We performed a longitudinal meta-analysis to synthesize repeated measurements reported across studies with heterogeneous follow-up durations. The primary aim was to (i) estimate pooled trajectories over time and (ii) quantify change-from-baseline patterns, including post-discontinuation dynamics (weight regain / rebound), while accounting for non-aligned follow-up schedules.
For each study and timepoint we extracted: sample size (n), mean and standard deviation (SD), follow-up timing (weeks from baseline), treatment status at each timepoint (on-treatment vs post-discontinuation), when applicable, study-level descriptors (design, population characteristics), If medians (IQR) were reported, they were converted to mean (SD) using validated approaches.
Because follow-up timing varied across studies, we did not force exact alignment into a single “T1/T2” grid when timepoints were non-comparable. Instead, follow-up time (in weeks) was modeled as a continuous moderator in meta-regression, allowing estimation of pooled trends and a more homogeneous comparison across heterogeneous schedules.
Two complementary longitudinal syntheses were performed.
At each observed follow-up time, study-specific means were synthesized using random-effects models.
For each study \(i\) at follow-up time \(t\), the change score was defined as: \[ \Delta_{it} = \bar{Y}_{it} - \bar{Y}_{i0} \]
where \(\bar{Y}_{it}\) is the mean outcome at time \(t\) and \(\bar{Y}_{i0}\) is the baseline mean. Because most studies did not report the standard deviation of within-subject change, conservative variance handling based on aggregate information was pre-specified.
The sampling variance of the mean at each study-timepoint was computed as:
\[ v_{it} = \frac{SD_{it}^2}{n_{it}} \]
Random-effects models (DerSimonian–Laird) were used to estimate pooled means at each timepoint. Between-study heterogeneity was quantified through: I² statistic, Tau² estimator
To address heterogeneous follow-up durations, mixed-effects meta-regression models were fitted with follow-up time as a continuous moderator:
\[ y_{it} = \beta_0 + \beta_1 \cdot time_{it} + u_i + \varepsilon_{it} \]
Non-linear temporal patterns were explored using restricted cubic splines:
\[ y_{it} = \beta_0 + f(time_{it}) + u_i + \varepsilon_{it} \]
When post-discontinuation data were available, a binary indicator was defined:
\[ discont_{it} = \begin{cases} 0, & \text{during active treatment} \\ 1, & \text{after treatment discontinuation} \end{cases} \]
The following interaction model was evaluated: \[ y_{it} = \beta_0 + \beta_1 \cdot time_{it} + \beta_2 \cdot discont_{it} + \beta_3 (time_{it} \times discont_{it}) + u_i + \varepsilon_{it} \]
Because multiple timepoints per study violate independence assumptions, analyses accounted for within-study dependence through random-effects structures at the study level.
All analyses were conducted in R (version 4.3) using
the packages metafor, ggplot2,
splines, and dplyr.
dados <- tribble(
~study, ~group, ~outcome, ~time_weeks, ~n, ~mean, ~sd, ~post_discontinuation,
"Andersen_2021", "Overall", "sperm_conc", 0, 47, 78.5, 74.9, 0,
"Andersen_2021", "Overall", "sperm_conc", 8, 47, 91.7, 70.8, 0,
"Gregoric_2024", "Overall", "sperm_conc", 0, 30, 79.2, 55.6, 0,
"Gregoric_2024", "Overall", "sperm_conc", 12, 30, 96.7, 62.3, 0
)
dados <- dados %>%
mutate(vi = (sd^2) / n)
kable(dados, caption = "Extracted longitudinal data")| study | group | outcome | time_weeks | n | mean | sd | post_discontinuation | vi |
|---|---|---|---|---|---|---|---|---|
| Andersen_2021 | Overall | sperm_conc | 0 | 47 | 78.5 | 74.9 | 0 | 119.3619 |
| Andersen_2021 | Overall | sperm_conc | 8 | 47 | 91.7 | 70.8 | 0 | 106.6519 |
| Gregoric_2024 | Overall | sperm_conc | 0 | 30 | 79.2 | 55.6 | 0 | 103.0453 |
| Gregoric_2024 | Overall | sperm_conc | 12 | 30 | 96.7 | 62.3 | 0 | 129.3763 |
dados_delta <- dados %>%
group_by(study, group, outcome) %>%
mutate(
baseline_mean = mean[time_weeks == 0][1],
delta = mean - baseline_mean
) %>%
ungroup()
res_meta_reg <- rma.uni(
yi = mean,
vi = vi,
mods = ~ time_weeks,
method = "REML",
data = dados
)
summary(res_meta_reg)##
## Mixed-Effects Model (k = 4; tau^2 estimator: REML)
##
## logLik deviance AIC BIC AICc
## -6.5633 13.1266 19.1266 15.2060 43.1266
##
## tau^2 (estimated amount of residual heterogeneity): 0 (SE = 112.3273)
## tau (square root of estimated tau^2 value): 0
## I^2 (residual heterogeneity / unaccounted variability): 0.00%
## H^2 (unaccounted variability / sampling variability): 1.00
## R^2 (amount of heterogeneity accounted for): 0.00%
##
## Test for Residual Heterogeneity:
## QE(df = 2) = 0.0074, p-val = 0.9963
##
## Test of Moderators (coefficient 2):
## QM(df = 1) = 2.1006, p-val = 0.1472
##
## Model Results:
##
## estimate se zval pval ci.lb ci.ub
## intrcpt 78.9776 7.3012 10.8171 <.0001 64.6676 93.2876 ***
## time_weeks 1.5166 1.0464 1.4494 0.1472 -0.5343 3.5675
##
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1All analyses and visualizations were performed in R (version
4.3) using the packages metafor,
ggplot2, splines, and dplyr.