Measures of Association 771 chi square value can be used to indicate a stronger relationship between two variables when two tables have the same sample size.Then the misleading nature of the chi square statistic when sample size differs will be shown. Opinion Male Female Total Agree 65 25 90 (60.0) (30.0) Disagree 35 25 60 (40.0) (20.0】 Total 100 50 150 X2=0.417+0.833+0.625+1.250=3.125 df =1 0.075<a<0.10 Table 11.1:Weak Relationship between Sex and Opinion Table 11.1 gives the chi square test for independence for the weak rela- tionship between sex and opinion,originally given in Table 6.9.The first entry in each cell of the table is the count,or observed number of cases. The number in brackets in each cell of the table is the expected number of cases under the assumption of no relationship between sex and opinion.It can be seen that the value of the chi square statistic for the relationship shown in Table 11.1 is 3.125.With one degree of freedom,this value is statistically significant at the 0.10 level of significance,but not at the 0.075 level.This indicates a rather weak relationship,providing some evidence for a relationship between sex and opinion.But the null hypothesis of no relationship between the two variables can be rejected at only the 0.10 level of significance. Table 11.2 gives much stronger evidence for a relationship between sex and opinion.In this table,the distribution of opinions for females is the same as in the earlier table,but more males are in agreement,and less in disagreement than in the earlier table.As a result,the chi square value for Table 11.2 gives a larger value,indicating a more significant relationship
Measures of Association 771 chi square value can be used to indicate a stronger relationship between two variables when two tables have the same sample size. Then the misleading nature of the chi square statistic when sample size differs will be shown. Opinion Male Female Total Agree 65 25 90 (60.0) (30.0) Disagree 35 25 60 (40.0) (20.0) Total 100 50 150 χ 2 = 0.417 + 0.833 + 0.625 + 1.250 = 3.125 df = 1 0.075 < α < 0.10 Table 11.1: Weak Relationship between Sex and Opinion Table 11.1 gives the chi square test for independence for the weak relationship between sex and opinion, originally given in Table 6.9. The first entry in each cell of the table is the count, or observed number of cases. The number in brackets in each cell of the table is the expected number of cases under the assumption of no relationship between sex and opinion. It can be seen that the value of the chi square statistic for the relationship shown in Table 11.1 is 3.125. With one degree of freedom, this value is statistically significant at the 0.10 level of significance, but not at the 0.075 level. This indicates a rather weak relationship, providing some evidence for a relationship between sex and opinion. But the null hypothesis of no relationship between the two variables can be rejected at only the 0.10 level of significance. Table 11.2 gives much stronger evidence for a relationship between sex and opinion. In this table, the distribution of opinions for females is the same as in the earlier table, but more males are in agreement, and less in disagreement than in the earlier table. As a result, the chi square value for Table 11.2 gives a larger value, indicating a more significant relationship
Measures of Association 772 Opinion Male Female Total Agree 75 25 100 (66.7)(33.3) Disagree 25 25 50 (33.3) (16.7) Total 100 50 150 X2=1.042+2.083+2.083+4.167=9.375 df =1 0.001<a<0.005 Table 11.2:Strong Relationship between Sex and Opinion than in Table 11.1.For Table 11.2,the chi square value is 9.375,and with one degree of freedom,this statistic provides evidence of a relationship at the 0.005 level of significance. When comparing these two tables,the size of the chi square value pro- vides a reliable guide to the strength of the relationship between sex and opinion in the two tables.The larger chi square value of Table 11.2 means a stronger relationship between sex and opinion than does the smaller chi square value of Table 11.1.In these two tables,the sample size is the same, with n =150 cases in each table. Now examine Table 11.3,which is based on the weak relationship of Table 11.1,but with the sample size increased from n=150 to n=600.In order to preserve the nature of the relationship,each of the observed numbers of cases in the cells of Table 11.1 are multiplied by 4.The new table again shows that females are equally split between agree and disagree,but males are split 260/140=65/35 between agree and disagree.The pattern of the relationship between sex and opinion is unchanged from Table 11.1.But now the chi square statistic is dramatically increased.In Table 11.3,the chi square statistic is 12.5,as opposed to only 3.125 in Table 11.1.The 12.5 of the new table is even larger than the chi square value of 9.375 of Table 11.2. The larger sample size in the new table has increased the value of the chi
Measures of Association 772 Opinion Male Female Total Agree 75 25 100 ( 66.7) (33.3) Disagree 25 25 50 (33.3) (16.7) Total 100 50 150 χ 2 = 1.042 + 2.083 + 2.083 + 4.167 = 9.375 df = 1 0.001 < α < 0.005 Table 11.2: Strong Relationship between Sex and Opinion than in Table 11.1. For Table 11.2, the chi square value is 9.375, and with one degree of freedom, this statistic provides evidence of a relationship at the 0.005 level of significance. When comparing these two tables, the size of the chi square value provides a reliable guide to the strength of the relationship between sex and opinion in the two tables. The larger chi square value of Table 11.2 means a stronger relationship between sex and opinion than does the smaller chi square value of Table 11.1. In these two tables, the sample size is the same, with n = 150 cases in each table. Now examine Table 11.3, which is based on the weak relationship of Table 11.1, but with the sample size increased from n = 150 to n = 600. In order to preserve the nature of the relationship, each of the observed numbers of cases in the cells of Table 11.1 are multiplied by 4. The new table again shows that females are equally split between agree and disagree, but males are split 260/140 = 65/35 between agree and disagree. The pattern of the relationship between sex and opinion is unchanged from Table 11.1. But now the chi square statistic is dramatically increased. In Table 11.3, the chi square statistic is 12.5, as opposed to only 3.125 in Table 11.1. The 12.5 of the new table is even larger than the chi square value of 9.375 of Table 11.2. The larger sample size in the new table has increased the value of the chi
Measures of Association 773 square statistic so that even the relatively weak relationship between sex and opinion becomes very significant statistically.Given the assumption of no relationship between sex and opinion,the probability of obtaining the data of Table 11.3 is less than 0.0005. Opinion Male Female Total Agree 260 100 360 (240.0) (120.0) Disagree 140 100 240 (160.0)(80.0)0 Total 400 200 600 X2=1.667+3.333+2.500+5.000=12.500 df=1 a<0.0005 Table 11.3:Weak Relationship-Larger Sample Size This example shows how the value of the chi square statistic is sensitive to the sample size.As can be seen by comparing Tables 11.1 and 11.3,mul- tiplying all the observed numbers of cases by 4 also increases the chi square statistic by 4.The degrees of freedom stay unchanged,so that the larger chi square value appears to imply a much stronger statistical relationship between sex and opinion. Considerable caution should be exercised when comparing the chi square statistic,and its significance,for two tables having different sample sizes.If the sample size for the two tables is the same,and the dimensions of the table are also identical,the table with the larger chi square value generally provides stronger evidence for a relationship between the two variables.But when the sample sizes,or the dimensions of the table differ,the chi square statistic and its significance may not provide an accurate idea of the extent of the relationship between the two variables. One way to solve some of the problems associated with the chi square
Measures of Association 773 square statistic so that even the relatively weak relationship between sex and opinion becomes very significant statistically. Given the assumption of no relationship between sex and opinion, the probability of obtaining the data of Table 11.3 is less than 0.0005. Opinion Male Female Total Agree 260 100 360 (240.0) (120.0) Disagree 140 100 240 (160.0) (80.0)0 Total 400 200 600 χ 2 = 1.667 + 3.333 + 2.500 + 5.000 = 12.500 df = 1 α < 0.0005 Table 11.3: Weak Relationship - Larger Sample Size This example shows how the value of the chi square statistic is sensitive to the sample size. As can be seen by comparing Tables 11.1 and 11.3, multiplying all the observed numbers of cases by 4 also increases the chi square statistic by 4. The degrees of freedom stay unchanged, so that the larger chi square value appears to imply a much stronger statistical relationship between sex and opinion. Considerable caution should be exercised when comparing the chi square statistic, and its significance, for two tables having different sample sizes. If the sample size for the two tables is the same, and the dimensions of the table are also identical, the table with the larger chi square value generally provides stronger evidence for a relationship between the two variables. But when the sample sizes, or the dimensions of the table differ, the chi square statistic and its significance may not provide an accurate idea of the extent of the relationship between the two variables. One way to solve some of the problems associated with the chi square
Measures of Association 774 statistic is to adjust the chi square statistic for either the sample size or the dimension of the table,or for both of these.Phi,the contingency coefficient and Cramer's V,are measures of association that carry out this adjustment, using the chi square statistic.These are defined in the following sections, with examples of each being provided. 11.2.1Phi The measure of association,phi,is a measure which adjusts the chi square statistic by the sample size.The symbol for phi is the Greek letter phi, written o,and usually pronounced fye'when used in statistics.Phi is most easily defined as n Sometimes phi squared is used as a measure of association,and phi squared is defined as In order to calculate these measures,the chi square statistic for the table is first determined,and from this it is relatively easy to determine phi or phi squared.Since phi is usually less than one,and since the square of a number less than one is an even smaller number,o2 can be extremely small.This is one the reasons that phi is more commonly used than is phi squared. Example 11.2.2 and o2 for Tables of Section 11.2 Table 11.4 gives the three two by two tables shown in the last section, without the row and column totals.The chi square statistic and sample size for each of the tables is given below the frequencies for each cell in the table.From these,2 and o are then determined.For the first table,with the strong relationship,having females equally divided on some issue,but with males split 75 agreeing and 25 disagreeing,x2=9.375.The sample size for this table is n 150 so that 2=2=9.375 =0.0625 150 and φ=n /9.37币=√0.0625=0.25
Measures of Association 774 statistic is to adjust the chi square statistic for either the sample size or the dimension of the table, or for both of these. Phi, the contingency coefficient and Cramer’s V, are measures of association that carry out this adjustment, using the chi square statistic. These are defined in the following sections, with examples of each being provided. 11.2.1 Phi The measure of association, phi, is a measure which adjusts the chi square statistic by the sample size. The symbol for phi is the Greek letter phi, written φ, and usually pronounced ‘fye’ when used in statistics. Phi is most easily defined as φ = s χ2 n . Sometimes phi squared is used as a measure of association, and phi squared is defined as φ 2 = χ 2 n . In order to calculate these measures, the chi square statistic for the table is first determined, and from this it is relatively easy to determine phi or phi squared. Since phi is usually less than one, and since the square of a number less than one is an even smaller number, φ 2 can be extremely small. This is one the reasons that phi is more commonly used than is phi squared. Example 11.2.2 φ and φ 2 for Tables of Section 11.2 Table 11.4 gives the three two by two tables shown in the last section, without the row and column totals. The chi square statistic and sample size for each of the tables is given below the frequencies for each cell in the table. From these, φ 2 and φ are then determined. For the first table, with the strong relationship, having females equally divided on some issue, but with males split 75 agreeing and 25 disagreeing, χ 2 = 9.375. The sample size for this table is n = 150 so that φ 2 = χ 2 n = 9.375 150 = 0.0625 and φ = s χ2 n = r 9.375 150 = √ 0.0625 = 0.25
Measures of Association 775 Nature of Relation and Sample Size Strong;n=150 Weak,n=150 Weak.n =600 Opinion Male Female Male Female Male Female Agree 75 25 65 25 260 100 Disagree 25 25 35 25 140 100 x2 9.375 3.125 12.500 n 150 150 600 2 0.0625 0.02083 0.02083 0.25 0.144 0.144 Table 11.4:02 and o for 2 x 2 Tables The values of o2 and o for the other tables are obtained in a similar manner.Note how small o2 is in each of the tables.Since a very small value might seem to indicate no relationship between the two variables,sex and opinion,it might be preferable to use o rather than o2.Note that o is 0.25 for the strong relationship,and only 0.144 for the weak relationship.By comparing the two values of o,you can obtain some idea of the association between sex and opinion.This indicates that the relationship of Table 11.2, for which o =0.25,is a stronger relationship than is the relationship of Table 11.1,where o is only 0.144.Also note in the two right panels of Table 11.4 that o for the weak relationship is the same,regardless of the sample size.As shown earlier,in Tables 11.1 and 11.3,the value of x2 is quite different for these two types of data,but o is the same.That is,the nature of the relationship is the same in the two right panels of Table 11.4, but the sample size is four times greater on the right than in the middle panel.This dramatically increases the size of the chi square statistic,but leaves the values of 2 and o unchanged. Example 11.2.3 Relationship Between Age and Opinion Concern- ing Male and Female Job Roles The Regina Labour Force Survey asked respondents the question
Measures of Association 775 Nature of Relation and Sample Size Strong, n = 150 Weak, n = 150 Weak, n = 600 Opinion Male Female Male Female Male Female Agree 75 25 65 25 260 100 Disagree 25 25 35 25 140 100 χ 2 9.375 3.125 12.500 n 150 150 600 φ 2 0.0625 0.02083 0.02083 φ 0.25 0.144 0.144 Table 11.4: φ 2 and φ for 2 × 2 Tables The values of φ 2 and φ for the other tables are obtained in a similar manner. Note how small φ 2 is in each of the tables. Since a very small value might seem to indicate no relationship between the two variables, sex and opinion, it might be preferable to use φ rather than φ 2 . Note that φ is 0.25 for the strong relationship, and only 0.144 for the weak relationship. By comparing the two values of φ, you can obtain some idea of the association between sex and opinion. This indicates that the relationship of Table 11.2, for which φ = 0.25, is a stronger relationship than is the relationship of Table 11.1, where φ is only 0.144. Also note in the two right panels of Table 11.4 that φ for the weak relationship is the same, regardless of the sample size. As shown earlier, in Tables 11.1 and 11.3, the value of χ 2 is quite different for these two types of data, but φ is the same. That is, the nature of the relationship is the same in the two right panels of Table 11.4, but the sample size is four times greater on the right than in the middle panel. This dramatically increases the size of the chi square statistic, but leaves the values of φ 2 and φ unchanged. Example 11.2.3 Relationship Between Age and Opinion Concerning Male and Female Job Roles The Regina Labour Force Survey asked respondents the question