The ACT study is a longitudinal cohort study of older adults whose aim is to better understand aging, brain aging, and dementia (Kukull et al., 2002). ACT enrolls dementia-free individuals over age 65 who were randomly selected from members of the Kaiser Permanente Washington integrated health care system (originally Group Health). The original cohort (N = 2,581) was enrolled between 1994 and 1996, with additional enrollment 2000–2003 (expansion cohort, N = 811), and beginning in 2004 the ACT Study began ongoing enrollment with a goal to maintain at least 2000 at-risk individuals by replacing those who discontinued, died, or developed dementia (Kukull et al., 2002; Gray et al., 2013). Participants are invited to return for evaluation at 2-year intervals for the ultimate purpose of identifying incident cases of dementia. The study procedures were approved by the institutional review boards of Kaiser Permanente Washington (formerly Group Health Cooperative) and the University of Washington, and participants provide written informed consent.

In April 2016, the ACT-AM sub-study began inviting ACT participants to wear activPAL accelerometers for 7 days following their regular biennial assessment visits (Rosenberg et al., 2020). Persons who were wheelchair dependent, living in a nursing home, receiving hospice or care for another critical illness, or who showed evidence of cognitive problems during the biennial visit were not asked to participate. Persons who consented were instructed how to wear the device and how to complete a daily log that provided information about device use and time in bed. They also completed an additional take-home survey that included questions about self-reported physical activity and sedentary behavior. The device, daily log, and questionnaire were returned to the research team by mail after 1 week. Between 2016 and 2018, 1,135 participants consented to wear the activPAL. For these analyses, 1,034 participants with at least 4 valid days of activPAL wear data were included.

Details of the activity monitoring device protocols for the ACT-AM sub-study were described previously (Rosenberg et al., 2020). Briefly, waking activity was measured with a thigh-worn activPAL3 micro (PAL Technologies, Glasgow, Scotland, United Kingdom) worn on the front central thigh of either leg 24-h/day for ~1 week. As a thigh-worn accelerometer, the activPAL detects both movement and posture (i.e., sitting/lying, with the thigh horizontal, vs. standing, with the thigh vertical). Consequently, activPAL classifies behaviors at the 1-s level as either sitting, standing, or stepping.

Daily time in bed was measured as a surrogate for sleep via participant self-report using a paper log to record in-bed and out-of-bed times each day of wear. Daily wake and sleep times were not constrained to the 24-h day, and instead were defined by a participant’s daily in-bed and out-of-bed schedule, which meant the specific length of any given wear day for a participant varied and could have been greater or <24-h. We defined each full day as out-of-bed time to the next day’s out-of-bed time. Based on best practices for free-living activity measurement, a minimum of 4 days with 10 or more hours of waking wear time, as defined by the presence of valid device data during participant self-reported waking periods, was required to be included in analyses (Donaldson et al., 2016; Migueles et al., 2017).

We used proprietary PAL Technologies software to extract event-level files. Events files were then processed by collapsing consecutive activities of the same activity type using a batch processing package activpalProcessing in the R software (Lyden, 2016). Self-reported time in bed, which the device captures as sitting/lying time, was removed from calculation of waking activity metrics. For simplicity, we refer to time in bed as “sleep.” Maintaining the activity labels from the activPAL device, we defined the waking 24-h activity cycle as time sedentary (i.e., sitting/lying down), standing (which also includes very light movement), and physical active (i.e., stepping). Specifically, we calculated daily measures of total sitting time (min/day), total standing time (min/day), and total stepping time (min/day) during the waking hours of each valid day.

ACT participants are evaluated using the Cognitive Abilities Screening Instrument (CASI) at study entry and at each biennial follow-up visit. The CASI consists of 40 items that evaluate attention/concentration, orientation, short- and long-term memory, language abilities, visual construction, verbal fluency, and executive functioning (abstract reasoning and judgment; Teng et al., 1994). Raw CASI scores range from 0 to 100, with higher scores indicating better cognitive performance. For these analyses, the CASI was scored using item response theory (CASI-IRT), which corrects for potential issues related to unequal interval scaling (Crane et al., 2008). CASI-IRT scores were standardized based on the larger ACT cohort enrollment scores of the study sample such that a 1-unit difference in CASI-IRT can be interpreted as approximately a 1 standard deviation (SD) unit difference in cognitive performance (Crane et al., 2008; Ehlenbach et al., 2010).

We examined several other participant characteristics. Objective physical function was derived from a composite of three physical performance tasks completed at the ACT-AM enrollment visit: gait speed (average of two 10-ft timed walks), chair stand time (time needed to move from a seated position in a chair to a standing position, repeated five times), and grip strength as measured by handheld dynamometer (average of three attempts in the dominant hand; Rosenberg et al., 2020). Each task was scored from 0 to 4 points (higher = better) and then summed to create a physical function score ranging from 0 to 12. For these analyses, two score categories (any impairment 0 to 10; no impairment 11–12) were included. Ability to walk half a mile was based on a single self-reported yes/no item “Because of health or physical problems, do you have any difficulty walking one-half mile (about 5 or 6 blocks)?” (McCurry et al., 2002) For persons endorsing walking problems, level of difficulty was further differentiated (no difficulty, some difficulty, a lot of difficulty, unable).

In addition to self-reported daily time in bed used in the 24HAC, descriptive measures of participants’ typical sleeping patterns were collected via self-report. Self-reported sleep quality (very poor, poor, fair, good, very good) was based upon a single item from the PROMIS 8-item sleep disturbance scale (Buysse et al., 2010; Yu et al., 2012). A summary of average time in bed (<6 h, 6–9 h, >9 h) was derived from the daily log kept by participants during the week that they wore the accelerometer devices (described above).

Other participant characteristics were collected from the ACT study visit most proximal to the date of device wear, which was typically the first day of device wear. These included: age (<74 years, 75–84 years, 85+ years), sex (male vs. female), race/ethnicity (non-Hispanic White vs. other), education (some college/post-secondary education vs. high school education or less), measured body mass index (BMI, kg/m²), depressive symptoms from the 10-item Center for Epidemiologic Studies Depression Scale (CES-D; Andresen et al., 1994, Mohebbi et al., 2018), and self-rated overall health (Kohout et al., 1993).

Table 1 provides a summary of the subject characteristics, including univariate summaries of the time in each activity considered for the 24HAC from 1,034 participants with at least 4 valid days of activPAL wear data (90.6% of participants had a full week of 7 valid days or more of activPAL wear). The sample had mean age 77 years (SD = 7 years, range = [65, 100] years), 55.8% were female, 90% were White, 1.4% were Hispanic. 92.1% of the subjects reported good to excellent self-rated health, 74.6% had no difficulty walking half a mile, and the mean (SD) of CASI-IRT score was 0.61 (0.69) SD units. The mean (SD) time spent on each of the four behaviors, sit, stand, step, and sleep, was, 10 (2), 4 (1.6), 1.4 (0.7), and 8.5 (1.1) hours per day. The median [inter-quartile range] total time per day is 1,440 [1,436, 1,445] mins.

The 24HAC paradigm imposes a statistical challenge that prevents application of linear regression models with the time spent on each behavior entered in the model simultaneously as a covariate, due to collinearity. By dropping the intercept term, it becomes possible to include all behaviors in one model, often called a “partition model,” which can be used to estimate the association of each behavior by holding time in all other behaviors constant. However, when total time per day is fixed, it is not sensible to estimate the effect of a specific behavior without considering other behaviors it displaces. To obtain more reasonable interpretation, a mathematically equivalent model called the ISM was adopted, which estimates the “substitution effect” associated with reallocating time from one behavior to another (Mekary et al., 2009). ISM has been widely used in physical activity epidemiology. For example, the ISM approach was applied to examine the association of physical activity intensities with physical health and psychosocial well-being in older adults (Buman et al., 2010), and with cardiovascular disease risk biomarkers using the cross-sectional U.S. National Health and Nutrition Examination Survey data (Buman et al., 2014). Mekary et al. (2013) applied the ISM approach with a time to event outcome to physical activity data from the Nurses’ Health Study.

Consider an example where each minute of the 24-h day is classified into one of four activities: sleeping, sitting, standing, and stepping. The ISM is formulated by including the total activity and all but one of the activity variables – the activity you will explore displacing – in the model. For example, with a continuous health outcome an ISM that leaves out the time stepping can be formulated, as below:

where E(Y) abbreviates the conditional mean of the health outcome given the time allocation variables (Sit, Stand, Sleep, Total measured on the same unit, e.g., minutes in a 24-h day), and other covariates X. In our ACT data, the total amount of time per day, captured by Total, slightly varies across individuals according to their sleep schedules and is only ~24 h (see Table 1), and hence a separate intercept term can be included in the model without causing perfect collinearity. When Total is exactly a constant 24 h/day for every subject, only one of the intercept or Total terms can be included in the model; however, this is often not necessary if day length varies, say due to using out-of-bed to out-of-bed time (or in-bed to in-bed time) to define a day. The ISM is a standard regression model (e.g., a linear regression model in Equation (1)), so it can be easily fit by statistical software. By omitting one behavior, e.g., stepping in model (1), and controlling for the total time a day, there is no longer perfect collinearity and the coefficients can be interpreted as the estimated effect of time reallocation. In particular, β̂1 can be interpreted as follows: when comparing two populations with the same values on the covariates, and the same amount of time spent on standing and sleeping, but with one population spending 1 unit of time/day (e.g., 1 min/day) more in sitting and the same amount of time less in stepping than the other population, the estimated difference in the mean health outcome is β̂1 and similarly for β̂2 and β̂3. Such comparisons are referred to as the “substitution effect.” The coefficients for Total and the intercept in this model are not necessarily meaningful. When total time per day is a constant and there is no intercept term, β̂4 functions as the intercept, with the interpretation as the estimated mean outcome for a hypothetical population with 0 mean times spent on sitting, standing, and sleeping, all activity as stepping, all other continuous covariates set at 0 and categorical covariates set at their baseline levels. Each of the variables Sit, Stand, Sleep, and Step could be centered at a set of values (c1,c2,c3,c4), respectively that add to the fixed Total (e.g., 24 h), so that β̂4 is the expected outcome for the profile (c1,c2,c3,c4). Generally, however, ISMs are designed to answer scientific questions about effects of time reallocations between activity behaviors, without a particular focus on the intercept coefficient. Importantly, the time reallocations implied by the values of the beta coefficients should not be interpreted as causal effects when the model is fit to observational data, particularly when data are cross-sectional as in our illustration below, despite the commonly used language of “time allocation” in the typical application of ISM that seems to imply causality or an actual change in behavior.

The linear model assumption in model (1) implies a constant substitution effect between any two behaviors regardless of baseline value of the displaced behavior and a symmetric result when the substitution is reversed. This assumption is sometimes too stringent and unrealistic. An easy way to explore a potential nonlinear substitution effect is to fit separate ISMs in groups defined by different intensity levels of an activity behavior. For example, when analyzing the effects of reallocating time from stepping to other behaviors, data can be divided into two groups defined by whether mean step time is above or below a meaningful cut-off. Differential effect estimates from ISMs fit to the two groups can signal potential nonlinearity. Alternatively, a more flexible ISM could be fit with each activity term modeled by a spline function, while keeping the total activity as a linear term, as in Foster et al. (2020). The flexible ISM still enjoys the interpretation of substitution effects but avoids making the linear assumption. The optimal trade-off between smoothness and goodness of fit can be determined by either performing cross validation or minimizing the generalized cross validation criteria. The significance of the association for each behavior in the nonlinear ISM model can be tested via a Wald like test (Wood, 2006). The nonlinear ISM analysis can be done in R with package “mgcv” Sample R code for this analysis is provided on GitHub.²

Lastly, through combinations of the regression coefficients, the ISM can also be used to estimate the expected difference in mean outcome between two populations that have different 24-HAC profiles (i.e., different mean time spent on each behavior), but the same values for all other model covariates, which may also be of scientific interest.

CoDA is another widely used analytic approach to handle 24HAC data and its associations with health outcomes (Chastin et al., 2015; Biddle et al., 2018; Dumuid et al., 2018b; McGregor et al., 2021). Janssen et al., 2020 conducted a systematic review of CoDA studies examining associations of 24HAC with diverse health outcomes in adults. For CoDA, the fundamental unit of observation is the multivariate vector of the proportions or percentages of the 24 h that are spent in each type of activity. In the ACT study, we will consider the 24HAC composed of the sleep, sit, stand and step behaviors. One scientific question of interest in the 24HAC paradigm is how different profiles of 24HAC can affect a health outcome. One advantage of CoDA is that it provides a natural way to compare health outcomes between two compositions, including substitution of one behavior for another. For example, given a hypothetical baseline composition, e.g., 10 h (41.7%) sitting, 3 h (12.5%) standing, 2 h (8.3%) stepping, and 9 h (37.5%) sleeping, reallocating 2.4 h (10%) time per day from sitting to standing corresponds to altering the baseline composition into another composition, i.e., 7.6 h (31.7%) sitting, 5.4 h (22.5%) standing, 2 h (8.3%) stepping, and 9 h (37.5%) sleeping.

Before illustrating how a regression-based CoDA works, we first introduce a fundamental feature of CoDA called “scale invariance” (Aitchison, 1994). Scale invariance means that the total, or the absolute value in a composition, is irrelevant in the analysis and only the relative proportions are of consequence in an outcome model. For example, the composition of sedentary behavior, light-intensity physical activity, and moderate-to-vigorous physical activity (MVPA) is often of interest in a physical activity study. The following two activity compositions: 5 h (50%) sedentary behavior, 3 h (30%) light-intensity physical activity, 2 h (20%) MVPA and 2.5 h (50%) sedentary behavior, 1.5 h (30%) light-intensity physical activity, 1 h (10%) MVPA, will be modeled to have the same mean outcome by CoDA because the two compositions are equivalent. This may not always be a reasonable assumption, particularly when the total activity varies across individuals. For example, in some studies of physical activity (Chastin et al., 2015; Biddle et al., 2018; McGregor et al., 2021), an accelerometry device is worn to capture the percentages of time spent on different types of activities, but the device may not be worn for the whole day. The total time the device is worn will vary by individuals and the sampled composition may not be representative due to selection bias, e.g., people are more likely to wear the device when they are more active. When 24HAC is of interest, the total amount of time per day is the same, or is approximately the same, for most subjects (e.g., in the ACT data, see Table 1), thus, the scale invariance assumption is considered reasonable. In other settings, where the total time may vary, the scale invariance assumption may still be reasonable (i.e., the composition is still of direct interest); however, it may also be of interest to include the total time a device was worn as an additional covariate.

Visualizations and compositional descriptive statistics of 24HAC can be helpful, before fitting any models. Figure 1 displays 24HAC compositions of sit, stand, step, and sleep for our ACT cohort through so-called ternary diagrams, a common tool to visualize composition with 3 parts. Since the 24HAC of interest here consists of four activity behaviors, we plotted four ternary diagrams (Figures 1A–D), with each graph representing a sub-composition of three activity behaviors. From Figure 1, we can see how sub-compositions are distributed and possibly associated with the outcome. For example, Figure 1A, shows the 3-part sub-compositions formed by sit, stand, and step. The data points are colored to indicate the CASI-IRT outcome, as an exploratory look at the unadjusted association with this outcome. Most data points are located around the lower left corner of the diagram, indicating that most individuals spend relatively more time sitting than standing and stepping, and there is a tendency for higher CASI-IRT scores (more orange points) to be observed from individuals with higher percentages of stepping time. Similar patterns can be observed from Figures 1C,D where the other sub-compositions involving stepping are considered.

Commonly used compositional descriptive statistics are the compositional mean (center) and variation matrix. The compositional mean can be created by rescaling the vector of geometric means of each behavior, so that the sum of the scaled components equal 100% and the resulting vector is still a composition (Aitchison, 1994). Supplementary Appendix A1.1 shows that the compositional mean defined this way has a natural interpretation of the center of a sample of compositions. A variation matrix for the log-ratio (Aitchison, 1994) is used to describe the interdependence between every pair of behaviors (Biddle et al., 2018; Dumuid et al., 2020; McGregor et al., 2020). An off-diagonal value close to 0 means the two parts are highly proportional in the observed data. Supplementary Table S1 shows an example of the variation matrix for the sleep, sit, stand, and step 24HAC composition in the ACT study.

CoDA relies on the isometric log-ratio (ilr) transformation, which transforms each D-part composition to a unique D-1 vector on a new coordinate system where each new coordinate is a log-ratio which falls along the real line (Egozcue et al., 2003). An example of the ilr-transformation is given in equations (2–4), where z1, z2, z3 defines the new coordinates for the transformed data; the numbers preceding the log-ratios are normalizing constants, necessary for the desirable mathematical properties of the transformed coordinates (e.g., distance preserving orthonormal basis; Egozcue et al., 2003). See Supplementary Figure S1 for a visualization of this transformation and Supplementary Appendix A1.2 for a more detailed description of the general procedure for this transformation. (2)z1=34lnSit(Stand×Step×Sleep)1/3 (3)z2=23lnStand(Step×Sleep)1/2 (4)z3=12lnStepSleep

The coordinate z1, usually called the pivot coordinate, can be interpreted as the log-ratio between the numerator behavior and the geometric mean of the other (denominator) behaviors. A specific set of ilr-coordinates can be chosen to capture a particular comparison of geometric means to aid interpretation of that coordinate. Compositional data represented using different ilr-coordinates contain essentially the same information and analysis results will be the same regardless of which set of coordinates is used (Pawlowsky-Glahn et al., 2015). Since the ilr-transformation creates a set of continuous variables that are no longer collinear, the transformed compositional data can then be analyzed with standard techniques, e.g., multivariate analysis of variance (James test) or regression analysis.

LPA assumes that there is a latent categorical variable that classifies individuals into different subpopulations, thereby identifying groups of individuals with distinct patterns (profiles) of responses for a given set of continuous “indicator” variables. For example, in the 24HAC context, if the three indicator variables were percent of time spent in sedentary behavior, standing, and stepping during the 24-h day, different profiles may have different mean levels for each of these variables. LPA also generally assumes the indicator variables follow a finite mixture of multivariate normal distributions with each profile having its own specific mean vector and covariance matrix. It is generally assumed that mean vectors differ across profiles, but additional assumptions can be made with respect to whether or not variance matrices of profile indicator variables vary across profiles. Further details are provided in Supplementary Appendix A2.1. Once the number of latent profiles and variance–covariance structures are fixed, the latent profile model parameters can be estimated using standard statistical software, such as the tidyLPA package in R (Rosenberg et al., 2019), Mplus (Muthén and Muthén, 2017) and LatentGold (Vermunt and Magidson, 2021a). The fitted model yields the probability of each individual belonging to each of the different profiles (a posterior probability). It is worth mentioning that the parameters are estimated via the expectation–maximization (EM) algorithm (Dempster et al., 1977), an iterative estimation procedure. The EM algorithm requires initial parameter values and can be sensitive to the chosen start values in settings where there may not be a unique solution (solution only a locally, not globally, optimal fit). Thus, running LPA using multiple starting values is recommended until the best solution based on log-likelihood value can be replicated (Berlin et al., 2014). Further details will be discussed in our illustration below.

It is not a straightforward task to decide on the number of latent profiles or the variance–covariance structure. One common strategy is to assume a flexible (i.e., varying by class) variance–covariance structure and fit a series of models with different specifications for the number of classes. A final decision for class number is based on the following commonly used criteria: (1) model fit statistics, including Akaike’s Information Criterion (AIC; Akaike, 1987), Bayesian Information Criterion (BIC; Schwarz, 1978), sample-size adjusted BIC (Sclove, 1987), and the consistent AIC (Bozdogan, 1987); (2) statistical comparison between model fit of k profiles and k-1 profiles with p-value < 0.05 favoring the model with k profiles, using bootstrap likelihood ratio test (McLachlan and Peel, 2000) or the Lo–Mendell–Rubin likelihood ratio test (Lo et al., 2001); (3) statistics that weight model fit, parsimony, and the performance of the classification, e.g., integrated complete likelihood-BIC (Biernacki et al., 2000) (4) profile sample size: although distinct rare groups can exist, a small profile sample size may indicate potential overfitting; and (5) interpretability or clinical utility of resulting profiles. Nylund et al., 2007 performed a simulation study and advocated BIC and the bootstrap likelihood ratio test for selecting the correct number of classes. The simulation study in Tein et al. (2013) showed that AIC and entropy were not reliable criteria. Sometimes, subjective judgment based on a mixture of criteria is necessary. For instance, a subjective judgment could be made when deciding between k and k-1 profiles, whether the larger number of profiles generated groups that appeared sufficiently distinct, or whether one group appeared too small.

Because of the normality assumption of LPA, it is recommended to always inspect the empirical distributions of profile indicator variables both before and after applying LPA. Although the distribution of a mixture of normal distributions is likely to look nonnormal, which makes it difficult to check the normality assumption at the data pre-processing stage, a highly skewed or heavy tailed distribution often signals a violation of the assumption. In such cases, performing appropriate transformations, e.g., natural logarithm or square root transformation, and/or an outlier analysis is worth considering (Vermunt and Magidson, 2002).

As mentioned before, a key feature of 24HAC data is the co-dependence between activity behaviors. This co-dependence prevents us from applying LPA to all 24HAC variables because it will lead to a degenerate (rank-deficient) covariance matrix. Two possible solutions to consider are (1) apply LPA to the same ilr-transformed variables used in CoDA (see Section 3.2 more details of ilr-transformation; see Gupta et al., 2020; von Rosen et al., 2020) or (2) drop one activity behavior variable from the LPA (see Jago et al., 2018). Although the first approach retains the full composition, one potential drawback is that the ilr-transformation, involving log-ratios, could generate skewed profile indicators unsuitable for LPA. For example, if one behavior often had a relatively small proportion of the composition, this could create a large ratio for the ilr-coordinates with this behavior in the denominator. For the second approach, determining which variable to leave out from the LPA model can be made based on scientific considerations, e.g., prior belief of which set of activity behaviors are the driving indicators of underlying profiles or dropping a behavior due to being highly correlated with another behavior.

After obtaining a final model for latent profiles (i.e., the fitted parameters for the underlying normal distributions that define the profiles) (step 1), it is often of interest to explore associations between profile categories and covariates (e.g., age, sex) or outcomes (e.g., CASI-IRT score), the latter often called external variables in the latent profile literature. Because the fundamental output of LPA is only a probability of belonging to each profile for each individual, additional steps are required to do this association. The most common next step in the literature is to assign each individual to the profile with the highest posterior probability (also known as modal assignment) (step 2), followed by association analyses between the assigned class membership and external variables, such as a health outcome (step 3). However, this three-step approach can lead to bias and invalid confidence intervals because the second step treats the class membership as an observed, perfectly measured grouping variable, ignoring the potential uncertainty, i.e., classification error introduced in step 2 (Bolck et al., 2004; Vermunt, 2010). Thus, it is always recommended to adopt an analytic approach that accounts for the uncertainty in class membership assignments. It should be noted that estimating the association of a latent variable with external variables is still an area of active research. There are several papers that review and compare different approaches proposed over the last two decades to deal with external outcomes in LPA (Dziak et al., 2016; Collier and Leite, 2017) and latent class analysis (Nylund-Gibson and Choi, 2018; Nylund-Gibson et al., 2019; Bakk and Kuha, 2021). Note that LPA and latent class analysis are similar, differing only in the nature of their indicator variables (latent class analysis applies to categorical indicator variables), not in the way the latent variable is related to an external variable. Introducing and discussing each method is beyond the scope of this article. Current recommended practice is to use either the improved three-step maximum likelihood (ML)-based method (Vermunt, 2010) or Bolck, Croons, and Hagenaars (BCH) method (Bolck et al., 2004; Vermunt, 2010; Asparouhov and Muthén, 2014), which adjust for the uncertainty in the latent profile assignment. These methods assume conditional independence of covariates and profile indicators given the latent variable. For recent advances and discussion of more complex models, such as assuming covariates have direct effects on observed profile indicators, multilevel or latent transition models, or more than one latent class variables, refer to Bakk and Kuha (2021), Nylund-Gibson et al. (2019), Bray and Dziak (2018), and Vermunt and Magidson (2021b).

In our data analysis, LPA was performed in the Latent GOLD 6.0 software (Vermunt and Magidson, 2021a). The improved BCH method, which has been advocated as a preferred method for continuous external outcomes (Bakk and Kuha, 2021), was used to relate latent profiles with the outcome CASI-IRT score adjusted for age, sex, race/ethnicity, BMI, education level, depressive symptoms, and self-rated health. Latent GOLD syntax for this analysis is provided in Supplementary Appendix A2.2. We compared these results to those of the biased approach of ignoring the uncertainty in the class assignment. As a sensitivity analysis, we repeated this analysis with the ML-based instead of the BCH adjustment for the uncertainty in the latent-profile assignment using the same software. We also performed analyses of using each of the covariates as a predictor for latent profiles with the ML-based method, which can be done using the Latent GOLD “Step-3” module. p-values for all associations were reported based on the Wald-type test using robust standard error estimates [using robust standard error estimates is necessary as shown in Vermunt (2010) and Bakk et al. (2014) and is the default in Latent GOLD 6.0].

The main research question that both ISM and CoDA address regarding 24HAC data is the associated effect of time reallocation on the study outcome. Typically, ISM is focused on substituting time spent in one activity for another, whereas CoDA more naturally considers differences between two compositions. Both ISM and CoDA estimate the substitution effect in a standard regression framework and the usual assumptions would apply, e.g., for our ACT study example, usual linear model assumptions apply. ISM is generally conducted on the original time scale (e.g., hours of activity); whereas, CoDA models the proportion of time spent in each activity. Hence, CoDA is scale invariant, which implicitly means that the time information of 24HAC is irrelevant in CoDA, e.g., the amount of time spent on each behavior, which is an additional assumption that researchers should evaluate. In contrast to ISM and CoDA, LPA is a more exploratory method used to identify distinct latent subgroups with respect to activity profiles based on observed 24HAC data. Another aim of LPA is to explore the correlates of identified profiles and the associations of the profiles with outcomes. LPA assumes the observed data are sampled from a population composed of distinct subpopulations with heterogeneous distributions of activity behaviors of a day and models the data using a mixture of multivariate normal distributions. Hence, LPA results can be sensitive to the distributions of activity behaviors, while for ISM and CoDA, this assumption is not necessary if the outcome model is correctly specified. Secondary research questions of interest may relate to comparisons of descriptive statistics of the 24HAC by different population characteristics. We discuss this as a part of exploratory data analysis in the following section.

In any regression setting, exploratory data analysis is carried out to provide a summary of observed data and justification of the modeling assumptions needed for further analysis. By considering each activity behavior of the 24HAC univariately, we can plot the distribution of each behavior, e.g., in Supplementary Figure S2. Further bivariate association/stratification analysis can also be done to explore possible correlates of each activity behavior. Under CoDA, there are unique tools and descriptive statistics available. For example, ternary diagrams, e.g., Figure 1, can be used to visualize the distribution of 24HAC component behaviors. A compositional mean and variation matrix can be calculated overall and in subgroups as descriptive statistics to summarize central tendency and co-dependences of activity behaviors. By applying the ilr-transformation, statistical comparisons of compositional means across subgroups can also be done using standard multivariate analysis of variance, which we think is superior to the bivariate association analysis that considers each activity behavior univariately and does not account for the co-dependence in 24HAC data. For LPA, profile-specific estimated means and variance matrices for time spent in each activity from a final fitted model, as well as a boxplot plot (e.g., Figure 4), can describe the distributions of each activity behavior across profiles. Bivariate associations of covariates with the profiles can be explored by calculating the summary statistics of each covariate across the profiles, e.g., Table 4. However, it should be noted that directly performing bivariate association analysis with the assigned class and ignoring the class uncertainty can lead to biased and inaccurate results and should be verified using an adjusted regression method (Bolck et al., 2004; Vermunt, 2010). Some adjustment methods are recommended. See LPA Section 3.3. This uncertainty in the profile assignment makes exploratory data analysis more challenging in the context of LPA, since one may rely on the modal assignment, which is subject to misclassification.

Both ISM and CoDA estimate the effects of time reallocations through standard multivariable regression models; however, they handle the collinearity between activity behaviors differently. ISM is formulated by including the total activity and all but one of the activity variables – the activity you will explore reallocating. In contrast, CoDA considers 24HAC as a composition and applies the isometric log-ratio (ilr) transformation to transform the compositional data into a set of continuous variables in a lower dimensional space that standard statistical approaches can work with. Although the procedure and the interpretation of the 24HAC model coefficients are more complicated, regarding 24HAC as a composition facilitates the comparison of health outcomes between any two compositions. Such comparisons need to be made within the main range of data to avoid extrapolation. Furthermore, although linear relationships are assumed between ilr-coordinates and an outcome, when results are anti-logged, the effects of time reallocations are nonlinear with respect to the amount of time reallocated, e.g., see Figures 2, 3 for nonlinear and asymmetric effects. ISM, in contrast, standardly estimates the linear effects of time reallocation; however, both ISM and CoDA have potential to be more flexible, e.g., by including higher order, spline terms, or extending to more complex semi-parametric or non-parametric models. Although not illustrated in detail in Section 3, standard model checking and diagnostic tools for multivariable regressions largely apply to both ISM and CoDA. For LPA, instead of estimating the effects of time reallocations, we can only estimate associations between an outcome and the identified profiles by using an approach that accounts for potential misclassification due to the uncertainty in the profile assignment, e.g., the improved three-step approach (Vermunt, 2010) as in Table 5. Lastly, although our illustrations were with a continuous outcome variable, all the three methods are applicable when other common types of outcomes are used, e.g., dichotomous or time-to-event (Mekary et al., 2013; Lythgoe et al., 2019; Mcgregor et al., 2020). Again, it should be noted that it is more challenging to do model checking for LPA due to the uncertainty in the profile assignment. Each model diagnostic statistic would need to be adjusted for the potential for misclassification.

If a zero value occurs for an individual for any behavior in the composition or it is of interest to predict an outcome based on a fitted CoDA regression model for a population with time spent in any behavior close to zero, CoDA will have numerical problems because of its reliance on log-transformations. Even for ISM and LPA, common zero values could lead to skewed distributions and could be influential on final results. If zero observations account for a very small proportion of the observed data for a given behavior, then one approach can be to consider these values as below the limit of detection. In such cases, it is common practice to replace the zero values with a small value relative to the observed data, e.g., ½ limit of detection, but this may not always be sensible. Alternatively, values could be imputed by statistical imputation tools (Martín-Fernández et al., 2003; Palarea-Albaladejo et al., 2007; Palarea-Albaladejo and Martín-Fernández, 2008). Rasmussen et al. (2020) compare different approaches for handling zeros in compositional physical activity data and found that the simplistic approach of replacing zeros with a small fixed value led to more distortion of the underlying composition than other imputation approaches (Martín-Fernández et al., 2003; Palarea-Albaladejo et al., 2007; Palarea-Albaladejo and Martín-Fernández, 2008). If the occurrence of zero values is expected for one or more behaviors, CoDA may not be an appropriate approach. For example, if MVPA was considered as a component of the 24HAC for a given cohort with limited physical function, many individuals could have zero min/day in MVPA. In this case, CoDA could be applied if the 24HAC is redefined as a composition of behaviors that all members typically engage in, such as by merging two or more activity behaviors into a single behavior (e.g., stepping, which would include most MVPA, along with lighter-intensity movement) before conducting CoDA.

Detecting and capturing non-linear effects of time reallocations could be a challenge to both ISM and CoDA. Developing models allowing for non-linear effects of the 24HAC exposure but with good interpretability is a direction of future research. LPA is a compelling method by which to summarize distinct patterns of activity behavior in a population, but this approach has a number of statistical challenges: including the uncertainty of the latent class assignment leading to non-standard procedures for statistical inference for the association of latent profiles with an outcome, reliance of current methods on multivariate normality, the need to drop one of the activity behaviors in the 24HAC due to the collinearity, and created profiles being unique to the specific cohort understudy.