Statistical Methods for Analyzing Multiple Race Response Data 

Tommi Gaines 

Abstract 

Collection of racial data is ubiquitous throughout research as an important measure of the 

demographic characteristics of the study population. However, the validity of racial data 

has been a concern, prompting several agencies to modify their measurements by 

allowing individuals to identify with multiple racial categories. This research aims to add 

to the current methodology for analyzing multiple race responses as well as single race 

categories for data generated from the California Health Interview Survey (CHIS). This 

paper explores three distinct methods for analyzing outcomes that indicate whether 

individual health behaviors are consistent with goals of the Healthy People 2010 

program. One approach uses supplementary data from the Census Bureau and the 

California Department of Finance to rake multiple-race respondents into single-race 

categories consistent with the 1977 OMB standards. The second method, following 

Schenker and Parker (2003), imputes a single race category for multiple race respondents 

to produce population health estimates. The third method, which we call multiple 

covariate adjustment, simultaneously controls for indicators of all self-identified race 

categories (using one group as the referent) in a regression analysis. The three methods 

are compared with attention focused on inference for the proportion of individuals who 

meet Healthy People 2010 goals, which has a common interpretation across methods, as 

well as inference about racial disparities in achieving those goals. 

Key Words: Multiracial data; Raking; Imputation; Regression 

Tommi Gaines 1 
 

 

 

1. Introduction 

Improving the health status of minority racial groups has been the focus of 

national health goals and planning in the United States over the past few decades (US 

Department of Health and Human Services, 1979; 1985; 1991; 2000). This is because

race is an important indicator of disparity in health care delivery and health outcomes 

such as excess mortality, morbidity, and disability. However the validity of racial data 

has been a concern since they were thought to incorrectly reflect the racial diversity of a 

wide variety of people. As a result several agencies have modified their standards of 

racial data collection to adjust for a changing racial and ethnic profile. Prior standards for 

the tabulation and presentation of racial data have typically followed federal guidelines 

discussed in the Office of Management and Budget (OMB) 1977 statistical policy 

directive (OMB, 1997). This policy defined racial categories as: White, Black, Native 

American or Alaska Native (AIAN) and Asian or Pacific Islander (API). The OMB 

revised these federal standards in 1997 allowing an individual to identify with more than 

one racial group, thus eliminating the idea of mutually exclusive single-race categories 

(OMB, 1997). Therefore under the revised standards a total of 31 possible racial 

categories exist. 

Although these revisions offer a wider choice for racial identification than 

previously available, the resulting data pose analytic dilemmas for the researcher. This is 

because the revised system inhibits compatibility between different data collection 

systems, presents difficulty with studying trends overtime, and can lead to insufficient 

sample sizes to generate statistically reliable estimates. The first two issues are directly 

related to other datasets that report statistics by single race category only, whereas the last 

is due to rare multiracial groups in a population. 

Tommi Gaines 2 
 

 

This paper will to add to the current methodology for analyzing data collected 

from multiracial individuals through comparing different statistical methods for 

analyzing multiple race responses as well as single race categories for data generated 

from the California Health Interview Survey (CHIS). This is a stratified sample that 

generates population based estimates of health outcomes across racial groups. The 

advantages and disadvantages of these methods will be explored by investigating how 

racial identification affects our understanding of health disparities among racial groups 

for all of California in context of health goals specified in Healthy People 2010 

(HP2010). This program provides a framework for health promotion and disease 

prevention. 

The next section will provide a description of the CHIS data set that is used for 

investigating the proposed methods. Section 3 discusses three methods to analyze 

multiple race response data, for which two approximate the size of the single race groups 

under the 1977 OMB standards to produce population health estimates and one 

simultaneously controls for the effects of all self-identified race categories on the 

estimation of health outcomes in a regression analysis. All three methods are used to 

estimate the proportion of Californians that experience a health outcome in context of 

HP2010 while adjusting for sociodemographic variables. The three proposed analyses 

differ in that two allocate multiracial individuals into a single race category creating a 

dataset with mutually exclusive racial categories whereas the third method preserves the 

multiracial status of the individual. Section 4 illustrates the application of these methods 

with the CHIS dataset by comparing the estimates generated under the three methods and 

conclude with a brief discussion in Section 5. 

Tommi Gaines 3 
 

2. Data Source 

The California Health Interview Survey is a biennial telephone interview survey 

with the first wave starting in 2001. The objective is to provide estimates of the health 

status of Californians through the collection of data on health, demographic, and 

economic characteristics. The sample design is a two-stage stratified random-digit-dial 

telephone survey. The first stage consists of randomly sampling telephone numbers 

generated for 44 predefined geographic areas that correspond to 41 individual California 

counties and 3 areas that are groupings of smaller California counties. The second stage 

involves the random sampling of one adult among all adults living in the household. A 

total of 56, 270 adults aged 18 years and older were sampled. 

3. Methods 

3.1 Raking Adjustment 

Raking is a statistical method that is primarily used is to adjust the survey 

estimates for undercoverage and response biases by attaching weights to the survey data 

using known population totals (Deming & Stephan, 1940; Deville & Sarndal, 1992; Brick 

2003). In general, this weighting procedure uses auxiliary data from a supplementary 

source, such as a larger survey or census. The advantages of this method are to reduce 

the bias and variance of the estimates, force totals to match external totals, and adjust for 

sources of error. 

Raking is performed by adjusting survey weights so that the marginal totals of the 

adjusted data agree with the population total from the marginal distribution of one 

dimension (or variable). The next step is to adjust the resulting weights to agree with the 

Tommi Gaines 4 
 

 

 

 

 

 

 

 

population totals for the second marginal distribution. This process continues by 

alternating between all dimensions in a cross-classification table. The algorithm iterates 

until convergence that is until the sum of the adjusted data simultaneously agree with the 

population totals for all the marginal distributions within a specified tolerance level. A 

formal mathematical description of computing the weights at each iteration t, in a two 

variable situation, is as follows: 

w ~ 

ij = Nˆ ij for t = 0, 

~(t−1) w N ~ ij i. 

w = for t = odd, ij ~(t −1) wi. 

~(t −1) w N ~ ij . j

w = for t = even, ij ~(t −1) w 

. j 

where Nˆ 

ij = ∑ d(ijk , )

is the unadjusted estimate of the population total in cell (i,j) and 

k ij ∈( , )

=

1 where π ijk d = is the probability of selecting unit k in cell (i,j) or the sum of (ijk , ) ( , ) π (ijk , )

the sampling weights for persons in the sample falling in the classification corresponding 

to cell (i,j). This process iterates until convergence. 

This procedure is implemented through raking multiple race respondents into 

single race categories consistent with the 1977 OMB standards that tabulate race as: 

White, Black, API, and AIAN. The CHIS sampling weights are adjust by introducing 

revised weights produced by the raking algorithm that uses demographic and countylevel data. These weights are constructed to sum to known California population totals 

that are obtained from the Census Bureau and the California Department of Finance. 

Tommi Gaines 5 
 

 

Specifically, the marginal counts for California’s resident population by race are obtained 

from the American Community Survey (ACS) that is an annual nationwide survey 

conducted by the Census Bureau to replace the decennial long from census. The 

categories for race are all inclusive in that the population totals are tabulated as ‘race 

alone or in combination with one or more other races’. Furthermore, it is assumed that 

the marginal totals for variables collected in ACS and used in the raking algorithm have 

negligible error. The California Department of Finance (DOF) is a secondary auxiliary 

source for population totals according to age. The data are publicly available through the 

DOF website for which the format of age is from 0 to 100 with 1 year increments. 

A total of two demographic variables are used in the raking process that includes 

race and age. The race variable has 5 levels and is aggregated as race alone or in 

combination involving the following groups: White, Black, AIAN, API, and other, 

whereas the age variable has 4 levels: 18-29, 30-44, 45-64, 65+. These variables form a 

cross-classification table of the CHIS sample for which the CHIS sample weights are 

adjusted by a factor so that the sum of the adjusted weights simultaneously agrees with 

the population totals of the demographic variables. To illustrate this method the 

following quantities are defined: 

• dij = the CHIS sample weighted proportion of Californians in racial category i (i 

= 1,..,5) and age-class j (j = 1,..,4). The weighted cell estimates are given in Table 

3.1.1. 

• pij

CHIS = the CHIS sample weighted proportion of Californian’s that visited the 

dentist in racial category i and age-class j. The weighted cell proportions are 

shown in Table 3.1.2. 

Tommi Gaines 6 
 

The marginal distributions of the cross-classification that forms dij are used to apply the 

raking algorithm within each combination of the race and age categories to obtain: 

• dij

rake = weighted cell estimates, dij , raked to the marginal totals obtained from 

ACS and DOF for race and age. The corresponding cell estimates are given in 

Table 3.1.3. 

The result is a modified weighted proportion of Californians in racial category i with 

outcome y = 1 (e.g., visited the dentist in the past 12 months) and is termed the CHIS 

Raked Adjusted proportion given by: 

填 認尼入賑·椎賦

日乳

頓那内縄

沈

寵眺凋 噺

∑乳転迭 鳥日乳

喧

∑

填 認尼入賑

乳転迭 鳥日乳

̂ (3.1.2)

3.2 Multiple Imputation 

The dilemma of attempting to combine or compare racial data when 

classification systems have been revised has been addressed in part by Schenker and 

Parker (2003) through missing-data methods. Their approach utilizes multiple 

imputation to generate a distribution of missing values for the single race category. 

Imputation is a common method for handing missing data by filling-in a value for the 

missing datum such that complete-data methods of analysis can be applied. With 

multiple imputation, two or more values are imputed rather than a single value in order to 

reflect the uncertainty about which value to impute. 

The general idea of multiple imputation, as discussed by Rubin (1987), is to 

replace each missing value with a vector composed of possible values that are 

independently drawn from a distribution. This distribution reflects assumptions about the 

Tommi Gaines 7 
 

 

data and the mechanisms creating the missing data. The result is the formation of two or 

more (usually five to ten) completed data sets for which each data set is analyzed with a 

complete-data method. The analyses are then combined in a simple way that reflects the 

extra uncertainty due to having imputed rather than having used actual data (Rubin & 

Schenker, 1991, 1997; Schafer, 1997). 

In terms of the imputation method proposed by Schenker and Parker, an estimated 

probability of each single-race response is imputed for each multiple-race respondent. 

This probability is then used to allocate a single race category to the multiracial 

respondent. Their method is summarized as a two-step procedure that creates a set of 10 

imputations for the missing single race category (Bernard et al, 1998). This procedure 

reflects the variability of primary race given the parameters of the imputation model and 

the variability due to estimating the parameters. 

The two-step procedure is applied to the largest multiple race groups in the CHIS 

sample as an illustration of the method due to small sample sizes of the other multiracial 

groups. This includes 3 groups that make up 83% of all multiracial respondents in the 

2001 sample involving the following: Black/White, API/White, and AIAN/White. 

However this approach can be applied to every multiracial combination of adequate 

sample size. In the first step a logistic regression model is fitted among respondents of a 

specific multiple race combination for which the outcome, y =1, is a single race category. 

This model is defined as: 

健剣訣 岾----

訂岫掴岻

峇噺紅待 髪 紅怠 髪橋髪紅椎捲椎 髪 綱

怠袋訂岫掴岻

(3.2.1)

The predictors, xj

 for j = 1,...,p, included in the model are Hispanic ethnicity, gender, 

born in the US versus foreign born, household income measured by federal poverty 

Tommi Gaines 8 
 

 

 

level (FPL) below 200% versus at least 200%, educational attainment of high school or 

less versus more than high school, median household income for county of residence, 

and racial composition of county of residence. The independent variable for racial 

composition measures the percentage of Black residents in the Black/White model, the 

percentage of API residents in the API/White model, and the percentage of AIAN 

residents in the AIAN/White model. 

Once that model has been generated logistic regression coefficients are drawn 

from their approximate posterior distribution. This distribution is a multivariate normal 

given by: 

紅~警撃軽盤紅實, Σ侮匪 (3.2.2) 

where the mean, β

ˆ , is the estimate of β , a vector of the logistic regression coefficients 

and whose covariance matrix, Σˆ , is the estimated variance-covariance matrix of β

ˆ . Both 

β

ˆ

 and Σˆ are estimated from the logistic regression model fitted in (3.2.1) to each 

particular multiracial combination, for example the model of all Black/White biracial 

respondents. 

The second step is to compute the probability of primary race category for every 

individual with a missing primary race by using the logistic regression coefficients drawn 

from the distribution specified in (3.2.2). The person specific probability of the ith

individual is given by: 

講沈 噺 怠袋勅

勅掴椎盤掴

掴椎盤掴

日

嫦

嫦

庭匪

日庭匪

(3.2.3)

After the 2-step procedure has been carried out a multiracial person is reallocated 

to a single race category and complete data methods of analysis are applied. The 

Tommi Gaines 9 
 

 

 

assignment of a multiracial individual to a particular single race group can be thought of 

as a Bernoulli coin flip where the chance of the coin randomly selecting a particular race 

is determined from π i calculated in (3.2.3). 

In the CHIS survey all multiracial persons were asked to select a single race 

category that best identified oneself. Responses included one of the single race 

categories that made up the multiracial combination, other, both/all/multiracial, refused, 

none of these, or don’t know. In the logistic regression model to predict primary race the 

binary outcome consisted of a specific single race category or ‘other’. The remaining 

responses were treated as missing and the 2-step procedure was applied to impute a single 

race category among these multiracial individuals. 

Details of the 2-step procedure can be illustrated, for example, by considering the 

Black/White respondents in the CHIS sample. A logistic regression model is fitted to this 

specific biracial combination to predict the probability of being Black, White, or Other. 

Initially a logistic regression model is fitted where the outcome is Other versus (Black or 

White). The probability of being Other is calculated for each person through the 2-step 

procedure. A subsequent logistic regression model is fitted where the outcome is Black 

versus White among all Black/White respondents that did not identify their primary race 

as Other. The probability of Black is then computed via the 2-step procedure. As a result 

each Black/White respondent with a missing primary race has a probability of being 

Black, White, or Other assigned to them. These probabilities are used to randomly assign 

a single race to that person. 

Table 3.2.1 displays the results of fitting separate logistic regression models to the 3 

multiracial groups. Many of the covariates in Table 3.2.1 are not significantly predictive 

Tommi Gaines 10 
 

 

of primary race as was found by Schenker and Parker. The covariates that are predictive 

differ across the racial groups with the exception of Hispanic ethnicity that has a positive 

association for all biracial models versus Other. 

Complete data methods of analysis are applied following the implementation of 

the 2-step procedure. This involves the utilization of CHIS sample weights to account for 

differential probabilities of selection in order to produce unbiased population estimates. 

For a survey of n subjects, the weighted proportion of individuals that have seen the 

dentist is calculated for each single race group. These groups consist of: White, Black, 

API, AIAN, and Other. For a particular racial group, the weighted proportion is defined 

as: 

喧̂賃

暢彫 噺

∑日転迭

韮 栂日入槻日入

∑

韮

日転迭 栂日入

(3.2.4)

where wik is the inverse probability of selecting individual i of racial group k and yik = 1 

if the ith individual of the kth racial group saw the dentist within the past 12 months and 

zero otherwise. The corresponding standard errors are calculated through the jackknife 

methods. 

3.3 Multiple Covariate Adjustment 

The methods discussed in the prior sections involve the allocation of multiracial 

persons into a single race group. This is followed by a separate analysis using the 

multiracial responses assigned to a single-race group as well as the single race responses 

to calculate the prevalence of a particular health outcome by racial subgroup. As a result, 

a weighted estimate of the crude or unadjusted proportion of Californians from group k 

Tommi Gaines 11 
 

 

 

with outcome y = 1 (e.g., the having seen the dentist at least once during the past year) is 

calculated. 

An alternative to the estimating the crude proportion is to use the conventional 

regression approach of including dummy variables in a multiple logistic regression 

analysis. In this situation the model would simultaneously control for the effects of all 

the self-identified race categories in the CHIS sample on the outcome y = 1. Table 3.3.1 

displays the estimated proportion of individuals that visited the dentist across different 

independent variables among 5 racial categories. Overall those who visited the dentist 

are younger, of higher income, more educated, carry insurance, and live in a metropolitan 

area. However differences between racial groups are evident with Whites having a 

greater proportion of individuals seeing the dentist in general across all categories of the 

independent variables. The variations of these socio-deomgraphic variables across racial 

groups can be adjusted for in a logistic regression model. The independent variables 

considered in the analysis include: (1) race: White, Black, API, AIAN, Other, (2) age (in 

years), (3) annual household income classified according to the federal poverty level 

(FPL): <100%, 100-199%, 200% - 299%, ≥300% of FPL , (4) education: less than high 

school, high school graduate, and any college education, (5) insurance status: currently 

uninsured, uninsured for any of the past 12 months, and insured for all of the past 12 

months, (6) OMB classification of a Metropolitan Statistical area (MSA): metropolitan 

and non-metropolitan. 

In addition to controlling for socio-demographic factors, the logistic model can 

also address the multiracial status of survey respondents. This is accomplished through 

incorporating dummy variables to reflect each self-identified racial group in the CHIS 

Tommi Gaines 12 
 

---

sample. For example, suppose that the jth independent variable, x j , has k j levels where 

the k j −1 design variables are denoted as Dl and the respective coefficients are denoted 

as β l , l = 1,2..., k j −1. The logit fitted to the data with p independent variables where the 

j

th independent variable is nominal is: 

k j −1

⎛ ⎞ log ⎜

μ

⎟ = β0 + β1

x1 + β 2 x2 +.... + ∑βl Dl +β x + ε (3.3.1) ⎝1− μ ⎠ l =1 

p p 

Therefore, to obtain the proportion of individuals by racial category with a 

particular health outcome involves the estimation of μ in model (3.3.1). For example, the 

proportion of Black individuals that visited the dentist is estimated from the logistic 

probability, μˆ = p(Y=1| race, age, education, income, insurance status, MSA). To do so 

by racial groups involves the incorporation of dummy variables where the jth independent 

variable is racial category with kj = 5 levels. Ignoring ethnicity, there are a total of 4 

single race categories in the 1977 OMB standards for racial data collection. However, an 

additional category considered in the model is “other” which can be thought of as a 

residual category for those that did not identify with one of the four. Therefore a total of 

4 design variables, k j −1, are included in the model with ‘other’ treated as the reference 

group. These design variables differ from the usual manner in that they are not mutually 

exclusive. Rather each racial category is defined as race alone or in combination and 

therefore will overlap one another. For example, the total sample size is n = 56,037 but 

from Table 3.3.1 we see that the sum of sample sizes across the 5 racial categories 

exceeds this total. 

Another aspect of the fitted model is to incorporate the complex sample design of 

CHIS involving stratification and sample weights. From the fitted logit we can predict 

Tommi Gaines 13 
 

 

the logistic proportion of individuals with outcome y = 1 among observations i = 1,..., n 

as: 

航̂沈 噺 結捲喧盤捲沈

嫗紅實匪 1 髪 結捲喧盤捲沈嫗紅實 斑 匪 (3.3.2) 

where β

ˆ is the value that maximizes the weighted likelihood estimator defined as: 

庭

庭

庭

庭

 

庭撫

健岫紅岻 噺 ∏ 釆

怠袋勅掴椎

勅掴椎盤掴

盤

日

嫦

掴

嫦

匪

庭撫

日

匪

挽

栂日槻日

津

沈退怠

庭撫

釆1 伐 怠袋

勅掴椎

勅掴椎

盤掴

盤

日

嫦

掴

嫦

匪

庭撫

日

匪

挽

栂日

岫怠貸槻日岻

(3.3.3)

and wi is the sample weight associated to each observation (Korn & Graubard, 1999). 

The predicted logistic proportions are then averaged across the k racial groups to 

produce the model adjusted pr portio n of o individuals with outcom e y = 1: 

韮

暢寵凋 噺

∑日転迭 栂日入禎赴日入 喧賃 ∑

韮

日転迭 栂日入

̂ (3.3.4)

The regression coefficients for the logistic regression model are given in Table 

3.3.2. After adjusting for the socio-demographic variables Black and AIAN are 

significant predictors in the model. An additional model was generated to investigate 

whether an additive effect of race appropriately represents the data. Under this situation, 

it is assumed that the impact of race on the proportion of Californians with outcome y = 1 

is independent of the multiracial status. However including interactive effects between 

each of the 4 design variables representing the four racial groups permits the effect of 

race on the outcome to vary across single and multiracial persons. This model included 

the same independent variables as shown in Table 3.3.2 however the interactive effects 

among the racial groups were included (model not shown). A likelihood ratio test 

between the two models did not support including these interactive effects ( χ

2

6 =10.37 ( )

and p-value = 0.11). Furthermore, the area under the ROC curve was compared between 

the two models and did not exhibit substantial improvement with either model but was 

Tommi Gaines 14 
 

 

 

approximately 0.67 for both. This area was used as another statistical comparison since 

it represents the probability that a randomly chosen subject with outcome y = 1 is 

correctly identified by the logistic model with greater likelihood of having outcome y = 1 

than a randomly chosen subject without outcome y = 1 (Hanley & McNeil, 1982). Of 

additional note is the non-linear transformation of age. The fractional polynomial 

method was used to investigate whether age is linear in the logit and it was determined 

that the squared inverse of age was significantly better than a linear term for age. 

3.4 Variance Estimation 

The methods presented in the previous section estimates the proportion of 

individuals with outcome y = 1 by accounting for the weights and stratification of the 

CHIS sample. The multiple imputation and covariate adjustment involve a non-linear 

function of the data through predictions from a logistic regression model to eventually 

generate a weighted proportion. The multiple imputation method does so through a 

logistic model to predict primary race that is subsequently used in the calculation of the 

weighted proportion; whereas the covariate adjustment utilizes a weighted logistic 

regression to derive a weighted predicted proportion from the model. The raking 

adjustment is less complex but involves a modification to the initial CHIS sampling 

weights. Therefore, replication methods are applied in estimating the variance for the 

proportion of individuals with outcome y = 1. 

The corresponding variance for the estimates produced by the multiple imputation 

(3.2.4) and multiple covariate adjustment (3.3.4) are calculated using the jackknife 

method. The jackknife variance estimator is calculated by excluding observations in the 

Tommi Gaines 15 
 

 

 

I 

i

th PSU of stratum h for the kth racial group. This increases the sample weights of the 

remaining observations in stratum h by a factor of mh (mh −1) , and is proceeded by 

calculating ˆp ( ) of the new data set. The variance estima tor is given by: hi

張 陳廿貸怠 陳廿

態

∑朕退怠 陳廿∑沈退怠盤喧̂ ̂ 岫朕沈岻 伐 喧匪 (3.4.1) 

The bootstrap is used for variance estimation in the raking adjustment. The 

bootstrap is calculated by generating artificial data sets that are of the same size and 

structure of the original where sampling is done repeatedly with replacement (Efron, 

1979; 1982). Korn and Graubard, (1999) illustrate the bootstrap estimate of the variance 

in context of survey data as follows: 

(3.4.2) 怠 喋

態

懸欠堅 阜 喋聴盤Θ撫匪 噺 ∑長退怠盤Θ撫茅

但 伐 Θ撫茅

匪

喋貸怠

where 

B

茅 Θ撫茅 噺

B

1布 Θ撫

但

但退怠

and B is the number of replicated data sets. 

4. Application 

The data for the applying the three methods is taken from the 2001 California 

Health Interview Survey (CHIS). The advantages of using CHIS are that survey 

participants are allowed to identify with more than one racial group and it can be used to 

measure the states progress towards achieving the national goals specified in HP2010. 

During the first wave approximately 4.6% identified as multiracial in the 2001 survey. 

Among the multiracial respondents, an additional question is asked requesting them to 

Tommi Gaines 16 
 

 

 

 

designate a race that best identifies them. The utility of CHIS is to whether California is 

meeting the goal of eliminating health disparities as outlined in Healthy People 2010. 

Therefore, the application of the three methods is illustrated in context of the oral health 

goal of Healthy People 2010. The current goal is that 56% of the population uses the oral 

health care system each year and their statistics document AIAN and Blacks see the 

dentist less frequently than other racial groups (US Department of Health, 2000). 

The proportions stratified by racial group were estimated from equations (3.1.2), 

(3.2.4), and (3.3.4). An additional estimate was generated to compare with the other 

three. This estimate is similar in that it is a weighted proportion of visiting the dentist 

where the weights are derived from the CHIS sample. However, it differs in that there is 

an additional racial group termed multiracial and this proportion does not account for the 

reallocation or adjusting of the multiracial status. Table 4.1 displays the estimates 

generated under each method. The standard errors for the CHIS weighted proportion, 

喧

寵張彫聴 喧暢彫 喧暢寵凋

̂ , multiple covariate adjustment, ̂ , and multiple covariate adjustment, ̂ , were 

obtained through the jackknife method. A bootstrap standard error was calculated for the 

喧

寵眺凋 raking adjustment, ̂ . We can see that the estimates generated by the different 

methods closely agree with each other. However noticeable differences are observed in 

the standard errors with the multiple covariate adjustment having smaller standard errors 

across all racial groups relative to the other methods. Overall, there is a disparity among 

the AIAN and Other racial group in the utilization of the oral health care system. These 

individuals see the dentist less frequently than the other groups. However, California has 

exceeded the HP2010 national goal across every racial group as estimated under each 

method. 

Tommi Gaines 17 
 

 

 

 

5. Discussion 

This paper described three distinct methodologies for analyzing multiple race as 

well as single race responses. The methods illustrated in this paper are best applied to 

analyses where the primary objective is to describe the role of race on the outcome of 

interest. For example, the raking adjustment and multiple imputation allocate a 

multiracial individual into a single race category in context of investigating health 

disparities across racial groups. Although these two methods do not preserve the 

multiracial status of the individual both attempt to avoid problems that can arise when 

aggregating all multiracial persons into a single racial category sometimes referred to as 

the “2+ Races” category. 

In Public Health, we perform analyses by racial groups with the goal of 

identifying potential problems and to gain an understanding of areas for improving health 

in the population. Therefore, we need data that not only identify groups suffering from 

health inequities but also identify the reasons behind these inequities. Yet understanding 

the mechanisms that lead to these disparities is highly complex in a category that 

aggregates all multiracial individuals. This is because of the heterogeneity of other 

demographic variables associated with health outcomes. It has previously been shown 

that the proportion of multiracial individuals with a particular educational attainment or 

labor force participation is dependent on the specific multiple race combination (Snipp, 

2006, personal communication). According to the 2004 American Community Survey 

(ACS), approximately 16% of non-Hispanic White/AIAN, 10% of non-Hispanic 

White/Black , 6% of non-Hispanic White/Asian, and 21% of non-Hispanic White/Other 

individuals have less than a high-school education. The 2004 ACS also estimates that 

Tommi Gaines 18 
 

 

 

 

approximately 64% of non-Hispanic White/AIAN, 75% of non-Hispanic White/Black , 

74% non-Hispanic White/Asian, and 70% of non-Hispanic White/Other individuals are 

participating in the civilian labor force. The diversity of demographic variables makes it 

difficult to produce meaningful inference on the multiracial population. The dilemma 

here is of obtaining accurate data needed for planning and policy making and to design 

interventions that target particular racial groups that improve their health. 

Comparison of the three methods was limited to the inference made in context of 

the proportion of individuals meeting the oral health goal of HP 2010. Future work 

entails a simulation to further evaluate the performance of each method. The sensitivity 

and robustness of the three methods are checked by fitting an empirical model to a 

simulated population developed from the CHIS data set. Particularly the simulation will 

address how each method is affected by varying characteristics of the population 

including age, income, education, and ratio of single race to multiple race individuals. 

This will allow for the determination of which method performs better over the situations 

considered in the simulation. 

Tommi Gaines 19 
Tables 

Table 3.1.1: Proportion of Californians 18+ by race and age before applying raking algorithm, dij 

Race 18-29 30-44 45-64 65+ Total 

White/combo 

Black/combo 

API/combo 

AIAN/combo 

Other/combo 

0.120 

0.016 

0.035 

0.009 

0.058 

0.186 

0.021 

0.045 

0.011 

0.066 

0.194 

0.019 

0.034 

0.010 

0.034 

0.108 

0.007 

0.016 

0.003 

0.009 

0.607 

0.063 

0.130 

0.033 

0.167 

Total 0.238 0.328 0.290 0.144 1 

CHIS Table 3.1.2: Weighted proportion of Californians that visited the Dentist, pij 

Total 18-29 30-44 45-64 65+ 

White/combo 

Black/combo 

API/combo 

AIAN/combo 

Other/combo 

0.70 

0.72 

0.69 

0.67 

0.55 

0.71 

0.69 

0.73 

0.65 

0.58 

0.77 

0.67 

0.71 

0.64 

0.59 

0.69 

0.51 

0.65 

0.52 

0.52 

Table 3.1.3: Proportion of Californians 18+ by race and age after applying 

raking algorithm, dij 

rake 

Total 18-29 30-44 45-64 65+ Total 

White/combo 

Black/combo 

API/combo 

AIAN/combo 

Other/combo 

0.132 

0.017 

0.034 

0.004 

0.043 

0.202 

0.022 

0.042 

0.005 

0.049 

0.217 

0.021 

0.033 

0.004 

0.026 

0.117 

0.008 

0.015 

0.001 

0.007 

0.668 

0.068 

0.124 

0.015 

0.125 

Total 0.230 0.320 0.302 0.148 1 

Tommi Gaines 20 
 

 

 

 

 

 

 

 

 

Table 3.2.1: Logistic Regression Predicting Primary Race. Estimated coefficients that are significantly different from zero at the 10 percent level 

are indicated by a + (positive coefficient) or – (negative coefficient) 

Variable Black/White API/White AIAN/White 

(Black or White) Black vs. (API or White) API (AIAN or White) AIAN 

 vs. Other White vs. Other vs. White vs. Other vs. White

Hispanic (Y = 1, N = 0) + + + + + 

Gender (F = 1, M = 0) 

Born in US (Y = 1, N = 0) + -

FPL (0-199% = 1, ≥ 200 = 0) + 

Education (≤ HS = 1, >HS = 0) + + 

Age (continuous) -

Median County 

 Household Income 

County percent Black, API, 

or AIAN + + 

Tommi Gaines 21 
 

 

 

 

 

 

 

 

 

 

 

 

 

Table 3.3.1: Univariate Descriptive Statistics of the proportion of Californian’s 18+ that have 

visited the dentist at least once in the past 12 months by racial group (n = 56,037) 

Variable White (se) Black (se) API (se) AIAN (se) Other (se)

n = 40,851 n = 3,002 n = 5,465 n = 2,882 n = 6,472 

Age 

18-29 

30-44 

45-64 

65+ 

FPL 

0.70 (.01) 

0.71 (.01) 

0.77 (.01) 

0.70 (.01) 

0.72 (.03) 

0.70 (.03) 

0.67 (.02) 

0.52 (.03) 

0.69 (.02) 

0.73 (.01) 

0.72 (.02) 

0.64 (.03) 

0.67 (.05) 

0.63 (.04) 

0.63 (.03) 

0.56 (.07) 

0.55 (.02) 

0.58 (.01) 

0.59 (.02) 

0.51 (.04) 

< 100% 

100-199% 

200-299% 

0.61 (.01) 

0.63 (.01) 

0.70 (.01) 

0.71 (.03) 

0.64 (.02) 

0.71 (.03) 

0.62 (.02) 

0.63 (.02) 

0.73 (.02) 

0.59 (.04) 

0.65 (.04) 

0.70 (.04) 

0.55 (.01) 

0.59 (.01) 

0.66 (.02) 

≥ 300% 

Education

0.81 (.00) 0.75 (.01) 0.78 (.01) 0.77 (.03) 0.76 (.01) 

< HS 

HS 

> HS 

Insurance 

0.57 (.01) 

0.70 (.01) 

0.80 (.00) 

0.59 (.04) 

0.70 (.02) 

0.74 (.01) 

0.54 (.02) 

0.71 (.02) 

0.76 (.01) 

0.62 (.04) 

0.69 (.03) 

0.72 (.03) 

0.54 (.01) 

0.68 (.01) 

0.71 (.01) 

Current unins 0.53 (.01) 

Unins 12 mo 0.62 (.01) 

Ins 12 mo 0.78 (.00) 

MSA 

0.50 (.04) 

0.67 (.04) 

0.74 (.01) 

0.55 (.02) 

0.62 (.05) 

0.76 (.01) 

0.53 (.04) 

0.62 (.07) 

0.72 (.02) 

0.49 (.01) 

0.55 (.02) 

0.68 (.01) 

Metro 0.74 (.00) 0.71 (.01) 0.72 (.01) 0.67 (.02) 0.61 (.01) 

Non-metro 0.70 (.01) 0.73 (.04) 0.68 (.04) 0.68 (.04) 0.60 (.02) 

Table 3.3.2: Logistic Regression Analysis for Visiting the Dentist of Californians (n = 56,037) 

Variable Coefficient (SE) p-value 

Race

White 

Black 

API 

AIAN 

Other (ref) 

Age

FPL 

欠訣結貸態

欠訣結貸怠

0.04 (0.04)

-0.11 (0.05) 

-0.02 (0.05) 

-0.09 (0.07) 

1.00 

-26.18 (7.16) 

524.1 (116.63) 

0.27 

0.03 

0.63 

0.22 

-

< 0.001 

< 0.001

< 100% 

100-199% 

200-299% 

-0.56 (0.05) 

-0.56 (0.04) 

-0.40 (0.04) 

< 0.001

< 0.001

< 0.001 

≥ 300% (ref)

Education

1.00 -

< HS 

HS 

> HS 

-0.65 (0.05) 

-0.32 (0.03) 

1.00 

< 0.001

< 0.001

-

Insurance 

Current unins

Unins 12 mo 

Ins 12 mo 

0.70 (0.04) 

0.56 (0.06) 

1.00 

< 0.001

< 0.001

-

MSA 

Metro 0.13 (0.03) < 0.001

Non-metro 1.00 -

Tommi Gaines 22 
Table 4.1: Proportion of individual s that visited the dentist in the past 12 months by racial g roup 

for the three proposed methods w ith standard errors in pare thesis, n = 56,270 

喧

寵眺凋

n 

喧

寵張彫聴 喧暢彫 喧暢寵凋 Race n ̂ ̂ ̂ ̂

White 38760 0.726 (0.003) 0.724 (0.007) 0.725 (0.001) 0.725 

(0.003) 

Black 2615 0.674 (0.011) 0.672 (0.013) 0.672 (0.003) 0.675 

(0.011) 

API 5053 0.705 (0.008) 0.703 (0.013) 0.702 (0.002) 0.704 

(0.008) 

AIAN 935 0.639 (0.023) 0.639 (0.016) 0.616 (0.005) 0.642 

(0.021) 

Other 6445 0.570 (0.008) 0.571 (0.008) 0.575 (0.002) 0.571

(0.008) 

2+ Races 2462 0.648 (0.016) - - -

Tommi Gaines 23 
 

 

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