Question

1. Define the following terms:

A. Contingency table

B. Chi-square test

2. List the assumptions required to perform a chi-square test?

Answer 1

solution:

1) Contingency Table

A two-way table (also called a contingency table) is a useful tool for examining relationships between categorical variables. The entries in the cells of a two-way table can be frequency counts or relative frequencies (just like a one-way table ).

	Dance	Sports	TV	Total
Men	2	10	8	20
Women	16	6	8	30
Total	18	16	16	50

The two-way table above shows the favorite leisure activities for 50 adults - 20 men and 30 women. Because entries in the table are frequency counts, the table is a frequency table .

B)

1) Chi-Square Goodness of Fit Test

A chi-square goodness of fit test attempts to answer the following question: Are sample data consistent with a hypothesized distribution?

The test is appropriate when the following conditions are met:

• The sampling method is simple random sampling .
• The population is at least 10 times as large as the sample.
• The variable under study is categorical .
• The expected value for each level of the variable is at least 5.

Here is how to conduct the test.

• Define hypotheses. For a chi-square goodness of fit test, the hypotheses take the following form.

  H0: The data are consistent with a specified distribution. Ha: The data are not consistent with a specified distribution.

  Typically, the null hypothesis specifies the proportion of observations at each level of the categorical variable. The alternative hypothesis is that at least one of the specified proportions is not true.

• Specify significance level. Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used.
  
• Find degrees of freedom. The degrees of freedom (DF) is equal to the number of levels (k) of the categorical variable minus one: DF = k - 1 .
  
• Compute expected frequency counts. The expected frequency counts at each level of the categorical variable are equal to the sample size times the hypothesized proportion from the null hypothesis

  E_i = np_i

  where E_i is the expected frequency count for the ith level of the categorical variable, n is the total sample size, and p_i is the hypothesized proportion of observations in level i.
  
• Find test statistic. The test statistic is a chi-square random variable (Χ²) defined by the following equation.

  Χ² = Σ [ (O_i - E_i)² / E_i ]

  where O_i is the observed frequency count for the ith level of the categorical variable, and E_i is the expected frequency count for the ith level of the categorical variable.
  
• Find P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.

If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

2)

Chi-Square Test for Independence

A chi-square test for independence is applied when you have two categorical variables from a single population. It is used to determine whether there is a significant association between the two variables.

The test consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

• State the hypotheses. A chi-square test for independence is conducted on two categorical variables. Suppose that Variable A has r levels, and Variable B has c levels. The null hypothesis states that knowing the level of Variable A does not help you predict the level of Variable B. That is, the variables are independent. The alternative hypothesis states that the variables are not independent.
  
• Formulate analysis plan. The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify a significance level and should identify the chi-square test for independence as the test method.
  
• Analyze sample data. Using sample data, find the degrees of freedom, expected frequencies, test statistic, and the P-value associated with the test statistic.
  
  • Degrees of freedom. The degrees of freedom (DF) is equal to:

    DF = (r - 1) * (c - 1)

    where r is the number of levels for one catagorical variable, and c is the number of levels for the other categorical variable.
    
  • Expected frequencies. The expected frequency counts are computed separately for each level of one categorical variable at each level of the other categorical variable. Compute r*c expected frequencies, according to the following formula.

    E_r,c = (n_r * n_c) / n

    where E_r,c is the expected frequency count for level r of Variable A and level c of Variable B, n_r is the total number of sample observations at level r of Variable A, n_c is the total number of sample observations at level cof Variable B, and n is the total sample size.
    
  • Test statistic. The test statistic is a chi-square random variable (Χ²) defined by the following equation.

    Χ² = Σ [ (O_r,c - E_r,c)² / E_r,c ]

    where O_r,c is the observed frequency count at level r of Variable A and level c of Variable B, and E_r,c is the expected frequency count at level r of Variable A and level c of Variable B.
    
  • P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.
• Interpret results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.

3)

Chi-Square Test for Homogeneity

The chi-square test of homogeneity is applied to a single categorical variable . It is used to compare the distribution of frequency counts across different populations. It answers the following question: Are frequency counts distributed identically across different populations?

The test procedure is appropriate when the following conditions are met:

• For each population, the sampling method is simple random sampling .
• The population is at least 10 times as large as the sample.
• The variable under study is categorical .
• If sample data are displayed in a contingency table (Populations x Category levels), the expected frequency count for each cell of the table is at least 5.

This approach consists of four steps: (1) state the hypotheses, (2) formulate an analysis plan, (3) analyze sample data, and (4) interpret results.

• State the hypothesis. Every hypothesis test requires a null hypothesis and an alternative hypothesis. Suppose that data were sampled from r populations, and assume that the categorical variable had c levels. At any specified level of the categorical variable, the null hypothesis states that each population has the same proportion of observations. Thus,
  

  H0: Plevel 1 of population 1 = Plevel 1 of population 2 = . . . = Plevel 1 of population r H0: Plevel 2 of population 1 = Plevel 2 of population 2 = . . . = Plevel 2 of population r . . . H0: Plevel c of population 1 = Plevel c of population 2 = . . . = Plevel c of population r

  
  The alternative hypothesis (H_a) is that at least one of the null hypothesis statements is false.
  
• Formulate an analysis plan. The analysis plan describes how to use sample data to accept or reject the null hypothesis. The plan should specify the significance level and the test method (i.e., the chi-square test of homogeneity).
  
• Analyze sample data. Using sample data from the contingency tables, find the degrees of freedom, expected frequency counts, test statistic, and the P-value associated with the test statistic. The analysis described in this section is illustrated in the sample problem at the end of this lesson.
  
  • Degrees of freedom. The degrees of freedom (DF) is equal to:

    DF = (r - 1) * (c - 1)

    where r is the number of populations, and c is the number of levels for the categorical variable.
    
  • Expected frequency counts. The expected frequency counts are computed separately for each population at each level of the categorical variable, according to the following formula.

    E_r,c = (n_r * n_c) / n

    where E_r,c is the expected frequency count for population r at level c of the categorical variable, n_r is the total number of observations from population r, n_c is the total number of observations at treatment level c, and n is the total sample size.
    
  • Test statistic. The test statistic is a chi-square random variable (Χ²) defined by the following equation.

    Χ² = Σ [ (O_r,c - E_r,c)² / E_r,c ]

    where O_r,c is the observed frequency count in population r for level c of the categorical variable, and E_r,c is the expected frequency count in population r for level c of the categorical variable.
    
  • P-value. The P-value is the probability of observing a sample statistic as extreme as the test statistic. Since the test statistic is a chi-square, use the Chi-Square Distribution Calculator to assess the probability associated with the test statistic. Use the degrees of freedom computed above.
  • Interpret results. If the sample findings are unlikely, given the null hypothesis, the researcher rejects the null hypothesis. Typically, this involves comparing the P-value to the significance level , and rejecting the null hypothesis when the P-value is less than the significance level.