The use of piecewise linear spline function on dose-response meta-analysis

Does-response meta-analysis is widely used in observational-based meta-analysis since proposed by Greenland et al. (1-4). The method works well in most of the situation, especially for the nonlinear trend approximating. Our argument is the linear slope insert within the function: by setting the parameters of quadratic and cubic term as zero. Linear relationship usually acts as main analysis when non-linear trend is non-significant or as additional analysis when non-linear trend is significant. Actually, only part of the linear slopes fit well (5). For J-shaped, U-shaped, S-shaped, and V-shaped curves, the linear slope fit bad since these slopes differ across pieces. We introduce a piecewise linear spline function that may be a solution for this issue (5,6). We assume log relative risks (LogRR) as dependent variable and exposure (X) as independent variable. Then a piecewise linear regression can be established as X against LogRR, with K indicates the cut points (knots) of the slopes.

If we take two knots as 30^th, 60^th of the distribution, which divide the function into three linear segments: 0^th–30^th; 30^th–60^th; 60^th–100^th.

In the Stata/SE 12.0 (Stata Corp., College Station, Texas, USA) software, mkspline procedure can be used to finish this work. Both one-step and two-step methods are available. Two-step method is recommended as it can reflect the heterogeneity across studies: first, estimating the coefficients of each segments of per unit increment of exposure within each study by mkspline procedure. Then summarizing the regression coefficients by random or fixed -effect model to produce a “mean” piecewise function.

We take the meta-analysis as an example which investigates the relationship between sleep duration and risk of all-cause mortality (data can be obtained from the corresponding author) (6). It is worth to note that sleep duration trend to be U-shaped association to health problems (7,8).

When fit with previous model (glst) by linear (1), we obtain the coefficient as 1.04 (95% CI: 1.02–1.06), which indicate that longer sleep duration is associated with higher risk of all-cause mortality; accordingly, the shorter one sleep, the lower risk he will be. This conclusion is obviously problematic. The correct way is to establish a piecewise linear slope (Figures S1-S3). With 1 knot (7 hours), we obtain two pieces of the slope: the coefficient at the left side of 7 hours is (−1.07; 95% CI: −1.03 to −1.11) and the coefficient of the right side of 7 hours is (1.12; 95% CI: 1.09–1.15), which suggest that, for those people who sleep less than 7 hours, shorter sleep duration is associated higher risk of all-cause mortality; while for people with more than 7 hours, longer sleep duration is also higher risk of all-cause mortality (6) (Figure S1).

Figure S1 Piecewise linear model with the cut point of 7 hours.

Figure S2 Results with the cut point of 4 hours.

Figure S3 The dose-response curve between sleep duration and risk of all-cause mortality with the cut point of 7 hours..

A simple method can be used to choose when to use linear model and piecewise one: the R² comes with Mkspline. When we set the cut point on the lowest dose (or highest), then there is only one piece (between the lowest to the highest distribution) with the slope since the other one (from zero to the lowest or from the highest to highest) is defaulted to be omitted (Figure S2). By comparing the adjusted R² between them we can get a more reasonable choice (9).

From the example, it is not hard to find that previous linear model may lead to incorrect conclusion. Actually, such incorrect use do occurred in published meta-analysis (10). In our opinion, the results of simple linear model should be treated with caution while piecewise linear model method should be widely considered in further dose-response meta-analysis.

Acknowledgements

None.

Stata code for piecewise model (one step method)

encode study, gen(studynum)
gen double logrr=log(adjrr)
gen double logub=log(ub)
gen double loglb=log(lb)
gen double se=(logub-loglb)/(2*invnormal(0.975))
mkspline linsp_dose1 7 linsp_dose2 = dose
reg logrr linsp*
lincom linsp_dose1*-1, eform
lincom linsp_dose2*1, eform

Stata code for simple linear model

encode study, gen(studynum)
gen double logrr=log(adjrr)
gen double logub=log(ub)
gen double loglb=log(lb)
gen double se=(logub-loglb)/(2*invnormal(0.975))
glst logrr dose, se(se) cov(n case) pfirst(id studynum) ts(r)
lincom dose*1, eform

Additional analysis

In our example, for the piecewise linear model, we got adjusted R² as 0.32. The results of piecewise linear model with the cut point of 7 hours were showed below (Figure S1).

We can set the cut point as the lowest dose (4 hours) to test the adjusted R² of simple linear model. As we can see that is omitted by the procedure and slope represents the slope of whole distribution (simple linear). The adjusted R² is 0.12, therefore, it is more reasonable to use the piecewise linear model here (Figure S2). The results show a wild different with the glst procedure (1.02 vs. 1.04), may be due to the procedure itself and a random effect model the former used but current one does not.

Footnote

Conflicts of Interest: The authors have no conflicts of interest to declare.

References

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