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Introduction Log-normal Regression with Random Effect Upper Tolerance Limits Final Remarks A generalized confidence interval for the mean response in log-regression models Miguel Fonseca Thomas Mathew Jo˜ ao Tiago Mexia CompStat’2010 Paris, France M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Upper Tolerance Limits Final Remarks Introduction 1 Log-normal Regression with Random Effect 2 Method 1 Method 2 Upper Tolerance Limits 3 Final Remarks 4 M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Upper Tolerance Limits Final Remarks Model � ′ � y = (log( w 1 ) , . . . , log( w n ) y = X β + Z τ + e X and Z are known design matrices of dimensions n × p and n × s τ ∼ N ( 0 , σ 2 τ I s ) e ∼ N ( 0 , σ 2 τ I n ) τ ZZ ′ + σ 2 y ∼ N ( X β , σ 2 e I n ) M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Upper Tolerance Limits Final Remarks Problem Y 0 ∼ N ( x ′ 0 β , σ 2 τ z ′ 0 z 0 + σ 2 e ) . The mean of W 0 is then given by 0 β , σ 2 τ z ′ 0 z 0 + σ 2 � � x ′ e E ( W 0 ) = E (exp( Y 0 )) = exp . 2 Thus the interval estimation of E ( W 0 ) is equivalent to the interval estimation of 0 β + σ 2 τ z ′ 0 z 0 + σ 2 θ = x ′ e . 2 M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Generalized Pivotal Quantities G ( y , y obs ; θ, η ) (i) given the observed value y obs , the distribution of G ( y , y obs ; θ, η ) is free of unknown parameters, (ii) the observed value of G ( y , y obs ; θ, η ) is free of the nuisance parameter η . M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Generalized Pivotal Quantities G ( y , y obs ; θ, η ) (i) given the observed value y obs , the distribution of G ( y , y obs ; θ, η ) is free of unknown parameters, (ii) the observed value of G ( y , y obs ; θ, η ) is free of the nuisance parameter η . When the above conditions hold, G 1 − α = { θ : G ( y obs , y obs ; θ, η ) ≤ G 1 − α } is a 100(1 − α )% one-sided generalized confidence interval for θ . A two-sided interval can be similarly defined. M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Generalized Pivotal Quantity τ × z ′ 0 β + G σ 2 0 z 0 + G σ 2 G θ = G x ′ e 2 0 β → x ′ G x ′ 0 β τ → σ 2 G σ 2 τ e → σ 2 G σ 2 e M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Parameters σ 2 e e = σ 2 ss e = ss e e G σ 2 U 2 SS e e U 2 e = SS e e ∼ χ 2 σ 2 n − r SS e = y ′ � � I n − P ( X , Z ) y P ( X , Z ) = ( X , Z ) [( X , Z ) ′ ( X , Z )] − ( X , Z ) ′ = QQ ′ M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Parameters σ 2 τ τ Q ′ ZZ ′ Q + G σ 2 V G = G σ 2 e I r QQ ′ = P ( X , Z ) , Q ′ Q = I r U 2 h V − 1 − V − 1 Q ′ X ( X ′ QV − 1 Q ′ X ) − 1 X ′ QV − 1 i 0 = y ′ Q ′ y obs obs Q G G G G U 2 0 ∼ χ 2 r − p x ′ 0 β 0 ( X ′ QV − 1 Q ′ X ) − 1 X ′ QV − 1 0 β = x ′ Q ′ y obs G x ′ G G 0 ˆ x ′ β V − x ′ 0 β rh 0 ( X ′ QV − 1 i Q ′ X ) − 1 x 0 x ′ − × G q + 0 ( X ′ QV − 1 Q ′ X ) − 1 x 0 x ′ rh i 0 ( X ′ QV − 1 Q ′ X ) − 1 X ′ QV − 1 0 ( X ′ QV − 1 = x ′ Q ′ y obs − Z Q ′ X ) − 1 x 0 x ′ G G G + Z ∼ N (0 , 1) M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Application The data were obtained from 34 licensed rural nursing facilities and 18 urban nursing facilities in the State of New Mexico. Y ij = β 0 + β 1 x 1 ij + β 2 x 2 ij + τ i + e ij x 1 number of beds x 2 medical in-patient days τ i Rural/non-rural W total patient-care revenue x 0 = (1 , 0 . 8368 , 1 . 8476) ′ → 90% confidence interval: [9 . 3241 , 9 . 5419] M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Simulations β = (1 , 1 , 1) ′ , (2 , 2 , 2) ′ , (3 , 3 , 3) ′ σ 2 τ = 0 . 1 , 0 . 25 , 0 . 5 , 1 , 2 , 5 σ 2 e = 1 1000 runs were performed, each with a pseudo-sample of size 1000. The confidence was 90%. Table: Coverage Probability β \ σ 2 0.1 0.25 0.5 1 2 5 1 (1 , 1 , 1) ′ 0.953 0.905 0.888 0.835 0.837 0.619 (2 , 2 , 2) ′ 0.952 0.922 0.877 0.836 0.822 0.630 (3 , 3 , 3) ′ 0.943 0.919 0.878 0.829 0.831 0.633 M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Simulations β = (1 , 1 , 1) ′ , (2 , 2 , 2) ′ , (3 , 3 , 3) ′ σ 2 τ = 0 . 1 , 0 . 25 , 0 . 5 , 1 , 2 , 5 σ 2 e = 1 1000 runs were performed, each with a pseudo-sample of size 1000. The confidence was 90%. Table: Average Length β \ σ 2 0.1 0.25 0.5 1 2 5 1 (1 , 1 , 1) ′ 2.306 2.584 2.891 3.160 3.487 3.733 (2 , 2 , 2) ′ 2.309 2.610 2.844 3.179 3.441 3.741 (3 , 3 , 3) ′ 2.338 2.622 2.847 3.146 3.480 3.765 M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Restricted Model Let P X = A X A ′ X be the orthogonal projection matrix (OPM) on ′ the OPM on the orthogonal complement R ( X ) an I − P = A o X A o X of R ( X ). Then ′ y ∼ N ′ ZZ ′ A o y 0 = A o 0 , σ 2 τ A o X + σ 2 � � e I X X ′ ZZ ′ A o Matrices σ 2 1 A o X and σ 2 e I span a commutative Jordan X algebra (CJA) with principal basis { Q 1 , . . . , Q w , Q w +1 } , where Q w +1 = I − � w j =1 Q j . Then, w +1 σ 2 ′ ZZ ′ A o X + σ 2 � 1 A o e I = c j Q j , X j =1 with c j = λ j σ 2 τ + σ 2 e for i = 1 , . . . , w and c w +1 . M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Generalized Pivotal Quantities σ 2 τ and σ 2 e 0 Q j y 0 ∼ c j χ 2 S j = y ′ g j c j = S j GPQ – ˙ U j e ) ′ = F + ˙ σ 2 σ 2 GPQs – ( ˙ τ , ˙ c , U j ∼ χ 2 g j x ′ 0 β τ ZZ ′ + ˙ ˙ σ 2 σ 2 V = ˙ e I �� � − 1 x 0 � X ′ ˙ 0 ˆ � V − 1 X G x ′ 0 β = x ′ β − Z x ′ 0 + Z ∼ N (0 , 1) M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Application The data were obtained from 34 licensed rural nursing facilities and 18 urban nursing facilities in the State of New Mexico. Y ij = β 0 + β 1 x 1 ij + β 2 x 2 ij + τ i + e ij x 1 number of beds x 2 medical in-patient days τ i Rural/non-rural W total patient-care revenue x 0 = (1 , 0 . 8368 , 1 . 8476) ′ → 90% confidence interval: [9 . 2527 , 9 . 9517] M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models

Introduction Log-normal Regression with Random Effect Method 1 Upper Tolerance Limits Method 2 Final Remarks Simulations β = (1 , 1 , 1) ′ , (2 , 2 , 2) ′ , (3 , 3 , 3) ′ σ 2 τ = 0 . 1 , 0 . 25 , 0 . 5 , 1 , 2 , 5 σ 2 e = 1 1000 runs were performed, each with a pseudo-sample of size 1000. The confidence was 90%. Table: Coverage Probability β \ σ 2 0.1 0.25 0.5 1 2 5 1 (1 , 1 , 1) ′ 0.909 0.893 0.887 0.887 0.887 0.898 (2 , 2 , 2) ′ 0.913 0.895 0.902 0.875 0.895 0.905 (3 , 3 , 3) ′ 0.907 0.913 0.884 0.877 0.885 0.886 M. Fonseca, T. Mathew, J.T. Mexia A generalized CI for the mean response in log-regression models