60 0.75. Therefore, the
  expected frequency is

18
Example 8-18 (Cont.)

• Since the expected frequency in the last cell is
  less than 3, we combine the last two cells

19
Example 8-18 (Cont.)

• 1. The variable of interest is the form of
  distribution of defects in printed circuit
  boards.
• 2. H0 The form of distribution of defects is
  Poisson
• H1 The form of distribution of defects is not
  Poisson
• 3. k 3, p 1, k-p-1 1 d.f.
• 4. At a 0.05, we reject H0 if X20 gt X20.05, 1
  3.84
• 5. The test statistics is
• 6. Since X20 2.94 lt X20.05, 1 3.84, we are
  unable to reject the null hypothesis that the
  distribution of defects in printed circuit boards
  is Poisson.

20
Contingency Table Tests

• Example 8-20
• A company has to choose among three pension
  plans. Management wishes to know whether the
  preference for plans is independent of job
  classification and wants to use a 0.05. The
  opinions of a random sample of 500 employees are
  shown in Table 8-4.

21
Contingency Table Test- The Problem Formulation
(I)

• There are two classifications, one has r levels
  and the other has c levels. (3 pension plans and
  2 type of workers)
• Want to know whether two methods of
  classification are statistically independent.
  (whether the preference of pension plans is
  independent of job classification)
• The table

22
Contingency Table Test- The Problem Formulation
(II)

• Let pij be the probability that a random selected
  element falls in the ijth cell, given that the
  two classifications are independent. Then pij
  uivj, where the estimator for ui and vj are
• Therefore, the expected frequency of each cell is
• Then, for large n, the statistic
• has an approximate chi-square distribution with
  (r-1)(c-1) d.f.

23
Example 8-20
24
(No Transcript)
25
Key Concepts and Formulas

• I. A Linear Probabilistic Model
• 1. When the data exhibit a linear relationship,
  the appropriate model is y a b x e .
• 2. The random error e has a normal distribution
  with mean 0 and variance s 2.
• II. Method of Least Squares
• 1. Estimates a and b, for a and b, are chosen to
  minimize SSE, The sum of the squared deviations
  about the regression line,

26

• 2. The least squares estimates are b Sxy / Sxx
  and
• III. Analysis of Variance
• 1. Total SS SSR SSE, where Total SS Syy
  and SSR (Sxy)2 / Sxx.
• 2. The best estimate of s 2 is MSE SSE / (n -
  2).
• IV. Testing, Estimation, and Prediction
• 1. A test for the significance of the linear
  regressionH0 b 0can be implemented using
  one of the two test statistics

27

• 2. The strength of the relationship between x
  and y can be measured using
• which gets closer to 1 as the relationship gets
  stronger.
• 3. Use residual plots to check for nonnormality,
  inequality of variances, and an incorrectly fit
  model.
• 4. Confidence intervals can be constructed to
  estimate the intercept a and slope b of the
  regression line and to estimate the average
  value of y, E( y ), for a given value of x.
• 5. Prediction intervals can be constructed to
  predict a particular observation, y, for a
  given value of x. For a given x, prediction
  intervals are always wider than confidence
  intervals.

28

• V. Correlation Analysis
• 1. Use the correlation coefficient to measure
  the relationship between x and y when both
  variables are random
• 2. The sign of r indicates the direction of the
  relationship r near 0 indicates no linear
  relationship, and r near 1 or -1 indicates a
  strong linear relationship.
• 3. A test of the significance of the correlation
  coefficient is identical to the test of the
  slope b.

29
Cause and Effect

• X could cause Y
• Y could cause X
• X and Y could cause each other
• X and Y could be caused by a third variable Z
• X and Y could be related by chance
• Bad (or good) luck
• Need careful examination of the study. Try to
  find previous evidences or academic explanations.