Signature
r = phi^d
| Inputs | Definition | Unit |
|---|---|---|
phi | Correlation between two measurements one time unit apart, above 0 and below 1 | correlation per chosen time unit, for example per year |
d | Absolute difference between the two measurement times | time, in the same unit as phi |
r | Correlation between the outcomes of one patient at two measurement times | correlation from 0 to 1 |
|---|
Function
Autoregressive correlation function for repeated health measurements
Maps the correlation between adjacent measurements on the same patient, and the distance between two measurements, to the correlation between them under a first-order autoregressive (AR(1)) structure. The correlation falls by the same factor with each step, so one parameter describes the whole within-patient correlation matrix. Combined with a common variance at every visit, the structure gives the variance of any weighted total or difference of the repeated measurements, such as a trial QALY total or a change in utility from baseline. In time-series notation the same pattern comes from the error process e_t = rho * e_(t-1) + u_t, which for a stationary series gives a correlation of rho^s between errors s periods apart.
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Implementations
Excel
Continuous-time AR(1) correlation from two visit times
With the correlation per time unit in a cell named CorrPerUnit and the two measurement times in TimeJ and TimeK, in the same unit, the formula returns the correlation.
=CorrPerUnit^ABS(TimeJ-TimeK)
Assumptions
Positive continuous-time AR(1) correlation
phi is above zero, because the elapsed time can be a fraction of a unit and a negative number has no real fractional power.
Elapsed time and phi in the same unit
The time unit of d matches the unit in which phi is defined. The correlation depends only on elapsed time, not on where in follow-up the two measurements fall.
Worked examples
Six-month gap at a continuous-time correlation of 0.49 per year
A correlation of 0.7 between visits six months apart is a correlation of 0.49 per year. A gap of half a year returns 0.7, and the 12-month gap from baseline returns 0.49, matching the visit-step form.
phi = 0.49; d = 0.5; r = 0.7
Three-month gap at a continuous-time correlation of 0.49 per year
For a schedule at 0, 3, 6 and 12 months, the 3-month gap gives about 0.837, above the 0.7 for a 6-month gap. Counting visit steps would give both gaps the same correlation. The figures are illustrative.
phi = 0.49; d = 0.25; r = 0.8367
Common errors
Counting visit steps for unequally spaced AR(1) visits
In an illustrative 0, 3, 6, 12-month schedule with 0.7 per visit step, the baseline and 12-month utilities are three steps apart and receive 0.343, whereas elapsed time at 0.49 per year gives 0.49. Because each step is given the correlation of a 6-month gap, step counting understates the correlation for every pair that spans either 3-month interval.
Entering months against a yearly AR(1) correlation
In an illustrative calculation, using a gap of 6 months with phi defined per year gives 0.49 raised to the power 6, about 0.0138, instead of 0.7. The time unit of d has to match the unit of phi.
Sources
Continuous-time AR-1 working correlation for arbitrary spacing
Liang KY, Zeger SL. Longitudinal data analysis using generalized linear models. Biometrika. 1986;73(1):13-22. Section 4, Example 4, which gives the correlation as alpha raised to the absolute time difference, the continuous-time analogue of the first-order autoregressive process, and notes that any number and spacing of observations can be accommodated.
Canonical Identity
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