Intensive Longitudinal Reliability

The fitted dynamic model

Fit the model in Mplus and keep the .out file.
Drop it on the box below. The page lists every parameter it read.
Say which parameter plays which part: the variance of the person means, the autoregressive effect, the innovation variance and, when the model has one, the measurement error variance. The page proposes a reading from the names and you correct it.

Drop the Mplus .out file here, or press Enter to choose one.

The file is read in your browser and is not sent anywhere.

Reliability at each distance in time

With moments nested in days, the agreement between two measurements depends on whether they belong to the same day and on how many moments or days separate them. Lin, Tuerlinckx and Vanbelle (2026) define a family of coefficients for that design from the variance components of a three-level model with an autoregressive process at each time scale. The components are typed in here, because the model that yields them is not one Mplus fits in this form.

Between persons, σ²_s

Between days, σ²_d

Day-level process, τ²_d

Moment-level process, τ²_m

Measurement error, σ²_ε

Moment-level autoregression, φ_m

Day-level autoregression, φ_d

Moments per day

Methods and formulas

The variance of a repeated measurement

In a two-level autoregressive model with measurement error the observed score of person i at occasion t is the person's mean, plus a within-person fluctuation that follows an autoregressive process, plus measurement error. Taken over all persons and occasions, its variance divides into three parts (Schuurman & Hamaker, 2019, Eq. 5):

V (y) = ψ_{μ}^{2} + E [τ_{y}^{2}] + γ_{σ}^{2}

Here ψ²_μ is the variance of the person means, that is the between-person or trait variance; E[τ²_y] is the expectation over persons of the variance of the autoregressive process, that is the systematic within-person fluctuation; and γ_σ² is the expectation of the measurement error variance. For a first-order process the variance of the fluctuation of person i is the stationary variance of an autoregression, $τ_{y}^{2} = ζ_{i}^{2} / (1 - φ_{i}^{2})$ , with ζ² the innovation variance and φ the autoregressive effect; the multivariate form is Equation 6 of the same paper.

Two reliabilities, and what each is for

For measuring differences between persons the true score is the person's mean, and everything that fluctuates within the person is noise (Schuurman & Hamaker, 2019, Eq. 7):

{rel}_{b} (y) = \frac{ψ_{μ}^{2}}{V (y)}

For measuring the fluctuations of one person the true score is that person's position in the process at that moment, and only measurement error is noise (Eq. 9):

{rel}_{w} (y_{i}) = \frac{τ_{y i}^{2}}{τ_{y i}^{2} + σ_{ε i}^{2}}

The second one is person-specific: two people with the same measure can have different within-person reliabilities, because their processes differ in how much systematic fluctuation they carry. The page computes it at the average parameters, which is the reliability of an average member of the group; the reliability of a particular person needs that person's own estimates, which Mplus writes with SAVEDATA when the model has random innovation variances.

What measurement error does to the dynamics

Disregarding measurement error does not only inflate the noise: it biases the autoregressive effect toward zero by exactly the factor the within-person reliability gives (Schuurman & Hamaker, 2019, Eq. 10):

{\hat{φ}}_{i} = {rel}_{w} (y_{i}) φ_{i}

Here φ̂_i is the autoregressive effect a model without measurement error returns and φ_i the one of the process itself. A within-person reliability of .70 therefore turns a true carry-over of .50 into an estimate near .35, and the inertia of a person is understated by the same factor. The page reports both directions: what the estimate in the file implies about the process, and what a process with the reported parameters would look like if the error were ignored.

Two time scales

When measurements are taken at several moments within a day and over several days, the agreement between two of them depends on how far apart they are and on whether they fall on the same day. Lin, Tuerlinckx and Vanbelle (2026) define the coefficients of that design from the components of a three-level model with an autoregressive process at each scale. Writing σ²_s for the variance between persons, σ²_d for the variance between days within a person, τ²_d and τ²_m for the stationary variances of the day-level and moment-level processes, σ²_ε for the measurement error and φ_d and φ_m for the two autoregressive effects, the between-person coefficients are

ρ_{B M} (Δ) = \frac{σ_{s}^{2} + σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} φ_{m}^{Δ}}{σ_{s}^{2} + σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} + σ_{ε}^{2}}, ρ_{B D} (Δ) = \frac{σ_{s}^{2} + τ_{d}^{2} φ_{d}^{Δ}}{σ_{s}^{2} + σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} + σ_{ε}^{2}}

The first is the agreement between two moments of the same day separated by Δ moments, the second between two days separated by Δ days. The limit of the first as Δ goes to zero is what the authors call instantaneity: the agreement between a measurement and a hypothetical parallel one taken at the same instant, which differs from it only by the realized measurement error.

The within-person coefficients drop the between-person variance from both parts of the ratio, since they describe one person:

ρ_{M}^{W} (Δ) = \frac{σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} φ_{m}^{Δ}}{σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} + σ_{ε}^{2}}, ρ_{D}^{W} (Δ) = \frac{τ_{d}^{2} φ_{d}^{Δ}}{σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} + σ_{ε}^{2}}

The first is the agreement between two moments of one person separated by Δ moments, the second between two days of one person separated by Δ days.

Daily averages have their own day-to-day coefficients at both levels. Averaging M moments within each day divides the moment-level process variance and measurement-error variance by M. Between persons, the numerator retains the stable person variance; within a person, it does not:

ρ̄_D^B(Δ) = [σ²_s + τ²_dφ_d^Δ] / [σ²_s + σ²_d + τ²_d + τ²_m/M + σ²_ε/M]
ρ̄_D^W(Δ) = [τ²_dφ_d^Δ] / [σ²_d + τ²_d + τ²_m/M + σ²_ε/M]

Here M is the number of moments averaged within a day. These are Equations 4.7 and 4.14 in Lin, Tuerlinckx and Vanbelle (2026). They compare averages from two distinct days, so their day lag starts at one.

The final coefficient is the limit of the between-person coefficient as the distance grows without bound, which takes the autoregressive terms to zero:

\lim_{Δ \to \infty} ρ_{B M} (Δ) = \frac{σ_{s}^{2}}{σ_{s}^{2} + σ_{d}^{2} + τ_{d}^{2} + τ_{m}^{2} + σ_{ε}^{2}}

What is left is the share of the variance that belongs to the stable differences between persons: what the measure retains of a person over the long run.

Self-test

Append ?selftest=1 to run checks of output parsing, stationary variance, variance decomposition, reliability, attenuation, and the lag-dependent coefficients.

References

Asparouhov, T., Hamaker, E. L., & Muthén, B. (2018). Dynamic structural equation models. Structural Equation Modeling, 25(3), 359–388. https://doi.org/10.1080/10705511.2017.1406803
Lin, T.-Y., Tuerlinckx, F., & Vanbelle, S. (2026). Single-item reliability for intensive longitudinal data: A three-level linear mixed-effect autoregressive modeling approach. Psychometrika. https://doi.org/10.1017/psy.2026.10133
McNeish, D., & Hamaker, E. L. (2020). A primer on two-level dynamic structural equation models for intensive longitudinal data in Mplus. Psychological Methods, 25(5), 610–635. https://doi.org/10.1037/met0000250
Schuurman, N. K., & Hamaker, E. L. (2019). Measurement error and person-specific reliability in multilevel autoregressive modeling. Psychological Methods, 24(1), 70–91. https://doi.org/10.1037/met0000188