How does the random coefficient model take care of autocorrelation?

Question

I’m working with time series data. Several variables among subjects are measured at intervals of two weeks over 1.5 years. The main goal is to estimate associations between different variables.

Consider a mixed model with AR(1) autocorrelation. In my sample, time points for measurements are irregular, so the spherical correlation between ordered observations in time may be imprecise. Alternately there is a random coefficient model. How does a random effect structure handle correlation between observations? If the autocorrelation between measurements within subject is constant no matter how far apart in time the measurements are, the model is not an option for me.

Answer 1

The question is,

How does a random effect structure handle correlation between observations?

The answer is revealed by analysing a simple example of the model you describe. It turns out the correlation does not depend on the time interval and therefore is unlikely to be appropriate in your application.

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Consider a numeric response $Y$ for subject $j$ (with characteristics $\mathbf x_{j} = (x_{j1},x_{j2}, \ldots, x_{jp})$) at time $t_i.$ Model this as a sum of

• fixed effects $\mathbf x_j\beta$ (for a constant but unknown parameter vector $\beta = (\beta_1,\beta_2,\ldots,\beta_p)^\prime)$),

• a random effect $U_j$ varying among subjects, and

• independent "error" $\varepsilon_{ij}$ associated with each observation of the response.

In mathematical notation this can be expressed as

$$Y_{ij} = \mathbf x_{j}\beta + U_j + \varepsilon_{ij}.$$

Standard assumptions are that all the random variables are uncorrelated and have constant variances and means so that (say)

$$\operatorname{Var}(U_j) = \tau^2$$

and

$$\operatorname{Var}(\varepsilon_{ij}) = \sigma^2.$$

(With no loss of generality, all means can be absorbed as an "intercept" in the fixed effect and thereby taken to be zero.)

Focus on a specific subject and compute the correlation among any two of its observations $(Y_{1j}, Y_{2j}, \ldots, Y_{nj})$ (which, with no loss of generality, we may take to be the first two). A standard formula uses the variance-covariance matrix, so let's begin there.

$$\operatorname{Var}(Y_{ij}) = \operatorname{Var}(\mathbf x_j \beta + U_j + \varepsilon_{ij}) = \tau^2 + \sigma^2$$

(because $U_j$ is assumed uncorrelated with $\varepsilon_{ij}$) and

$$\operatorname{Cov}(Y_{1j}, Y_{2j}) = \operatorname{Cov}(\mathbf x_j \beta + U_j + \varepsilon_{1j},\ \mathbf x_j \beta + U_j + \varepsilon_{2j}) = \tau^2$$

(because $U_j,$ $\varepsilon_{1j},$ and $\varepsilon_{2j}$ are uncorrelated).

Consequently the correlation between any two pairs of observations of the same subject is

$$\rho_j = \operatorname{Cor}(Y_{1j}, Y_{2j}) = \frac{\operatorname{Cov}(Y_{1j},Y_{2j})}{\sqrt{\operatorname{Var}(Y_{1j})\operatorname{Var}(Y_{2j})}} = \frac{\tau^2}{\tau^2 + \sigma^2}.$$

This does not depend on the time interval $\tau_2 - \tau_1$ and therefore does not depend on any time interval.

Notice, too, that this model implies strictly positive correlations among the observations of any given subject. Intuitively, this arises from the common (but random) influence $U_j$ pertaining to subject $j.$

Answer 2

1. You could just ignore this problem. You can show through simulation that, with some extreme exceptions, an approximate but not exact answer to the variance covariance structure usually leads to reasonable estimation. You can safeguard this approach even more by using a generalized estimating equation (GEE) model.
2. If AR1 is the right model, treat the unmeasured time series observations as missing data and impute them. If your X, Y processes are asynchronous, it could be as simple as LOCF. Otherwise, multiple imputation could be used. Alternately, you can edit the generalized least squares fitting process to account for imbalanced design and populate the variance-covariance matrix according to the observed design.
3. Random effects models contain random intercepts or random slopes. In your example, including time as a random slope nested in subject induces a spherical-type correlation between adjacent observations. As always, fit a variogram to inspect the shape and nature of the temporal correlation.

Answer 3

Would be the MMRM model an option for you? It is a flavour of mixed models. If you are only interested in the fixed effects and just want to account for the random effects you could define your fixed effects, "shovel" your random effects in the error part of the function and set the covariance structure to 'unstructured'. This could work as long you are not interessted in the random effects and you have enough data.

I hope I could help somewhat.

Answer 4

You could use glm in combination with the corCAR1 correlation structure from the nlme package in R. This is for continuous irregular time measurements as you have. Although this is not exactly an answer to your question, because you do not have/need a random effect in gls models.

To answer your question, you could use a random linear (or quadratic,cubic, ...) slope of "week". With a random linear slope, you e.g. model a quadratic pattern in the variances over the weeks. And also a particular pattern for the correlations between any pair of weeks you select. E.g. Singer and Willet show the formula's in their book Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence.