longitudinal data with time-varying binary exposure in linear mixed model

Question

Good morning, i am looking for some help with longitudinal data with time-varying binary exposure in linear mixed model. Outcome is continuous variable. I am going to use R and the lme4 package. I want to identify association between time varying smoking status and hemoglobin A1c with random intercept and random slope.

Here is a sample dataset and R code that i conducted. Is this right analysis for my hypothesis?

id	smoking	HbA1c	visit
1	0	6.5	1
1	0	6.7	2
1	0	6.8	3
1	1	7	4
1	1	7.5	5
2	1	6.8	1
2	1	7.4	2
2	1	7.6	3
2	1	7.5	4
3	0	6.4	1
3	0	6.5	2
3	0	6.6	3
4	0	6.7	1
4	0	6.6	2
4	0	6.7	3
4	0	6.8	4
4	0	6.8	5
5	0	7	1
5	0	7	2
5	1	7	3
6	0	7.1	1
6	0	7	2
6	0	7.1	3
6	0	7.2	4

data1 <- read.xlsx("lmm_test.xlsx")
model1 <- lmer(a1c ~ smk + smk * visit + (visit | id), data = data1)
a <- tidy(model, conf.int = TRUE)
a %>% select(term, estimate, conf.low, conf.high)

# A tibble: 8 x 4
term                   estimate conf.low conf.high
smk                     -0.779   -1.22     -0.336
visit                    0.0373  -0.0218    0.0964
smk:visit                0.268    0.160     0.377 

When the results of the analysis are as above mentioned, can i interpret that glycated hemoglobin increases by 0.268 in the presence of smoking status over time (1 visit unit) than in the absence of smoking?

Answer 1

The model you have fitted has the following mathematical form: $$\texttt{HbA1c}_i(t) = \beta_0 + \beta_1 \texttt{smoking}_i(t) + \beta_2 t + \beta_3 \{\texttt{smoking}_i(t) \times t\} + b_{i0} + b_{i1} t + \varepsilon_i(t),$$ where $b_i = (b_{i0}, b_{i1})$ are the random effects.

The presence of the interaction terms complicates the interpretation. If, at a particular time point $t$, the smoking status changes from zero to one, then the HbA1c at $t$ is expected to change by $\beta_1 + \beta_3 t$.