var(β 1 ) For ”Reference Level” Coding - var(β 1 ) Under AN Unbalanced Design

D. var(β 1 ) Under AN Unbalanced Design

2. var(β 1 ) For ”Reference Level” Coding

No matter how the independent variables are coded, the multilevel generalized linear model’s linearization is the same as (D.13). In this section, I will derive the variance of race which is coded as 1 for black and 0 for white veterans.The same unbalance design scenarios as discussed in Appendix D.D.2will be considered, here.

a. In RI Model

Adapting the general linearized 2-level logistic model (D.13) to the 2 level RI model, adjusted by one binary variable and coded as 1 in black and 0 for white race, yields the following results:

y^∗_ij = β0+ xijβ1+ µ0j+ ε^∗_ij (D.102)

To identify the black effect, I can rewrite the model (D.14) as the race specific models

y^∗_ijA = β0+ β1+ µ0j+ ε^∗_ij for black

y_ijW^∗ = β0+ µ0j+ ε^∗_ij for white (D.103)

Therefore, the average linearized race specific response at jth site are shown as (D.104) and (D.105) for black and white, respectively.

¯ y^∗_.jB =

PnjB

i=1(β0+ β1+ µ0j+ ε^∗_ijB) njB

for black

Pn_jB

i=1(β0) +Pn_jB

i=1(β1) +Pn_jB

i=1(µ0j) +Pn_jB i=1(ε^∗_ijB) njB

= njBβ0+ njBβ1+ njBµ0j +Pn_jB i=1(ε^∗_ijB) njB

= β0+ β1+ µ0j+ ¯ε^∗_.jB (Eε^∗_.jB = 0)

= β0+ β1+ µ0j

(D.104)

Similarly, we can get an average outcome as shown below.

Therefore, the GLS estimate of race can be obtained by

y_..B^∗ − ¯y..W^∗ = β0+ β1− β⁰ = β1 (D.108)

By applying the corresponding f(H) to (D.12) and (D.11), the modified y^∗_ij and ε^∗_ij results in y^∗_ij = (yij− ˜πij)[˜πij(1 − ˜πij)]⁻¹+ β0+ xijβ1

ε^∗_ij = εij[˜πij(1 − ˜πij)]⁻¹ (D.109)

Then, applying Vj=σ_µ²₀ to var(Y^∗_ij) as (D.17) for RI model yields

var(Y^∗_ij) = σ_µ²₀ + [˜πij(1 − ˜π^ij)]⁻¹ (D.110) Based on the linearized 2-level logistic model (D.13) and by applying the GLS estimate of β1 as (D.108), I get

I have presented yijB and yijW as (D.103). Plugging in (D.111), I will get

βˆ1 =

β0 and β1 will be constant everywhere.

βˆ1 = β0+ β1+ µ0+

As known, var(ε^∗_ijB) or var(ε^∗_ijW) are the same as σ_ε^2∗_ijB and σ_ε^2∗_ijW in different cells. I can write the above equation as

var( ˆβ1) = (PN

2: The related sample size from (D.21) shows as n1 = n2 = · · · = n^j = n, and PN

For condition Bw¯: The related sample size from (D.21) shows as

I can write (D.113) as

var( ˆβ1) = σ_ε^2∗_ijB

For condition Bwj: The related sample size from (D.20) shows as

I can write (D.114) as

var( ˆβ1) = σ^2∗_ε_ijB

2: The related sample size from (D.20) shows as n1 = n2 = · · · = n^j = n, and PN

The related sample size from (D.20) shows as

Equation (D.113) can be written as

The related sample size from (D.20) shows as

b. In RC Model

Adapting the general linearized 2-level logistic model (D.13) to the 2 level RC model, yields the following rsults: adjusted by race, and coded as 1 for black and 0 fir white race.

y^∗_ij = β0+ xijβ1 + µ0j+ µ1j+ ε^∗_ij (D.124)

To identify the fixed effect of black race, I can rewrite the model (D.14) as the race specific models

y_ijA^∗ = β0+ β1 + µ0j+ µ1j+ ε^∗_ij for black

y^∗_ijW = β0+ µ0j+ ε^∗_ij for white (D.125)

Therefore, the average linearized race specific responses at jth site are shown as (D.126) and (D.127) for black and white, respectively.

Similarly, the average outcome yields

Therefore, the GLS estimate of race can be obtained by

y_..B^∗ − ¯y..W^∗ = β0+ β1− β⁰ = β1 (D.130) By applying corresponding f(H) to (D.12) and (D.11), the modified y^∗_ij and ε^∗_ij result in

y^∗_ij = (yij− ˜πij)[˜πij(1 − ˜πij)]⁻¹+ β0+ xijβ1

ε^∗_ij = εij[˜πij(1 − ˜πij)]⁻¹ (D.131)

Then, applying Vj=σ_µ²₀ to var(Y^∗_ij) as (D.17) for the RI model yields

var(Y^∗_ij) = σ_µ²₀ + σ²_µ₁ + [˜πij(1 − ˜π^ij)]⁻¹ (D.132)

Based on the linearized 2-level logistic model (D.13) and by applying the GLS estimate of

I have presented yijB and yijW as (D.125). Plugging in (D.133), I will get

βˆ1 =

β0 and β1 will be constant everywhere. µ0j will be constant at jth hospital.

βˆ1 = β0+ β1+

For condition B¹

2: The related sample size from (D.21) shows as n1 = n2 = · · · = n^j = n, and PN

For condition Bw¯: The related sample size from (D.20) shows as

I can write (D.136) as

var( ˆβ1) = σ_µ²₁

For condition Bwj: The related sample size from (D.20) shows as

I can write (D.136) as

2: The related sample size from (D.20) shows as n1 = n2 = · · · = n^j = n, and PN

The related sample size from (D.20) shows as

Equation (D.136) can be written as

The related sample size from (D.20) shows as

APPENDIX E

IMAGE OF MLWIN RESULTS USING MLWIN VERSION 2.02

Figure E1: RI Model for Pneumonia Patient Younger than 65, Using RIGLS PQL2