Goodness-of-Fit Tests and Contingency Tables

1
Chapter 14

• Goodness-of-Fit Tests and Contingency Tables

2
The Chi-Square Random Variable

• A random sample of n observations each of which
  can be classified into exactly one of K
  categories is selected. Denote the observed
  numbers in each category by O1, O2, . . ., OK.
  If the null hypothesis (H0) specifies
  probabilities ?1, ?2, . . ., ?K for an
  observation falling into each of these
  categories, the expected numbers in the
  categories under H0 would be
• If the null hypothesis in true and the sample
  size is large enough so that the expected values
  are at least five, then the random variable
  associated with
• has, to a good approximation, a chi-square
  distribution with (K 1) degrees of freedom.

3
A Goodness-of-Fit Test

• A goodness-of-fit test, of significance level ?,
  of H0 against the alternative that the specified
  probabilities are not correct is based on the
  decision rule
• where ?2 K-1, ? is the number for which
• And the random variable ?2 K-1 follows a
  chi-square distribution with (K 1) degrees of
  freedom.

4
Goodness-of-Fit Tests When Population Parameters
are Estimated

• Suppose that a null hypothesis specifies category
  probabilities that depend on the estimation (from
  the data) of m unknown population parameters.
  The appropriate goodness-of-fit test of the null
  hypothesis when population parameters are
  estimated is the same as that previously
  mentioned, except that the number of degrees of
  freedom for the chi-square random variable is
• Where K is the number of categories.

5
Bowman-Shelton Test for Normality

• The Bowman-Shelton Test for Normality is based on
  the closeness to 0 of the sample skewness and the
  closeness to 3 of the sample kurtosis. The test
  statistic is
• It is known that as the number of sample
  observations becomes very large, this statistic
  has, under the null hypothesis that the
  population distribution is normal, a chi-square
  distribution with 2 degrees of freedom. The null
  hypothesis is, of course, rejected for large
  values of the test statistic.

6
r x c Contingency Table(Table 14.8)
Attribute B Attribute B Attribute B Attribute B Attribute B Attribute B
Attribute A 1 2 . . . C Totals
1 2 . . . r Totals O11 O21 . . . Or1 C1 O12 O22 . . . Or2 C2 O1c O2c . . . Orc Cc R1 R2 . . . Rr n
7
Chi-Square Random Variable for Contingency Tables

• It can be shown that under the null hypothesis
  the random variable associated with
• has, to a good approximation, a chi-square
  distribution with (r 1)(c 1) degrees of
  freedom. The approximation works well if each of
  the estimated expected numbers Eij is at least 5.
  Sometimes adjacent classes can be combined in
  order to meet this assumption.

8
A Test of Association in Contingency Tables

• Suppose that a sample of n observations is cross
  classified according to two attributes in an r x
  c contingency table. Denote by Oij the number of
  observations in the cell that is in the ith row
  and the jth column. If the null hypothesis is
• The estimated expected number of observations in
  this cell, under H0, is
• Where Ri and Cj are the corresponding row and
  column totals. A test of association at a
  significance level ? is based on the following
  decision rule

9
Key Words

• Bowman-Shelton Test for Normality
• ?2 Random Variable
• Goodness-of-Fit Tests
• Specified Parameters
• Unknown Parameters
• Poisson Distribution
• Normal Distribution
• Kurtosis
• Skewness
• Test of Association