Contingency Tables

Contents

Objective   Contingency Tables   Chi-Square Statistic  Chi Square Test of Independence  Example-Undergraduate Students  Chi Square Test of Homogeneity  Example-No Tax Increase

^ Objective

Upon completion of this module you should be able to perform two chi-square tests, the chi-square test of independence and the chi-square test of homogeneity, through the use of contingency tables.

^ Contingency Tables

Contingency tables are two-way frequency tables used to organize data that has been collected on two categorical random variables. We are interested in discovering whether the two variables are dependent upon one another.

There are two types of contingency tables that are most often used. The first type organizes the data we collect or observe. We call this the table of observed counts.

The second type, the table of expected counts, organizes our expected counts in the same way as the table of observed counts. The expected counts are the values that we would expect to obtain for each cell under the assumption that the categorical variables are independent.

The expected counts can be computed by multiplying the row and column totals of our table of observed counts and then dividing by the grand total or total number observations. The e_ij are the expected frequencies for the cell located in row i and column j.

Notice in the tables above that the contingency tables should always have row and column labels along with the type of table, either observed or expected.

^ Chi-Square Statistic

This statistic measures the amount of disagreement between the observed frequencies and the expected frequencies.

Degrees of freedom = df = (r - 1)*(c - 1) where r = number of rows and c = number of columns in the contingency table

Critical value: χ² _α,df where α = the chosen significance level and df = the degrees of freedom

^ Chi Square Test of Independence

• Application
  To test the independence of two categorical variables in a simple random sample
• Assumptions--These are conditions that must be satisfied in order for the results of this test to have validity.
  • The observations are independent.
  • No expected cell count is less than 5.
  • The values in the contingency table are counts, not percents or proportions.
• Hypotheses
  H₀: The two categorical variables are independent.
  H₁: The two categorical variables are dependent.
• Significance level
  The significance level, α corresponds to the size of the rejection region.
  It determines how small the p-value should be in order to reject the null hypothesis.
  The suitable choice for a significance level usually depends the severity of a Type I error in relation to the severity of a Type II error (for more information on selection of a significance level, see the hypothesis test module or the hypothesis test on mu module in IDEAL). However, common choices for α are 0.05, 0.01, or 0.10.
• Test Statistic
  χ²
  Refer to the introduction material above on the χ² statistic for the form of the test statistic
• P-value
  The p-value is the probability of getting a value for the test statistic as large or larger than the observed value of the test statistic just by random chance. This can be shown symbolically as
  p-value = P(χ² ≥ test statistic).
  To determine a p-value look at the χ² table with df degrees of freedom and find where the observed value of the χ² statistic falls on this table.
  The p-value can be obtained from the applet by clicking the CALC button after filling in the cells in the contingency table.
• Critical value
  χ² _α,df where α = the chosen significance level and df = the degrees of freedom
• Decision--there are two ways to make a decision in this test
  • Classical
    Reject null hypothesis if χ² ≥ χ² _α,df
    Fail to reject null hypothesis if χ² < χ² _α,df
    OR
    
    
  • P-value: this method is preferred by researchers currently conducting research
    Reject null hypothesis if p-value ≤ α
    Fail to reject null hypothesis if p-value > α
• Conclusion

  The conclusion is a statement written to convey the results of the research. The conclusion should avoid statistical terminology if possible and should be written in a form that can be easily understood by non-statisticans.

  • If the null hypothesis is rejected then conclude that the two variables are not independent at the specified significance level.
  • If the null hypothesis is not rejected then conclude that the variables are independent at the specified significance level.

^ Example-Undergraduate Students

A random sample of 400 undergraduate college students was classified according to GPA classification and study habits (Good, Average, Bad). An A corresponds to a GPA of 3.5 or greater, a B is a GPA between 2.5 and 3.5, a C is a GPA between 1.5 and 2.5, and a D or F corresponds to a GPA of 1.5 or less. From the data in the contingency table below, test to see whether the two classifications are independent.

In this applet the observed counts in the ovals will be determined from the problem description and set so that they cannot be changed. Select the Expect Radio Button in the bottom panel to display the expected values. Click in the rectangular regions and fill in the missing counts. This is like a cross-word puzzle in which the row and column sums must equal the marginal totals. By clicking the Calc Button the value of the test statistic and its associated p-value are computed. The hypotheses to be tested and the results of this test are also displayed.

H₀: GPA and Study Habits are independent
H₁: GPA and Study Habits are dependent

• Expected Cell Frequencies
  
  The expected cell frequenices for the first row are computed below. Expected cell frequencies for the cells in rows 2 and 3 are left for you to calculate. You can compare your calculations with those given by the applet.
  
  Cell in Row 1, Column 1:
  
  Cell in Row 1, Column 2:
  
  Cell in Row 1, Column 3:
  
• Test Statistic
  
  df = (r - 1)*(c - 1) = (4 - 1)*(3 - 1) = 6
  α=0.05
• Critical Value
  χ² _0.05,6=12.6
• Decision--there are two ways to make a decision in this test
  • Classical
    χ² _obs ≥ χ² _0.05,6
    15.221 > 12.6
    Reject null hypothesis
    
    OR
    
    
  • P-Value
    From the χ² table with 6 degrees of freedom we see that 14.45 < 15.221 < 16.81
    therefore 0.01 < p-value < 0.025
    p-value < 0.05
    (From the applet the p-value=0.0191)
    
    Therefore, reject H₀
• Conclusion
  The study habits of students are related to their GPA classification at the α=0.05 significance level.

^ Chi Square Test of Homogeneity

Homogeneity means that all units within a group are alike in their characteristics (here we are interested in whether the two populations sampled are the same with respect to the classification levels of a single categorical variable). In this test we have two samples, one from each of two populations. These two populations will be compared according to some criteria (these criteria will be categorical variables).

• Application
  Contingency table with fixed row or column totals and with observations from two independent samples
• Assumptions
  • A random sample is selected from each of the row category populations. The sample sizes (row totals) are fixed prior to sampling.
  • The observations are independent.
  • No expected cell count is less than 5.
  • Values in contingency table are counts, not percents or proportions.
• Hypotheses
  H₀: The populations are homogeneous with respect to the variable of classification
  H₁: The populations are not homogeneous with respect to the variable of classification
• Significance level
  The significance level, α corresponds to the size of the rejection region.
  It determines how small the p-value should be in order to reject the null hypothesis.
  The suitable choice for a significance level usually depends the severity of a Type I error in relation to the severity of a Type II error (for more information on selection of a significance level, see the hypothesis test module or the hypothesis test on mu module in IDEAL). However, common choices for α are 0.05, 0.01, or 0.10.
• Test Statistic
  χ²
  Refer to the introduction material above on the χ² statistic for the form of the test statistic
• P-value
  The p-value is the probability of getting a value for the test statistic as large or larger than the observed value of the test statistic just by random chance. This can be shown symbolically as
  p-value = P(χ² ≥ test statistic).
  To determine a p-value look at the χ² table with df degrees of freedom and find where the observed value of the χ² statistic falls on this table.
  The p-value can be obtained from the applet by clicking the CALC button after filling in the cells in the contingency table.
• Critical value
  χ² _α,df where α = the chosen significance level and df = the degrees of freedom
• Decision--there are two ways to make a decision in this test
  • Classical
    Reject null hypothesis if χ² ≥ χ² _α,df
    Fail to reject null hypothesis if χ² < χ² _α,df
    OR
    
    
  • P-value: this method is preferred by researchers currently conducting research
    Reject null hypothesis if p-value ≤ α
    Fail to reject null hypothesis if p-value > α
• Conclusion

  The conclusion is a statement written to convey the results of the research. The conclusion should avoid statistical terminology if possible and should be written in a form that can be easily understood by non-statisticans.

  • If the null hypothesis is rejected then conclude that the two variables are not homogeneous at the specified significance level.
  • If the null hypothesis is not rejected then conclude that the variables are homogeneous at the specified significance level.

^ Example-No Tax Increase

Statistics for Research, 2nd edition, Dowdy and Wearden, pages 122-123

(Used with permission from the authors)

A political scientist is interested in determining if the promise of no tax increase is of the same level of importance for voters of different political affiliations. Using voter registration lists, she chooses a random sample of 100 voters from each of the groups, Democrats, Republicans and Independents, and she asks the subjects to rate the importance of no tax increases according to the following scale; very important, somewhat important, not too important, not at all important. The results are listed in the table below.

H₀: Members of the three parties agree on the importance of no tax increase (homogeneity)
H₁: Members of the three parties do not agree on the importance of no tax increase (lack of homogeneity)

• Test Statistic
  
  df = (r - 1)*(c - 1) = (3 - 1)*(4 - 1) = 6
  α=0.05
• Critical Value
  χ² _0.05,6=12.6
• Decision--there are two ways to make a decision in this test
  • Classical
    χ² _obs ≥ χ² _0.05,6
    7.425 < 12.6
    Fail to reject null hypothesis
    
    OR
    
    
  • P-Value
    From the χ² table with 6 degrees of freedom we see that 2.20 < 7.425 < 8.56
    therefore 0.90 < p-value < 0.200
    p-value > 0.05
    (From the applet the p-value=0.2840)
    
    Therefore, fail to reject H₀
• Conclusion
  The evidence indicates that the three political parties are not significantly different with respect to their opinions on the importance of no tax increase at the α=0.05 significance level.

Self-test