Chapter 11 Contingency Table Analysis. Nonparametric Systems Another method of examining the...

Chapter 11Chapter 11

Contingency Table AnalysisContingency Table Analysis

Nonparametric SystemsNonparametric Systems

• Another method of examining the Another method of examining the relationship between independent (X) relationship between independent (X) and dependant (Y) variables is and dependant (Y) variables is contingency table analysis. contingency table analysis.

• Up to this point we have used Up to this point we have used parametric statistics. These methods parametric statistics. These methods make a number of assumptions about make a number of assumptions about the way that the population that serves the way that the population that serves as the basis for your research sample is as the basis for your research sample is distributed.distributed.


• Correlation and regression assumes Correlation and regression assumes that the independent and dependent that the independent and dependent variables are linearly related. variables are linearly related.

• Other assumptions behind the use of Other assumptions behind the use of parametric statistics include:parametric statistics include:

– Independent observations: the measurement or Independent observations: the measurement or selection of one case does not affect the selection of one case does not affect the measurement or selection of another casemeasurement or selection of another case

– The level of measurement for the variables is at The level of measurement for the variables is at least interval in natureleast interval in nature


• If these assumptions can not be If these assumptions can not be meet then the researcher has the meet then the researcher has the option of using nonparametric option of using nonparametric statistics. statistics.

• They are useful when the data is They are useful when the data is nominal or ordinal, and they require nominal or ordinal, and they require no assumptions about the population no assumptions about the population parametersparameters


• Parametric statistics are usually preferred Parametric statistics are usually preferred because they are more powerful, the power because they are more powerful, the power of a statistic involves the acceptance of a of a statistic involves the acceptance of a false null hypothesis (reaching the false null hypothesis (reaching the conclusion that there are no differences conclusion that there are no differences between the sample and the population between the sample and the population when in fact there are)when in fact there are)

• The greater power of the stats the less likely The greater power of the stats the less likely the researcher is to commit a Type II errorthe researcher is to commit a Type II error


• In this chapter we will consider one particular In this chapter we will consider one particular type of nonparametric measure, the chi-square type of nonparametric measure, the chi-square ((XX

22), ), and some of the measures of association and some of the measures of association that are used with nominal and ordinal data. that are used with nominal and ordinal data.

• Chi-squared is most appropriate when the data Chi-squared is most appropriate when the data is divided into mutually exclusive categories is divided into mutually exclusive categories that can not be legitimately summed up- data that can not be legitimately summed up- data at the nominal or ordinal levelat the nominal or ordinal level

• Chi-squared tells us whether the observed Chi-squared tells us whether the observed distribution is significantly different from the distribution is significantly different from the one that we would expect to occur by chance.one that we would expect to occur by chance.

Constructing Contingency Constructing Contingency TablesTables

• A contingency table is a joint frequency A contingency table is a joint frequency distribution- a frequency distribution with distribution- a frequency distribution with two categorical variables. two categorical variables.

• Again we are concerned with the Again we are concerned with the relationship between the independent and relationship between the independent and dependant variables. dependant variables.

• The contingency table is also known as a The contingency table is also known as a crosstabulation, because it counts the crosstabulation, because it counts the cases that fall into each pairing of the cases that fall into each pairing of the tabletable


• The cells contain those cases that fall into each The cells contain those cases that fall into each pairing of the variables- the number of cases pairing of the variables- the number of cases that fit the categories described by the cross that fit the categories described by the cross listing of the variables. listing of the variables.

• The joint frequencies fall within the cells of the The joint frequencies fall within the cells of the table under the categories for the independent table under the categories for the independent and dependant variables. It is called a and dependant variables. It is called a contingency table because the cases contained contingency table because the cases contained along the rows (the categories of the dependent along the rows (the categories of the dependent variable) are contingent upon what is contained variable) are contingent upon what is contained along the columns( the independent variables).along the columns( the independent variables).


• Consider a crosstabulation of race and Consider a crosstabulation of race and attitudes toward capital punishment from attitudes toward capital punishment from the National Crime Survey data set. Our the National Crime Survey data set. Our research hypothesis is that whites are more research hypothesis is that whites are more likely to favor capital punishment than are likely to favor capital punishment than are minorities.minorities.

• The examination begins with a look at the The examination begins with a look at the frequency distribution of a variable, frequency distribution of a variable, including the percentages within each including the percentages within each categories as it relates to the entire group. categories as it relates to the entire group.


• Marginals are the total frequency column, Marginals are the total frequency column, because they are presented at the margins of because they are presented at the margins of the table. the table.

• However we are usually concerned with However we are usually concerned with subgroup analysis. We examine the subgroup analysis. We examine the breakdown of frequencies and percentages of breakdown of frequencies and percentages of the dependent variables as they are the dependent variables as they are categorized under the independent variable, categorized under the independent variable, as with correlation, the assumption is that the as with correlation, the assumption is that the independent variable produces an effect on independent variable produces an effect on the dependent variable. the dependent variable.


• The table lists the independent and dependant The table lists the independent and dependant variables, the usual procedure is to construct variables, the usual procedure is to construct the table so that the independent variable is the table so that the independent variable is listed along the columns and the dependant listed along the columns and the dependant variable follows the rows. We are interested in variable follows the rows. We are interested in the impact of the independent variable follows the impact of the independent variable follows the rows. So we read the table by comparing the rows. So we read the table by comparing the percentage value of the column the percentage value of the column (independent variable) for the subgroups (independent variable) for the subgroups under the dependent variable (rows) under the dependent variable (rows)


• The examination of a relationship The examination of a relationship typically begins with a look at the typically begins with a look at the frequency distribution of a variable, frequency distribution of a variable, including the percentages within each including the percentages within each category as it relates to the entire category as it relates to the entire group.group.

• The total frequencies are called The total frequencies are called marginals, because they are found in marginals, because they are found in the in the margins of the table.the in the margins of the table.


• With this method we usually concerned With this method we usually concerned with the subgroup analysis, the with the subgroup analysis, the breakdown of frequencies and breakdown of frequencies and percentages of the dependent variable percentages of the dependent variable as they are categorized under the as they are categorized under the independent variable, as with correlation independent variable, as with correlation the assumption is that the independent the assumption is that the independent variable produces and effect on the variable produces and effect on the dependent variable. dependent variable.


• The usual procedure is to construct the The usual procedure is to construct the table so the independent variable is listed table so the independent variable is listed along the columns and the dependent along the columns and the dependent variable is listed along the rows.variable is listed along the rows.

• We read the table by comparing the We read the table by comparing the percentage value of the column percentage value of the column (independent variable) for the subgroups (independent variable) for the subgroups under the dependent variable (rows) under the dependent variable (rows)

Rules for the Construction and Rules for the Construction and Interpretation of TablesInterpretation of Tables• 1. Divide the sample into categories based upon the 1. Divide the sample into categories based upon the

values of the independent variable. values of the independent variable. • 2. The table should be fully labeled. The categories of 2. The table should be fully labeled. The categories of

the independent and dependent variable should be the independent and dependent variable should be clearly presented. The variable headings should describe clearly presented. The variable headings should describe what is contained in the table.what is contained in the table.

• 3. The independent variable follows the columns of the 3. The independent variable follows the columns of the table. The dependent variable follows the rows of the table. The dependent variable follows the rows of the table.table.

• 4. Each subgroup is described in terms of the categories 4. Each subgroup is described in terms of the categories of the dependent variable.of the dependent variable.

• 5. To read the table, compare the percentages of the 5. To read the table, compare the percentages of the independent variable subgroups in terms of the independent variable subgroups in terms of the percentages of the subgroups of the dependent variable.percentages of the subgroups of the dependent variable.

Chi-Square Test for Chi-Square Test for Independent SamplesIndependent Samples

• In statistical analysis, conclusions typically In statistical analysis, conclusions typically result from a description of the findings. result from a description of the findings. Inferential statistics then allow us to make a Inferential statistics then allow us to make a decision about the null hypothesis and decision about the null hypothesis and whether this finding would hold true if we whether this finding would hold true if we had the data from the entire population. had the data from the entire population.

• In the previous example a statistical test is In the previous example a statistical test is needed to determine whether we can needed to determine whether we can assume that this difference in attitudes on assume that this difference in attitudes on capital punishment between racial groups capital punishment between racial groups also exists in the entire population.also exists in the entire population.

Chi-Square Test for Chi-Square Test for Independent SamplesIndependent Samples• The data are at the nominal (race) and The data are at the nominal (race) and

ordinal (support for capital punishment) level ordinal (support for capital punishment) level of measurement .of measurement .

• The groups and the choices fall into different The groups and the choices fall into different categories. We can use chi-squared to tell us categories. We can use chi-squared to tell us the probability that the frequencies we the probability that the frequencies we observed in our survey results (observed observed in our survey results (observed frequencies) differ from an expected frequencies) differ from an expected (hypothesized) set of frequencies.(hypothesized) set of frequencies.

• With chi-squared, these expected frequencies With chi-squared, these expected frequencies represent what we could expect to occur by represent what we could expect to occur by chancechance

Chi-Square Test for Chi-Square Test for Independent SamplesIndependent Samples

• Chi-squared is based upon the differences Chi-squared is based upon the differences between observed and expected frequencies. between observed and expected frequencies. It tells us the level of probability of obtaining It tells us the level of probability of obtaining the differences between the observed and the differences between the observed and expected frequencies.expected frequencies.

• If the observed frequencies (the survey If the observed frequencies (the survey results) differ greatly (.05 level) then the null results) differ greatly (.05 level) then the null hypothesis can be rejected. hypothesis can be rejected.

• If they do not substantially differ, the If they do not substantially differ, the difference between the two sets of frequencies difference between the two sets of frequencies could be due to a sampling error.could be due to a sampling error.

Limitations of Chi-SquaredLimitations of Chi-Squared

• 1. The sample must be randomly selected 1. The sample must be randomly selected

• 2. Each category must be independent – the 2. Each category must be independent – the way in which one response is categorized way in which one response is categorized does not influence the way that another does not influence the way that another response is listed. In our example, the opinion response is listed. In our example, the opinion of one respondent did not affect another in of one respondent did not affect another in terms of his/her attitude toward the death terms of his/her attitude toward the death penalty.penalty.

• 3. Each cell must have an expected frequency 3. Each cell must have an expected frequency of no less than fiveof no less than five

Calculation of Chi-SquaredCalculation of Chi-Squared

• Chi-squared is relatively easy to Chi-squared is relatively easy to calculate by hand with a calculator. calculate by hand with a calculator. In table 11.2 we show how to In table 11.2 we show how to calculate chi-squared by hand in our calculate chi-squared by hand in our example of the relationship between example of the relationship between race and attitude toward capital race and attitude toward capital punishment.punishment.

• Insert table 11.2Insert table 11.2

Calculation of Chi-Squared Calculation of Chi-Squared using SPSSusing SPSS

• To calculate chi-squared using SPSS, To calculate chi-squared using SPSS, take the following steps.take the following steps.

• 1. On the Menu bar, click on “Analyze”1. On the Menu bar, click on “Analyze”

• 2. On the drop down menu, click on 2. On the drop down menu, click on “Descriptive Statistics” “Descriptive Statistics”

• 3. On the next menu, click on 3. On the next menu, click on “Crosstabs”. These steps are on figure “Crosstabs”. These steps are on figure 11.111.1


• 4. In the Crosstabs menu, the variables are 4. In the Crosstabs menu, the variables are listed in the left-hand window. Highlight listed in the left-hand window. Highlight “Favor: Death Penalty for Murderers” and “Favor: Death Penalty for Murderers” and paste it into the “Row” window by clicking paste it into the “Row” window by clicking on the arrow button. This is your dependent on the arrow button. This is your dependent variable (Y) – the respondents attitude variable (Y) – the respondents attitude toward capital punishment. Remember that toward capital punishment. Remember that Y is always the row variable in a contingency Y is always the row variable in a contingency table.table.

• This is shown in Figure 11.2This is shown in Figure 11.2


• 5. In the same window, highlight the 5. In the same window, highlight the independent variable (X), “Race Recode” and independent variable (X), “Race Recode” and paste it into the “Columns” window by paste it into the “Columns” window by clicking on the arrow button. clicking on the arrow button.

• 6. In the “Crosstabs” window, click in the 6. In the “Crosstabs” window, click in the “Statistics” button. The “Crosstabs: “Statistics” button. The “Crosstabs: Statistics” menu then appears. Click on the Statistics” menu then appears. Click on the box next to “Chi-square” to include a box next to “Chi-square” to include a checkmark. Then, click on the “Continue” checkmark. Then, click on the “Continue” buttonbutton

• This is shown in figure 11.3 This is shown in figure 11.3


• 7. When you return to the “Crosstabs” 7. When you return to the “Crosstabs” window, click the “Cells” button. The window, click the “Cells” button. The “Crosstabs: Cell Display” window appears. In “Crosstabs: Cell Display” window appears. In this window, in the “Counts” section, click the this window, in the “Counts” section, click the box next to “Observed” to make a checkmark. box next to “Observed” to make a checkmark. Then in the “Percentages” section, click on the Then in the “Percentages” section, click on the box next to “Column”. Now your contingency box next to “Column”. Now your contingency table will give you the observed frequencies table will give you the observed frequencies for each cell. The table will contain the for each cell. The table will contain the percentages for the independent variable.percentages for the independent variable.

• This is shown in figure 11.4This is shown in figure 11.4


• 8. Click on the “Continue” button. 8. Click on the “Continue” button. You then return to the “Crosstabs” You then return to the “Crosstabs” window. Click on “OK” to generate window. Click on “OK” to generate your contingency table and the chi-your contingency table and the chi-squared statistic. squared statistic.

• This is shown in figure 11.5This is shown in figure 11.5

Figure11.5Figure11.5

• The Crosstabs printout contains the The Crosstabs printout contains the contingency table and statistics. The first table contingency table and statistics. The first table tells us the number of cases in the sample tells us the number of cases in the sample that had valid information for these variables. that had valid information for these variables. The second table mirrors our Table 11.1.The second table mirrors our Table 11.1.

• Our conclusion is that a higher percentage of Our conclusion is that a higher percentage of whites favor the death penalty in murder whites favor the death penalty in murder cases by a difference of 30 percentage points cases by a difference of 30 percentage points (whites, 78.5 percent; minorities, 48.5 (whites, 78.5 percent; minorities, 48.5 percent).percent).

SPSS OutputSPSS Output

• What we still need is a decision as to whether What we still need is a decision as to whether this conclusion would be true if we had data this conclusion would be true if we had data from the entire U.S. population. This is where from the entire U.S. population. This is where chi-squared as an inferential statistic, comes chi-squared as an inferential statistic, comes in.in.

• One major limitation of chi-squared is that no One major limitation of chi-squared is that no cell can have a expected frequency of less cell can have a expected frequency of less than five, in our case our lowest expected than five, in our case our lowest expected frequency is 17.46, so we can assume that our frequency is 17.46, so we can assume that our chi-squared statistic is valid. chi-squared statistic is valid.

SPSS OutputSPSS Output

• In the chi-squared tests table, we see In the chi-squared tests table, we see that with two degrees of freedom the that with two degrees of freedom the Pearson Chi-Squared value of 86.304 Pearson Chi-Squared value of 86.304 is significant at .000. is significant at .000.

• Because .000 is less than .05, we Because .000 is less than .05, we reject the null hypothesis. Our reject the null hypothesis. Our research conclusion is statistically research conclusion is statistically significant .significant .

Measures of Association with Measures of Association with Chi-SquaredChi-Squared

• Another aspect of chi-squared analysis Another aspect of chi-squared analysis involves measures of association. These involves measures of association. These measures indicate the strength of a measures indicate the strength of a relationship between the independent and relationship between the independent and dependent variable.dependent variable.

• The measures of association available under The measures of association available under SPSS Studentware are listed in the SPSS Studentware are listed in the “Crosstabs: Statistics” screen. The following “Crosstabs: Statistics” screen. The following measures are listed under the “Nominal” measures are listed under the “Nominal” section.section.

Cramer’s V Cramer’s V

• Is useful with nominal data. Is useful with nominal data.

• It is probably the most used popular of It is probably the most used popular of the three measures we have discussed the three measures we have discussed because it has a lower limit of 0 (no because it has a lower limit of 0 (no relationship) and an upper limit of 1 relationship) and an upper limit of 1 (perfect relationship). Unlike C and Phi, (perfect relationship). Unlike C and Phi, there is no need to do further there is no need to do further calculations to determine the upper limit calculations to determine the upper limit of Cramer’s V. of Cramer’s V.

Introducing a third VariableIntroducing a third Variable

• We are introducing a third variable as a We are introducing a third variable as a second independent or control variable. second independent or control variable.

• We reexamine the relationship We reexamine the relationship between the original two variables (X between the original two variables (X and Y) within each of the categories of and Y) within each of the categories of the control variable and then compare the control variable and then compare the results across the categories of the the results across the categories of the control variable.control variable.

Introducing a third VariableIntroducing a third Variable

• Returning to our examination of the Returning to our examination of the forces that influence attitudes toward forces that influence attitudes toward capital punishment, another key capital punishment, another key independent variable is sex. independent variable is sex.

• Table 11.5 shows this reexamination.Table 11.5 shows this reexamination.

ConclusionConclusion

• Categorical data measured at the nominal and ordinal Categorical data measured at the nominal and ordinal level are very common in criminal justice research. level are very common in criminal justice research.

• A contingency table is an excellent method to A contingency table is an excellent method to summarize and highlight research findings. summarize and highlight research findings. Conclusions are drawn from the table and its results.Conclusions are drawn from the table and its results.

• Chi-Squared and its accompanying measures of Chi-Squared and its accompanying measures of associations provide a method to determine statistical associations provide a method to determine statistical significance. significance.

• In order to address complex problems such as crime, In order to address complex problems such as crime, multivariate analysis must be conducted.multivariate analysis must be conducted.

• Usually there is more than one contributing factor to Usually there is more than one contributing factor to social problems such as crime. social problems such as crime.

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