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A statistical approach to distinguish telomere elongation from error in longitudinal datasets

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Telomere length and the rate of telomere attrition vary between individuals and have been interpreted as the rate at which individuals have aged. The biology of telomeres dictates shortening with age, although telomere elongation with age has repeatedly been observed within a minority of individuals in several populations. These findings have been attributed to error, rather than actual telomere elongation, restricting our understanding of its possible biological significance. Here we present a method to distinguish between error and telomere elongation in longitudinal datasets, which is easy to apply and has few assumptions. Using simulations, we show that the method has considerable statistical power (>80 %) to detect even a small proportion (6.7 %) of TL increases in the population, within a relatively small sample (N = 200), while maintaining the standard level of Type I error rate (α ≤ 0.05).

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MJPS is supported by the Natural Environment Research Council (J024597/1) (United Kingdom). SN is supported by the Rutherford Discovery Fellowship (New Zealand). GS is supported by a grant by The Netherlands Organisation for Scientific Research (452-10-012), granted to M. Mills.

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Correspondence to Mirre J. P. Simons.

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The derivation of Equation 5

A two-level regression which model telomere length (TL) can be expressed as:

$$y_{ij} = \beta_{0} + \gamma_{j} + (\beta_{1} + \varphi_{j} )t_{ij} + \varepsilon_{ij} ,$$
$$\left( {\begin{array}{*{20}c} {\gamma_{j} } \\ {\varphi_{j} } \\ \end{array} } \right)\sim N\left( {\left( {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right),\left( {\begin{array}{*{20}c} {\sigma_{\gamma }^{2} } & {\rho \sigma_{\gamma } \sigma_{\varphi } } \\ {\rho \sigma_{\gamma } \sigma_{\varphi } } & {\sigma_{\varphi }^{2} } \\ \end{array} } \right)} \right) ,$$
$$\varepsilon_{ij} \sim N(0, \, \sigma_{\varepsilon }^{2} )$$

where t ij is the ith time point at which TL, y ij is measured for the jth individual (i = 1, 2,…, n; n is the number of TL measurements and n > 2; j = 1, 2,…, N; N is the number of individuals in a study), β 0 is the grand intercept (TL at t = 0), β 0 is the grand slope (regression coefficient for t), γ j is the deviation from β 0 for the jth individual, φ j is the deviation from β 0 for the jth individual, γ j and φ j has a multivariate normal distribution with the variance–covariance structure specified in A2, and ε ij is the ith residual value and residuals are normally distributed with a variance of \(\sigma_{\varepsilon }^{2}\).

When we consider A1 at the time points 1 and n (i.e. i = 1 and i = n), TL can be written as:

$$y_{1j} = \beta_{0} + \gamma_{j} + (\beta_{1} + \varphi_{j} )t_{1j} + \varepsilon_{1j} ,$$
$$y_{nj} = \beta_{0} + \gamma_{j} + (\beta_{1} + \varphi_{j} )t_{nj} + \varepsilon_{nj} .$$

When we have two measurements in time, 1 and m (the final time point) of telomeres the difference in telomere length is described by:

$$y_{nj} - y_{1j} = \beta_{1} (t_{nj} - t_{1j} ) + \varphi_{j} (t_{nj} - t_{1j} ) + \varepsilon_{nj} - \varepsilon_{1j} .$$

By setting d j  = y nj  − y 1j , the variance of d j can be expressed as:

$${\text{Var}}(d_{j} ) = (t_{nj} - t_{1j} )^{2} \sigma_{\varphi }^{2} + 2\sigma_{\varepsilon }^{2} .$$

Note that the constant β 1(t nj  − t 1j ) disappears. Using the definition of variance and further rearranging;

$$\frac{1}{(N - 1)}\sum\limits_{j = 1}^{N} {(d_{j} - \bar{d})^{2} } = (t_{nj} - t_{1j} )^{2} \sigma_{\varphi }^{2} + 2\sigma_{\varepsilon }^{2} ,$$
$$\frac{1}{2(N - 1)}\sum\limits_{j = 1}^{N} {d_{j}^{2} } = \sigma_{\varepsilon }^{2} + \frac{{(t_{nj} - t_{1j} )^{2} \sigma_{\varphi }^{2} }}{2} + \frac{{\bar{d}^{2} }}{2},$$

where \(\bar{d}\) is the mean value of dj. As \(\bar{d} = \beta_{1} (t_{nj} - t_{1j} )\) and setting (t nj  − t 1j ) = u;

$$\frac{1}{2(N - 1)}\sum\limits_{j = 1}^{N} {d_{j}^{2} } = \sigma_{\varepsilon }^{2} + \frac{{u^{2} }}{2}(\sigma_{\varphi }^{2} + \beta_{1}^{2} ).$$

When we assume that TL does not increase or decrease, i.e. \(\left( {\sigma_{\varphi }^{ 2} + \beta_{1}^{ 2} } \right) = 0\), A10 reduces to:

$$\sigma_{\varepsilon }^{2} = \frac{1}{2(N - 1)}\sum\limits_{j = 1}^{N} {d_{j}^{2} } .$$

If we estimate \(\sigma_{\varepsilon }^{2}\) in A11 only from individuals that show an increase of TL, or d j  > 0 (set such d j as D j ), we have Eq. 5 from the main text;

$$\sigma_{\varepsilon }^{\prime 2} = \frac{1}{2(m - 1)}\sum\limits_{k = 1}^{m} {D_{k}^{2} } ,$$

where \(D_{k}^{2}\) is the difference in TL between the initial and last measurements in the kth individuals that showed an increase in TL (k = 1, 2,…, m; m is the number of individuals whose TL elongated). Note that we assume \(\sigma_{\varepsilon }^{\prime 2}\) is also normally distributed as with \(\sigma_{\varepsilon }^{2}\) (A3). Due the symmetric nature of the normal distribution, \(\sigma_{\varepsilon }^{\prime 2}\) can be correctly estimated from restricted data, D j under our assumption.

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Simons, M.J.P., Stulp, G. & Nakagawa, S. A statistical approach to distinguish telomere elongation from error in longitudinal datasets. Biogerontology 15, 99–103 (2014).

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