Estimating the reference interval from a fixed effects meta-analysis

Abstract

A reference interval provides a basis for physicians to determine whether a measurement is typical of a healthy individual. It can be interpreted as a prediction interval for a new individual from the overall population. However, a reference interval based on a single study may not be representative of the broader population. Meta-analysis can provide a general reference interval based on the overall population by combining results from multiple studies. Methods for estimating the reference interval from a random effects meta-analysis have been recently proposed to incorporate the within and between-study variation, but a random effects model may give imprecise estimates of the between-study variation with only few studies. In addition, the normal distribution of underlying study-specific means, and equal within-study variance assumption in these methods may be inappropriate in some settings. In this article, we aim to estimate the reference interval based on the fixed effects model assuming study effects are unrelated, which is useful for a meta-analysis with only a few studies (e.g., ≤5). We propose a mixture distribution method only assuming parametric distributions (e.g., normal) for individuals within each study and integrating them to form the overall population distribution. This method is compared to an empirical method only assuming a parametric overall population distribution. Simulation studies have shown that both methods can estimate a reference interval with coverage close to the targeted value (i.e., 95%). Meta-analyses of women daytime urination frequency and frontal subjective postural vertical measurements are reanalyzed to demonstrate the application of our methods.

Keywords: fixed effects model; meta-analysis; reference interval; very few studies.

Figure 1:. Simulation Results:

The median (line), 2.5%, and 97.5% (shaded area) of the proportion of the true population distribution captured by the estimated 95% reference interval, for different numbers N of studies. The horizontal axis, proportion of between-study variance to the total variance, represent the degree of heterogeneity across studies. Three distributions are assumed: (a) normal distribution; (b) log-normal distribution; (c) gamma distribution.

Figure 2:. An illustration of the 95% reference interval estimated by the mixture distribution method:

The blue dashed curves are the estimated densities for 5 studies weighted by the sample sizes, and the solid black curve represents the pooled population distribution density. The 95% reference interval is the region of x-axis between two vertical lines, and the sum of area under each blue curve outside the vertical line on each side is equal to 0.025

Figure 3:. A Meta-analysis of Daytime Frequency:

Mean (95% CI) and 95% prediction interval for a new individual for each study; Overall is the 95% CI for pooled mean estimated by the fixed effects model; 95% reference ranges are estimated from the mixture distribution and the empirical methods under: (a) the log-normal distribution; (b) the gamma distribution.

Figure 4:. A Meta-analysis of Sagittal Plane SPV:

Mean (95% CI) and 95% prediction interval for a new individual for each study; Overall is the 95% CI for pooled mean estimated by the fixed effects model; 95% reference ranges are estimated from the mixture distribution and the empirical methods under the normal distribution.