Hi Daniel,

this is an extremely interesting question which bothers me, too, at the moment. I am starting to dig myself into sensor-based data which always result in intensive longitudinal data.

I have no absolut answer yet, but I would not combine the daily measures into weekly ones. Rather I would start with individual time series models which can be expanded to multilevel models. The experience sampling literature provides some knowledge how do it (see the references)

With regard to the time variable: not as a factor (see the recommended book on time series). May be that you already know all of that. I am still at the beginning and have to do too many other things. The following references reflect my current strategy.

Best,

Holger

Bolger, N., & Laurenceau, J.-P. (2013). Intensive longitudinal methods: An introduction to diary and experience sampling research: Guilford Press.

Conner, T. S., Tennen, H., Fleeson, W., & Barrett, L. F. (2009). Experience sampling methods: A modern idiographic approach to personality research. Social and personality psychology compass, 3(3), 292-313.

Cowpertwait, P. S., & Metcalfe, A. V. (2009). Introductory time series with R. Dordrecht: Springer.

Csikszentmihalyi, M., & Larson, R. (2014). Validity and reliability of the experience-sampling method. In Flow and the foundations of positive psychology (pp. 35-54): Springer.

Dominici, F., McDermott, A., Zeger, S. L., & Samet, J. M. (2002). On the use of generalized additive models in time-series studies of air pollution and health. American Journal of Epidemiology, 156(3), 193-203. doi:10.1093/aje/kwf062

Dziak, J. J., Li, R., Tan, X., Shiffman, S., & Shiyko, M. P. (2015). Modeling intensive longitudinal data with mixtures of nonparametric trajectories and time-varying effects. Psychological Methods, 20(4), 444-469. doi:10.1037/met0000048

Hamaker, E., Asparouhov, T., Brose, A., Schmiedek, F., & Muthén, B. (2018). At the frontiers of modeling intensive longitudinal data: Dynamic structural equation models for the affective measurements from the COGITO study. Multivariate Behavioral Research, 1-22. doi:10.1080/00273171.2018.1446819

Jebb, A. T., Tay, L., Wang, W., & Huang, Q. (2015). Time series analysis for psychological research: examining and forecasting change. Frontiers in Psychology, 6, 727.

Shumway, R. H., & Stoffer, D. S. (2017). Time series analysis and its applications: with R examples: Springer.

Suk, H. W., West, S. G., Fine, K. L., & Grimm, K. J. (2018). Nonlinear growth curve modeling using penalized spline models: A gentle introduction. Psychological Methods, 24(3), 269-290.

Hi Holger,

thanks for the literatur recommendation and your advice. I think I'm gonna read a little more literature.

Best,

Daniel

Your question: Should I combine the daily measurements into weekly measurements

NO you are loosing valuable information; yo need to understand between individual variation and within individual between occasion variation.

Your question: Should I use the time variable (measurement oder week) as continous or as a factor?

A bit of both!

In a repeated measures random effects multilevel model you have occasions at level 1 nested within individuals at level 2.. The residual at the individual level will give 21 differences one for each individual around an overall intercept and the 98 by 21 occasion residuals will give differences from the individual overall mean (intercept plus individual residual) for each and every occasion.

You can then put in variables in the fixed part of the model to try and account for this differences in the fixed which may be a function of continuous time ( linear or quadratic etc to get the overall trends), categorical time (a dummy for weekend or morning) and dummies distinguishing the three groups. You may also need interactions eg a linear rend by group interaction but 21 is not many for this type of modelling.

It is also usual to fit a random-slope model whereby the differences between individual may increase over time. Finally if you have very close measurements with high temporal dependency you may need something more complicated that the standard compound symmetry of the standard mixed model, such as an autoregressive structure see

Book Developing multilevel models for analysing contextuality, he...

A potential sequence of models is given in my answer to this question:

https://www.researchgate.net/post/Repeated_measures_mixed_effects_model_How_to_interpret_SPSS_estimates_of_fixed_effects_for_treatment_vs_control_gender_interaction

These are two very good sites with lots of resources:

http://www.bristol.ac.uk/cmm/software/mlwin/mlwin-resources.html#multipleresp

http://www.pilesofvariance.com/

A note on the syntax, you should include your time variable both as fixed effect and random slope, if you have a "growth"-model (repeated measurements):

model1 = number of observations (you can omit that check, though). Rather, you could even think if you model time (at least in the fixed effects) as quadratic, cubic or spline:

library(splines)

model1

Thanks Kelvyn for the great advice. I think 21 participants is too small to fit a random-slope. I think so, because I get different results when fitting with lme or lmer (is this possible?).

fit1

When were the interventions? You seem to treat "week" as it has some special significance (as opposed to other cycles). Do you expect cyclical effects by week? Also, I assume you expect day k measures to "influence" day k+1 measures. Do you expect other things, like day k to influence day k+2 beyond the effects of day k+1. nlme/lme has a lot for these types of models.

Daniel Groß

> I think 21 participants is too small to fit a random-slope. I think so, because I get different results when fitting with lme or lmer (is this possible?).

Not necessarily. Different results are mostly due to different defaults (optimizer, correlation structure, ...). While lme gives you more options for correlation structures, lme4 can handle more complex random effects structures. A good alternativ that offers both is the "glmmTMB" package (also from some of the lme4-authors), which is currently probably one of the most advanced mixed modelling package in R (for frequentist framework).

In most longitudinal settings, it makes sense to assume that the dependent variable varies over time differently for each subject, thus I would suggest using time as random slope. Else, you assume that the "time trend" of your dependent variable is the same for each subject.

In case of convergence, you find some remedies here: ?lme4::convergence

For the second warning, standardizing your variable should help (either base::scale() or sjmisc::std() to standardize your variables).

With random slopes model failing to converge, it is often helpful to centre the continuous time variable around its overall mean. the Issue often is an impossible correlation between the random intercepts and slopes outside the range of +/- 1 , the random intercept being the value for when time is zero, centering puts this in the information-rich middle of the data, not at extreme lower edge.

if this does not work, let me know as there are other things that you can do.