Chapter 26: Comparing Counts

1
Chapter 26Comparing Counts
2
Goodness-of-Fit Test

• Involves testing a hypothesis.
• There is no single parameter to estimate.
• Considers all categories to give an overall idea
  of whether the observed distribution differs from
  the hypothesized one.

• All creatures have their determined time for
  giving birth and carrying fetus, only a man is
  born all year long, not in determined time, one
  in the seventh month, the other in the eighth,
  and so on till the beginning of the eleventh
  month.
• Aristotle

3
Assumptions and Conditions

• Counted Data Condition
• Check that the data are counts for the categories
  of a categorical variable.
• Independence Assumption
• Check that the individuals counted in the cells
  are sampled independently from some population.
• If not, check the randomization condition the
  individuals who have been counted should be a
  random sample from some population.
• Sample Size Assumption
• Expected cell frequency condition expect to
  observe at least 5 individuals in each cell.

4
A Chi-Square Test for Goodness-of-Fit

• Compare the observed counts in each cell with the
  expected counts.
• Look at the differences between the observed and
  expected counts.
• The test is always one-sided.
• There is no direction to the rejection of the
  null model we know it just doesnt fit.

• Chi- Square statistic refers to a family of
  sampling distribution models.
• Number of degrees of freedom is n 1, where n is
  the number of categories.

5
Whats Your Sign?

• Check Conditions
• Counted data condition there are counts of the
  number of executives in categories.
• Randomization condition this is a convenience
  sample, but no expectation of bias.
• Expected cell frequency condition the null
  hypothesis expects that of the 256 should
  occur in each sign.

• Hypothesis
• Ho Births are uniformly distributed over zodiac
  signs. (pAriespTaurus)
• HA Births are not uniformly distributed over
  zodiac signs.

• The sampling distribution of the test statistic
  is ?2 with 12 1 11 degrees of freedom.
• Use a Chi-Square goodness-of-fit test.

6
Whats Your Sign?

• The chi-square procedure
• Find the expected values.
• Values come from the null hypothesis.
• Multiply the total number of observations by the
  hypothesized proportion.
• Compute the residuals, Observed Expected.
• Square the residuals.
• Compute the component for each cell,
• Find the sum of the components.
• Find the degrees of freedom, the number of cells
  minus 1.
• Test the hypothesis find the P-value.

7
Whats Your Sign?TI-84 Calculator for
chi-square goodness of fit test

• Enter counts in L1 and expected percentages in
  L2.
• Convert expected percentages to expected counts.
• Calculate chi-square in L3.

8
Whats Your Sign?TI-84 Calculator for
chi-square goodness of fit test

• Find the sum of L3.
• Find the P-value
• The probability of finding a ?2 value at least as
  high as the one calculated from the data.
• DISTR menu, ?2 cdf

9
Whats Your Sign?

• P-value
• Test is one-sided, only consider the right tail.
• Large ?2 values correspond to small P-values,
  leading to rejection of the null hypothesis.
• The P-value is the area in the upper tail of the
  ?2 model for 11 degrees of freedom above the
  computed ?2 value.
• Conclusion
• The P-value of 0.926 means that an observed
  chi-square value of 5.08 or higher would occur
  about 93 of the time.
• There is virtually no evidence that the
  distribution of zodiac signs among executives is
  not uniform.

10
Comparing Observed Distributions

• Chi-square test for homogeneity
• Assumptions and Conditions
• Counted data condition
• Check that the data are counts for the categories
  of a categorical variable.
• Independence Assumption Randomization condition
• When we test for homogeneity, we often are not
  interested in some larger population so we dont
  need to check the randomization condition.
• Sample Size Assumption
• Expected cell frequency condition expected
  count in each cell must be at least 5
  individuals.

11
Post-Graduation Plans

• Who High school graduates
• What Post-graduation activities
• When 1980, 1990, 2000
• Why Regular survey for general information

12
Post-Graduation Plans

• Hypothesis
• Have the choices made by high school graduates in
  what they do after graduation changed?
• Ho The post-high school choices made by the
  classes of 1980, 1990, and 2000 have the same
  distribution (homogeneous).
• HA The post-high school choices made by the
  classes of 1980, 1990, and 2000 do not have the
  same distribution.

• Check the conditions
• Counted data condition there are counts of the
  number of students in categories.
• Randomization condition No inference will be
  drawn to other high schools or other classes, so
  no need to check for a random sample.
• Expected cell frequency condition The expected
  values are all at least 5 (see table, later).

• Under these conditions, the sampling distribution
  of the test statistic is ?2 with (4 1) X (3
  1) 6 degrees of freedom.
• Perform a chi-square test of homogeneity.

13
Post-Graduation Plans

• TI-84 Steps
• Enter data in a matrix.
• Do the chi-square test of homogeneity.
• Matrix Edit B
• Note that all expected counts are at least 5.

14
Post-Graduation Plans

• Conclusion
• The P-value is very small.
• Observed pattern is very unlikely to occur by
  chance.
• Reject the null hypothesis.
• The choices made by high school graduates have
  changed over the two decades examined.

15
Post-Graduation Plans

• Examine the Residuals
• Standardized Residuals
• Divide the cells residual by the square root of
  its expected value.
• Values are the square root of the components
  calculated for each cell, with or to show
  whether we observed more or less cases than
  expected.

• What trends do you see?

16
Independence

• Chi-Square Test for Independence
• Data categorize subjects from a single group on
  two categorical variables.
• Contingency Tables
• Categorize counts on two or more variables.
• Decide whether the distribution of counts on one
  variable is contingent on the other.

• Assumptions and Conditions
• Counted data condition
• Check that the data are counts for the categories
  of a categorical variable.
• Independence Assumption Randomization condition
• When we test for independence, we are interested
  in generalizing to some larger population.
• Sample Size Assumption
• Expected cell frequency condition expected
  count in each cell must be at least 5
  individuals.

17
Hepatitis C Related to Tattoos?

• Who Patients being treated for non-blood-related
  disorders
• What Tattoo status and hepatitis C status
• When 1991, 1992
• Where Texas

18
Hepatitis C Related to Tattoos?

• Hypothesis
• Are the categorical variables tattoo status and
  hepatitis C status statistically independent?
• H0 Tattoo status and hepatitis C status are
  independent.
• HA Tattoo status and hepatitis C status are not
  independent.

• Check the conditions
• Counted data condition there are counts of
  individuals in categories of two categorical
  variables.
• Randomization condition Although not an SRS, the
  data were selected to avoid biases and should be
  representative of the general population.
• Expected cell frequency condition The expected
  values do not meet the condition that all are
  greater than 5. Continue with caution be sure
  to check the residuals.

• Under these conditions, the sampling distribution
  of the test statistic is ?2 with (3 1) X (2
  1) 2 df.
• Perform a chi-square test for independence.

19
Hepatitis C Related to Tattoos?

• TI-84 Steps
• Enter data in a matrix.
• Do the chi-square test of independence.
• Matrix Edit B
• Note that not all expected counts are at least 5.

20
Hepatitis C Related to Tattoos?

• Conclusion
• The P-Value is very small, indicating that if
  these variables were independent, the pattern
  seen would be very unlikely to occur by chance.
• The hepatitis C status is not independent of the
  tattoo status.
• HOWEVER, check the two cells with the small
  expected counts to determine if they did or did
  not influence the result too greatly.
• Remember A complete solution must include
  additional analysis, recalculation, and a final
  conclusion.

21
Hepatitis C Related to Tattoos?

• Analysis of Residuals
• Too small an expected frequency can arbitrarily
  inflate the residual, leading to an inflated
  chi-square statistic.
• In this case, the standardized residual for the
  hepatitis C and Tattoo, Parlor cell is large ?
  Inflated chi-square statistic?

• Standardized Residuals

22
Hepatitis C Related to Tattoos?

• Options
• Based upon concerns, choose not to report the
  results.
• Include a warning when reporting the results.
• Combine the appropriate categories to larger
  sample size and expected frequencies.
• Recalculation

• Recalculation (continued)
• Conclusion
• The tattoo status and hepatitis C status are not
  independent. The data suggest that tattoo parlors
  may be a particular problem, but we do not have
  enough data to draw that conclusion.

23
What Can Go Wrong?

• A failure of independence between two categorical
  variables does not show a cause-and-effect
  relationship between them.
• There is no way to differentiate the direction of
  any possible causation from one variable to
  another.
• Lurking variables could be responsible for the
  observed lack of independence.

• Dont use chi-square methods unless the data are
  counts.
• Data reported as proportions or percentages can
  be used if they are converted to counts.
• Just because data are reported in a two-way table
  does not mean they are suitable for chi-square
  procedures.
• Beware large samples.
• The degrees of freedom for the chi-square tests
  do not grow with sample size.
• With a sufficiently large sample size, a
  chi-square test can always reject the null
  hypothesis.
• There are no confidence intervals to help in
  determining the effect size.