3. METHODOLOGY
3.6 Individual-Level Analysis
3.6.5 Regression Standardization and Components Analyses,
In this section I discuss how I use regression standardization and component analysis to determine how (a) residential outcome vary by group means and coefficients and (b) the separate and joint components that contribute to white-black segregation. Regression standardization involves white-black coefficients and white-black means on income to compute neighborhood outcomes (Fossett 2017; Fox 2014). In other words, this method generates neighborhood outcomes by manipulating white-black coefficients and means, separately. I use this method to answer two substantive questions. First, what would black neighborhood outcomes look like if they had the same distributions as whites? Second, what would black neighborhood outcomes look like if they could
36
convert their distributions into more residential contact with whites at the same rate as whites? Of note, this exercise not directly assessing the out-migration thesis, it does however relate to Wilson’s race-specific policies in TTD. Under this hypothetical scenario, matching the white rates of return in the black regression model reflects black residential outcomes when discrimination does not exist. Matching whites’ distributions in the black regression model represents black residential outcomes if an economic policy was implemented in a city. Findings from the standardization exercise allows for assessing white-black difference in “rates of returns,” “difference in “distributions,” and “joint impact” have in white-black residential segregation (Fossett 2017; Fox-Crowell and Fossett 2017; Fox 2014).
The first step is to obtain regression results. Once obtained, regression
standardization is performed. In the past, scholars often rely on ordinary least squares regression, as cited and quoted from (Fox 2014:69):
“(a) Yw = Bw0 + B1Xw1 (b) Yb = Bb0 + B1Xb1 (c) Yb = Bb0 + (Bb1 * Xb1) + (Bb2 * Xb2) (d) Yb = Bb0 + (Bb1 * Xw1) + (Bb2 * Xw2) (e) Yb = Bw0 + (Bw1 * Xb1) + (Bw2 * Xb2) (f) Yb = Bw0 + (Bw1 * Xw1) + (Bw2 * Xw2)”
Where equation (a) is the regression equation for whites, equation (b) is the regression equation for blacks, equation (c) is the expansion of equation (b) for blacks
37
with black distributions and black rates of return, equation (d) calculates the neighborhood outcomes blacks would have with whites if they had the same
distributions as whites, equation (e) calculates neighborhood outcomes for blacks when they are equalized to white rates of return, and equation (f) calculates neighborhood outcomes for black when they are equalized to both whites’ distributions and rates of return. The difference between (c) and (f) equals the city segregation score, D or S. For this study, the standardization equation mentioned above is not appropriate to use because neighborhood outcomes are non-linear and non-additive (see fractional regression discussion below).
For linear additive models like ordinary least squares regression, the components analysis involves inserting whites’ distributions and rates of return in the black
regression equation and vice versa (Jones and Kelley 1984;Fossett 2017; Fox 2014). This convenient option does not hold in the case of non-linear, non-additive
neighborhood outcomes ranging from 0-1 (Fossett 2017). Values obtained by manipulating white-black ordinary least squares equation is often very close to the “mean on (y)” (Fossett 2017). As a result, values could fall outside of 0-1 bounds and can vary by a large amount (Fossett 2017).
Instead of ordinary least squares to model neighborhood outcomes, I use
fractional regression (Kieschnick and McCullough 2003). The logic for using fractional regression is fairly straightforward: non-linear, non-additive segregation scores are bounded by 0 to 1. In the past, researchers have relied on logit transformations where the S-shaped regression curve is bounded by 0 to 1. This option becomes complicated
38
because it has the potential of violating the linear regression assumption of linearity and additivity, and normality. For example, if nativity, limited English language, and
educational attainment all negatively and additively affect parity contact with whites for blacks, then the regression line can be taken out of bound. The logic for using fractional logit regression is as follows: “individuals are assigned scores based on whether or not their neighborhood is “at or above parity” (see Fossett 2017:90) with MSA proportion white (1) or not (0)” (Fossett 2017:97; Fox 2014).
As previously mentioned, the regression standardization and components analysis of fractional regression analyses require a more involved approach than previous studies (see Althauser and Wigler 1972; Jones and Kelley 1984). Following work by Fox- Crowell and Fossett (2017) and Fossett (2017), I calculate the “observed group means” and “standardized group means” for whites and blacks, respectively. Equations for the two observed group means below are cited and quoted from (Fox-Crowell and Fossett 2017: equation section):
“ 𝑌̅𝑊𝐷𝑊𝑅 = The observed white mean (the average of predicted values for whites in the model for whites)
𝑌̅𝐵𝐷𝐵𝑅 = The observed black mean (the average of predicted values for blacks in the model for blacks)”
Equations for the two standardized group means below are cited and quoted from (Fox- Crowell and Fossett 2017: equation section):
39
“ 𝑌̅𝑊𝐷𝐵𝑅 = The black mean standardized to whites’ distributions (the average of predicted values for whites in the model for blacks)
𝑌̅𝐵𝐷𝑊𝑅 = The black mean standardized to whites’ rates (the average of predicted values for blacks in the model for whites)”
The overall level of segregation is derived by the difference of between the observed means for whites and blacks (Fossett 2017). Again, I follow Fox-Crowell and Fossett (2017) and Fossett (2017) studies by using similar equations for obtaining the value of overall white-black segregation and the components below cited and directly quoted from (Fox-Crowell and Fossett 2017: equation section):
“(DR)
𝑌̅
𝐵𝐷𝑊𝑅
- 𝑌̅
𝐵𝐷𝐵𝑅 Rate of return component of segregation(DD)
𝑌̅
𝑊𝐷𝐵𝑅
- 𝑌̅
𝐵𝐷𝐵𝑅 Distributions component of segregation(DJ) 𝐷 − (𝐷𝐷+ 𝐷𝑅) Joint impact component of segregation
(D)
𝑌̅
𝑊𝐷𝑊𝑅
-𝑌̅
𝐵𝐷𝐵𝑅 Total difference”In sum, findings from the standardization and components analyses allow me to directly measure hypothesis 3—whether the role of income is increasing or not for white- black group differences in overall level in segregation over time.
40