Lecture 7:Parametric Models for Covariance

Structure (Examples)

1. Model for the mean

2. Model for the covariance matrix

2. Model for the covariance matrix (cont’d)

Which model to pick?

Which model to pick? (cont’d)

Example: CD4+ Level

HIV attacks CD4+ cells, which regulate the body’s immuneresponse against infectious agents

We have 2376 values of CD4+ cell numbers plotted against time since seroconversion for 369 infected men enrolled in the MAC Study

Example: CD4+ Level (cont’d)

Goals:

1.Estimate the average time course of CD4+ cell depletion

2.Identify factors which predict CD4+ cell changes

3.Estimate the time course for an individual man taking into account the measurement error in CD4+ cell determinations

4.Characterize the degree of heterogeneity across men in the rate of progression

Example: CD4+ LevelGoal 1: Estimate average time course of CD4+ cell

depletion

The model for the covariance matrix is a model of serial correlation.

Example: CD4+ LevelGoal 2: Identify factors predictive of CD4+ cell

changes

The model for the covariance matrix is still a model of serial correlation, however we have changed the model for the mean.

Example: CD4+ LevelParameter Interpretation

Example: CD4+ LevelGoal 3: Estimate time course for an individual man,

accounting for measurement error in CD4+ cell counts

This is a model with a random intercept and

slope + serial correlation + measurement error.

Example: CD4+ LevelParameter Interpretation

Random Effects Model:Interpretation of coefficients

Heterogeneity between subjects at baseline

Heterogeneity between subjects in rate of change

Example: CD4+ LevelGoal 4: Characterize degree of heterogeneity across

men in progression rate

(From the previous slide)

Example: Protein contents of milk samples

• Barley (25 cows)

• Mixed (27 cows)

• Lupins (27 cows)

Example: Protein contents of milk samples (cont’d)

0.02

Example: Protein contents of milk samples Model for the Mean

Example: Protein contents of milk samples Model for the Covariance Matrix

Table 5.1 (b)

Example: Protein contents of milk samples Does diet affect the mean response profile?

Example: Protein contents of milk samples Is there a rise in the mean response towards the end of

the experiment?

Example: Protein contents of milk samples Is there a rise in the mean response towards the end of

the experiment? (cont’d)

Example: Protein contents of milk samples Is there a rise in the mean response towards the end of

the experiment? (cont’d)

0.02

Example: Body weight of 26 cows

Dataset consists of body weights of 26 cows, measured at 23 unequally-spaced times over a period of about 22 months.

Treatments were allocated in a 2x2 factorial design:

Control (4)Iron-dosing (4)Infection (9)Iron + Infection (10)

Control

N=4

Iron

N=4

Infection

N=9

Iron + Infection

N=10

Log Y:

Variance-stabilizing transformation

Example: Body weight of 26 cows (cont’d)

• Look at your data

• Estimate empirical variogram

• What do you see?

• Measurement variance small

• Substantial between-cow variability

• Gaussian correlation model appropriate

•Small measurement error

•Experimental correlation

•Random effects

Empirical variogram of the OLS residuals from a saturated model for the mean response

Example: Body weight of 26 cowsModel for the Mean

Example: Body weight of 26 cowsModel for the Covariance Matrix

Example: Body weight of 26 cowsQuestions

Q1: Can we conclude linear (vs. quadratic) growth?

…The quadratic curve is appropriate.

Q2: Is there a main effect for iron?…NO

Example: Body weight of 26 cowsQuestions (cont’d)

Q3: Is there a main effect for infection?…YES

Q4: Is there an interaction between iron and infection?…NO

Example: Body weight of 26 cowsQuestions (cont’d)

We re-fit the model with only the infection term:

Conclusions:

• Highly significant effect of infection

• No significant effect of iron

• No significant effect of interaction