The racial wage gap in the US Counterfactual outcomes from a purging procedure

P.M. Brink, BSc.

Supervisor: Dr. Manfred te Grotenhuis

Master's thesis

Research Master Social and Cultural Science Radboud University Nijmegen Nijmegen, August 2015

Abstract The main goal of this thesis is to employ the alternative method of purging to assess racial wage disparities in the United States, using data from the 2013 American Community Survey. A distinction was made between discriminatory and non-discriminatory causes for the racial wage gap. The most important wage determinants from relevant theory were used to describe how racial groups differ from the group of whites regarding distributions of the determinants (non-discriminatory), and associations between the determinants and wages (discriminatory). Based on multivariate regression models, a purging procedure was performed to come to counterfactual outcomes. For the purpose of this paper, a package was developed to apply this method using generalized linear models, featuring a decomposition method, bootstrapping techniques to enable hypothesis testing, and plotting functions. Applying associations of whites to disadvantaged racial groups produced varying results. Applying whites' distributions to the underprivileged groups revealed that these groups are generally speaking less advantageously composed regarding the main determinants, and therefore earn on average less than whites. The higher average educational levels and occupational prestige levels held by Asians seem to be the main reason for their higher wages, compared to whites.

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Acknowledgements It is apparently considered good practice to inflict upon those who make the effort of reading a master’s thesis, a collection of acknowledgements, personal dramas and, perhaps, a carefully formulated joke. Let me be clear: feel free to turn the page right now, if you are not interested in this mumbo-jumbo. Over the past seven years, I have been granted the possibility to develop into a wiser and more educated person, for which I have my parents to thank above all others. Vainly bragging about one’s own share after achieving a master’s degree is – at least in many cases – unwise. Especially for sociology students, it would be a real ‘faux pas’. Your support, in all its forms, has never gone unnoticed. One of the greatest joys of studying sociology in Nijmegen has been to meet lots and lots of people through Den Geitenwollen Soc. Undoubtedly, my studying could have been much more efficient without you guys, but I could never resist fooling around and drinking coffee until my stomach hurt. Leaving the university in Nijmegen does not mean that all friendships are coming to an end! Overall, the one who I want to thank in particular is Manfred te Grotenhuis, for giving me the opportunity to write a thesis that was a little out of the ordinary. There aren’t many other supervisors who would have appreciated a thesis subject that does not fit the general assembly line approach of quantitative sociology papers. Thanks for your everlasting enthusiasm and drive. Etched in my memory are your words: geen spoor zonder dwarsliggers.

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Table of contents Abstract ................................................................................................................................. iii Acknowledgements ................................................................................................................ v 1

Background .................................................................................................................... 1

2

Theory ............................................................................................................................ 3 Discriminatory and non-discriminatory effects on wages ............................................ 3 Disadvantaged groups ............................................................................................... 4 Wage determinants .................................................................................................... 5

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Data and methods .......................................................................................................... 7 Data ........................................................................................................................... 7 Measurements ........................................................................................................... 7 Methods and the purging procedure........................................................................... 9

4

Results ..........................................................................................................................13 Bivariate results ........................................................................................................13 Trivariate results .......................................................................................................16 Multivariate results ....................................................................................................19 Counterfactual outcomes ..........................................................................................20

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Conclusion and discussion ............................................................................................25

References ...........................................................................................................................27 Appendix A: The method of purging .....................................................................................31 Appendix B: Results from multivariate models and purging procedures ...............................45 Appendix C: Complete code of all functions in the package .................................................49

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1

Background

In first decades following World War II, the USA went through a transition from a racially segregated society towards a society with equal rights. The efforts of individuals and organizations such as the National Association for the Advancement of Colored People and the African-American Civil Rights Movement gave rise to shifting relations between white and colored Americans and ultimately resulted in a series of Civil Rights Acts passed by the US Congress. In these acts equal rights were secured concerning various topics, such as voting rights, housing, public space, but also jobs and payment. Under Title VII of the Civil Rights Act of 1964, employers are prohibited to discriminate against people because of their race, color, religion, sex, or national origin. This also includes the ban on difference in wages based on race, color, religion, sex or national origin. Since the passing of this law up to the present, there have been numerous studies trying to assert the possible persistence of the so-called wage gap and its size. The racial wage gap is the difference in average earnings or wages between people from different races. In a study by the CEAPIR (1998), it was reported that the wage gap between white and black males hardly changed in the period from 1970 to 1990. After 1990, this gap became somewhat smaller. The gap between white and Hispanic males even widened between 1985 and 1997. In 1997, the median black and Hispanic males earned respectively 74 and 63 percent of the median white male’s wage. The wage gap for women was reported to be smaller, with median black females earning approximately 82 percent and median Hispanic females earning approximately 71 percent of the median white female’s wage. Yet, by 1997 the gap for non-white females had increased considerably since 1979. The abovementioned results are an indication for disadvantages of non-white racial groups compared to whites. Still, in academia these disadvantages are often – at least partially – ascribed to compositional differences between the racial groups. Mincer (1974) found that, to estimate individual white male workers’ wages in 1959, a combination of the years of (potential) experience and the years of schooling should be introduced in the equation. Others also take the type of occupation (industry) or the occupation prestige (e.g. Oaxaca, 1973; Bergmann, 1974; Cotton, 1988) or even cognitive skills (Farkas & Vicknair, 1996) into account. By analyzing wage levels in relation to these determinants, it is possible to distinguish between discriminatory and non-discriminatory effects. Non-discriminatory differences in wages are due to differences in composition, because they imply that people with equal characteristics are paid equally. Differences in composition mean that characteristics are not equally distributed across the racial groups. Discriminatory differences are present if the associations between wages and wage determinants differ across racial

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groups, because this implies that people from different groups with equal characteristics are not rewarded equally in terms of wages. The racial wage gap has been approached in different manners by many scholars. Often, the estimation techniques have been subject of discussions. In so-called BlinderOaxaca decompositions, based on the work of Blinder (1973) and Oaxaca (1973), the main issue has been to assess the extent or percentage to which wage differences can be ascribed to discrimination based on race. The present paper’s main goal is not to examine the ratio between discriminatory (associational) and non-discriminatory (compositional or distributional) components of the wage gap, but aims to present counterfactual outcomes as a result of applying either equal distributions or equal associations. The Blinder-Oaxaca decompositions are different because they consider the gap a given fact, which can be decomposed into the two components. Conversely, the approach employed in the present study considers if the gap would still exist if either one of the components (distributions or associations) would be held constant (set to zero). Furthermore, a technique is presented which allows to demonstrate the decomposition of effects within each of the components and to test for significance of differences between observed wages and counterfactual outcomes by using bootstrapping. All in all, this is captured in the following research questions: (1) Providing that people from different racial groups in the US in 2013 would profit equally to whites, from characteristics that determine wages, how high would their wages then be, compared to their actual wages? And (2) Providing that people from different racial groups in the US in 2013 would be equal in composition to whites, regarding main determinants of wages, how high would their wages then be, compared to their actual wages?

The next chapter will give an overview of relevant theory regarding important determinants of wages and provide hypotheses deduced from that theory. The following chapter will consist of an overview of the data used and an explanation of the method of purging used for the analysis. A usable and generalized way of applying this method was not yet available. For the purpose of the paper, a package for the program R was developed, enabling other researchers to profit from the method. The newly developed package is aimed at spreading the knowledge built up so far. Moreover, it should enable researchers to apply the purging method in a uniform and consistent way. The paper will be concluded with chapters containing results from these analyses and conclusions regarding the formulated hypotheses and methods used.

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2

Theory

There is a vast amount of literature and studies on the racial wage gap in the USA. The racial wage gap is the difference in average earnings or wages between people from different races. Before being able to compare these groups, it is first and foremost important to know the most important determinants of persons’ wages, that is, to come to a good model to estimate persons’ wages. This chapter will present an overview of relevant theory and results from the scientific work on this subject, which will be used to come to hypotheses on counterfactual outcomes provided either constant distribution or constant association of the relevant determinants of wages. Discriminatory and non-discriminatory effects on wages In order to formulate hypotheses on counterfactual outcomes of wage differences, one has to distinguish between discriminatory and non-discriminatory effects of wage determinants on the different races’ wages. A discriminatory effect is present if people from a disadvantaged race group do not profit as much from wage determinants as people from other race groups do. In other words, a discriminatory effect would be present if people from disadvantaged race groups earn lower wages than people from other race groups despite being equal in terms of wage determinants. This is traced by comparing the actual wages of the disadvantaged groups to their counterfactual wages based on constant association. This means that counterfactual outcomes for the disadvantaged groups are calculated by assuming that their wages are equally related to the wage determinants as is the case for the advantaged group, and then compared to the actual wages of the disadvantaged group. In other words, it is examined how much more or less the disadvantaged groups would have earned, if they would have profited from wage determinants equally to white. A non-discriminatory effect of wage determinants on wages is found when the disadvantage in wages is due to the unequal distribution of wage determinants between the members of different race groups. These differences would appear when comparing the actual wages to the counterfactual wages based on constant distribution of wage determinants. Becker (1971) called such differences non-discriminatory because of equal returns to equal productivity. In a full equilibrium of productivity characteristics between different racial groups, and in the absence of discrimination, workers from the different groups would earn the same. Provided the condition of absent discrimination, but with different productivity characteristics between the groups, differences in wages between racial

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groups cannot be labeled as discriminative1. To point out the difference with constant association, take the example of wages related to educational level. If education would be non-discriminatory, people from different racial groups would profit equally from being highly educated. These groups can still have different average wage levels if, for example, low educated are overrepresented in the one group, while highly educated are overrepresented in the other group. Across the racial groups, people with the same educational level would earn the same. Yet, the different composition with respect to educational levels causes an overall difference in average wage size across the racial groups. The counterfactual wages are calculated by assuming that the underprivileged racial group is composed equally to the privileged group, i.e. by assuming that within the racial groups, the determinants of wages are equally distributed. The counterfactual outcomes based on constant association and constant distribution are part of the method of purging described by Clogg (1978) and developed further by Te Grotenhuis, Eisinga and Scheepers (2004). The basics are explained in the section Methods and the purging procedure in chapter 3. The details and technical specifics are found in Appendix A. Disadvantaged groups In accordance with the study by the CEAPIR (1998, p.24), the present study considers Hispanics and blacks (or African Americans) as the main disadvantaged groups when it comes to wage differences. Whites are considered the established advantageous group, to which all other groups are compared. In the study by the CEAPIR (1998), it was found that people from the Asian minority were reasonably similar to the majority of whites and could therefore not be deemed a disadvantaged group. They are however included in the analyses of the present research, and compared to whites. Asians may not be a disadvantaged minority, based on their average earnings alone. However, it is still possible that they profit differently from wage determinants, compared to whites. For example, they might be more advantageous when it comes to their group composition regarding wage determinants, but still profit less from these characteristics than whites do. Furthermore, based on the analysis of occupations in the abovementioned study, the group of American Indians should also be considered as a disadvantaged minority. An important note in the distinction of (racial) groups, is that Hispanic people were distinguished as a group independent of their race. People from different racial groups were considered a member of their racial group, only if 1

It should be noted that unequal distributions of productivity characteristics as such might be a result of discrimination within a given context. For example: discrimination within the context of the educational system resulting in inequality in educational levels between racial groups could ultimately result in a racial wage gap. However, within the context of the relation between educational levels and wages, the gap could be exclusively a result of differences in educational levels and not of the way education pays off differently between racial groups.

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they were not previously classified as Hispanic. For convenience, Hispanics will be referred to as one of the racial groups in the remainder of this paper. Following Smelser, Wilson & Mitchell (2001), American Indians and Alaska natives were seen as one group, as were Pacific Islanders together with Hawaiian natives. Wage determinants A basic model is the human capital earnings function by Mincer (1974), who used data from workers in the US in 1959 on (potential) experience and years of schooling to estimate one’s wages. Working hours were also incorporated, because under equal circumstances, a person who works more hours earns more. Mincer (1974) found that the more potential experience workers had, the higher their wages. The potential experience could be expressed as the years a worker lived after leaving school, or as the age of the worker. The underprivileged racial groups are expected to have different demographical structures than whites. Because of lower life expectancy and higher shares of young immigrants in underprivileged racial groups, the average age of persons in those groups is expected to be lower. The younger a person is, the less (potential) working experience and therefore, the lower this person’s expected wages. Except for possible differences in demographical structures, the racial groups can differ regarding the way (potential) experience pays off. If the underprivileged groups do not profit from experience as much as whites do, it is considered a discriminatory effect. Hence the hypotheses: provided that people from disadvantaged racial groups would profit from (potential) experience equally to whites, their average wages would be higher than their average actual wages (H1a) and provided that people from disadvantaged racial groups would on average be equal to whites regarding (potential) experience, their average wages would be higher than their average actual wages (H1b). Mincer (1974) also found that higher levels of education were associated with higher wage levels, an idea deeply rooted in human capital theory (see for example: Becker, 1964; Becker & Chiswick, 1966; Mincer, 1958) and is often included in analyses on the racial wage gap (e.g. Blinder, 1973; Cotton, 1988; Neumark 1988). From earlier findings, it is unclear if educational attainment should (remain to) be seen as a discriminatory factor affecting wage levels of different racial groups. Couch and Daly (2002) observed convergence between the racial groups from the 1970s until the 1990s, which was mostly due to equalizing distributions of education, but also of equalizing gains of educational attainment. However, Cancio, Evans and Maume (1996) measured an increase in the wage gap between 1976 and 1985 due to increasing discrimination. Hence the hypotheses: provided that people from disadvantaged racial groups would profit from higher levels of educational attainment equally to whites, their average wages would be higher than their average actual wages (H2a) and 5

provided that people from disadvantaged racial groups would on average be equal to whites regarding their levels of educational attainment, their average wages would be higher than their average actual wages (H2b). Besides the association with educational level and experience, wages also differ across occupations. In light of the racial wage gap, Bergmann (1974) considered that wages differ between whites and blacks for two reasons: the jobs are unequally distributed across the races, and whites earn more than blacks, even if they are equally employed. Grodsky and Pager (2001) also distinguished between racial disparities between and within occupations. The differences between occupations can account for the distribution part of the wage gap. However, they also found that the higher the occupational status of black men, the greater the wage disparities with their white counterparts. These two insights lead to the following hypotheses: provided that people from disadvantaged racial groups would profit from more prestigious jobs equally to whites, their average wages would be higher than their average actual wages (H3a) and provided that people from disadvantaged racial groups would on average be equal to whites regarding their levels of occupational prestige, their average wages would be higher than their average actual wages (H3b). Besides race, sex is often discussed related to wage gaps (see for example Bergmann, 1974; Corcoran & Duncan, 1979). Because sex is related to wage levels, and because the effects of sex may differ across racial groups, it is included as a control variable in the multivariate analyses. Furthermore, some racial groups are concentrated in parts of the USA, for example Hispanic immigrants in the south and Alaska Natives mainly living in the north. Also, members of the different groups may be differently concentrated in urban areas. Following previous studies (e.g. Cotton, 1988; Neumark, 1988), these characteristics are included as controls.

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3

Data and methods

Data Appropriate data to test the hypotheses were found in the Public Use Microdata Sample (PUMS) of the 2013 American Community Survey (ACS) (United States Census Bureau, 2015a). These data include information on various demographic characteristics like occupation, education and wages from persons in a nationwide representative 1 percent sample of nearly 3.6 million US households and over 200,000 group quarters2. Persons from group quarters were excluded from further analysis, because many of these people do not partake in society as working members or are excluded from the labor market (such as psychiatric patients, elderly etc.). Respondents from Puerto Rico were excluded from further analysis, because Puerto Rico is only United States territory and not a US state, which means the economy and legal system may differ too much from the rest of the United States to analyze the racial wage gap in an equal way. Excluding people from group quarters and Puerto Rice resulted in a total number of 2,983,885 respondents in the initial dataset. Respondents were interviewed via e-mail and telephone initially, and eventually personally. For people from selected households, participation is mandatory by law, resulting in high response rates (89.9 percent)3. Missing data because of item non-response or invalid answering were imputed (United States Census Bureau, 2015b). Coverage rates were calculated by dividing the number of potentially sampled housing units in a certain state by an independent official measure of the number of housing units in that state, controlled for sex, age and race. The overall coverage rate was 98.8 percent. Abnormalities were corrected for by applying weights. Data from IPUMS (Ruggles et al., 2015) provided additional geographic features and it also included occupational prestige scores. These data were matched to the original PUMS data. Measurements Race was administered by asking for self-identification with either of the following categories: White alone; Black or African American alone; American Indian alone; Alaska Native alone; American Indian and Alaska Native tribes specified, or American Indian or Alaska Native, not specified and no other races; Asian alone; Native Hawaiian and Other Pacific Islander alone; Some other race alone; Two or more races. These were consolidated to the following 5 categories to be compared in the analyses: White alone; Black or African American alone; Asian alone; American Indian or Alaska Native or Pacific Islander; and Some other race or 2

Group quarters are institutions for people to stay, such as homeless shelters, residential treatment centers, correctional facilities and college residence halls. 3 Response rates are usually around 97 percent. However, due to the government shutdown in 2013, the ACS was unable to perform a second mailing.

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more than one race. Respondents were asked for their possible Hispanic origin in a separate question. These two questions were combined by defining the group of Hispanics first, regardless of their race. All non-Hispanic respondents were then distinguished by race. Because of group sizes, American Indians, Alaska natives, Hawaiian natives and Pacific Islanders were combined into one group. Income components were measured with separate questions. The main measure was recorded as the wages or salary income in US dollars in the past 12 months. Other income components were not used in the analyses, because of various reasons. The height of income from self-employment cannot be discriminatorily affected by employers, and analyzing it as a result of possible discrimination by customers calls for a different and more comprehensive research approach. The height of income from retirement is a result of past events and is more or less invariable through time and therefore not relevant. Income from social security is mostly intended for people with non-competing labor market positions and will relate differently to the established wage determinants. Income from interest, dividend and rental is also expected to be associated to the wage determinants differently. Therefore, all analyses are limited to the working respondents having earned at least some salary during the 12 months prior to the interview. Of all 2,983,885 initial respondents 1,542,397 (51.7%) did not have any monthly salary, leaving 1,441,488 units for further analyses. Respondents were also asked for their usual working hours per week in the prior 12 months, using a ratio measurement level. Respondents were presented a list of educational levels to indicate their educational attainment. The levels ranged from 1 (no schooling completed) to 24 (doctorate degree) and were limited to three dummy variables: No high school, at least high school completed, and at least a bachelor’s degree. Occupational prestige scores were retrieved from IPUMS (Ruggles et al., 2015). The classification of Nakao and Treas (1994) was used, because it is based on occupations’ prestige assessed by respondents of the 1989 General Social Surveys. Most other measures of occupational prestige incorporate the associated average wages in their calculations, and are therefore inadequate for estimating wages. 6,959 (0.5%) of the records with a valid score for wages did not have a valid score for occupational prestige. These were excluded from further analysis, leaving 1,434,529 units for further analysis. The potential working experience of respondents was operationalized by using their age. The PUMS data don’t include information on respondents working history. Neither did it include the age at which they finished education. Therefore, age was used as a proxy for potential working experience. The multivariate analyses are estimated including sex, marital status, region and metropolitan areas as control variables. Sex was reported by ACS respondents and was 8

answered by all respondents. Marital status was measured by asking respondents to choose on the following: Married, widowed, divorced, separated, and never married. Region was included to control for differences between the north east, mid-west, south and west. It was also measured if respondents did or did not live in a metropolitan area. Because of data restrictions due to privacy regulations, some areas could not be assigned to either of the two groups. The final number of 1,434,529 respondents is by far sufficient for complex analyses. For computational reasons, a 10% random sample was drawn from the data and used throughout the rest of the paper. Methods and the purging procedure Chapter 4 will present different ways to analyze the racial wage gap and to test the hypotheses. First, an overview will be given of the bivariate associations between different determinant and wages. Next, multivariate (regression) models will be presented. The third part will consist of the outcomes of a purging procedure based on the multivariate models. This method, initially developed by Clogg (1978) is explained in short in the following paragraph. For the purpose of the following example, we distinguish between the following groups: high wages versus low wages, high education versus low education, and whites versus African Americans or blacks. For each of the racial groups, we can show the association between wages and educational level in a cross table. Table 1 and Table 2 display the data based on the 10% sample of the 2013 ACS PUMS data. Respondents with a degree were assigned to the high education group. All others were assigned to the low education group. Those earning more than the general average wage were classified as earning high wages, while all others were classified as earning low wages. The tables present both absolute and relative frequencies. To properly read the tables, note that the inner cells and the row totals include column percentages representing the association between education and wages, while the column totals per category of education include row percentages representing the distribution of education within the two racial groups.

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Table 1. Association between educational level and wages in the United States in 2013. Group: African Americans Low wages

High wages

Total

Low education

High education

Total

7441

2405

9846

85.5%

56.4%

75.9%

1262

1861

3123

14.5%

43.6%

24.1%

8703

4266

12969

67.1%

32.9%

100.0%

Source: ACS PUMS Data 2013

Table 2. Association between educational level and wages in the United States in 2013. Group: Whites Low wages

Low education 41257

21732

76.2% High wages

46.1%

12897

25345

23.8% Total

High education

53.8%

54154

47077

53.5%

46.5%

Total 62989 62.2% 38242 37.8% 101231 100.0%

Source: ACS PUMS Data 2013

In this example, a total of 24.1 percent of blacks earned high wages. For the low educated, 14.5 percent earned high wages, while of the highly educated 43.6 percent earned high wages. Table 2 shows that in the group of whites 37.8 percent earned high wages. Within the group of whites 23.8 percent of the low educated and 53.8 percent of the highly educated earned high wages. It was hypothesized that if blacks would profit from education equally to whites, they would have higher wages than they actually do (H2a). To test this, the association between the two variables for whites is applied to the marginal counts in the table for blacks. To calculate the so-called ‘purged’ percentage that results from it, the percentages of the inner cells in Table 2 are applied to the marginal counts of low and highly educated blacks in Table 1, following Te Grotenhuis et al. (2004). In other words, the inner cell percentages in Table 1 are purged of their deviation from their counterparts in Table 2. Next, the counts in the inner cells are recalculated based on the original column totals. These counts are then summed to new row totals. The results are found in Table 3, in which the transferred percentages are underlined. The table shows that as a result of the purging procedure, the total purged proportion using constant association is 0.238 * 0.671 + 0.538 * 0.329 = 33.7 percent, which is 9.6 percentage points higher than the actual percentage of blacks in the high wage group.

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Table 3. Simulated association between educational level and wages in the United States in 2013, with the association for whites and the distribution of education and sample of blacks. Low wages

High wages

Total

Low education

High education

Total

6630

1969

8599

76.2%

46.1%

66.3%

2073

2297

4370

23.8%

53.8%

33.7%

8703

4266

12969

67.1%

32.9%

100.0%

Source: ACS PUMS Data 2013

Calculating the purged proportion using constant distribution is calculated as follows. The bottom row in Table 1 shows that within the sample of blacks, 67.1 percent is in the group of low education and 32.9 percent in the group of high education. Table 2 shows that education is differently distributed in the sample of whites: 53.5 percent is low educated and 46.5 percent is highly educated. Under hypothesis H2b, it was presumed that if education was distributed evenly among blacks and whites, blacks would earn higher wages than they actually do. To calculate this, the method by Te Grotenhuis et al. (2004) is applied and the marginal row percentages of whites from Table 2 are applied to the sample of blacks in Table 1, preserving the total sample size and maintaining the relative weights of the inner cells (the association between education and wages for blacks). The row total counts are then recalculated based on the transferred margin row percentage. The inner cell counts are recalculated based on the new row totals and the old inner cell percentages. These lead to new row total counts and a new purged total rate. The results are found in Table 4, in which the transferred percentages are again underlined. It shows that, under the condition of equal distribution of education, 12969 * 0.535 = 6938 of blacks would be low educated and 12969 * 0.465 = 6031 of blacks would be highly educated. Assuming this distribution and combining these frequencies with the inner cell weights, the number of blacks earning high wages are calculated. Under the condition of constant distribution of education, the purged proportion of blacks earning high wages is 0.145 * 0.535 + 0.436 + 0.465 = 28.0 percent, which is 3.9 percentage points higher than the actual percentage of blacks earning high wages.

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Table 4. Simulated association between educational level and wages in the United States in 2013, with the sample and association for blacks and the distribution of education of whites. Low education Low wages

High wages

Total

High education

Total

5932

3400

9332

85.5%

56.4%

72.0%

1006

2631

3637

14.5%

43.6%

28.0%

6938

6031

12969

53.5%

46.5%

100.0%

Source: ACS PUMS Data 2013

This example shows the basic principles of the method of purging, which can also be executed by estimating regression models and combining either the coefficients belonging to blacks with the distributions of explanatory variables belonging to whites (constant distribution) or combining the distributions of explanatory variables belonging to blacks with the coefficients belonging to whites (constant association). For the present paper, an R package was developed to apply this method using generalized linear models. Using this package, it is possible to calculate purged counterfactual outcomes using any link-function, including linear and logistic regression. Also, a decomposition method was implemented to purge the distributions of a subset of the independent variables while correcting for nonlinearity in the associations between explanatory and response variables (when using a link function other than linear). Also, the package allows to control for characteristics outside of the purging equation, which is useful when one wants to control for variables which are not measured for each of the groups under study. Finally, the package offers the possibility to use a bootstrapping strategy to calculate confidence intervals for the difference between observed and purged outcomes. This is especially useful if the shift in distributions between groups is a result of survey outcomes and can therefore not be considered as fixed. To test if blacks would indeed earn more than they actually do, if certain purging conditions are applied, the difference between the observed outcome and the purged outcomes is repeatedly calculated for each bootstrap sample to form confidence intervals. If the confidence intervals of these difference scores do not include the value 0, the null hypothesis is rejected. The results from the cross tables of wages and education can also be calculated using the R package. The specifics of the procedures and the way the R package was programmed can be found in Appendix A. The results of this example are shown for all racial groups in Figure A2. Based on these preliminary results, both hypothesis 2a and 2b are supported. The results of a more thorough approach are found in the following chapter.

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4

Results

Bivariate results Throughout this section, descriptive statistics of the determinants of wages are presented separately for each racial group. These outcomes give a first indication of differences in average wages between the racial groups and differences in the composition of these groups regarding the most important wage determinants. These descriptives show how (un)equal the distributions are, but not if possible inequality in distributions accounts for the differences in average wage levels. Table 5. Average wages by racial group and number of observations. Mean

SD

N

Hispanic

31752

36780

18271

White non-Hispanic

49097

59509

101231

Black or African American non-Hispanic

33617

35940

12969

American Indian / Alaska Native / Pacific non-Hispanic

30742

32246

1293

Asian non-Hispanic

56984

65561

7231

Some other race / more than one race non-Hispanic

38961

47963

2440

Total

45558

55768

143435

Source: ACS PUMS Data 2013

Table 5 shows the frequencies of each racial group and the average wages earned by the people in the group. Asians earn more on average than any other racial group. Blacks, Hispanics and American Natives all earn considerably lower wages than Whites and Asians. Wage scores ranged from 4 to 660,000 US dollars. The wage distributions for all racial groups are displayed in Figure 1. The scores are positively skewed in all racial groups, hence the high standard deviations in Table 5. Considering the skewness of the income distribution, and following relevant prior research (e.g. Blinder, 1973, Corcoran & Duncan, 1979, Mincer, 1974), the natural logarithm of wages is used for the multivariate analyses. The distributions of the transformed scores are found in Figure 2.

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Figure 1. Distribution of wages by racial group.

Source: ACS PUMS Data 2013

Figure 2. Distribution of log-transformed wages by racial group.

Source: ACS PUMS Data 2013

The average amount of working hours does not differ much across the racial groups. Scores ranged from 1 to 99 hours per week. Table 6 shows that whites and Asians work somewhat more than people from the other racial groups. As the number of working hours is a logical determinant of wages, the difference in distributions of working hours could account for differences in average wages between racial groups.

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Table 6. Average working hours by racial group. Mean

SD

N

Hispanic

38

11.6

18271

White non-Hispanic

39

13.0

101231

Black or African American non-Hispanic

37

12.0

12969

American Indian / Alaska Native / Pacific non-Hispanic

38

12.3

1293

Asian non-Hispanic

39

12.6

7231

Some other race / more than one race non-Hispanic

37

13.7

2440

Total

38

12.7

143435

Source: ACS PUMS Data 2013

Table 7 shows that the average age of whites in the sample is slightly higher (44) than the age of people in other racial groups. The ages in the sample range from 16 to 95. Higher age is associated with more potential working experience and can therefore be a cause of higher wages for whites. Table 7. Average age by racial group. Mean

SD

N

Hispanic

39

13.3

18271

White non-Hispanic

44

14.8

101231

Black or African American non-Hispanic

42

14.3

12969

American Indian / Alaska Native / Pacific non-Hispanic

41

13.8

1293

Asian non-Hispanic

42

13.2

7231

Some other race / more than one race non-Hispanic

37

14.0

2440

Total

43

14.6

143435

Source: ACS PUMS Data 2013

The occupational prestige scores in the final sample range from 17 to 86. The results displayed in Table 8 show that Asians hold the highest average prestige scores (49), followed by whites (46), while Hispanics hold the lowest average prestige scores. Table 8. Average occupational prestige by racial group. Mean

SD

N

Hispanic

39

13.3

18271

White non-Hispanic

46

14.6

101231

Black or African American non-Hispanic

42

13.8

12969

American Indian / Alaska Native / Pacific non-Hispanic

41

12.8

1293

Asian non-Hispanic

49

16.4

7231

Some other race / more than one race non-Hispanic

44

14.9

2440

Total

45

14.7

143435

Source: ACS PUMS Data 2013

15

Asians have the highest chance to hold a degree, as shown in Table 9 (64.1%). Of all whites in the sample, 46.5 percent has a degree. In all other groups, this portion is lower. Within the group of Hispanics, 23.6 percent has a degree. The groups of Native Americans (25.2%) and blacks (32.8%) are also less represented on the highest educational level. Given a positive effect of holding a degree, the unequal distribution will be a limiting factor in the disadvantaged groups’ wages. Table 9. Absolute and relative frequencies of educational levels by racial group. < High school Hispanic

White non-Hispanic

9678

4293

23.5%

53.0%

23.5%

4.9%

Asian non-Hispanic

Some other race / more than one race non-Hispanic

Total

49204

47077

48.6%

46.5%

7688

4266

7.8%

59.3%

32.9%

124

867

302

9.6%

67.1%

23.4%

528

2068

4635

7.3%

28.6%

64.1%

201

1248

991

8.2%

51.1%

40.6%

1015

American Indian / Alaska Native / Pacific non-Hispanic

Degree

4300

4950

Black or African American non-Hispanic

High school

11118 7.8%

70753 49.3%

61564 42.9%

Source: ACS PUMS Data 2013

Trivariate results Whereas the bivariate results provided insight into the possible differences between groups in their compositions, trivariate results show how the groups may differ when it comes to the associations between the wages and wage determinants. In other words, the trivariate results may show the relevance of purging the differences in associations. As both the respondents’ working hours and wages are ratio level variables, linear OLS regression is used to show how both variables are related. A quadratic function proved to fit the data best. The coefficients (all significant at 0.001 level) for these models are found in Appendix B, Table B1.

16

Figure 3. Quadratic models of wages regressed on working hours, by racial group.

Source: ACS PUMS Data 2013

Figure 3 displays a visual representation of the model of the relation between working hours and the natural logarithm of wages. The models’ R2 values ranged from 0.35 to 0.46. Whites and Asians hold steeper functions, which means they profit more from working more hours than people from other racial groups do. Figure 4. Quadratic models of wages regressed on age, by racial group.

Source: ACS PUMS Data 2013

17

Quadratic OLS models were also estimated for the relation between age (or potential working experience) and the natural logarithm of wages. The R2 values ranged from 0.18 to 0.28. The coefficients (all significant at 0.001 level) are found in found in Appendix B, Table B2. The effects are displayed in Figure 4. Higher age seems most beneficial for Asians, Whites and people with other or multiple races. Their functions are steeper between the ages of 20 and 50, and their peaks are higher than those of other racial groups. The relation between occupational prestige and the natural logarithm of wages was modeled using OLS, without quadratic effect, which proved to fit the data best. The corresponding coefficients (all significant at 0.001 level) are found in Appendix B, Table B3. The R2 values ranged from 0.14 to 0.22. Figure 5. Linear models of wages regressed on occupational prestige, by racial group.

Source: ACS PUMS Data 2013

As displayed in Figure 5, the steepness of the regression lines did not vary much across the racial groups. Hispanics profit least from having a more prestigious job, while persons from the group ‘Other race or more than one race’ profit most from having a better job. Mean wage levels for each of the three educational levels within the racial groups are displayed in Figure 6. The figure shows that differences between the racial groups are rather small for the low educated, while these differences are greater for people with a degree. In accordance with the other trivariate analyses, the data were modeled using OLS regression analysis, by using dummy variables for the educational levels. Coefficients (all significant at the 0.001 level, model R2 varying from 0.08 to 0.15) are found in Appendix B, Table B4. The

18

differences in wages between educational levels are greatest for whites and the mixed racial group. Figure 6. Mean wages by educational level, by racial group.

Source: ACS PUMS Data 2013

Multivariate results To properly asses the relations between the explanatory variables and the wages, multivariate regression models were estimated using OLS. For a better fit, and similar to the simpler models used for the trivariate models, variables with squares of age and working hours of respondents were added to the model. Because of the skewed nature of the wage distributions, the log of wages was used as dependent variable. Also, a dummy variable was added indicating that respondents’ scores for their working hours were imputed. Detailed results of these models are found in Appendix B, Table B5. The proportion of variance explained by the models varied from 0.52 for the group of Native Americans to 0.61 for the groups of whites and unclassified races or multiple races. In all models, positive effects were observed for working hours, age and occupational prestige, all of comparable size. Respondents who finished high school or held a degree earned significantly more than respondents with less schooling, in all racial groups. Furthermore, women in all racial groups earned significantly less than males in their groups. As for the other controls, married people earned in general more than others in their racial groups, however not all differences were significant in each of the groups. Respondents living in metropolitan areas in all racial groups earned more than others. Lastly, within all racial groups, respondents from the northeast region earned higher wages than respondents from other regions, yet differences were not significant in all racial groups.

19

Counterfactual outcomes Using the multivariate regression models explained in the previous section, counterfactual outcomes were calculated in accordance with the procedures described in Appendix A. The procedure was repeated multiple times to firstly come to counterfactual outcomes based on purging of all model variables, and subsequently outcomes based on purging of only differences in education, only differences in occupational prestige, and only differences in age. Lastly, this is also done for the influence of only the amount of working hours, since this is so closely and naturally related to one’s wages. All exact counterfactual outcomes are found in Appendix B. Outcomes purged of association differences are found in Table B6, and those purged of distribution differences in Table B7. In the following section, the outcomes are displayed in figures. Figure 7. Counterfactual outcomes, all variables purged, with bootstrapped confidence intervals.

Source: ACS PUMS Data 2013 Confidence intervals and significance tests based on 10,000 bootstrap samples, alpha = .05

The results based on constant association found in Figure 7 show that, if the total impact of all explanatory variables on wages is purged of the deviance from whites, all groups except Asians would be better off than they actually are. Purging the differences in distributions of all explanatory variables resulted in higher counterfactual wages for all groups except Asians. For Asians, this resulted in significantly lower wages. The results indicate that the underprivileged groups are on average less fortunately composed in the sample used, regarding the determinants included in the models. The difference in associations between whites and other groups is of less impact than the difference in composition. Applying 20

constant distributions accounts for larger differences with the actual average wages than applying constant associations does. Applying only the effects of age estimated for whites to the other groups (as shown in Figure 8) results in higher wages for Asians and blacks, who thus seem to profit less from potential working experience than whites do. This was predicted by hypothesis 1a, yet these differences did not prove to be statistically significant based on the bootstrapping procedure. Therefore, the hypothesis is not generally corroborated. Hispanics would earn significantly less than whites, under the condition of purged associations. This counterintuitive result is most likely caused by the more flattened age effect curve of Hispanics, including a higher intercept. If the differences in age distributions found in Table 7 are purged, the counterfactual outcomes are higher than the observed outcomes for all groups except the group of Asians, for which no difference was found. It was hypothesized under 1b, that differences in average actual and average counterfactual wages should occur, if the distributions of age would be purged. The results are therefore in favor of the hypothesis, bearing in mind that it was already foreseen that differences would not necessarily occur for the advantageous group of Asians. Figure 8. Counterfactual outcomes, only age purged, with bootstrapped confidence intervals.

Source: ACS PUMS Data 2013 Confidence intervals and significance tests based on 10,000 bootstrap samples, alpha = .05

The models in which the differences in the association between education and wages were purged suggest that some racial groups profit less and some groups profit more from a higher educational level than whites do, as shown by Figure 9. It is suggested that Asians profit less from education than whites, and that Hispanics profit more from education that 21

whites do. These ambiguous results are clearly not in favor of hypothesis 2a. A possible explanation for this is the difference in intercepts found in Table B5. Applying the education distribution of whites to the other groups, keeping all other distributions and associations equal, results in significantly higher average counterfactual wages for all disadvantaged groups. For Asians, an opposite result was found. This suggests that Asians’ higher wage levels may be due to their higher educational levels, while the lower actual wage levels of the other groups may be due to their lower educational levels. This is in accordance with hypothesis 2b. Figure 9. Counterfactual outcomes, only education purged, with bootstrapped confidence intervals.

Source: ACS PUMS Data 2013 Confidence intervals and significance tests based on 10,000 bootstrap samples, alpha = .05

In hypothesis 3a, it was predicted that the disadvantaged groups would have higher average wages, it they would profit from occupational prestige equally to whites. Purging the differences in effects of occupational prestige for the different racial groups compared to the effect in the model for whites results in an increase in wages for blacks, Asians and the group of American Natives, yet none of these differences proved to be of significance based on the bootstrapping procedure (as shown in Figure 10). For Hispanics, a lower average wage level is found when applying the effects of whites. Hypothesis 3a is therefore not supported by the results and consequently not generally applicable. Figure 10 shows that the counterfactual outcomes of applying equal distributions of occupational prestige. In other words, it simulates that people from the underprivileged racial groups would work in jobs that are on average associated with the same prestige levels as the jobs of whites. The results 22

are significantly higher than the actual average wages of these groups. For Asians, a contrary result is found. Purging the differences between their higher levels of occupational prestige and the lower level held by whites (as found in Table 8) accounts for their higher average wages. Hypothesis 3b applies to each of the clearly disadvantaged racial groups and is therefore not rejected. Figure 10. Counterfactual outcomes, only occupational prestige purged, with bootstrapped confidence intervals.

Source: ACS PUMS Data 2013 Confidence intervals and significance tests based on 10,000 bootstrap samples, alpha = .05

To complete this section, additional purging procedures were performed in which the association and distribution of only the working hours variables were purged of differences. The results are displayed in Figure 11. The counterfactuals based on the association among whites imply that the groups of Native Americans and Asians profit less than whites from working more hours. The counterfactual outcomes based on the working hours distributions of whites result in significantly higher wages for people in the group of blacks and the group of other and mixed races, which is in line with the lower average working hours found in Table 6. For Asians, the opposite result was found.

23

Figure 11. Counterfactual outcomes, only working hours purged, with bootstrapped confidence intervals.

Source: ACS PUMS Data 2013 Confidence intervals and significance tests based on 10,000 bootstrap samples, alpha = .05

24

5

Conclusion and discussion

The present study employed the method of purging to come to counterfactual outcomes in a comparison of average wages earned by people from different racial groups in the US in 2013. Bivariate analysis showed that Hispanics, blacks (or African Americans), the group of American Indians, Alaska natives and Pacific islanders, and the group of people with another race or multiple races earned lower wages than whites, while Asians earned higher wages. In order to answer the two research questions, average simulated wages for these disadvantaged groups were calculated by (1) applying the associations of main determinants of wages for whites to the subsamples of these groups and (2) applying the distributions of main determinants of wages for whites to the subsamples of the other groups. A resampling bootstrapping technique was used to assess the credibility of possible differences. As for the answer to the first research question, the results across the racial groups under study were found to be ambivalent. It was found that, for the disadvantaged groups (all but the Asians), profits of either having more experience or holding higher educational levels or being employed in better jobs do generally result in higher wages proportionately to whites, i.e. there were no unambiguous differences in associations between the underprivileged groups and whites. This leaves little suggestion for the presence of discriminatory effects. However, such differences compared to whites were more evident when all the differences in profits combined were cancelled out. In other words, the payoffs of being generally more advantageously endowed than others within one’s disadvantaged group is lower, compared to whites in a similar situation. With respect to the second research question, bivariate statistics showed that the racial groups earning lower wages were less prestigiously occupied, were on average younger and were less often highly educated. Relying on the results from the second part of the counterfactual analyses, strong evidence was brought forth that if these disadvantaged groups would be equal to whites with respect to the distribution of the given wage determinants, their wage levels would be higher. Asians have played a remarkable role in the counterfactual analyses employed as part of this study. First of all, their average wage level is higher than that of whites. Still, they seem to profit less than whites from either being higher educated or working more hours. Be that as it may, the total package of whites’ profits from wage determinants would not imply higher wages for Asians. Lastly, it was found that the cause for the advantage in average wages of Asians is most probably found in the favorable composition of their group, regarding wage determinants. Considering the racial wage gap in the US as a societal challenge, the main conclusion of this paper could be that, irrespective of the possible presence of discriminatory 25

gaps, differences in wages are most likely – and for the greater part – due to nondiscriminatory factors. While differences in the age compositions of groups cannot be altered, differences in educational levels and the level occupation can more easily be influenced by targeted policy. Approaching the racial wage gap as it was done in this study is not suitable for a complete assessment of wage discrimination in the US. Chandra (2003) stressed that unequal incarceration rates between racial groups and selective withdrawal from the labor market in disadvantaged groups are relevant phenomena when studying the racial wage gap. Given the fact that the present paper only considered differences in the working population, such factors were not taken into account. A possible downside of the counterfactual analysis employed in this paper is that the models used cannot take into account hierarchical clustering in the data. Huffman and Cohen (2004) analyzed individual nested in jobs across labor markets and concluded that black population size plays a role in discrimination against workers. The R functions that were developed as a part of this study seem to have successfully withstood a first hands-on test. Appendix A may serve as a decent and fairly straightforward explanation of how the purging procedure is applied to GLMs. In addition to this explanation, it may serve as a manual for those who want to use the source code, and should provide insight for those who want to examine or verify the procedures. A last remark regarding the present study and regarding purging analyses in general should be made. Concurrently with relevant known research the natural logarithm of wages was used to come to better model fits. Because of this transformation, outcomes may seem less appealing and difficult to translate to their real-world equivalents, albeit the most appropriate way to model wages technically. With respect to purging analyses, it should be noticed that although the outcomes can be appealing, they require a lot of computational work. When using more and more complex techniques such as the purging procedure, or when working with transformed data, researchers must continually stay in touch with the original relations and straightforward comparisons. It is important to consider the modeling and bootstrapping techniques used in this paper strictly as a tool to come to valid comparisons of groups and observations, and to eventually answer a real-life question.

26

References Becker, G. S. (1971). The Economics of Discrimination (2nd ed.). Chicago, IL: University of Chicago Press. Becker, G. S., & Chiswick, B. R. (1966). Education and the Distribution of Earnings. The American Economic Review, 56(1/2), 358-369. Bergmann, B. R. (1974). Occupational segregation, wages and profits when employers discriminate by race or sex. Eastern Economic Journal, 1(2), 103-110. Blinder, A. S. (1973). Wage discrimination: reduced form and structural estimates. Journal of Human resources, 8(4), 436-455. Cancio, A. S., Evans, T. D., & Maume Jr, D. J. (1996). Reconsidering the declining significance of race: Racial differences in early career wages. American Sociological Review, 61(4), 541-556. Chandra, A. (2003). Is the convergence of the racial wage gap illusory? (NBER Working Paper No. 9476). Cambridge, MA: National Bureau of Economic Research. Clogg, C. C. (1978). Adjustment of rates using multiplicative models. Demography, 15(4), 523-539. Clogg, C. C., & Eliason, S. R. (1988). A flexible procedure for adjusting rates and proportions, including statistical methods for group comparisons. American Sociological Review, 53(2), 267-283. Clogg, C. C., Shockey, J. W., & Eliason, S. R. (1990). A general statistical framework for adjustment of rates. Sociological Methods & Research, 19(2), 156-195. Corcoran, M., & Duncan, G. J. (1979). Work history, labor force attachment, and earnings differences between the races and sexes. Journal of Human Resources, 14(1), 3-20. Cotton, J. (1988). On the decomposition of wage differentials. The review of economics and statistics, 70(2), 236-243. Couch, K., & Daly, M. C. (2002). Black-white wage inequality in the 1990s: a decade of progress. Economic Inquiry, 40(1), 31-41. Council of Economic Advisers for the President’s Initiative on Race [CEAPIR] (1998). Changing America: Indicators of Social and Economic Well-being by Race and Hispanic Origin. Washington DC: CEAPIR.

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Farkas, G., & Vicknair, K. (1996). Appropriate tests of racial wage discrimination require controls for cognitive skill: comment on Cancio, Evans, and Maume. American Sociological Review, 61(4), 557-560. Fox, J., & Weisberg, S. (2012). Bootstrapping Regression Models in R. Retrieved August 22, 2015, from http://socserv.mcmaster.ca/jfox/Books/Companion/appendix/Appendixbootstrapping.pdf Grodsky, E., & Pager, D. (2001). The structure of disadvantage: Individual and occupational determinants of the black-white wage gap. American Sociological Review, 66(4), 542567. Grotenhuis, M. Te, Eisinga, R., & Scheepers, P. (2004). The method of purging applied to repeated cross-sectional data. Quality and Quantity, 38(1), 1-16. Huffman, M. L., & Cohen, P. N. (2004). Racial Wage Inequality: Job Segregation and Devaluation across US Labor Markets. American Journal of Sociology, 109(4), 902936. McCullagh, P., & Nelder, J. A. (1989). Generalized linear models (Vol. 37). Boca Raton, FL: CRC Press. Mincer, J. (1974). Schooling, Experience and Earnings. New York: National Bureau of Economic Research. Nakao, K., & Treas, J. (1994). Updating occupational prestige and socioeconomic scores: How the new measures measure up. Sociological methodology, 24(24), 1-72. Neumark, D. (1988). Employers' discriminatory behavior and the estimation of wage discrimination. Journal of Human resources, 23(3), 279-295. Oaxaca, R. (1973). Male-female wage differentials in urban labor markets. International economic review, 14(3), 693-709. Ruggles, S., Genadek, K., Goeken, R., Grover, J., & Sobek, M. (2015). Integrated Public Use Microdata Series (Version 6.0) [Machine-readable database]. Retrieved June 16, 2015 from https://usa.ipums.org/usa/index.shtml. Smelser, N. J., Wilson, W. J. & Mitchell, F. (Eds.). (2001). America Becoming: Racial Trends and Their Consequences (Vol. 1). Washington DC: National Academies Press. United States Census Bureau (2015a). American Community Survey PUMS Data. Retrieved June 24, 2015, from http://www.census.gov/programs-surveys/acs/data/pums.html

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United States Census Bureau (2015b). American Community Survey Methodology. Retrieved June 24, 2015, from http://www.census.gov/programssurveys/acs/methodology.html

29

Appendix A: The method of purging The counterfactual outcomes of wage levels for the different racial groups in the present paper, were calculated using the purging procedure, which was explained in short in chapter 3. This appendix serves as a more comprehensive guide to the specifics of the procedure. The first part will go deeper into the way in which the purging produce is applied when using outcomes from generalized linear models. The second part provides an explanation of how the purged outcomes from multivariate models can be decomposed into the partial contributions of a subset of predictors. The third part will show how the R package was programmed that calculates the counterfactual outcomes for multiple groups at once, using bootstrap resampling in order to calculate confidence intervals.

Using generalized linear models The crucial point of calculating purged outcomes is to compare the actual mean (the observed mean) within a group with the purged outcomes, providing an answer to what-if questions. Within the context of the conventional way of estimating and interpreting statistical models, such as generalized linear models, the attention is often directed at interpreting effects, effect sizes and statistical significance of those effects of independent variables on a dependent variable, and eventually estimating interaction effects to compare groups. The questions raised in such situations can be summarized as: given a characteristic 𝑌, with a given amount of variance, in what way and how strong are other (independent) variables 𝑋 related to this variance? The answer is then found in the estimated coefficients 𝛽 and their standard errors. Interaction effects can serve as a way to estimate separate relations for different subgroups in a sample and assess differences between these groups. Comparing subsamples using the purging method is in some way equal to estimating interaction effects for the subsamples for all other terms in the model, by estimating separate models. The technique is however not aimed at assessing the direction, strength and statistical significance of model parameters within the model as such. Within the context of purging, the question asked comes down to: a characteristic 𝑌, with a certain amount of variance, and related in a certain way with variables 𝑋, has means 𝐸(𝑌) that are different across groups. How high would the sample mean of a certain group be, if this group would be identical to another group with regard to its composition and/or effects on the dependent variable? Te Grotenhuis et al. (2004) demonstrated how the method of purging can be applied to estimates of both logistic regression and OLS regression. In essence, these two types of models can both be estimated using generalized linear models (GLMs), as explained by McCullagh & Nelder (1989). The generalized linear model consists of three parts. The first part is the random component, or the dependent variable 𝒀, with the expected value 𝐸(𝑌) or 31

𝜇. The second part is the systematic part, which is called the linear predictor 𝜂. The last part is the link between 𝝁 and 𝜂, which is called the link function 𝑔(𝜇) and which defines how 𝝁 depends on the linear predictor. The dependent variable in the equation is expressed as 𝑌𝑖 , with 𝑖 = 1, . . . , 𝑛. The linear predictor consists of explanatory variables 𝑋𝑗 with 𝑗 = 1, . . ., 𝑝 and their coefficients 𝛽𝑗 : 𝜂𝑖 = 𝛽0 + 𝛽1 𝑋1𝑖 + … + 𝛽𝑝 𝑋𝑝𝑖

(1)

The link function defines the way 𝑌𝑖 depends on 𝜂𝑖 : 𝑔(𝜇𝑖 ) = 𝐸(𝑌𝑖 ) = 𝜂𝑖

(2)

If the variable under study is a normally distributed scale variable, the identity link function is used, meaning that the response variable has a direct linear relation with the linear predictor. Examples of other link functions are the log function for Poisson-distributed response variables and the logit function for binomially distributed response variables.

The generalized linear model can thus be expressed as: 𝐸(𝑌𝑖 ) = 𝛽0 + 𝛽1 𝑋1𝑖 + … + 𝛽𝑝 𝑋𝑝𝑖

(3)

The mean is expressed as: 𝐸(𝑌) =

1 ∙ ∑ 𝐸(𝑌𝑖 ) 𝑛

(4)

In order to come to the untransformed value of 𝜇 (when using a link function different from the identity link), the inverse function of 𝑔(𝜇) is applied: 𝜇 = 𝑔−1 (𝜂)

(5)

To generate counterfactual purged outcomes, two GLMs are estimated: one for each subsample that is used. The equations are equal to equation (3), yet they are estimated separately using subsamples. The subscripts 𝑟𝑒𝑓 and 𝑖𝑛𝑡 refer to the subsample of interest and the subsample of reference. 𝐸(𝑌𝑖 𝑟𝑒𝑓 ) = 𝛽0 𝑟𝑒𝑓 + 𝛽1 𝑟𝑒𝑓 𝑋1𝑖 𝑟𝑒𝑓 + … + 𝛽𝑝 𝑟𝑒𝑓 𝑋𝑝𝑖 𝑟𝑒𝑓

(6)

𝐸(𝑌𝑖 𝑖𝑛𝑡 ) = 𝛽0 𝑖𝑛𝑡 + 𝛽1 𝑖𝑛𝑡 𝑋1𝑖 𝑖𝑛𝑡 + … + 𝛽𝑝 𝑖𝑛𝑡 𝑋𝑝𝑖 𝑖𝑛𝑡

(7)

It was pointed out earlier that there are two types of purging procedures. The first type is to calculate purged outcomes using constant association, the second type is to use constant distribution. Constant association means that a subsample’s mean (for the subsample of interest) is calculated under the assumption that this sample does not differ from the alternative subsample (reference subsample) regarding the associations (𝛽’s) between the independent variables and the dependent variable. Constant distribution implies the opposite of the constant association: the subsample’s mean (for the subsample of interest) is calculated under the assumption that the distributions of the independent variables (𝑋’s) do 32

not differ from the distributions in the alternative subsample (reference subsample). To illustrate this, we rewrite the GLM equation in (6) and (7) as a combination of the column vector of the intercept and the coefficients 𝜷 and the matrix of the intercept and the observations for the independent variables 𝑿: 𝜷 = [𝛽0

𝛽1 ⋯ 𝛽𝑝 ]𝑇

1 𝑋1𝑖 ⋯ 𝑋𝑝𝑖 𝑿 = [1 ⋮ ⋱ ⋮ ] 1𝑋1𝑛 ⋯𝑋𝑝𝑛

(8) (9)

The counterfactual outcomes using constant association and constant distribution are then calculated by multiplying the matrix 𝑿 of the subsample of interest with vector 𝜷 of the reference subsample (for constant association) and multiplying the matrix 𝑿 of the subsample of reference with the vector 𝜷 of the subsample of interest (for constant distribution). The resulting matrices are either purged of the shift in associations, or purged of the shift in distributions. The purged outcomes are ultimately calculated by averaging the row sums (the fitted values) of those matrices: 𝐸(𝑌𝐶𝐴 ) =

1 ∙ ∑ ∑(𝑿𝑖𝑛𝑡 ∙ 𝜷𝑟𝑒𝑓 ) 𝑛𝑖𝑛𝑡 𝑖

𝐸(𝑌𝐶𝐷 ) =

1 𝑛𝑟𝑒𝑓

𝑝

∙ ∑ ∑(𝑿𝑟𝑒𝑓 ∙ 𝜷𝑖𝑛𝑡 ) 𝑖

(10)

(11)

𝑝

Equation (10) shows the calculation method for purged outcomes using constant association and (11) shows the method when applying constant distribution. The subscripts 𝑟𝑒𝑓 and 𝑖𝑛𝑡 indicate whether parameters are taken from the reference subsample or the subsample of interest. 𝑖 refers to the observations within the subsamples or the rows of the matrix resulting from 𝑿 ∙ 𝜷. The symbol 𝑝 refers to the different parameters within the models or the different columns of the matrix resulting from 𝑿 ∙ 𝜷.

Decomposition of effects Suppose the purging procedure is applied to a set of models in which the column vector of coefficients 𝜷 consists of multiple coefficients and there is a difference between the observed mean in the subsample of interest and either of the purged means. In such a situation, researchers may be interested in the difference between the observed mean and the purged mean when purging only the difference in distribution or association of a subset of the predictors. When it comes to purging the difference in associations of a subset of the coefficients in 𝜷, the decomposition procedure is straightforward. In equation (10), 𝜷𝒓𝒆𝒇 and 𝑿𝒊𝒏𝒕 are split up into 𝜷𝒓𝒆𝒇 and 𝑿𝒊𝒏𝒕 (referring to variables to be purged) and 𝜸𝒊𝒏𝒕 and 𝒁𝒊𝒏𝒕 (referring to variables not to be purged): 33

𝐸(𝑌𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝐶𝐴 ) =

1 ∙ ∑ ∑([𝑿𝑖𝑛𝑡 , 𝒁𝑖𝑛𝑡 ] ∙ [𝜷𝑟𝑒𝑓 , 𝜸𝑖𝑛𝑡 ]) 𝑛𝑖𝑛𝑡 𝑖

(12)

𝑝

Purging differences in distribution of a subset of characteristics can be a somewhat more intricate matter, depending on the link function used in the model estimation. When an identity link function is used, the method to be employed is concurrent with the method of purging the difference in association of a subset of the predictors. The variables to be purged of the change in distribution 𝑿𝑟𝑒𝑓 from equation (11) should be split into the subset to be purged 𝑿𝑟𝑒𝑓 and the subset not to be purged 𝒁𝑖𝑛𝑡 . However, this can result in matrices of different sizes, which cannot form one matrix to be multiplied with the vector of coefficients. Therefore, these values in the rows must first be summed and divided by the appropriate number of rows as follows, resulting in a vector of matrix column means equal in size to the column vector of coefficients: 1 1 ̅ 𝑟𝑒𝑓 , 𝒁 ̅ 𝑖𝑛𝑡 ] = [ ∙ ∑ 𝑿𝑟𝑒𝑓 , ∙ ∑ 𝒁𝑖𝑛𝑡 ] [𝑿 𝑛𝑟𝑒𝑓 𝑛𝑖𝑛𝑡 𝑖

(13)

𝑖

These subsample means are then introduced into the equation: ̅ 𝑟𝑒𝑓 , 𝒁 ̅ 𝑖𝑛𝑡 ] ∙ [𝜷𝑖𝑛𝑡 , 𝜸𝑖𝑛𝑡 ]) 𝐸(𝑌) = 𝜇 ⟶ 𝐸(𝑌𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝐶𝐷 ) = ∑([𝑿

(14)

𝑝

̅ 𝑟𝑒𝑓 refer to the variables to be purged, while 𝜸𝑖𝑛𝑡 and 𝒁 ̅ 𝑖𝑛𝑡 refer to In equation (13) 𝜷𝑖𝑛𝑡 and 𝑿 the variables not to be purged. The condition 𝐸(𝑌) = 𝜇 denotes all situations in which an identity link function is used. Whenever there is a non-linear relation between 𝜇 and 𝜂, the link function is not equal to identity and therefore 𝐸(𝑌) ≠ 𝜇. In such a case, the method presented in equations (13) and (14) is not suitable, since the result of the multiplication of the column vector of ̅ ̅ coefficients [𝜷𝑖𝑛𝑡 , 𝜸𝑟𝑒𝑓 ] with the vector of observation means [𝑿 𝑟𝑒𝑓 , 𝒁𝑖𝑛𝑡 ] is a vector of

values of 𝐸(𝑌𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝐶𝐷 ) in which the non-linearity is ignored. The fitted values based on equation (13) are assigned according to the observations 𝑖 which are ultimately weighed by their fraction of 𝑛𝑟𝑒𝑓 or 𝑛𝑖𝑛𝑡 to come to the mean that is partially purged of distribution ̅ 𝑟𝑒𝑓 can deviations. This implies that the part of 𝜇 as a result of applying the variable means 𝑿 ̅ 𝑖𝑛𝑡 . However, within the two original models, an be added to the unchanged part based on 𝒁 equal shift in means of the independent variables 𝑿𝑟𝑒𝑓 and 𝑿𝑖𝑛𝑡 would not have led to equal increases in 𝐸(𝑌)𝑟𝑒𝑓 and 𝐸(𝑌)𝑖𝑛𝑡 , because the relations are not linear and steepness of the model curve differs across its range. Following Liao (1989), whose work was built upon earlier papers on the purging method (Clogg, 1978; Clogg & Eliason, 1988; Clogg, Shockey, and Eliason, 1990), Te Grotenhuis et al. (2004) defined a way to decompose the impact of applying a constant 34

distribution of a subset of variables in models with a link function other than identity. This decomposition method requires calculating 𝐸(𝑌𝐶𝐷 ) purged of distribution changes for all variables, but within groups for each unique combination of values for the variables in 𝒁𝑟𝑒𝑓 . The outcomes are subsequently multiplied by weights 𝑊 to correct for difference in relative group size with the subsample of interest. The weights are calculated as follows, with 𝑓 denoting absolute frequencies: 𝑊𝑖 =

𝑓(𝑍𝑖 )𝑟𝑒𝑓 𝑓(𝑍𝑖 )𝑖𝑛𝑡 ⁄ 𝑛𝑟𝑒𝑓 𝑛𝑖𝑛𝑡

(15)

𝑓(𝒁𝑖 ) in equation (15) denotes the frequency of entries in matrix 𝒁 with the combination of values in 𝒁𝑖 . The weight for each entry is calculated as a correction for the difference in relative frequency of the group in the reference subsample compared to its relative frequency in the subsample of interest. Introducing the weights into equation (11) results in: (16) 𝐸(𝑌) ≠ 𝜇 ⟶ 𝐸(𝑌𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝐶𝐷 ) = ∑ (∑([𝑿𝑖𝑛𝑡 , 𝒁𝑖𝑛𝑡 ] ∙ [𝜷𝑟𝑒𝑓 , 𝜸𝑖𝑛𝑡 ]) ∗ 𝑊𝑖 ) 𝑖

𝑝

This decomposition method relies heavily on the restriction that each unique combination of values present in matrix 𝒁𝑟𝑒𝑓 should also be present in the matrix 𝒁𝑖𝑛𝑡 , and vice versa. Using controlled coefficients In some cases, a researcher may want or need to estimate effects controlled for other characteristics. When using the abovementioned decomposition technique, the number of variables that can be included in the purging procedure is in practice often restricted, because of the condition that each unique combination of categories in 𝒁 should be present in each of the samples. One way to make sure the core coefficients are controlled for relevant other characteristics, is to first estimate unrestricted models and extracting subsets of 𝜷 and 𝑿 to replace the coefficients in the restricted uncontrolled models. The observed outcomes 𝐸(𝑌) will not necessarily (probably not) reflect the real sample means 𝜇, because of the altered intercepts. Therefore, all further calculations of 𝐸(𝑌), purged and not purged of differences, must subsequently be calculated (fitted) using the controlled coefficients and the controlled intercept. In order to be able to properly assess the differences between constant distribution outcomes and observed outcomes, the constant distribution outcome must be compared to the fitted observed outcome, rather than the observed sample mean.

35

Generating bootstrapped results using the R package Calculating the purged means using the procedure explained in the previous sections requires quite some computational work, especially when a researcher wants to perform the analyses for multiple combinations of subsamples. A second difficulty is found in transferring distributions and coefficients from one to another subsample or group, while these are in fact random parameters, when applying the procedure to survey data. Treating them as fixed parameters implies that occurrence of differences between the observed and purged means cannot be subjected to significance testing. Both of these problems were overcome by programming an R package allowing for easy and consistent execution of the purging analyses, including the ability to bootstrap the results to obtain confidence intervals. Based on work by Fox and Weisberg (2012), a resampling bootstrap (or random-x) technique was used. This section will explain how this package works and how one can come to the purged outcomes. The package dependencies are lme4, data.table and ggplot2. The lme4 package is used to estimate the GLMs and related functions from the package are used to generate predicted values. data.table is used to speed up time-consuming tasks such as resampling and calculating aggregate means for the decomposition of purging differences. ggplot2 is used to generate plots of bootstrapped results. The main function in the package to be called is purgemul, which is called as follows: purgemul(formula, purFormula, contrFormula, pur, purIntCat, purRefCat, data, family = gaussian, linear = TRUE, nboot = 0, strip = TRUE, noStrip = "none", verbose = FALSE, ...)

Users must specify the following arguments: formula

A formula in the form of LHS ~ RHS, which is used as the formula to estimate GLMs.

pur

The quoted name of the variable used to identify the subsamples in the data used. If the variable is not a factor, it is coerced to a factor.

purIntCat

A vector of character strings identifying the levels in the factor pur that should be used as the subsample(s) of interest. Minimum length is 1.

purRefCat

A vector of character strings identifying the levels in the factor pur that should be used as the subsample(s) of reference. Minimum length is 1.

data

The object containing the data.

36

All other arguments are optional: purFormula

A formula in the form of LHS ~ RHS, indicating the subset of model terms in the subsample of interest that should be purged of changes in association and distribution, compared to the models in the subsample of reference. The LHS should be equal to the LHS of the argument formula, and all terms on the RHS should be present in the RHS of formula.

contrFormula A formula in the form of LHS ~ RHS, used to replace the coefficients in the

RHS of formula by coefficients controlled for additional terms. The LHS should be equal to the LHS in formula. The RHS should contain all terms in the RHS of formula and one ore multiple additional terms. family

The specification of the link function used when estimating the GLMs. When omitted, gaussian(link = "identity"). Valid arguments are either of the link functions compatible with the glm function.

linear

If TRUE (default), and link = "identity", and a valid purFormula is specified, linear decomposition of distribution effects is used. If FALSE the weighting decomposition method is used.

nboot

If specified and greater than 0, estimated results are bootstrapped using bootstrap resampling, and results for each bootstrap sample are returned.

strip

If TRUE (default), all glm objects used in the function are stripped of excessive

big

elements

not

used

in

the

process,

such

as

glm.object[["data"]]. This is done by calling on the function strip, also

present in the package. noStrip

If "none", all glm objects used in the procedure are stripped of the elements c("y", "residuals", "effects", "linear.predictors", "data") . A

vector of a single or multiple of these elements can be specified to preserve them. This can be useful when additional arguments are passed to the function. verbose

Default is FALSE. If TRUE, and if nboot > 0, a signal is printed in the console after each 100th bootstrap iteration that is processed. This can be useful when handling big sets of data or complex glm models, to monitor the progress.

37

The result of the function is an object of class "purgeobj", which is a list containing the following: results

A three-dimensional array of outcomes with the first dimension being of the length of purIntCat, the second dimension of the length of purRefCat. The third dimension denotes the type of result. At least the following entries have values: Observed reference, Observed, CA, CD, CA Diff, CD Diff. If a contrFormula is present, the entries Fitted reference and Fitted have

values. In that case, CD Diff represents the difference between CD and Fitted, instead of CD and Observed. If the results are bootstrapped, the

mean of the bootstrapped results are also present, along with the 95% confidence intervals for the difference scores. bootstrap

A four-dimensional array of outcomes with the first dimension being of the length of purIntCat, and the second dimension of the length of purRefCat. The third dimension is of length 8, indicating the type of result, equal to the first entries in the result array. The fourth dimension is of the length of nboot, indicating the bootstrap sample number.

The purgemul function starts by converting the presented data object to a data.table object. Next, a number of checks are performed to make sure a valid purging procedure can be applied. This involves checking the levels of the variable pur and comparing them to the arguments purRefCat and purIntCat. If the conditions are satisfied, subsamples are defined for each unique level of purRefCat and purIntCat. For each of these subsamples, a glm object is created in which the GLM is saved. If needed, controlled coefficients are inserted in these models. Then, a number of additional variables are created, if a purFormula is supplied. These are used to decompose the purged outcomes. Afterwards,

the models are stripped of excessive information, using the strip function. After these rudimentary operations, a set of nested loops is used to calculate purged outcomes for each combination of reference subsamples and interest subsamples. To calculate these outcomes, the underlying function purge is called. Firstly, this is done for each combination of waves, and the results are stored in the object results. Then, the operations are reiterated for each instance of nboot (if asked) and written to the bootstrap object. After the last iteration, the results object is completed with the mean of the bootstrapped results, including 95% confidence intervals for the differences between observed or fitted means on the one hand and CA and CD purged means on the other hand. The purge function receives the following arguments when called upon from the function purgemul: 38

formula

A formula in the form of LHS ~ RHS, which is used as the formula to estimate GLMs.

intWave

A subsample of interest. An object of class "data.frame" and "data.table".

refWave

A reference subsample. An object of class "data.frame" and "data.table".

modelIntWave

A GLM using the formula and family on the data from intWave.

modelRefWave

A GLM using the formula and family on the data from refWave.

control

A length-one logical vector, serving as a flag to indicate if the coefficients are controlled for additional characteristics.

purgeAll

A length-one logical vector, serving as a flag to indicate if the outcomes should be partially purged (FALSE) or not (TRUE).

aggVar

Only supplied if purgeAll = FALSE, a vector of variable names that should be purged of changes in association and distribution.

aggVarList

A valid expression supplying a list of the entries in aggVar.

purTermsLabels

A vector of the labels of all the terms in the model matrix to be purged of changes.

notPurTermsLabels A vector of the labels of all the terms in the model matrix not to be

purged of changes. linear

A length-one logical vector, indicating if (and only if necessary) linear decomposition of effects should be used.

The function purge is suitable to be called independently, but serves for economical use of operations. The result is a list of 6 results that are returned to the purgemul function. The function starts by calculating observed means in the reference subsample and the sample of interest, making use of the predict.glm function. They are calculated using the fitted.values element of the glm objects. If control == TRUE, fitted means are also

calculated using the predict.glm function and passing the argument newdata = refWave or newdata = intWave, to force the function to recalculate the fitted values based on the

controlled coefficients. If purgeAll == TRUE, equations (10) and (11) are computed using the code:

39

if (purgeAll) { CA