CoTiMA: Continuous-Time Meta-Analysis Services
When studies measure the same longitudinal effect but at different time lags, their coefficients are not comparable—and pooling them directly is misleading. CoTiMA solves this by estimating an underlying continuous-time model, reconciling every study on a common time scale. It is among the most rigorous ways to synthesise longitudinal evidence.
CoTiMA (continuous-time meta-analysis) pools longitudinal effects from studies that used different time intervals between measurements. Instead of averaging lag-specific coefficients—which depend on the lag—it estimates an underlying continuous-time model (a drift matrix) that all studies share, making effects measured over different intervals directly comparable.
The problem CoTiMA solves
Longitudinal studies estimate how one variable affects another over time—a cross-lagged effect from an earlier measurement to a later one. But there is a hidden trap in synthesising them: the size of a cross-lagged coefficient depends on the time lag between measurements. A study measuring an effect over six months and one measuring it over two years will report different coefficients even if the underlying process is identical, simply because they used different intervals.
This means that pooling lag-specific coefficients in an ordinary meta-analysis compares things that are not comparable—averaging a six-month effect with a two-year effect produces a number that describes no real interval and can be seriously misleading. CoTiMA fixes this at the root. Instead of pooling the observed coefficients, it estimates the underlying continuous-time model (characterised by a drift matrix) that generated them. Because that model is defined independently of any particular lag, every study—whatever interval it used—informs the same parameters, and effects can then be recovered for any time interval you choose.
When to use it
CoTiMA is the right tool when your evidence base is longitudinal or cross-lagged panel studies that used different time intervals between waves, and you want to synthesise their dynamic effects properly. It is the honest answer to a question that trips up many longitudinal reviews: how do we combine studies whose time lags differ?
It is especially valuable when the process unfolds over time and the timing matters—reciprocal relationships, developmental or organisational change, and any theory expressed as effects that accumulate or decay. If all your studies happen to use the same interval, a conventional meta-analysis of the lag-specific effect may suffice; CoTiMA earns its sophistication precisely when the lags differ—which, in real literatures, they almost always do.
Conventional meta-analysis vs CoTiMA
| Conventional meta-analysis | CoTiMA | |
|---|---|---|
| What it pools | Lag-specific coefficients | The underlying continuous-time model |
| Different time lags | Treated as comparable (they aren’t) | Explicitly reconciled |
| Result | An average of non-comparable effects | Effects recoverable at any interval |
| Timing of the process | Ignored | Modelled directly (drift matrix) |
| Best for | Same-lag studies | Studies with varying intervals |
How CoTiMA works—and what it needs
CoTiMA fits a continuous-time structural equation model to the longitudinal information from each study simultaneously, estimating the parameters of an underlying continuous process rather than discrete lag-specific effects. The core quantity is the drift matrix, which describes how the system changes at every instant; from it, the model can generate the expected cross-lagged and auto-regressive effects for any chosen time interval. This is what makes studies with a six-month lag and a two-year lag contribute to the same estimate.
The method sits at the intersection of meta-analysis and continuous-time structural equation modeling, and it inherits requirements from both. It needs the longitudinal statistics from each study (typically the correlations or covariances among the repeated measures, with the exact time lags), a genuinely dynamic model that theory supports, and enough studies to estimate the continuous-time parameters. As with any pooled synthesis, heterogeneity across studies must be examined rather than assumed away.
A cross-lagged coefficient is meaningless without its time lag. Pooling six-month and two-year effects as if they were the same quantity is the error CoTiMA exists to prevent—it models the process, not the interval-bound snapshot.
Advanced, but not a black box
CoTiMA is a specialist method, and part of delivering it well is interpreting and communicating it clearly—reporting the drift parameters, the effects at interpretable intervals, and the model’s assumptions in language a reviewer and a reader can follow. We fit it in established, reproducible tools (R’s ctsem and the CoTiMA package) with versioned code.
How we deliver a CoTiMA
CoTiMA draws on both our meta-analysis and longitudinal & panel practices, run on a full systematic-review workflow—so the continuous-time model is estimated on a sound, reproducible synthesis.
We begin with a registered protocol and a clearly specified dynamic model, a comprehensive search, and careful extraction of each study’s longitudinal statistics with its exact time lags—the detail conventional reviews often miss. We then fit the continuous-time meta-analytic model across all studies and examine heterogeneity.
Reporting follows PRISMA standards alongside continuous-time modelling conventions, with the drift model, assumptions, and lag handling reported transparently.
You receive the estimated continuous-time (drift) parameters, the cross-lagged and auto-regressive effects recovered at interpretable time intervals, tests of heterogeneity, a clear interpretation of the dynamics, and reproducible code and data. The result is a longitudinal synthesis that respects time—defensible in the most methodologically demanding review.
CoTiMA across Management & Allied Studies
Reciprocal, time-dependent processes are central to social-science theory—and cross-lagged panel studies with differing intervals are common—so CoTiMA is widely applicable across the disciplines we serve.
Management & Organisational Studies
Reciprocal effects over time—such as engagement and performance—synthesised across studies with different follow-up intervals.
Applied & Organisational Psychology
A leading CoTiMA setting: dynamic constructs like stress, motivation, and well-being measured at varying lags.
Marketing & Consumer Research
Time-dependent processes such as satisfaction and loyalty, or advertising and behaviour, pooled across intervals.
Economics & Public Policy
Dynamic relationships whose estimated strength depends on the interval studied, reconciled on a common time scale.
Education & Learning Sciences
Developmental and reciprocal effects—such as motivation and achievement—measured over different periods.
Health & Behavioural Science
Longitudinal reciprocal processes where timing matters and study lags vary widely.
CoTiMA: common questions
ctsem framework and the CoTiMA package—and deliver versioned code so the analysis can be checked and rerun. We also report the drift parameters and the recovered effects at interpretable intervals so the results are usable, not just technically correct.Longitudinal studies with different time lags?
If your evidence base is cross-lagged panel studies measured over varying intervals, CoTiMA is the rigorous way to synthesise them. We design and deliver it—and translate the continuous-time results into findings you can report.