Linear '
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General Model for Correlated Data
Objectives: Weighted least squares methods Moment estimation Sandwich variance Linear mixed models. Models for covariance Maximum likelihood and REML Empirical Bayes estimation &
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General Linear Model for Correlated Data
Example: Longitudinal FEV in cystic fibrosis patients. Example: Longitudinal CD4 count in HIV patients. Example: Depression score in a community randomized trial. Example: Lung function, asthma, and air pollution.
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General Linear Model for Correlated Data
Consider a sample of N randomly Yi1 Y i2 Yi = ... Yini selected units: i = 1, 2, . . . , N
where the Y i are independent vectors and ni may or may not be the same for all units i.
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General Linear Model for Correlated Data
Associated with the jth measurement on the ith unit is a 1 p vector of covariates X ij (1 p) = (Xij1 , Xij2 , . . . , Xijp ) X i2 = ... X ini X i1
Xi
(ni p) In the design matrix X i the rows correspond to different times of measurement, and the columns are different variables. &
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' Covariates may be: 1. Cluster specific, time invariant, or between-subject, so that Xi1k = Xi2k = . . . = Xini k for some 1 k p. Examples include sex and race in a longitudinal study, and fixed experimental conditions in a longitudinal clinical trial. 2. Subject specific, time varying, or within-subject, i.e., covariate k varies with j so that Xijk = Xij k . Examples include time since baseline, experimental condition in crossover or repeated measures designs, smoking status or height in a longitudinal study, or individual characteristics in a clustered sample survey. In some cases (pure repeated measures designs, or longitudinal studies with fixed time points), Xijk = Xi jk for all j. &
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' 3. Fixed by design, e.g., treatment group indicator, time since baseline, or individual characteristics in a sample survey. 4. Stochastic, e.g., height, current smoking status, or pollution exposure in a longitudinal survey.
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General Linear Model
The model assumes: 1. If X i is stochastic, (Y 1 , X 1 ), . . . , (Y N , X N ) are independently distributed. If X i is fixed by design, then Y i are independent. 2. Given X i E(Y i | X i ) (ni 1) cov(Y i | X i ) = X i (ni p) (p 1) = i (ni ni )
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General Linear Model
(1) simply states that the sample consists of independently selected units. (2) says that given X i1 , X i2 , . . . , X ini , the mean of Yij is linear and depends on X ij : E(Yij | X i ) = 0 + 1 Xij1 + 2 Xij2 + . . . + p Xijp
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' Note: The model may not hold with certain stochastic time varying covariates. The model implies E(Yij | X i1 , X i2 , . . . , X ini ) = E(Yij | X ij ) but if current outcomes predict future values of the covariates then the mean of the outcome at a given occasion may depend on future covariates. For example, if Yij is a symptom measure, and Xij is an indicator of drug treatment then past symptoms may influence current treatment (usually a good idea!). Formally, if f (Xi,j+1 | Yij , Xi,j ) = then f (Yij | Xi,j , Xi,j+1 ) = f (Xi,j+1 | Xij ) f (Yij | Xi,j )
$
So that the conditional expectation E(Yij | X i ) may not be correct. &
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' Note: The covariance i allows for dependencies among measurements on the same unit. Covariance may vary with covariates, e.g., across treatment group, or the covariance may be a function of time. Alternatively, i may depend on i only through ni .
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Covariance Matrix
We will consider two approaches where i is unstructured (only for "balanced" data), and where i has a specified structure. Define: A Balanced and complete design means that all subjects are measured at the same n occasions. Balanced only means that subjects should be measured at the same occasions, but some subjects are not observed at all occasions (ni < n).
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GLMCD using stacked notation
The GLMCD can be written as: E(Y | X) = X (
i
ni 1)
(
i
ni p) (p 1) 1 0 2 ... ... .. . ... 0
cov(Y | X i )
0 = 0
i
0
0 n
If Balanced and complete then
ni = n N .
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Examples
One sample repeated measures ANOVA N subjects are measured repeatedly under n different experimental conditions. The goal is to quantify differences in experimental conditions. 1 0 ... 0 1 0 1 ... 0 2 E(Y i | X i ) = .. ... . n 0 0 ... 1 Here X i = I n , = , and p = n. &
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Example: Repeated Measures ANOVA
It is often assumed that the covariance of Y i has a compound symmetric form which arises from the model: Yij = j + i +
ij
where the i 's and the ij 's are independent of each other, with var(i ) = 2 and var( ij ) = 2 .
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Example: Repeated Measures ANOVA
We can use vector notation to represent the model as: Yi = + i 1 +
i
cov(Y i | X i ) =
2 11T + 2 I n +
2 2 2 2
=
2 2
... ... .. . ...
2 2
+
2
2
2 + 2
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Examples
One way multivariate ANOVA (MANOVA) We assume G treatment groups, and n measurements are obtained in each of Ng subjects in treatment group g. Goal is to test if the mean vector is the same for all G groups, where we assume the mean model: E(Y ig | X i ) (n 1) = g g = 1, 2, . . . , G (n 1)
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Example: MANOVA
1 E(Y ig | X i ) =
2 (0, . . . , 0, I n , 0 . . . , 0) ... G
The usual MANOVA assumes i = is unstructured: 11 12 . . . 1n 21 22 . . . 2n cov(Y ig ) = .. . n1 n2 . . . nn &
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Examples
One group polynomial growth curve model N subjects are observed at the same times t1 , t2 , . . . , tn ; for example a group of children from the same cohort is observed yearly at ages 6, 7, 8, ..., 12. A linear model of the average response can be expressed as a polynomial in tj E(Yij | X i ) = 0 + 1 tj + 2 t2 j E(Y i | X i ) 1 = 1 1 t1 t2 ... tn t2 n t2 1 t2 2
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&
203
' One model for i arises from assuming that each subject has his or her own growth curve with parameters i : E(Y i | i , X i ) and E( i | X i ) cov( i | X i ) then E(Y i | X i ) = X cov(Y i | X i ) i = cov[E(Y i | i )] + E[cov(Y i | i )] = XDX T + 2 I n = = D = X i
$
cov(Y i | i , X i ) = 2 I n
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Examples
Family studies: Hypertension Here i indexes family. For N families the outcome is blood pressure and the covariates include age, gender, weight, height, physical activity, diet, smoking, etc. We include all known risk factors in the mean model, and then study the residual correlation. i may be structured for genetic models as follows:
M F C1 C2 C3 MF CP CP CP
2 A
2 A
CP CP CP
2 C
CC CC
2 C
2 C
CC
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Longitudinal Studies
In many cases the primary focus of a study is the change in the mean response over time. This can be modelled by X i and then i represents parameters of secondary interest (or quite possible nuisance parameters). If ni is large and/or the design is inherently unbalanced then it may be desirable to impose some strucure on i . Method 1: Random effects models Method 2: Serial correlation models
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' Banded: When measurements are equally spaced is that the correlation only depends on the distance 1 1 2 ... 1 1 1 ... 2 1 1 ... i = 2 .. . (n-1) This implies var(Yij ) = corr(Yi,j , Yi,j+k ) = 2 k j, k (n-2) (n-3) ... one assumption
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(n-1) (n-2) (n-3)
1
&
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' Autoregressive: For data that arise over time it is often reasonable to assume that correlation between measurements that are close in time is greater than the correlation of measurements that are widely separated in time. One model is (here for unit spaced observations):
2 1
1
$
1
2 1
... ... ... .. . ...
(n-1) (n-2)
1 1
i
=
2
1
(n-3)
(n-1)
(n-2)
(n-3)
1
One construction is given by var(Yi0 ) = Yi,j+1 | Yij var( &
208 ij )
2 Yij +
ij
= =
2 (1 - 2 ) independent %
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rho = 0.4
ID = 1
1 2 3 1 2 3
ID = 2
1 2 3
ID = 3
1 2 3
ID = 4
y
y
y
-1
-1
-1
y
20
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
-3
-1
5
10 x
15
20
ID = 5
1 2 3 1 2 3
ID = 6
1 2 3
ID = 7
1 2 3
ID = 8
y
y
y
y
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
-1
5
10 x
15
20
ID = 9
1 2 3 1 2 3
ID = 10
1 2 3
ID = 11
1 2 3
ID = 12
y
y
y
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
-1
y
5
10 x
15
20
ID = 13
1 2 3 1 2 3
ID = 14
1 2 3
ID = 15
1 2 3
ID = 16
5 10 x 15
5
y
y
y
-1
-1
-1
-3
-3
-3
20
5
10 x
15
20
10 x
15
20
-3
-1
y
5
10 x
15
20
ID = 17
1 2 3 1 2 3
ID = 18
1 2 3
ID = 19
1 2 3
ID = 20
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
-1
y
y
y
y
5
10 x
15
20
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rho = 0.9
ID = 1
1 2 3 1 2 3
ID = 2
1 2 3
ID = 3
1 2 3
ID = 4
y
y
y
-1
-1
-1
-1
y
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
5
10 x
15
20
ID = 5
1 2 3 1 2 3
ID = 6
1 2 3
ID = 7
1 2 3
ID = 8
y
y
y
y
-1
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
5
10 x
15
20
ID = 9
1 2 3 1 2 3
ID = 10
1 2 3
ID = 11
1 2 3
ID = 12
y
y
y
-1
-1
-1
-1
y
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
5
10 x
15
20
ID = 13
1 2 3 1 2 3
ID = 14
1 2 3
ID = 15
1 2 3
ID = 16
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
-1
y
y
y
y
5
10 x
15
20
ID = 17
1 2 3 1 2 3
ID = 18
1 2 3
ID = 19
1 2 3
ID = 20
y
y
y
y
-1
-1
-1
-1
-3
-3
-3
5
10 x
15
20
5
10 x
15
20
5
10 x
15
20
-3
5
10 x
15
20
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Cross-sectional versus Longitudinal Effects
In our simple growth curve model we assumed that all subjects were measured at the same times, tj , and were from the same cohort (i.e. same age at baseline). This is rarely the case in observational studies. Individuals enter at different ages, and measurements may be taken at different times. This design provides the opportunity to obtain information about differences between cohorts, as well as differences due to aging.
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Cross-sectional versus Longitudinal Effects
Ware et al. (1990) discuss a study of pulmonary function where PF was measured every three years for baseline and two follow-up visits on a sample of never-smoking adults. One PF measure is FEV1 (forced expiratory volume in 1 second). They found:
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Cross-sectional versus Longitudinal Effects
Possible reasons: Cohort Effects younger cohorts are exposed to higher levels of pollution. Attrition we may only see older subjects that are healthy.
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Cross-sectional versus Longitudinal Effects
We can partition age, Xij , into two components: Cross-sectional comparisons: E(Yi1 | X i ) = 0 + C Xi1 Longitudinal comparisons: E(Yij - Yi1 | X i ) = L (Xij - Xi1 ) Putting these two models together we have: E(Yij | X i ) = 0 + C Xi1 + L (Xij - Xi1 )
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Missing Data Issues
With longitudinal data we must consider the reasons for missing data since the missing data mechanism (MDM) can impact the validity of estimates and tests. Examples: 1. Repeated measures experiment HIV patients are given an anti-viral therapy and viral load is measured monthly for 6 months. Some subjects do not comply or drop-out. 2. Validation study Food frequency questionnaires (FFQ) are obtained on all subjects; a validation (more costly but accurate instrument) is obtained for a subset only. We may randomly sample subjects for validation or select them based on the FFQ data. &
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' 3. Longitudinal study Children are measured for FEV through the schools. Children may move in and out of study schools. 4. Quality of life study Many clinical trials now routinely collect self-reported information on quality of life. Patients may be to ill to give an evaluation.
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Missing Data Issues
To formulate different missing data mechanisms we introduce additional notation: Rij = 1 Rij = 0 if subject i is observed at time j if subject i is not observed at time j
MCAR
Missing completely at random if f (Ri | Y i , X i , ) = f (Ri | X i , )
This implies that E(Yij | Rij = 1, X i ) = E(Yij | X i ).
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Missing Data Issues
MAR Missing at random if f (Ri | Y O , Y M , X i , ) = f (Ri | Y O , X i , ) i i i Here the probability of missing data only depends on the observed values and not the missing values. Trouble starts here since this implies E(Yij | Rij = 1, X i ) = E(Yij | X i ) (possibly). NI Non-ignorable if f (Ri | Y O , Y M , X i , ) depends on Y M i i i
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Summary
GLMCD is flexible. Covariate models / issues. Covariance models. Missing data issues. Estimation semiparametric. Estimation parametric (Maximum likelihood).
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General Linear Model for Correlated Data
Estimating with known Weighted least squares: In univariate regression, WLS yields estimates of that minimize the objective function
N
Q() =
i=1
wi (Yi - X i )2
Analogously, the multivariate version of WLS finds the value of the parameter (W ) that minimizes
N
QW () =
i=1
(Y i - X i )T W i (Y i - X i ) %
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&
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' where W i is an (ni ni ) positive definite symmetric matrix. It's straight forward to see that U () = QW () = -2 X T W i (Y i - X i ) i i=1
N
$
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GLMCD: WLS
The solution to the minimization solves U () = 0 and yields
N -1 N
(W ) =
i=1
X T W iX i i
i=1
X T W iY i i
Example 1: When W -1 = 2 I ni then i
N ni
QI () =
i=1 j=1
1 (Yij - X ij )2 2
and (I) is the OLS estimator assuming observations are independent both within and between clusters. &
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GLMCD: WLS
Example 2: If X i = X and W i = W for all i (e.g. complete and balanced polynomial growth curve data) then, (W ) = X W X
T -1
XT W
1 N
Yi
i
This implies that is the regression of the averages. (Q: Is it also the average of the regressions?).
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Properties of (W )
Given X 1 , X 2 , . . . X N and W 1 , W 2 , . . . W N
N -1 N
E (W )
=
i=1
X T W iX i i
i=1
X T W i E[Y i ] i
=
N
var (W )
=
A-1
i=1 N
X T W i i W i X i i
-1
A-1
where A-1
=
i=1
X T W iX i i
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' Thus, W i = I ni var (I) W i = -1 i var (-1 )
-1
$
-1
=
i
XT Xi i
i
X T i X i i
i
XT Xi i
-1
=
i
X T -1 X i i i
&
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' Bio/Stat Lemma: var (-1 ) var (W )
$
Notice that W i = I ni gives OLS where all observations are treated as independent (weighted equally). It follows that E( OLS ) = , but OLS may not be very efficient. For maximum efficiency we must estimate i . Q: How do we compare "efficiencies"? Answer: We compare the ratio efficiency = &
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var (-1 ) var (W ) %
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Estimating
In general i is unknown. With balanced and complete data we can construct a simple estimator of and use this to obtain ( Lemma: Under regularity conditions on the covariate space X i , if W i is a consistent estimator of W i then (W ) and (W ) have the same asymptotic distribution. N (W ) - N ( 0, C W ) N (W ) - N ( 0, C W )
-1
).
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' where CW AN = =
i N
$
lim N A-1 N
i
X T W i i W i X i i
A-1 N
X T W iX i i
&
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Estimating
In the case where cov(Y i ) = for all i, we can construct a consistent estimator of the optimal weight matrix W i = -1 : 1 = N
N
Y i - X i (In )
i=1
Y i - X i (In )
T
.
In fact any consistent estimator of will suffice. Two-step estimator: Step 1: Obtain (In ) and . Step 2: Obtain ( &
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). %
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Estimating
As we shall show, if these two steps are iterated, with balanced and complete data where i = , then ( assuming multivariate normality.
-1
) and are also the MLE's
We can estimate the asymptotic variance of (
N
-1
) using
C -1 = N
T -1 Xi Xi
i=1
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Modelling
In n is large (ie. n = dim(Y i )) then we estimate a large number of parameters in . In that case we may require large sample sizes, N , before the distribution of (
-1
) and (-1 ) approximately agree.
Q: Can't we adopt some simple structure for ? Answer: Yes! With small to moderate samples we may use our substantive knowledge about Y i and exploratory data analysis to guide selection of a covariance model. This permits to be modeled in terms of , a smaller number of parameters.
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Modelling
For example, we may use the compound symmetric covariance model: var(Yij ) = cov(Yij , Yik ) = 1 2
and we may use simple moment estimators to obtain estimates 1 = 1 N 1 N
N i=1 N i=1
1 n
n
Yij - X ij
j=1
2
2
=
1 n(n - 1)
Yij - X ij
j=k
Yik - X ik
Q: What if it's not really compound symmetric? &
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' Answer:
1
$
1 2
1 = n
2 j j
2 =
1 n(n - 1)
jk
j=k
These are simple moment estimators and therefore a (general) WLLN implies that these will converge to their limit mean.
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Modelling
Again, we obtain asymptotic normality for the WLS estimator that uses W i = ()-1 . Again, there is no difference (asymptotically) between use of W i and W i = ( )-1 . Again, in general the asymptotic covariance of (W ) is given in "sandwich" form: AN BN &
232
= =
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Another Empirical Sandwich!
We can use the independent replication across subjects to estimate the matrix B N .
N
BN BN
=
i=1
T -1 -1 X i i var(Y i )i X i
=
Note: The key property of this estmator is consistency requires large number of subjects (N ). &
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Testing Hypotheses
Finally, consider testing hypotheses of the form: H0 : QT = 0 (q p) (p 1) (q 1)
Under the null hypothesis we have (asymptotically): N QT (W ) N ( 0, QT C W Q ) So that we can use N Q (W ) Q C W Q as a general Wald statistic. &
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(W )T Q 2 (q)
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General Linear Model
Comments: 1. The theory sketched here can be considered semi-parametric in the sense that estimation and inference for a parameter can be achieved based solely on specification of the mean. 2. DHLZ (2002) call W -1 the "working covariance model" since i inference using the sandwich variance estimator doesn't require W i = -1 . The matrix W i is used to improve efficiency. i 3. It is possible to allow i to depend on i we'll see examples. 4. The estimates of and the variance of outlined above are special cases of GEE. 5. DHLZ take a slightly different approach to specifying () by assuming var(Yij | X i ) = 2 and then cov(Yi | X i ) = 2 R() &
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Efficiency and WLS
Q: If using W = -1 is optimal for WLS estimation of , then how suboptimal is OLS ? Recall:
-1
var( opt )
=
i
X T -1 X i i i
var( OLS )
=
A-1
i
X T i X i i
-1
A-1
where
A-1
=
i
XT Xi i
See: Bloomfield and Watson (1975); Watson (1967) &
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' Comment on notation here: Y = Y1
$
Y2 stack(Y i ) = . . . Yn X2 stack(X i ) = . . . Xn X1
X
=
&
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' Comment on notation here: = 1 0 2 ... ... .. . ... 0 0
$
0 block(i ) = . . . 0
0
N
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Efficiency of OLS estimators
Y N (X, ) Theorem: Let C be an estimable function for the linear model Y = X + where E( ) = 0, and E(
T
) = . Then
M(X) = M(X) implies that the BLUE and LS estimators of C are equivalent. Note: M(X) denotes the column space of X. Proof: Watson (1967) &
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Efficiency of OLS
Example 1: N n tj = = = 10 5 (-2, -1, 0, 1, 2) 0 + 1 tj 2 {(1 - )I + J }
E(Yij ) = =
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' Then, x = = = 2 {(1 - )I + J } x 2 (1 - ) x + nx 1 ax+b1 M(X)
$
Therefore, OLS = ().
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Efficiency of OLS
Theorem: Rao (1965) X if and only if = XX T + ZZ T + 2 I where Z T X = 0, and , , 2 are arbitrary. This result implies that for any balanced random effects model (with conditional independence) we will have OLS = ().
T -1
X
-1
X
T
-1
= X X
T
-1
XT
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Efficiency of OLS
Example 2: consider the same mean model as in Example 1 but now assume AR(1) errors: 1 2 3 4 1 2 3 2 i = 1 1 1
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' some algebra yields var( OLS ) V11 V22 =
2
$ V11 0 0 V22
= =
0.004(5 + 8 + 62 + 43 + 24 ) 0.002(5 + 4 - 2 - 43 - 44 ) 0.01(5 - 8 + 32 )-1 0 0 0.005(5 - 4 + 2 )-1
var[()]
=
2
&
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Heagerty, Bio/Stat 571
' ...and for certain values of we obtain:
$
e(0 ) e(1 ) e(0 ) e(1 )
0.1 0.998 0.997 0.6 0.955 0.952
0.2 0.992 0.989 0.7 0.952 0.955
0.3 0.983 0.980 0.8 0.956 0.952
0.4 0.973 0.970 0.9 0.970 0.955
0.5 0.963 0.962 0.99 0.996 0.961
Comparisons of this kind (and earlier results) suggest that in many circumstances the OLS estimator is satisfactory. This is not always the case. Consider... &
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Heagerty, Bio/Stat 571
'
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Efficiency of OLS
Example 3: again, assume that we have AR(1) errors and now assume n = 3 and that subjects crossover from treatment A to treatment B (and from B to A). Assume that subjects have observed treatment paths in equals numbers: AAA AAB ABA ABB BAA BAB BBA BBB
This is a form of "crossover" design. &
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Heagerty, Bio/Stat 571
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$
Efficiency of OLS: Example 3
Now assume that the predictor of interest is treatment group, xij E(Yij | X i ) = 1(TXi = B, at time j) = 0 + 1 xij
&
247
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Heagerty, Bio/Stat 571
' ...now for certain values of we obtain:
$
e(0 ) e(1 ) e(0 ) e(1 )
0.1 0.993 0.987 0.6 0.846 0.571
0.2 0.974 0.947 0.7 0.815 0.438
0.3 0.946 0.883 0.8 0.788 0.297
0.4 0.914 0.797 0.9 0.766 0.150
0.5 0.880 0.692 0.99 0.751 0.015
Discuss: Why the efficiency difference now? &
248
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Heagerty, Bio/Stat 571
'
$
EDA and Covariance Models
Q: What are the appropriate EDA techniques for longitudinal data? Lines plot (spaghetti plot) Average & distribution plots (boxplot, quantiles) Empirical covariance Residual "pairs" plot Standard deviation plot Variogram
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Heagerty, Bio/Stat 571
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$
EDA and Covariance Models
Q: What are some parametric covariance models? Mixed model ( 2 I + XDX T ) Nested models (bi + bij + eijk ) Autoregressive models Moving average models General isotropic correlation models Combined Random effects and Serial (Diggle 1988)
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250
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Heagerty, Bio/Stat 571
' Example: (DLZ Example 1.1) CD4+ Cell Counts
$
HIV attacks CD4+ cells. Data from the MACS study. N = 369 infected men incident cases. Analysis focuses on characterizing the process. CD4+ cell counts are used to monitor patient status, and characteristics of the longitudinal process within a patient are thought to be predictive of clinical course. More recently HIV research has focused on longitudinal measures of viral load.
&
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Heagerty, Bio/Stat 571
' Descriptives:
time Min. :-2.9900 1st Qu.:-0.3922 Median : 0.7296 Mean : 0.8284 3rd Qu.: 2.1920 Max. : 5.4590 drugs Min. :0.0000 1st Qu.:1.0000 Median :1.0000 Mean :0.7559 3rd Qu.:1.0000 Max. :1.0000 cd4 Min. : 10.0 1st Qu.: 482.8 Median : 701.5 Mean : 765.1 3rd Qu.: 964.0 Max. :3184.0 partners Min. :-5.00000 1st Qu.:-3.00000 Median :-1.00000 Mean :-0.03409 3rd Qu.: 5.00000 Max. : 5.00000 age Min. :-11.290 1st Qu.: -2.760 Median : 1.510 Mean : 2.636 3rd Qu.: 6.950 Max. : 29.080 cesd Min. : -7.000 1st Qu.: -5.000 Median : 0.000 Mean : 2.496 3rd Qu.: 6.000 Max. : 49.000 packs Min. :0.0000 1st Qu.:0.0000 Median :0.0000 Mean :0.9891 3rd Qu.:2.0000 Max. :4.0000 id Min. :10000 1st Qu.:11200 Median :30050 Mean :26190 3rd Qu.:40360 Max. :41840
$
Number of subjects = 369 Number of observations = 2376 Number of subjects with a given observations-per-subject: 1 2 3 4 5 6 7 8 9 10 11 12 5 24 25 47 43 52 40 41 38 21 23 10
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Heagerty, Bio/Stat 571
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MACS CD4 Data
$
CD4+ cell...