Contingency Tables

1
Contingency Tables

• Chapters Seven, Sixteen, and Eighteen
• Chapter Seven
• Definition of Contingency Tables
• Basic Statistics
• SPSS program (Crosstabulation)
• Chapter Sixteen
• Basic Probability Theory Concepts
• Test of Hypothesis of Independence

2
Contingency Tables (continued)

• Chapter Eighteen
• Measures of Association
• For nominal variables
• For ordinal variables

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Basic Empirical Situation

• Unit of data.
• Two nominal scales measured for each unit.
• Example interview study, sex of respondent,
  variable such as whether or not subject has a
  cellular telephone.
• Objective is to compare males and females with
  respect to what fraction have cellular telephones.

4
Crosstabulation of Data

• Prepare a data file for study.
• One record per subject.
• Three variables per record subject ID, sex of
  subject, and indicator variable of whether
  subject has cellular telephone.
• SPSS analysis
• Statistics, summarize, crosstabs
• Basic information is the contingency table.

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Two Common Situations

• Hypothesized causal relation between variables.
• No hypothesized causal relation.

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Hypothesized Causal Relation

• Classification of variables
• Independent variable is one hypothesized to be
  cause. Example sex of respondent.
• Dependent variable is hypothesized to be the
  effect. Example whether or not subject has
  cellular telephone.
• Format convention
• Columns to categories of independent variable
• Rows to categories of dependent variable

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Association Study

• No hypothesized causal mechanism.
• Whether or not subject above median on verbal SAT
  and whether or not above median on quantitative
  SAT.
• No convention about assigning variables to rows
  and columns.

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Contingency Table

• One column for each value of the column variable
  C is the number of columns.
• One row for each value of the row variable R is
  the number of rows.
• R x C contingency table.

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Contingency Table

• Each entry is the OBSERVED COUNT O(i,j) of the
  number of units having the (i,j) contingency.
• Column of marginal totals.
• Row of marginal totals.

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Example Contingency Table (Hypothetical)
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Example Contingency Table (Hypothetical)

• Entry 60 in the upper left hand corner means that
  there were 60 male respondents who owned a
  cellular telephone.
• ASSUME marginal totals are known
• THEN, knowing entry of 60 means that you can
  deduce all other entries.
• This 2 x 2 table has one degree of freedom.
• R x C table has (R-1)(C-1) degrees of freedom.

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Row and Column Percentages

• Natural to use percentages rather than raw
  counts.
• Remember that you want to use these numbers for
  comparison purposes.
• The term rate is often used to refer to a
  percentage or probability.
• Can ask for column percentages, row percentages,
  or both.
• Percentage in the direction of the independent
  variable (usually the column).

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Relation of Percentages to Probabilities

• ASSUME that the column variable is the
  independent variable.
• THEN the column percentages are estimates of the
  conditional probabilities given the setting of
  the independent variable.
• The basic questions revolve around whether or not
  the conditional distributions are the same for
  all settings of the independent variable.

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Bar Charts

• Graphical means of presenting data.
• SPSS analysis
• Graphs, bar chart.
• Can use either count scale or percentage scale
  (prefer percentage scale).
• Can have bars side by side or stacked.

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Generalization of the R x C contingency table

• Can have three or more variables to classify each
  subject. These are called layers.
• In example, can add whether respondent is student
  in college or student in high school.

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Chapter Sixteen Comparing Observed and Expected
Counts

• Basic hypothesis
• Definitions of expected counts.
• Chi-squared test of independence.

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Basic Hypothesis

• ASSUME column variable is the independent
  variable.
• Hypothesis is independence.
• That is, the conditional distribution in any
  column is the same as the conditional
  distribution in any other column.

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Expected Count

• Basic idea is proportional allocation of
  observations in a column based on column total.
• Expected count in (i, j ) contingency E(i,j)
  total number in column j total number in row
  i/total number in table.
• Expected count need not be an integer one
  expected count for each contingency.

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Residual

• Residual in (i,j) contingency observed count in
  (i,j) contingency - expected count in (i,j)
  contingency.
• That is, R(i,j) O(i,j)-E(i,j)
• One residual for each contingency.

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Pearson Chi-squared Component

• Chi-squared component for (i, j) contingency
  C(i,j) (Residual in (i, j) contingency)2/expecte
  d count in (i, j) contingency.
• C(i,j)(R(i,j))2 / E(i,j)

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Assessing Pearson Component

• Rough guides on whether the (i, j) contingency
  has an excessively large chi-squared component
  C(i,j)
• the observed significance level of 3.84 is about
  0.05.
• Of 6.63 is about 0.01.
• Of 10.83 is 0.001.

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Pearson Chi-Squared Test

• Sum C(i,j) over all contingencies.
• Pearson chi-squared test has (R-1)(C-1) degrees
  of freedom.
• Under null hypothesis
• Expected value of chi-square equals its degrees
  of freedom.
• Variance is twice its degrees of freedom

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Special Case of 2 x 2 Contingency Table
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Chi-squared test for a 2x2 table

• 1 degree of freedom (R-1)(C-1)1
• Value of chi-squared test is given by
• N(AD-BC)2 /(AB)(CD)(AC)(BD)
• There is a correction for continuity

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Computer Output for Chi-Squared Tests

• Output gives value of test.
• Asymptotic significance level (p-value)
• Four types of test
• Pearson chi-squared
• Pearson chi-squared with continuity correction
• Likelihood ratio test (theoretically strong test)
• Fishers exact test (most accepted, if given.

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Example Problem Set

• The independent variable is whether or not the
  subject reported using marijuana at time 3 in a
  study (time 3 is roughly in later high school).
  The dependent variable is whether or not the
  subject reported using marijuana at time 4 in a
  study (time 4 is roughly in middle college or
  beginning independent living). The contingency
  table is on the next slide.

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Marijuana Use at Time 4 by Marijuana Use at Time 3
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Example Question 1

• Which of the following conclusions is correct
  about the test of the null hypothesis that the
  distribution of whether or not a subject uses
  marijuana at time 3 is independent of whether the
  subject uses marijuana at time 4?
• Usual options.

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Solution to question 1

• Find the significance level in the chi-square
  test output. Pearson chi-square (without and with
  continuity correction), likelihood ratio, and
  Fishers exact had significance levels of 0.000.
• Option A (reject at the 0.001 level of
  significance) is the correct choice.

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Example Question 2

• How many degrees of freedom does the contingency
  table describing this output have?
• Solution (R-1)(C-1)(2-1)(2-1)1.

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Example Question 3

• Specify how the expected count of 97.8 for
  subjects who did use marijuana at time 3 and
  time 4 was calculated?
• Solution
• Total number using at time 3 was 151.
• Total number using at time 4 was 237.
• Total N was 366.
• Expected Count151237/366.

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Example Question 4

• Compute the contribution to Pearsons chi-square
  statistic from the cell used marijuana at time 3
  and used marijuana at time 4.
• Solution
• Observed count was 142
• Expected count was 97.8
• Component(142-97.8)2/97.819.97

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Example Question 5

• Describe the pattern of association between these
  two variables.
• Solution. There was a strong dependence between
  the two variables. About 44 percent of nonusers
  at time 3 used at time 4, compared to 94 percent
  of users at time 3. That is, marijuana usage
  increases very consistently over time.

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Review

• Basic introduction to contingency tables.
• Study Chapter 18 for next lecture.