Direction Estimation in a General Regression Model with Discrete Predictors

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1 Direction Estimation in a General Regression Model with Discrete Predictors Yuexiao Dong and Zhou Yu Abstract Consider a general regression model, where the response Y depends on discrete predictors X only through the index β T X. It is well-known that the ordinary least squares (OLS) estimator can recover the underlying direction β exactly if the link function between Y and X is linear. Li and Duan (989) showed that the OLS estimator can recover β proportionally if the predictors satisfy the linear conditional mean (LCM) condition. For discrete predictors, we demonstrate that the LCM condition generally does not hold. To improve the OLS estimator in the presence of discrete predictors, we model the conditional mean E(X β T X) as a polynomial function of β T X and use the central solution space (CSS) estimator. The superior performances of the CSS estimators are confirmed through numerical studies. Introduction Consider univariate response Y R and q-dimensional predictor X R q. For β R q, let θ = β T X be a linear combination of the predictor X. Suppose the probability distribution function of Y depends on X only through θ, such that Y F θ (Y ), where θ = β T X. () Model () allows Y to be continuous or discrete, and we will refer to it as the general regression model. Many popular regression models can be formulated under this framework, such as linear regression, single index model, transformation models, and generalized linear regression. Yuexiao Dong Temple University, Philadelphia, USA, 9. ydong@temple.edu Zhou Yu East China Normal University, Shanghai, China, zyu@stat.ecnu.edu.cn

2 Yuexiao Dong and Zhou Yu As a special case of the general regression model (), single index model (SIM) considers Y = g(β T X) + ε for continuous response Y and unknown link function g : R R. SIM is a semiparametric model that allows for flexibility through the nonparametric link function and mitigates the curse of dimensionality through the parametric index structure. The estimation problem in SIM involves estimation of the index β and the link function g( ) simultaneously. Some classical references include Powell et al. (989), Härdle et al. (993), Ichimura (993), Horowitz and Härdle (996), Carroll et al. (997), etc. Cui et al. (0) considered a more general model, where the response can be continuous or discrete, and the single index appears in both the mean component and the variance component. Since nonparametric smoothing is involved in all the aforementioned methods, theses methods require that the link function g( ) is smooth and at least one predictor is continuous. To avoid nonparametric smoothing, methods based on distance covariance and Hilbert-Schmidt Independence Criterion are proposed, respectively, in Sheng and Yin (03) and Zhang and Yin (05). In this paper, we aim to estimate the index θ = β T X in the general regression model (), where all the predictors are discrete. To the best of our knowledge, Cook and Li (009) and Sheng and Yin (03) are the only methods in the literature that allow all the predictors to be discrete. Let X = (X,...,X q ) T. Cook and Li (009) assumed that the conditional distribution of X i Y follows a one-parameter exponential family distribution for i =,..., q, and used the maximum likelihood estimator of the inverse regression model to recover the direction β in the original forward regression. On the other hand, Sheng and Yin (03) argued that β can be recovered through maximizing the distance covariance between η T X and Y over η R p. The estimator in Sheng and Yin (03) is desirable as its asymptotic normality has been established under some technical conditions. We provide an alternative method for direction recovery with discrete predictors. Denote var(x) = Σ and assume E(X) = 0 without loss of generality. The classical ordinary least squares (OLS) estimator from the linear regression model is β OLS = Σ E(XY ). A surprising fact revealed by Li and Duan (989) is that β OLS is proportional to the true direction β in model () under the following linear conditional mean (LCM) condition E(X β T X) is a linear function of β T X. () For continuous X, it is well-known that if () holds for all β R, then X has to follow an elliptically-contoured distribution. Variable transformation and elliptical trimming (Cook and Nachtsheim, 994) are thus useful data preprocessing tools for continuous predictors. For discrete predictors, variable transformation and elliptical trimming no longer work, and new direction recovery method is needed when the LCM condition () is violated. This motivates us to consider the OLS estimator based on the central solution space (CSS). Although the CSS estimators have been systematically studied in Li and Dong (009), Dong and Li (00), and Dong and Yu (0), we are the first to study CSS estimators in the presence of exclusively discrete predictors.

3 Direction Estimation in a General Regression Model with Discrete Predictors 3 The rest of the paper is organized as follows. In Section, we examine the OLS estimator with discrete predictors and examine the role of the LCM condition. In Section 3, we demonstrate that the LCM condition is oftentimes violated with discrete predictors, and introduce the central solution space estimator as an extension of OLS. Numerical studies are performed in Section 4 and we conclude the paper with some discussions in Section 5. Two scenarios that OLS works We explore the two scenarios when β OLS can be used to recover the true direction β. In the first scenario, the response Y and the discrete predictor X have a linear relationship. In the second scenario, the discrete predictor X satisfies the LCM condition (). We assume E(X) = 0 unless specified otherwise.. OLS with linear link function In the case with continuous Y, suppose Y and X follow the linear model Y = α + β T X+ε, where α R, β R q, and ε is independent of X. Then obviously we have β OLS = Σ E(XY ) = ασ E(X) + Σ E(XX T )β + Σ E(X)E(ε) = β, where the last equality holds because E(X) = 0 and E(XX T ) = Σ. In the case with discrete Y, β OLS can be a good estimator of β when Y and X satisfy Y = α + β T X, which will be referred to as the no-error linear model. To fix the idea, consider the following two examples. We will see that the response and the predictors exactly follow the no-error linear model in Example, while the no-error linear model is a good approximation in Example. Example. Consider the zoo data from the UCI machine learning repository htt p : //archive.ics.uci.edu/ml/datasets/zoo. This data set contains 0 animals with 6 attributes. Except for the number of legs, all the other attributes (aquatic or not; feathers or not; milk or not; etc.) are binary. The response is the type of the animal, with mammals (4 cases) and birds (0 cases) being the two dominant types. We collapse the other 40 cases into the third category, which includes insects, fish, reptiles, molluscs, and amphibians. The sample OLS estimator has only two coefficients that are significantly different from zero. Not surprisingly, the corresponding attributes are feathers and milk. For this particular data set, all the animals that milk are mammals, and all the animals with feathers are birds. Example. Consider the congressional voting records from the 98th Congress, nd session in 984, which is available at the UCI machine learning repository htt p : //archive.ics.uci.edu/ml/datasets/congressional + Voting + Records. The data set consists of votes from 435 U.S. House of Representatives Congressmen on

4 4 Yuexiao Dong and Zhou Yu 6 subjects, with each vote being one of the three categories: yea, nay, and unknown disposition. The response here is whether the Congressman is democrat or republican. From the sample OLS estimator, we see that the most dominant coefficient corresponds to a bill related to physician fee freeze. It turns out that 45 of 67 democrats voted against this bill, while 63 of 68 republicans voted for the same bill.. OLS with the LCM condition For nonlinear link functions, Li and Duan (989) showed that β OLS is proportional to the true direction β in model () under the LCM condition (). For a,b R q, the Σ-inner product between a and b is a,b Σ = a T Σb. Let P Σ (β) = Σβ(β T Σβ) β T be the projection matrix onto the column space of β under the Σ-inner product. The LCM condition () then implies that E(X β T X) = P Σ (β)x. Under model (), we have E(XY ) = E{XE(Y X)} = E{XE(Y β T X)} = E{E(X β T X)Y }. (3) Plug in E(X β T X) = P Σ (β)x and pre-multiply Σ, we get β OLS = Σ E(XY ) = Σ P Σ (β)e(xy ) = β(β T Σβ) β T E(XY ) β, where means proportional to. REMARK. A related concept is sufficient dimension reduction (SDR) (Cook, 998). SDR aims to find B = {β,...,β d } R q d with the smallest possible column space, such that Y X B T X, (4) where means statistical independence. Correspondingly, the smallest column space is called the central space for the regression between Y and X, and is denoted as S Y X. (4) implies that Y depends on X only through B T X. Under the general regression model (), Y is independent of X given the index θ = β T X, which guarantees the second equality in (3). Model () can thus be viewed as a special case of (4) with the number of indices d =. Using the SDR terminology, the finding in Li and Duan (989) becomes that β OLS S Y X. REMARK. Sheng and Yin (03) proposed direction recovery in model () through distance covariance. They require a technical condition that P Σ (β)x Q Σ (β)x, (5) where Q Σ (β) = I q P Σ (β) with I q being the q q dimensional identity matrix. Note that Y Q Σ (β)x P Σ (β)x under model (). Together with (5), we have {Y,P Σ (β)x} Q Σ (β)x. It follows that

5 Direction Estimation in a General Regression Model with Discrete Predictors 5 E(XY ) = E{P Σ (β)xy } + E{Q Σ (β)xy } = E{P Σ (β)xy } = P Σ (β)e(xy ), where the second equality holds because E{Q Σ (β)xy } = Q Σ (β)e(x)e(y ) = 0. Under the same technical condition (5) as in Sheng and Yin (03), we have shown that β OLS β. Next we present an example with discrete predictors and demonstrate that β OLS β when the LCM condition () is satisfied. The square link function is used for the ease of demonstration. Other types of link functions lead to similar results. We do not require E(X) = 0, and the OLS estimator β OLS = Σ E{(X E(X))Y } is used in this example. Example 3. Consider X = (X,X ) T, β = (,) T, and Y = (β T X). Case i: X Poisson(l ), X Poisson(l ), and X X. It is easy to see X (X + X = m) Binomial(m,ρ) with ρ = l /(l + l ). It follows that E(X X + X ) = ρ(x + X ) is linear in X + X. Similarly E(X X + X ) is linear in X + X. Together the LCM condition () holds. On the other hand, we have l 0 l (l Σ = and E{(X E(X))Y } = + l + ). 0 l l (l + l + ) It follows that β OLS = Σ E{(X E(X))Y } = l + l + l + l + = β. Case ii: X Binomial(n, p), X Binomial(n, p), and X X. Then X (X + X = m) Hypergeometric(n + n,m,n ). It is easy to see that the LCM condition is satisfied. We also have n p 0 n p( p)( p + n Σ = and E{(X E(X))Y } = p + n p). 0 n p n p( p)( p + n p + n p) It follows that β OLS = Σ E{(X E(X))Y } = ( p)( p + n p + n p) ( p)( p + n p + n p) Case iii: (X,X,W) T Multinomial(n,(p, p, p 3 )). We have X (X + X = m) Binomial(m,π) with π = p /(p + p ). = β. Thus the LCM condition is satisfied. The explicit calculation of β OLS shows that it is proportional to β, and the details are provided in the Appendix.

6 6 Yuexiao Dong and Zhou Yu 3 Central solution space with discrete predictors We have seen in Section that the LCM condition plays an important role for β OLS to recover β in model (). We will see that the LCM condition is more often than not violated with discrete predictors. Transformation or trimming, which are popular methods for continuous predictors, are not applicable for discrete predictors. This motivates us to consider the central solution space estimator (Li and Dong, 009), which does not require the LCM condition. 3. Beyond the LCM condition Consider X = (X,X ) T, X Binomial(n =, p ), X Binomial(n = 3, p ), and X X. We are interested in E(X β T X) as a function of β T X for different combinations of p, p and β. The explicit form of E(X β T X) may not be easily derived. Instead, we generate N = 0 6 random samples and calculate the sample conditional mean E N (X β T X) as an approximation of E(X β T X). Specifically, let {X (i),i =,...,N} be an i.i.d. sample of X, where X (i) = (X (i) (i),x )T. Suppose Ω is the support of β T X. For any ω Ω, let I(β T X = ω) be the indicator function of β T X = ω. Then E(X β T X = ω) can be approximated by E N (X β T X = ω) = E N{X I(β T X = ω)} E N {I(β T X = ω)} = N i= X (i) I(β T X (i) = ω) N i= I(β T X (i) = ω) We plot E N (X β T X) versus β T X in Figure. The linear, quadratic and cubic fit between E N (X β T X) and β T X are also included. The upper left panel confirms the finding in case ii of Example 3. Namely, E(X β T X) is a linear function of β T X and the LCM condition is satisfied when p = p and β = (,) T. In the first row of Figure, we fix β = (,) T and modify p and p. It is clear that E(X β T X) is a nonlinear function of β T X when p p. In the second row of Figure, we fix p = p =. and modify β. We see that the LCM condition will not hold when β is not proportional to (,) T. Similar results hold true for cases i and iii of Example 3, which are not reported here. We see from Figure that the LCM condition could easily be violated with discrete predictors. The next example demonstrates that without the LCM condition, β OLS may not be able to recover β. Example 4. Let X = (X,X ) T, β = (,) T, and Y = (β T X) 3. Suppose X Bernoulli(p ), X Bernoulli(p ), and X X. Similar to Figure, one can check that the LCM condition is violated if p p. On the other hand, we have ( p 0 p p Σ = and E{(X E(X))Y } = + 6p p 6p p ) 0 p p p + 6p p 6p p..

7 Direction Estimation in a General Regression Model with Discrete Predictors 7 p =., p =., β T =(,) p =., p =.5, β T =(,) p =., p =.9, β T =(,) E N(X X + X) linear quadratic cubic E N(X X + X) linear quadratic cubic E N(X X + X) linear quadratic cubic X + X X + X X + X p =., p =., β T =(,) p =., p =., β T =(,) p =., p =., β T =(5,) E N(X X + X) linear quadratic cubic E N(X X + X) linear quadratic cubic E N(X 5X + X) linear quadratic cubic X + X X + X 5X + X Fig. E N (X β T X) v.s. β T X. N = 0 6, X Binomial (, p ), X Binomial (3, p ), and X independent of X. It follows that β OLS = Σ p + 6p E{(X E(X))Y } = 6p p, p + 6p 6p p which is not proportional to β = (,) T unless p = p. 3. The central solution space estimator The OLS-based central solution space (CSS) estimator was first proposed in Li and Dong (009), which does not require the LCM condition. Although Li and Dong (009) focused on CSS with continuous predictors, the CSS method can be applied to discrete predictors as well. Suppose E(X) = 0. Under model (), we observe the following equality E(XY ) = E{E(X β T X)Y }. (6)

8 8 Yuexiao Dong and Zhou Yu Denote the solution of equation (6) as β CSS. The following result is a special case of Theorem 4. in Li and Dong (009). Proposition. Suppose Pr(E{E(X η T X)Y } = E{E(X β T X)Y }) > 0 whenever η is not proportional to β. Then under model (), the solution of equation (6) is unique up to a scalar multiplication, and satisfies β CSS β. Furthermore, if the LCM condition () holds, then β CSS β OLS. The first part of Proposition states that one can use β CSS to recover β without the LCM condition (). The second part reveals that β CSS coincides with β OLS when the LCM condition is true. Ichimura (993) provided an example that the parameters in the index β are unidentifiable when the predictors are all discrete. Our assumption in Proposition eliminates such pathological situations, and guarantees that the CSS estimator is unique even with exclusively discrete predictors. For a = (a,...,a q ) T R q, let a = q i= a i. To find the solution of equation (6), Li and Dong (009) suggested solving the following optimization problem minimize L(η) over η R q, where L(η) = E{XY E(X η T X)Y }. (7) Denote the minimizer of L(η) as η 0. As long as all the moments involved exist, Theorem 4. in Li and Dong (009) states that η 0 is proportional to β CSS under the same condition as in Proposition. The sample level CSS estimator is based on minimizing L N (η), the sample objective function of L(η) in (7). Motivated from the observations in Section 3., we model E(X η T X) as a polynomial function of η T X. As we have seen in Figure, for discrete predictors, using quadratic and cubic functions of η T X may be more suitable than using linear functions of η T X. For any function r(x,y ) and an i.i.d. sample {(X (i),y (i) ),i =,...,N}, we denote the sample estimator of E{r(X,Y )} by E N {r(x,y )} = N N i= r(x(i),y (i) ). The step-by-step algorithm is as follows.. Center Y (i) and X (i) as Ŷ (i) = Y (i) E N (Y ), ˆX (i) = X (i) E N (X).. Set k = or k = 3. Denote G(η T ˆX (i) ) as {,η T ˆX (i),...,(η T ˆX (i) ) k } T. The sample estimator of E( ˆX η T ˆX) becomes E N ( ˆX η T ˆX) = E N { ˆXG T (η T ˆX)} ( E N {G(η T ˆX)G T (η T ˆX)} ) G(η T ˆX). 3. Estimate β by solving the following optimization problem minimize L N (η) over η R q, where L N (η) = E N { ˆXŶ E N ( ˆX η T ˆX)Ŷ }, where the minimizer is denoted as ˆβ (k) CSS. A Newton-Raphson type estimator can be implemented to find the minimizer in step 3 of the above algorithm. See, for example, Cook and Li (009), Dong and Zhu (0).

9 Direction Estimation in a General Regression Model with Discrete Predictors 9 4 Numerical studies In this section, we demonstrate the effectiveness of CSS estimator with discrete predictors via simulation studies. For X = (X,...,X q ) T and β R q, consider three settings of β T X as follows, Case (i): β = (,,,0,...,0) T,X Binomial(,.),X Binomial(3,.9), X 3,...,X q Bernoulli(.5), and X i X j for i j. Case (ii): β = (,,3,0,...,0) T,X Poisson(),X Poisson(.5), X 3,...,X q Bernoulli(.5), and X i X j for i j. Case (iii): β = (,0,,0,...,0) T,(X,X,W) T Multinomial(4,(.,.4,.4)), X 3,...,X q Bernoulli(.5), and X i X j for i j except for X and X. For each combination of β and X, consider the following models Model I: Y =.exp(β T X + ) +.ε, Model II: Y =.(β T X) 3 +.ε, { if β T X 0, Model III: Y = 0 if β T X < 0. In Model I and II, we have continuous response with error ε N(0,) and ε X. The response in Model III is discrete. All three models are special cases of the general regression model (). Denote Σ N as the sample covariance matrix of X. Then the classical ordinary least squares estimator is ˆβ OLS = Σ N E N{(X E N (X))Y }. Two () (3) CSS estimators ˆβ CSS and ˆβ CSS are also included for the comparison, where we estimate E(X β T X) as a quadratic or a cubic function of β T X. Let R ( ˆβ,β) be the squared sample Pearson correlation between β T X and ˆβ T X, which will be used to measure the performances of different estimators. We fix predictor dimension q = 0, sample size N = 00, and report the simulation results in Table based on 00 repetitions. Each entry of Table is formatted as a(b), where a is the average of R ( ˆβ,β) across the 00 repetitions, and b is the standard error of the average. We see that settings, while ˆβ () CSS outperforms ˆβ OLS in six out of nine ˆβ (3) CSS consistently outperforms ˆβ OLS in all nine settings. ˆβ (3) CSS enjoys the best overall performances, as it is better than CSS in all but one setting. This simulation study confirms our intuition in Section 3. Namely, when the LCM condition is violated and E(X β T X) can not be modeled by linear functions of β T X, the CCS estimators can improve the OLS estimator as long as we can model E(X β T X) properly. ˆβ ()

10 0 Yuexiao Dong and Zhou Yu Table Comparison of OLS and CSS estimators based on 00 repetitions. Case Model R ( ˆβ OLS,β ) R ( ˆβ () CSS,β) R ( ˆβ (3) CSS,β) I.96 (.003).956 (.00).97 (.008) (i) II.964 (.00).9984 (.000).9994 (.000) III.8038 (.0084).7976 (.044).8700 (.0099) I.97 (.005).9547 (.000).9803 (.003) (ii) II.8997 (.0035).980 (.0034).9430 (.0040) III.9593 (.000).9543 (.00).9803 (.0009) I.960 (.008).9855 (.000).9977 (.0004) (iii) II.9389 (.003).9647 (.005).9880 (.0008) III.90 (.009).9846 (.0009).9794 (.005) 5 Discussion To recover the direction β in the general regression model () with discrete predictors, we examine the OLS and the CSS-based OLS estimators in this paper. In the case when the link function is linear, or when the predictor distribution satisfies the LCM condition, OLS can recover the true direction up to a multiplicative scalar. If the link function is nonlinear and the LCM condition is not satisfied, OLS is no longer suitable. By relaxing the LCM condition and modeling E(X β T X) as polynomial functions of β T X, the CSS-based estimators improve OLS and lead to better accuracy at recovering β. Our development addresses a gap in the existing CSS literature, which mainly focuses on continuous predictors (Li and Dong, 009; Dong and Li,00; Dong and Yu, 0). In the case of continuous predictors, Theorem 6. in Li and Dong (009) provides the consistency results of the CSS estimators. From Figure, it is obvious that E(X β T X) can not be consistently estimated by polynomial functions of β T X. Finding consistent CSS estimators with discrete predictors is a topic worth future investigation. Appendix Proof of β OLS β in Example 3, case iii. For (X,X,W) T Multinomial (n,(p, p, p 3 )), it is well-known that X i Binomial(n, p i ) for i =,, and cov(x,x ) = np p. It follows that E(X X ) = n(n )p p. It can be shown that E(X 3 ) = np + 3n(n )p + n(n )(n )p 3. Next we denote k 3 = n k k and calculate E(X X ) as follows

11 Direction Estimation in a General Regression Model with Discrete Predictors E(X X ) = k n! k k,k k!k!k 3! pk pk pk 3 3 (n )! = n(n )p p (k + ) k,k (k )!(k )!k 3! pk p k p k 3 3 = n(n )(n )p (n 3)! p k,k (k )!(k )!k 3! pk p k p k 3 3 (n )! + n(n )p p k,k (k )!(k )!k 3! pk p k p k 3 3 = n(n )p p ( + np p ). For Y = (X + X ), we thus have E{(X E(X ))Y } = E(X 3 ) E(X )E(X ) + E(X X ) + E(X X ) E(X )E(X ) + E(X X ) E(X )E(X X ) = np { + (n 3)p (n )p + p (n 3 (n )p (4n 4)p )}. Similarly, E{(X E(X ))Y } is equal to np { + (n 3)p (n )p + p (n 3 (n )p (4n 4)p )}. Recall that X = (X,X ) T and var(x) = Σ. Let Σ be the determinant of Σ. Since the first row of Σ is Σ {np ( p ),np p }, the first component of β OLS becomes Σ [np ( p )E{(X E(X ))Y } + np p E{(X E(X ))Y }] = Σ n p p { + (n 3)(p + p ) (n )(p + p )}. Due to the symmetry between p and p in the expression above, the second component of β OLS is exactly the same as the first component. Thus we have β OLS (,) T = β. PROOF OF PROPOSITION. Assume E{E(X η T X)Y } = E{E(X β T X)Y } with probability for some η, such that η is not proportional to β. Then both η and β will satisfy equation (6), which means the solution of (6) is not unique up to a s- calar multiplication. Thus under the assumption that Pr(E{E(X η T X)Y } = E{E(X β T X)Y }) > 0 whenever η is not proportional to β, the solution of (6) is unique. Because β satisfies (6) and the solution of (6) is unique up to a scalar multiplication, we have β CSS β. Under the additional LCM condition (), β OLS is also proportional to β. Consequently we have β CSS β OLS.

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