The Measure of Exogamous Marriage through Disagreement Scaling

In this paper we used the weighted kappa through disagreement scaling proposed by Cohen (1968) to measure the exogamous marriage. It is the interest of sociologist to investigate the trend of exogamous or mixed marriage between different ethnic groups, as the upward trend of mixed marriage can be view as degree of assimilation of particular ethnic groups. We are able to measure the strength of exogamous marriage directly. We found that the upward trend of mixed marriage among Americans of different ethnicity tend to increase from 1980 to 2000. We also used the estimated large sample variance of weighted kappa given by Fleiss et al . (1969) to build the Wald confidence interval and hence testing the null hypothesis of nonexistence of exogamous marriage.


Introduction
In American society with multiethnic immigrants, the issue of exogamous marriage has been widely studied.The kappa statistics (Cohen, 1960) were introduced to measure the agreement between two raters for nominal scales.Strauss (1977) has applied the kappa statistics by Cohen to study the endogamous marriage, followed by Rust and Seed (1985) and further developed by Ahmad Mahir (1993).This issue refers to the intersection of race and class.Lessard (2001) has used the definition given by Chang (1999) who stated that "Intermarriage refers to marriage between members of different groups, identified by one or more socially important dimensions of differentiation".In this paper, we studied the intermarriage data which have been arranged in the form of a square two-ways contingency table.This arrangement was originally introduced by Ahmad Mahir (1991) as shown in table 1.The ethnic groups for both men and women are given by E 1 , E 2 ,.., E k .The diagonal cells refers to the endogamy marriage that is the marriage between the brides and grooms of the same ethnic groups, whereas off diagonal cells give the couples of exogamous marriage that is the marriage between the brides and grooms of different ethnic groups.The main objectives of this paper are using the weighted kappa to measure marriage the strength of the exogamous marriage and to study the trend of the exogamous marriage.
For simplicity, we assigned the weight zero and one to the diagonal and off diagonal cells respectively.Then, weighted kappa was calculated followed by the construction of the 95% confidence interval with the estimated variance given by Fleiss et al. (1969).
We extracted the five percent of the 1980, 1990 and 1990 U.S. censuses, made available through the Integrated Public Use Microdata Sample (IPUMS, 2003).By using cross tabulation procedure of SPSS software, we obtained two-ways table shown in table 2, 3 and 4. Cohen (1968) proposed a basic formula for weighted kappa which is given by 1-( / )

Theories and the calculation
where Let oij p be the observed proportion of marriage placed in the (i,j) cells and eij p be the expected proportion in the (i,j) cells.The investigator chose the disagreement weighted scale as follow: Assume that we have x ij which is the number of observation in the (i,j) cell with a total of n observations.Let i x + and j x + refer to the total sum of the ith row and jth column respectively.The estimates of oij p and eij p are given by 0 and assuming that the column and the row variables are independent.
ˆ/ eij i j p x x n By substituted Eq. ( 2.2) to Eq. (2.6) into Eq.( 2.1), we obtained the estimated weighted kappa as below.p w q w w q q q q q n q with weighted average of the weights in the ith row, a weighted average of the weights in the jth column.
We introduced the overall exogamy measure as lambda, λ where the estimate is given by We can show that the variance of λ is approximately equals to the variance of ˆw κ since 1 is a constant.We have ˆˆˆˆvar( )= var(1-) var( )

Result and Finding
We computed the strength of exogamy estimate λ and var( ) λ using marriage data sets given in table 2, 3, and 4. The 95 % confidence limit also been constructed to test the hypothesis ˆ0 λ = .The results are shown in the table 5.
According to Cohen (1968) both measures are chance corrected proportion.Weighted kappa is a special case of kappa.The value of weighted kappa w κ is smaller than the κ as it gives weight to the disagreement cells.Normally the value of weighted kappa falls the in the range (0,1), so we can deduce overall exogamy measure is 0 1 λ < < .If ˆ1 λ = , it said to be perfect exogamy.Where as ˆ0 λ = implied that exogamy does not happen.
By comparing data set 1, 2 and 3 we can conclude that the exogamous marriage in the U.S society has increased over the past two decades as the value of λ has increased from 0.0967 to 0.1764.The increase of λ for year 1980 to 1990 is 0.0335 compare to the year 1990 to 2000 is 0.0462.From Figure 1 indicate the upward exogamous marriage trend of American society.
For testing the null hypothesis ˆ0 λ = , we can reject the null hypothesis for the two decades that we chosen. Rust

Table 4 .
Data set 3 of marriage by race of U.S. Women 2000 census

Table 5 .
Parameters estimate, standard errors and 95% confident limit for λ Figure 1.The trend of exogamous marriage in United States for the year 1980, 1990 and 2000 census.