Inference for the marginal model

Chapter 1 Introduction

3.7 Inference for the marginal model

Inference for the fixed effects βcan be based on the Wald test, t-test,F-test, robust inference or the likelihood ratio (LR) test. Inference for the variance components is based on the Wald test and the LR test. The information cri- teria can generally be useful for making inference about the marginal model.

The estimate ofβ is β(α) = (ˆ

n

X

i=1

X_i⁰W_iX_i)⁻¹

n

X

i=1

X_i⁰W_iy_i (3.13) withαbeing replaced by its ML or REML estimate according Harville (1974) and Laird and Ware (1982). Conditional on α, ˆβ(α) is multivariate normal

with mean β and covariance Var( ˆβ) = (

n

X

i=1

X_i⁰W_iX_i)⁻¹(

n

X

i=1

X_i⁰W_i(VarY_i)W_iX_i)(

n

X

i=1

X_i⁰W_iX_i)⁻¹

=

n

X

i=1

X_i⁰W_iX_i

!−1

(3.14) provided that W_i =V_i⁻¹ where V_i = Var(Y_i) =Z_iGZ_i+ Σ_i.

3.7.1 Approximate Wald test

LetLbe a known contrast or transformation matrix and consider testing the hypothesis

H₀ :Lβ=0 versus

H_A :Lβ 6=0 (3.15)

Then the Wald test statistic is given by W_T =βˆ⁰L⁰[L

n

X

i=1

X_i⁰V_i⁻¹( ˆα)X_i

!

L⁰]⁻¹Lβˆ

The asymptotic sum distribution ofW_T is chi-square distributed with rank(L) degrees of freedom. Thus using the statisticW_T inference on fixed effects can be made via the transformation Lβ.

3.7.2 Approximate t-test and F-test

It should be noted that the Wald test is based on var( ˆβ) =

n

X

i=1

X_i⁰Wi(α)Xi

!⁻¹

The deficiency with the Wald test statistic is that the variability introduced by replacingα by some estimate (ML or REML) is not taken into account in

the subsequent test. Therefore Wald tests will only provide valid inferences in sufficiently large samples. In practice, this is often resolved by replacing the χ² distribution by an appropriate F distribution. Thus to test the hypothesis H₀ versus H_A in Eq. (3.14), the above statistic becomes

F_T =

βˆ⁰L⁰[L Pn

i=1X_i⁰V_i⁻¹( ˆα)Xi

L⁰]⁻¹Lβˆ rank(L)

The approximate null distribution of F_T is F with numerator degrees of freedom equal to rank(L). The denominator degrees of freedom have to be estimated from the data using common methods such as the containment method, the Sattherwaite approximation and the Kenward and Roger ap- proximation. In the context of longitudinal data, all methods typically lead to large degrees of freedom, and therefore also very similar p-values. For univariate hypotheses, rank(L)=1 and in this case the F-test is equivalent reduces to a t-test. Linear hypotheses of the form given by Eq. (3.14) can be tested in SAS using a CONTRAST statement. The option “chisq” in the CONTRAST statement is needed in order to obtain a Wald test. SAS Proc Mixed also allows the estimation and testing of linear combinations of the elements in β using an ESTIMATE statement. Using similar arguments as for approximate Wald tests, t-tests, and F-tests, approximate confidence intervals can be obtained for such linear combinations, also implemented in the ESTIMATE statement. Specification of L remains the same as for the CONTRAST statement.

3.7.3 Robust Inference

Given the estimate for β in Eq. (3.13) with αreplaced by its ML or REML estimates then conditional on α, ˆβ has the expected value given by,

E[ ˆβ(α)] =

n

X

i=1

X_i⁰W_iX_i

!⁻¹ _n X

i=1

X_iW_iE(Y_i)

=

n

X

i=1

X_i⁰W_iX_i

!−1 n

X

i=1

X_iW_iX_iβ

= β

provided that theE(Y_i) =X_iβ. Hence in order for ˆβto be unbiased, it is only sufficient that the mean of the response is correctly specified. Conditional on α, ˆβ has covariance, var( ˆβ) =Pn

i=1(X_i⁰W_iX_i)⁻¹ as derived in Eq. (3.14) Var( ˆβ) = (

n

X

i=1

X_i⁰WiXi)⁻¹

n

X

i=1

(X_i⁰WiVar(Yi)WiXi)(

n

X

i=1

X_i⁰WiXi)⁻¹

= (

n

X

i=1

X_i⁰WiXi)⁻¹

Note that this assumes that the covariance matrix is correctly modelled as Var(Y_i) = V_i = Z_iGZ_i⁰ + Σ_i and W_i = V_i⁻¹. This form of the covari- ance estimate is therefore often called the ‘naive’ estimate. The so-called robust estimate for Var( ˆβ) which does not assume the covariance matrix to be correctly specified is obtained by replacing Var(Y_i) by

Var(Y^_i) = [Y_i−X_iβ][Y_i−X_iβ]⁰

rather than Vi . The only condition for Var(Y^i) to be unbiased for Var(Yi) is that the mean is correctly specified. The ‘robust’ variance estimate also called the sandwich estimate is now given by

Var( ˆβ) = (

n

X

i=1

X_i⁰W_iX_i)⁻¹(

n

X

i=1

X_i⁰W_iVar(Y^_i)W_iX_i)(

n

X

i=1

X_i⁰W_iX_i)⁻¹

Based on this sandwich estimate, robust versions of the Wald test as well as of the t-test and the F-test can be obtained. This signifies the point that as long as interest is only in the inferences in the mean structure, little effort should be spent in modelling the exact covariance structure, provided that the data set is sufficiently large. An extreme point of view involves the use of OLS with robust standard errors. Nevertheless appropriate covariance modelling may still be of interest, firstly for the purpose of interpretation of random variation in the data, secondly for gaining efficiency and thirdly because in the presence of missing data, robust inference is only valid under very severe assumptions about the underlying missingness process. Issues of missingness were discussed briefly in Chapter 2 and will be revisited in more detail in Chapter 9. Robust inference for the fixed effects can be obtained by adding the option ‘empirical’ in the PROC MIXED statement in SAS namely

proc mixed data=data1 method=reml empirical;

assuming the data set is ‘data 1’. It is quite possible that for some parameters, the robust standard error is smaller than the naive, model based one. For others the opposite can be true. Thus interpretation of both standard errors should be done with caution.

3.7.4 Likelihood ratio test

The likelihood ratio tests are used to compare nested models with different mean structures, but equal covariance structures. The null hypothesis of interest can therefore be stated as

H0 :β∈Θβ,0

for some subspace Θ_β,0 of the parameter space Θ_β of the fixed effects β.

Let the notations L_{M L},ˆθ_{M L,0} and ˆθ_{M L} respectively denote the maximum likelihood (ML) function, the maximum likelihood estimator(MLE) underH₀ and under the general model. Then the test statistic under the LR method is

−2lnλ_n =−2ln

"

L_{M L}(ˆθ_{M L,0}) L_{M L}(ˆθ_{M L})

# .

The asymptotic distribution of the statistic under the null distribution is χ² with degrees of freedom (df) equal to the difference in dimension of Θ_β and Θ_β,0 that is

dimΘ_β−dimΘ_β,0.

It should be noted that LR tests for the mean structure are not valid under REML. A negative LR test statistic is a very possible outcome under REML.

The reason is as follows: under REML the response Y is transformed into error contrasts U =A⁰Y, for some matrixA withA⁰X = 0. Afterwards ML estimation is performed based on error contrasts. Models with different mean structures lead to different sets of error contrasts. Hence the corresponding REML likelihoods are based on different observations, which makes them no longer comparable.