Appendix S1: Description of Algorithms A.1 and A.2. Algorithm A.1. An E-M type algorithm for constrained estimation in linear mixed models As noted earlier in this article, our testing procedure is flexible as one could any constrained estimator (e.g. RMLE). Along the lines of [1] we propose a simple E-M type algorithm to obtain a constrained estimator as described below. If the unconstrained estimator of is either exactly or asymptotically multivariate normally distributed then following arguments in [2], under some suitable conditions, the resulting constrained estimator of A would converge to the restricted maximum likelihood estimator. Thus, in the following we do not assume that the underlying data are normally distributed. However, since we are using the MINQUE for the variance components, the unconstrained estimator of (under suitable regularity conditions) is asymptotically normally distributed. Let ˆ1( m) , ˆ2( m) , ˆ 2( m ) , ˆ 2( m ) denote the mth iterate estimates of 1 , 2 , 2 , and 2 , respectively. 1 Step 0 (m = 0). Let X X1 : X2 , 1, 2 . Compute (0) XX XY , the ordinary least squares estimator for . Compute i2(0) 1 Yi Xi (0) ni 2 for i 1, , k . For ˆi2(0) we use MINQUE [3]. Step 1. Set m = m + 1. Fix 1 , 2 , and 2 at ˆ1( m1) , ˆ2( m1) , ˆ 2( m1) respectively, and iteratively estimate 2 (for m = 1, 2, …): 1 4( m 1) ˆ i ni ˆ i2( m ) i2( m 1) ˆ ( m 1) tr Ψ ˆ ( m 1) Ψ Y X ˆ 1 1 ( m 1) 1 1 , i 1, ii X 2ˆ2( m 1) Y X1ˆ1( m 1) X 2ˆ2( m 1) Ψˆ ( m 1) 1 (A1) , k, ˆ ( m1) UT ˆ ( m1) U Σ ˆ , Τˆ ( m1) diag ˆ2( m1) I : where Ψ 1 c1 : ˆq2( m1) Icq , ˆ 2(0) 2( m1) , and Aii indicates the (i, i)th block of A. Step 2. (a) Fix 2 at ˆ 2( m1) and iteratively estimate 1 , 2 , and 2 using the following equations (for m = 1, 2, …): ˆ1( m) ˆ1( m1) X1 Σˆ 1X1 ˆ2( m) ˆ2( m 1) X2 Σˆ 1X2 ˆi2( m ) ˆi2( m1) ˆi4( m1) ci Y X1ˆ1( m 1) X2ˆ2( m 1) where ˆ(0) ( m1) , diag 12( m1)Ic1 : ˆ ( m 1) tr U i Ψ 1 1 Y X ˆ X2ˆ2( m1) , (A2) Y X ˆ X2ˆ2( m 1) , (A3) ˆ ( m 1) X2 Ψ 1 1 ( m 1) 1 1 Ψˆ 1 ( m 1) ( m 1) 1 1 1 Y X ˆ ˆ ( m 1) Ψ ˆ 2(0) 2( m1) , ˆ ( m1) X1 Ψ 1 ( m 1) 1 1 X2ˆ2( m 1) U , i 1, i ˆ ( m1) UT ˆ ( m1) U Σ ˆ, Ψ (A4) , q, and ˆ ( m1) Τ is : q2( m1)Icq . Note that although we could have combined equations (A2) and (A3) and use ( XΣ1X) XΣ1Y to obtain a combined updated estimate for ˆ( m ) ˆ1( m) : ˆ2( m) , our proposed approach is computationally more stable and simpler. (b) Using the estimates obtained in (a), we apply the PAVA type methodology along the lines of [4] with weights proportional to the inverse of the sample sizes to obtain ˆ1( m) under the desired inequality constraints. Alternatively, one could perform the following constrained optimization to obtain ˆ1( m) : ˆ -1(i ) (Y - X i X ˆi ) . min(Y - X11i X2ˆ2i ) ' Ψ 1 1 2 2 A 0 (A5) Steps 1 and 2 are iterated until convergence. The resulting constrained estimators are denoted by ˆ ˆ1,ˆ2 . Since it is well-known that the RMLE can perform poorly, even for simple order restriction when the data are correlated (see [4]), therefore as an alternative to (A5), we consider PAVA type methodology of [4]. Thus the point estimators derived here are very generally applicable for any order restriction. In Step 2(b), if one uses (A5) for obtaining a constrained 2 estimator then applying the general theory established in [5], we note that ˆ ˆ1,ˆ2 is consistent. To describe the MINQUE methodology for estimating 12 , 22 ,..., q2 , 12 , 22 ,..., k2 we rewrite the linear model (1) as Y X , where 1, 2 , X X1 : X2 with qk E ( ) 0 , Var ( ) G i Fi , Fi Ui Ui , i 1, 2, ..., q, and Fi Diag[0 : 0 :...: I : 0...0] , i 1 i q 1, 2, ..., q k with the identity matrix of order ni q ni q located at the ith location. Each Fi is N N . Let F F1 : F2 :...: Fq k , W G XX , R W W X(X ' W X) X ' W , z (Y R F1R Y , Y R F2 R Y ,..., Y R Fq k R Y ) , and S Tr (R Fi R F j ), i, j 1, 2,..., q k . In the above expression, A denotes a generalized inverse (or g-inverse) of A . The above expressions are invariant to the choice of g-inverse. Hence without loss of generality one may use the Moore-Penrose inverse A . From [3], the MINQUE of is then obtained by solving the system of linear equations S ( 0) z ( 0) , where (0) denotes an initial estimate of . Since the MINQUE depends upon the initial estimate, Rao and Kleffe [6] recommend iterating as above until convergence. The resulting estimator is known as the iterated MINQUE (or IMINQUE). Denote the I-MINQUE of by ˆ . As discussed in [7], estimated parameters can be negative. As is commonly done, in such cases we replace them with 0.01. Since A is an estimable linear function of , its weighted least squares estimator is given by ˆ A( XWˆ X) XWˆY . Let R1: 0 min 1 , 2 , R2: E R3: 4 , q k min 1 , 2 , , q k . . d Tr W F W F Tr W Fi W Fj i ij , where the matrix D dij , i, j 1, 2, i 3 , q k is non-singular. R4: max W Fi max W Fj Tr W Fi W Fi 0 , where max H is the largest eigenvalue of a matrix H . R5: ( X' W X) M ( ) , where M ( ) is a positive definite matrix. R6: max (XXWG W XX) 0 . Theorem A1: For any estimable linear function A in a linear mixed model (1) satisfying the regularity conditions R1 to R6, A(θˆ - θ ) asymptotically ~ N (0, AM( ) -1 A' ) Proof: Under the regularity conditions R1 to R4, from Theorem 10.2.3 in [6] we deduce that the MINQUE ˆ is consistent for . Appealing to Noether’s conditions (R5 and R6) we deduce the asymptotic normality of Aˆ from the discussion in Chapter 10.7 in [6]. Algorithm A.2. The EBLUP based bootstrap methodology To derive the p-values of the above test statistics, we now describe the non-parametric EBLUP bootstrap methodology for deriving the null distribution of the test statistic (9). We begin by constructing bootstrap sample Y * as follows. Step 1: Obtain the point estimator of 1, 2 under the null hypothesis. Denote it by 0 10 , 20 . Step 2: Let θˆ ˆ1,ˆ2 , T̂ , and Σ̂ denote the unconstrained point estimators of 1, 2 , T and Σ . Compute ˆ UT ˆ UTU ˆ 1X) 1 XΨ ˆ 1 , ˆ, Ψ ˆ Σ ˆ , Pˆ X( XΨ C ˆ ˆ 1 Ψ (I Pˆ )Y . ˆ (I Pˆ )Y , ˆ ˆ1 : ˆ2 :...: ˆq C Step 3: Let ˆi ˆi ˆi , i 1, 2,..., k where sd (.) represents the , i 1, 2,..., q and let ˆi ˆ sd (ˆi ) sd (i ) 4 usual sample standard deviation of the elements in the vector. Step 4: Let ˆi* , i 1, 2,..., q , denote a random vector obtained by taking a random sample (with replacement) of size ci 1, i 1, 2, i 1, 2, , q from the components of ˆi . Similarly, letˆi* , , k denote a random vector obtained by taking a random sample (with replacement) of size ni 1 from the components of ˆi . Finally, let ˆ* ˆiˆi* , i 1, 2, i 1, 2, , q and ˆi* ˆ iˆi* , , k , then the EBLUP bootstrap sample Y * is constructed using the following equation Y * X 0 Uˆ* ˆ* . The above model honors the null hypothesis regarding the parameter 1, 2 as well as honors the underlying variance components structure. REFERENCES 1 Hoferkamp CL, Peddada SD (2002). Parameter estimation in linear models with heteroscedastic variances subject to order restrictions. Journal of Multivariate Analysis 82: 6587. 2 Davidov O, Rosen S (2011). Constrained inference in mixed-effects models for longitudinal data with application to hearing loss. Biostatistics 12: 327-340. 3 Rao CR (1972). Estimation of variance and covariance components in linear models. Journal of the American Statistical Association 67: 112-115. 4 Hwang JTG and Peddada SD (1994). Confidence Interval Estimation Subject to Order Restrictions. Annals of Statistics, 22: 67-93. 5 Nettleton, D (1999). Convergence properties of the em algorithm in constrained parameter spaces. The Canadian Journal of Statistics / La Revue Canadienne de Statistique 27: 639-648. 6 Rao CR, Kleffe J (1988). Estimation of variance components and applications. Amsterdam; New York; New York, N.Y., U.S.A.: North-Holland. 7 Rao JNK, Subrahmaniam K (1971). Combining independent estimators and estimation in linear regression with unequal variances. Biometrics 27: 971-990. 5