Signature
Var_D = 2 * sigma2 * (1 - rho^s)
| Inputs | Definition | Unit |
|---|---|---|
sigma2 | Variance of the outcome at each of the two visits, assumed equal | outcome units squared |
rho | Correlation between outcomes at adjacent visits | correlation |
s | Number of equally spaced visit steps between the two visits; 2 for baseline to 12 months with 6-monthly visits | count of visits |
Var_D | Variance of one patient's change in outcome between the two visits | outcome units squared, for example utility squared |
|---|
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
Variance of an AR(1) change in one cell
With named cells Sigma2, AdjacentCorr and VisitSteps, the formula returns the variance of the change between the two visits.
=2*Sigma2*(1-AdjacentCorr^VisitSteps)
Assumptions
Equal variance at both visits of an AR(1) change
The outcome has the same variance at the two visits. With unequal variances the variance of the change is the sum of the two variances minus twice their covariance.
Correlation for the change taken from the AR(1) structure
The correlation between the two visits is rho^s. If stable differences between patients hold the correlation above a floor, the correlation between the two visits is taken from that combined structure instead of from rho^s.
Worked examples
Variance of the baseline to 12-month change at an AR(1) correlation of 0.7
Baseline and 12 months are two steps apart with correlation 0.49, so the variance of the change is 2 × 0.04 × 0.51 = 0.0408.
sigma2 = 0.04; rho = 0.7; s = 2; Var_D = 0.0408
Variance of a six-month change at an AR(1) correlation of 0.7
For adjacent visits the correlation is 0.7 and the variance of the change is 0.024, smaller than for the 12-month change because the two measurements are more alike.
sigma2 = 0.04; rho = 0.7; s = 1; Var_D = 0.024
Common errors
Using an exchangeable correlation for a change between distant visits
An exchangeable structure assumed to share the adjacent correlation of 0.7 gives a baseline to 12-month change variance of 0.024, about 41 per cent below the AR(1) value of 0.0408. A fitted exchangeable model would estimate roughly the average pairwise correlation, 0.63, giving 0.0296, still about 27 per cent below. The choice of structure matters most for changes from baseline and baseline-adjusted effects on final utility.
Sources
AR(1) covariance behind the variance of a change between visits
Fitzmaurice GM, Laird NM, Ware JH. Applied Longitudinal Analysis. 2nd ed. Hoboken, NJ: Wiley; 2011. Covariance pattern models, including the first-order autoregressive pattern, under which the covariance between occasions j and k is sigma2 * rho^abs(j-k); the variance of a difference follows from it.
Canonical Identity
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