# Time Varying Mediation Function: Continuous Outcome and Three Treatment Groups

yajnaseni.chakraborti@temple.edu dcoffman@temple.edu harryzobeljr@gmail.com

## Introduction

The purpose of this vignette is to provide users with a step-by-step guide for performing and interpreting the results of a time varying mediation analysis with three treatment (exposure) groups and a continuous outcome. Please note, that this package has been built considering the structure of panel data, where each subject/participant has repeated responses (collected over time) for the outcome and mediator variables. We do not address dynamic treatment regimens in this package. Therefore, we assume the scenario where the treatment (exposure) is time-invariant (i.e., does not change over time). For more details, refer to Cai et al., 2020 (the paper can be obtained at https://arxiv.org/abs/2008.11797).

### Data

For illustration, we rely on an example dataset, simulated based on the Wisconsin Smokers’ Health Study 2 data (Baker et. al., 2016) which includes 1086 individuals assigned to one of three treatment conditions. One-third of participants received only a nicotine patch; another one-third received varenicline, and the final third of participants received a combination nicotine replacement therapy (NRT) which included nicotine patch + nicotine mini-lozenge. The outcome of interest is cessation fatigue; that is, how tired a participant felt of trying to quit smoking (7-point Likert scale). In addition, mediator variables were measured by asking participants if they felt a negative mood in the last fifteen minutes, and whether they wanted to smoke in the last 15 minutes, also recorded on a 7-point Likert scale. Both the outcome and mediator variables were assessed two times per day for the first two weeks post-quit day (rendering 30 time points of response since assessments start on day 0 post target quit day), and every other day (2x per day) for weeks 3 and 4 (rendering 14 time points of response).

A traditional approach to analyzing this type of data would be to use mediation analysis in which the effects are assumed to not vary as a function of time. First, a single (i.e., time-invariant) direct effect would be calculated by regressing the outcome on the treatment condition and mediator. Next, a time-invariant indirect effect would be computed by multiplying the effect of treatment condition on the mediator by the effect of the mediator on the outcome. However, this method potentially misses important information about the dynamic effect that a mediator may have over time. Specifically, we hypothesize that mood changes across and within days and thus, its mediating effect on one’s success of quitting smoking is likely to vary over time. We therefore propose a time varying mediation analysis which estimates the mediation effect as a function that varies over time.

## Getting started

To use the time varying mediation analysis package in R, you must first install the package and load it. Before that, make sure you have R version 4.0.3 or greater. There are two ways to install the package from the CRAN (Comprehensive R Archive Network) repository, by using install.packages or the devtools function.

install.packages("tvmediation", dependencies = TRUE)
library(tvmediation)

The equivalent code using devtools is:

devtools::install_cran("tvmediation", dependencies = TRUE)
library(tvmediation)

If you do not have devtools installed and loaded, you can do so using the following code:

install.packages("devtools", dependencies = TRUE)
library(devtools)

Alternatively, if you want to install the package directly from the GitHub repository to access new or revised functions in development, use the following code:

devtools::install_github("dcoffman/tvmediation", dependencies = TRUE)
library(tvmediation)

## Formatting your data before calling the tvma_3trt function

Once installed, you can type ?tvmediation in the console to view the package documentation, as well as links to the important functions and data included in the package. The time-varying mediation analysis for continuous outcomes and three exposure groups, relies on two user functions tvma_3trt and LongToWide as well as a number of internal functions of the tvmediation package.

The tvma_3trt function has five required and six optional arguments.

1. T1 A vector indicating assignment to treatment 1
2. T2 A vector indicating assignment to treatment 2
3. t.seq The numeric vector of the time sequence of the measures
4. mediator The matrix of mediator values in wide format
5. outcome The matrix of outcome values in wide format

The optional arguments are:

1. t.est The time sequence for estimation. This is by default equal to t.seq.
2. plot TRUE or FALSE for plotting the mediation effect. The default value is “FALSE”.
3. CI “none” or “boot” method of deriving confidence intervals (CIs). The default value is “boot”.
4. replicates Number of replicates for bootstrapping CIs. The default value is 1000.
5. grpname Name of the treatment (exposure) groups to be displayed in the results. The default value is “T”.
6. verbose TRUE or FALSE for printing results to screen. The default value is “FALSE”.

The dataset we will use for our illustration is named smoker and is included in the package.

To load the simulated dataset smoker.rda, type:

data(smoker)

The smoker data frame is organized in long format with SubjectID repeating over multiple rows for each participant. The tvma_3trt function requires that the data be in wide format to estimate the time varying coefficients. The tvmediation package includes a useful function LongToWide to help users properly format their data for analysis.

LongToWide has three required arguments and a fourth optional argument.

1. subject.id specifies the column of subject identifiers.
2. time.sequences specifies the column of measurement times.
3. outcome specifies the column for the variable (either the outcome or the mediator) that will be transposed.
4. verbose is an optional argument that if “TRUE” prints the output of LongToWide to the console. The default value is “FALSE”.

The output of LongToWide is a matrix of data in wide format where columns represent the subjects and rows represent the time sequence. Thus, each cell contains the j-th subject’s response at the i-th time point.

The tvma_3trt function requires two matrices, one for the mediator, and one for the outcome. Thus, we use the LongToWide function twice as illustrated below:

mediator <- LongToWide(smoker$SubjectID, smoker$timeseq,
smoker$NegMoodLst15min) outcome <- LongToWide(smoker$SubjectID, smoker$timeseq, smoker$cessFatig)
class(mediator)
## [1] "matrix" "array"
mediator[1:16, 1:10]
##     1 2  3  4  5  6  7  8  9 10
## 0   7 1  1  1  1 NA  4  1  1  3
## 0.5 2 1  1  6  1 NA NA  1  1  1
## 1   1 1  2 NA NA  1  3  1  1  1
## 1.5 1 1  1  1  1  1  2  1 NA  1
## 2   1 1  1  1 NA  1  1  1  1  1
## 2.5 3 1  1  1  1  3  1  1  1  1
## 3   1 6 NA  1  1  1  1  2 NA  1
## 3.5 1 1  2  6  3  7  1  6  1  1
## 4   1 1  1  1  1  1  1  1  2  4
## 4.5 1 1  1  5  1  5  2 NA  1  3
## 5   4 1  2  1 NA  5  2  1  1  1
## 5.5 1 1  2  1  1  2 NA  1  1  2
## 6   2 1  1  4  1  7  1 NA  2  3
## 6.5 1 1  1  1  1  1  1  5  3  1
## 7   1 2  1 NA  1  1 NA  1  1  4
## 7.5 2 2  1  1  1 NA  1  1  1  6
class(outcome)
## [1] "matrix" "array"
outcome[1:16, 1:10]
##     1 2  3  4  5  6  7  8  9 10
## 0   1 6  1  2  1 NA  1  6  7  3
## 0.5 1 3  1  1  1 NA NA  1  7  1
## 1   1 1  1 NA NA  1  1  7  7  1
## 1.5 6 7  6  1  1  6  6  4 NA  7
## 2   7 7  1  1 NA  1  7  1  1  1
## 2.5 1 2  1  7  1  6  1  2  1  7
## 3   4 1 NA  1  7  2  1  4 NA  5
## 3.5 5 3  7  7  6  1  1  7  5  1
## 4   6 3  5  4  7  2  7  5  7  5
## 4.5 7 7  1  7  7  1  7 NA  4  1
## 5   7 7  1  1 NA  1  1  7  1  1
## 5.5 6 5  1  1  1  3 NA  5  7  1
## 6   1 1  1  7  5  1  7 NA  2  1
## 6.5 5 7  5  7  7  1  1  1  7  7
## 7   1 1  1 NA  7  1 NA  1  7  7
## 7.5 5 1  2  1  1 NA  1  1  1  7

If your data are already in wide format, there is no need to use the LongToWide function and you can simply subset your dataset. However, mediator and outcome must be of class matrix; hence make sure you convert the class of the subsetted mediator and outcome objects to matrix before proceeding. This can be done using the R function as.matrix.

The tvma_3trt function requires three more variables that we have not yet created:

1. T1 A binary numeric vector with treatment1 schedule
2. T2 A binary numeric vector with treatment2 schedule
3. t.seq A numeric vector of the time sequence of the measures

When there are three treatment groups, we can create two dummy variables NRT1 and NRT2 corresponding to the treatment (exposure) groups other than the placebo or, in this case, the nicotine patch only standard of care. NRT1 = 1 indicates that the participant has been given varenicline and NRT1 = 0 for no varenicline. Similarly, NRT2 = 1 indicates that the participant received combination NRT and NRT2 = 0 indicates that the participant did not receive the combination treatment (comboNRT). Naturally, NRT1 = 1 and NRT2 = 1 are mutually exclusive; that is, a participant cannot have NRT1 = 1 and NRT2 = 1 simultaneously. If NRT1 = 1 then for that participant NRT2 must be equal to 0. However, NRT1 = 0 and NRT2 = 0 are not mutually exclusive. When NRT1 = 0 and NRT2 = 0 then the participant has received the placebo or the standard of care, in this case nicotine patch only. Thus if there are three treatment groups, two dummy variables need to be created indicating NRT1 or treatment1 and NRT2 or treatment2. In the smoker.rda dataset, three columns indicating the assignment of the three treatments are present, such that we can use any two of the columns and the omitted column becomes the reference group.

We create two dummy variables to indicate whether a participant was given varenicline or combination NRT. Because treatment is time-invariant, we need only one instance of each subject’s response for the treatment group (e.g. varenicline or combination NRT). We then convert it to a numeric value and subtract 1 to yield a vector of zeros and ones, as shown below.

# Step 1: Since each subject has multiple rows of data, extract the unique response of each subject to receiving varenicline. The data is still in dataframe format.
trtv <- unique(smoker[ , c("SubjectID","varenicline")])
trtv[1:10,]
##    SubjectID varenicline
## 1          1          No
## 2          2         Yes
## 3          3         Yes
## 4          4          No
## 5          5         Yes
## 6          6         Yes
## 7          7         Yes
## 8          8         Yes
## 9          9         Yes
## 10        10         Yes
# Step 2: 2 to those subjects who received varenicline and 1 to the rest. The data is now in vector format.
trtv2 <- as.numeric(trtv[ , 2])
trtv2[1:10]
##  [1] 2 1 1 2 1 1 1 1 1 1
# Step 3: subtract 1 from these numeric responses and procure a vector of zeros and ones
NRT1 <- trtv2 -1
NRT1[1:10]
##  [1] 1 0 0 1 0 0 0 0 0 0
# Step 1: Since each subject has multiple rows of data, extract the unique response of each subject to receiving comboNRT. The data is still in dataframe format.
trtc <- unique(smoker[ , c("SubjectID","comboNRT")])
trtc[1:10,]
##    SubjectID comboNRT
## 1          1      Yes
## 2          2       No
## 3          3      Yes
## 4          4      Yes
## 5          5       No
## 6          6      Yes
## 7          7      Yes
## 8          8      Yes
## 9          9      Yes
## 10        10      Yes
# Step 2: 2 to those subjects who received comboNRT and 1 to the rest. The data is now in vector format.
trtc2 <- as.numeric(trtc[ , 2])
trtc2[1:10]
##  [1] 1 2 1 1 2 1 1 1 1 1
# Step 3: subtract 1 from these numeric responses and procure a vector of zeros and ones
NRT2 <- trtc2 -1
NRT2[1:10]
##  [1] 0 1 0 0 1 0 0 0 0 0

This steps can be alternatively collated into a single step and written as follows:

NRT1 <- as.numeric(unique(smoker[ ,c("SubjectID","varenicline")])[,2])-1
NRT2 <- as.numeric(unique(smoker[ ,c("SubjectID","comboNRT")])[,2])-1
NRT1[1:10]
##  [1] 1 0 0 1 0 0 0 0 0 0
NRT2[1:10]
##  [1] 0 1 0 0 1 0 0 0 0 0

To generate t.seq we found the unique instance of each time point and then sorted from smallest to largest. There are 44 unique time points in the dataset where 0 after the decimal indicates the morning measurement and 5 after the decimal indicates the evening measurement recorded for that day.

t.seq <- sort(unique(smoker$timeseq)) t.seq ## [1] 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 ## [16] 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.5 13.0 13.5 14.0 14.5 ## [31] 16.0 16.5 18.0 18.5 20.0 20.5 22.0 22.5 24.0 24.5 26.0 26.5 28.0 28.5 We are now ready to perform the time varying mediation analysis. ## Calling the tvma_3trt function As discussed earlier, the tvma_3trt function has five required and six optional arguments. 1. T1 A vector indicating assignment to treatment 1 2. T2 A vector indicating assignment to treatment 2 3. t.seq The numeric vector of the time sequence of the measures 4. mediator The matrix of mediator values in wide format 5. outcome The matrix of outcome values in wide format The optional arguments are: 1. t.est The time sequence for estimation. This is by default equal to t.seq. 2. plot TRUE or FALSE for plotting the mediation effect. The default value is “FALSE”. 3. CI “none” or “boot” method of deriving CIs. The default value is “boot”. 4. replicates Number of replicates for bootstrapping CIs. The default value is 1000. 5. grpname Name of the treatment (exposure) groups to be displayed in the results. The default value is “T”. 6. verbose TRUE or FALSE for printing results to screen. The default value is “FALSE”. The argument grpname is important if the user wants the results from the bootstrapped CIs for the mediation effects to be displayed with a particular name in the Estimates dataframe and plots. For example, if the user defines grpname = "exposure", instead of the default value T, the effect of this choice will be reflected in the column names; CI.upper.T1 will be CI.upper.exposure1 and so forth. The plot titles will also change accordingly. For our illustration we use NRT since the example is drawn from a Nicotine Replacement Therapy study. We will call the function with additional optional arguments plot=TRUE and replicates = 250. We decreased the number of bootstrap replicates so that this vignette compiles faster but we suggest using at least 500 replicates in an actual analysis. The remaining optional arguments are left to their respective default values. results_tvma_3trt <- tvma_3trt(NRT1, NRT2, t.seq, mediator, outcome, plot = TRUE, replicates = 250, grpname = "NRT") ## Results The tvma_3trt function returns a list of results that include: 1. hat.alpha1 estimated time-varying effect of treatment 1 on the mediator 2. CI.lower.alpha1 CI lower limit for hat.alpha1 3. CI.upper.alpha1 CI upper limit for hat.alpha1 4. hat.alpha2 estimated time-varying effect of treatment 2 on the mediator 5. CI.lower.alpha2 CI lower limit for hat.alpha2 6. CI.upper.alpha2 CI upper limit for hat.alpha2 7. hat.gamma1 estimated time-varying direct effect of treatment 1 on the outcome 8. CI.lower.gamma1 CI lower limit for hat.gamma1 9. CI.upper.gamma1 CI upper limit for hat.gamma1 10. hat.gamma2 estimated time-varying direct effect of treatment 2 on the outcome 11. CI.lower.gamma2 CI lower limit for hat.gamma2 12. CI.upper.gamma2 CI upper limit for hat.gamma2 13. hat.tau1 estimated time-varying total effect of treatment 1 on the outcome 14. CI.lower.tau1 CI lower limit for hat.tau1 15. CI.upper.tau1 CI upper limit for hat.tau1 16. hat.tau2 estimated time-varying total effect of treatment 2 on the outcome 17. CI.lower.tau2 CI lower limit for hat.tau2 18. CI.upper.tau2 CI upper limit for hat.tau2 19. hat.beta estimated time-varying effect of the mediator on the outcome 20. CI.lower.beta CI lower limit for hat.beta 21. CI.upper.beta CI upper limit for hat.beta 22. hat.mediation1 time varying mediation effect of treatment 1 on the outcome 23. hat.mediation2 time varying mediation effect of treatment 2 on the outcome Optional returns based on argument CI = "boot" include: 1. CI.lower.T1 CI lower limit for the time varying mediation effect of treatment 1 2. CI.upper.T1 CI upper limit for the time varying mediation effect of treatment 1 3. CI.lower.T2 CI lower limit for the time varying mediation effect of treatment 2 4. CI.upper.T2 CI upper limit for the time varying mediation effect of treatment 2 5. SE_MedEff1 estimated standard error of the time varying mediation effect for treatment 1 6. SE_MedEff2 estimated standard error of the time varying mediation effect for treatment 2 The above estimates are compiled in a single dataframe which can be accessed using nameOfStoredResultsObj$Estimates. The following line of code displays only the estimates at the first 6 time-points.

head(results_tvma_3trt$Estimates) ## timeseq hat.alpha1 CI.lower.alpha1 CI.upper.alpha1 hat.alpha2 CI.lower.alpha2 ## 1 0.0 -0.7157486 -0.4737295 -1.1250982 -0.8871672 -0.3263855 ## 2 0.5 -0.7000508 -0.4793805 -0.9592556 -0.7883056 -0.3891667 ## 3 1.0 -0.6924259 -0.5339295 -0.8747141 -0.7115165 -0.4572214 ## 4 1.5 -0.6913309 -0.5706236 -0.8181397 -0.6621873 -0.5053024 ## 5 2.0 -0.6949645 -0.5864469 -0.7899621 -0.6423338 -0.5178458 ## 6 2.5 -0.7014576 -0.6204193 -0.7776869 -0.6489869 -0.5475813 ## CI.upper.alpha2 hat.gamma1 CI.lower.gamma1 CI.upper.gamma1 hat.gamma2 ## 1 -1.2391513 0.5192259 0.8417433 0.02158184 0.6184312 ## 2 -1.0387556 0.5979069 0.8577297 0.25797668 0.6632065 ## 3 -0.8786651 0.6577964 0.8452876 0.43418577 0.6978976 ## 4 -0.7912214 0.6981700 0.8496592 0.54920643 0.7219649 ## 5 -0.7522936 0.7199237 0.8482852 0.59725258 0.7358780 ## 6 -0.7417643 0.7253630 0.8444562 0.61209312 0.7408749 ## CI.lower.gamma2 CI.upper.gamma2 hat.tau1 CI.lower.tau1 CI.upper.tau1 ## 1 0.8835678 0.2723078 -0.271779695 0.07342616 -1.2035293 ## 2 0.8555217 0.4031040 -0.174314576 0.09232692 -0.9369578 ## 3 0.8569076 0.5000175 -0.097974873 0.12396106 -0.6221263 ## 4 0.8718653 0.5799488 -0.042454107 0.17282037 -0.3785643 ## 5 0.8557381 0.6132280 -0.006592278 0.17755868 -0.2111034 ## 6 0.8513722 0.6315880 0.011546099 0.17519121 -0.1271759 ## hat.tau2 CI.lower.tau2 CI.upper.tau2 hat.beta CI.lower.beta ## 1 -0.379043226 0.2352917 -0.8929108 0.1854940 0.2763794 ## 2 -0.258942738 0.1434451 -0.6782298 0.1919473 0.2576013 ## 3 -0.149034271 0.1220041 -0.4756705 0.1949410 0.2500705 ## 4 -0.057886622 0.1544505 -0.2943409 0.1965561 0.2461197 ## 5 0.008665267 0.1886272 -0.1673289 0.1983277 0.2442402 ## 6 0.048551637 0.2303431 -0.1051564 0.2012113 0.2466614 ## CI.upper.beta hat.mediation1 SE_MedEff1 CI.lower.NRT1 CI.upper.NRT1 ## 1 0.08180988 -0.1327670 0.04486634 -0.2253446 -0.05091982 ## 2 0.12391649 -0.1343729 0.03023095 -0.1970266 -0.08124524 ## 3 0.14569337 -0.1349822 0.02156411 -0.1786543 -0.09905334 ## 4 0.15695781 -0.1358853 0.01704610 -0.1677488 -0.10611347 ## 5 0.16345260 -0.1378307 0.01496486 -0.1653311 -0.10765707 ## 6 0.16958086 -0.1411412 0.01416942 -0.1672116 -0.10959277 ## hat.mediation2 SE_MedEff2 CI.lower.NRT2 CI.upper.NRT2 ## 1 -0.1645642 0.05248490 -0.2553042 -0.06603817 ## 2 -0.1513131 0.03489355 -0.2131060 -0.08326861 ## 3 -0.1387037 0.02403887 -0.1879364 -0.09289220 ## 4 -0.1301569 0.01835298 -0.1667433 -0.09594698 ## 5 -0.1273926 0.01578373 -0.1573869 -0.09571622 ## 6 -0.1305835 0.01450536 -0.1594585 -0.10194035 At each time point of interest timeseq = t.est, which in this case is equal to t.seq, the effects of the treatment on the mediator, the treatment on the outcome (adjusted and not adjusted for mediator), and the mediator on the outcome are estimated along with the respective 95% CIs. The CIs are computed via a non-parametric bootstrap method (Efron and Tibshirani, 1986), drawing samples of size 1086 from the original sample with replacement, estimating the sample mean and then applying the percentile method to compute the 95% CIs. Note that the CIs for the alpha, gamma, beta and tau coefficients (hat.alpha.1, hat.alpha.2, hat.gamma1, hat.gamma2, hat.tau1, hat.tau2, hat.beta) are computed regardless of the value of CI argument in the function. hat.mediation1 and hat.mediation2 are the estimated mediation effects of NRT1 (varenicline) and NRT2 (combination NRT) compared to the placebo (nicotine patch only) that varies over t.est. For CI = "boot" (which is the default option unless the user chooses otherwise) the standard errors of the estimated mediation effects and 95% CIs are estimated via a similar bootstrapping technique described earlier for the coefficients. If plot = TRUE, the results will also include the following figures: 1. plot1_a1 plot for hat.alpha1 with 95% CIs across timeseq 2. plot2_a2 plot for hat.alpha2 with 95% CIs across timeseq 3. plot3_g1 plot for hat.gamma1 with 95% CIs across timeseq 4. plot4_g2 plot for hat.gamma2 with 95% CIs across timeseq 5. plot5_t1 plot for hat.tau1 with 95% CIs across timeseq 6. plot6_t2 plot for hat.tau2 with 95% CIs across timeseq 7. plot7_b plot for hat.beta with 95% CIs across timeseq 8. MedEff_T1 plot for hat.mediation1 across timeseq 9. MedEff_T2 plot for hat.mediation2 across timeseq 10. MedEff_CI_T1 plot for hat.mediation1 with 95% CIs across timeseq 11. MedEff_CI_T2 plot for hat.mediation2 with 95% CIs across timeseq We recommend using the plots to interpret your findings as it may be difficult to derive meaning from the numerical values alone. To display the plots, use nameOfStoredResultsObj$ followed by the name of the plot to access the required plot accordingly. For example:

results_tvma_3trt$plot1_a1 In the above plot, the effect of varenicline on subjects’ feeling of negative mood in the last fifteen minutes varies somewhat over time. The effect is negative compared to the effect of nicotine patch only. That is, those assigned to varenicline have a less negative mood in the last 15 min. compared to those assigned to the nicotine patch only group. results_tvma_3trt$plot2_a2

In the above plot, the effect of combination NRT on subjects’ feeling of negative mood in the last fifteen minutes varies over time. The effect is negative compared to the effect of nicotine patch only. That is, those assigned to combination NRT have a less negative mood in the last 15 min. compared to those assigned to the nicotine patch only group.

results_tvma_3trt$plot3_g1 The above plot shows the time-varying direct effect of varenicline on the outcome cessation fatigue is positive compared to the direct effect of nicotine patch only. That is, those assigned to varenicline have greater cessation fatigue (due to factors other than negative mood) compared to those in the nicotine patch only group. results_tvma_3trt$plot4_g2

The above plot shows the time-varying direct effect of combination NRT on the outcome cessation fatigue is positive compared to the direct effect of nicotine patch only. That is, those assigned to combination NRT have greater cessation fatigue (due to factors other than negative mood) compared to those in the nicotine patch only group.

results_tvma_3trt$plot5_t1 The above plot shows the time-varying total effect of varenicline on the outcome cessation fatigue, which is not statistically significant over much of the time. The estimated 95% CI covers 0 (no effect), except during days 6-9. results_tvma_3trt$plot6_t2

The above plot shows the time-varying total effect of combination NRT on the outcome cessation fatigue, which is not statistically significant over time. The estimated 95% CI covers 0 (no effect).

results_tvma_3trt$plot7_b In the above plot, the time-varying effect of subjects’ negative mood in the last fifteen minutes on the outcome cessation fatigue increases slightly until around day 6-7 (end of week 1) and then decreases beginning around day 20-21 (end of week 3). results_tvma_3trt$MedEff_T1

results_tvma_3trt$MedEff_CI_T1 In the above plots, the time-varying effect of varenicline on cessation fatigue that is mediated by the negative mood in the last fifteen minutes is negative indicating that in comparison to the nicotine patch only group, varenicline reduces cessation fatigue due to a decrease in negative mood. This effect decreases during the first week, increases slightly from day 8-10, and then continues to weaken further until week 3. The effect increases after week 3 with a slight decrease around days 24-26. Given that the CIs do not cross zero, we can conclude that this effect is statistically significant. results_tvma_3trt$MedEff_T2

results_tvma_3trt\$MedEff_CI_T2

In the above plots, the effect of combination NRT on cessation fatigue that is mediated by the negative mood in the last fifteen minutes is weaker than the effect of using the nicotine patch only. This effect is sinusoidal over time, with occasional spikes at days 2, 11, 24 and 28. The effect takes a decreasing trend between week 2 and week 3. The effect increases after week 3 and continues its upward trend with occasional decrease. Given that the confidence intervals do not cross zero, we can conclude that this effect is statistically significant.

The tvma_3trt function computes bootstrap confidence intervals by default. Therefore, if the user decides to not bootstrap CIs for the mediation effect by specifying CI = "none", but by mistake also specifies replicates = 500, the function will not display an error, but will simply execute without computing the CIs for mediation effect. Note that the CIs for the effects of the treatment on the mediator and the mediator on the outcome, and for the direct and total effects are computed even if the user passes the argument CI = "none".

## Summary

The tvmediation package provides a set of functions for estimating mediation effects that vary over time for both binary and continuous time-varying outcomes. Currently, the package only allows for a time-invariant treatment. The mediator and outcome are assumed to be time-varying, such as intensive longitudinal measurements obtained via Ecological Momentary Assessment or via wearable and mobile devices. The development of this tool has widespread application for use in human behavior research, clinical trials, addiction research, and others by allowing specification of mediation effects that vary as a function of time.

## References

1. Cai, X., Piper, M. E., Li, R., & Coffman, D. L. (2020). Estimation and inference for the mediation effect in a time-varying mediation model. https://arxiv.org/abs/2008.11797

2. Baker, T. B., Piper, M. E., Stein, J. H., Smith, S. S., Bolt, D. M., Fraser, D. L., & Fiore, M. C. (2016). Effects of nicotine patch vs varenicline vs combination nicotine replacement therapy on smoking cessation at 26 weeks: A randomized clinical trial. JAMA, 315(4), 371-379.}

3. Efron, B. & Tibshirani, R. Bootstrap Methods for Standard Errors, Confidence Intervals, and Other Measures of Statistical Accuracy. Statist. Sci. 1 (1986), no. 1, 54–75.}