6.1 Simple Temporal Effects

We might have measured a variable over the course of a certain time period (e.g. 20 years). We might expect that there is stochastic year-to-year variation, which we can simulate already. However we might also want to simulate patterns in that temporal data. We can treat the levels associated with a particular grouping factor (e.g. year) as both a factor and continuous.

To treat a grouping factor as continuous, we use covariate=TRUE in the parameter list. In this way we can simulate a linear effect of year:

squid_data <- simulate_population( 
  data_structure= make_structure(structure = "year(20) + sex(2)/individual(50)",repeat_obs=20),
  parameters=list(
    year_cont = list(
      group="year",
      names= "year_cont",
      covariate=TRUE,
      beta=0.3
    ),
    year = list(
      vcov = 0.8
    ),
    residual=list(
      vcov = 1
      )
    )
)

note we have specified group in the parameter list. This enables us to link a set of parameters to the grouping factor in the data structure. This doesn’t have to be specified and defaults to the name of the list item.

data <- get_population_data(squid_data)
head(data)

##           y year_cont year_effect   residual year sex individual squid_pop
## 1 1.8129692         1    1.715485 -0.2025162    1   1          1         1
## 2 0.8407636         1    1.715485 -1.1747219    1   1          1         1
## 3 0.6603346         1    1.715485 -1.3551508    1   1          1         1
## 4 2.2919631         1    1.715485  0.2764777    1   1          1         1
## 5 2.3573981         1    1.715485  0.3419127    1   1          1         1
## 6 1.0391202         1    1.715485 -0.9763652    1   1          1         1

plot(y ~ year_cont, data)

Here we can see there is within year variation, year to year variation, as well as a linear directional year effect.

lmer(y ~ year_cont + (1|year), data)

## Linear mixed model fit by REML ['lmerMod']
## Formula: y ~ year_cont + (1 | year)
##    Data: data
## REML criterion at convergence: 113790.7
## Random effects:
##  Groups   Name        Std.Dev.
##  year     (Intercept) 0.9352  
##  Residual             1.0015  
## Number of obs: 40000, groups:  year, 20
## Fixed Effects:
## (Intercept)    year_cont  
##      0.1705       0.3114

In a similar way we can also simulate a quadratic effect of time.

squid_data <- simulate_population(
  data_structure = make_structure(structure = "year(20) + sex(2)/individual(50)",repeat_obs=20),
  parameters=list(
    year_cont = list(
      group="year",
      names= c("year_cont"),
      covariate=TRUE,
      beta=c(0.3)
    ),
    interactions=list(
      names= c("year_cont:year_cont"),
      beta=c(-0.05)
    ),
    year = list(
      vcov = 1
    ),
    residual=list(
      vcov = 0.8
    )
  )
)

data <- get_population_data(squid_data)

plot(y~year_cont,data)