The statistical model:

\(y_t = \beta_0 + \beta_1 * (Elevation_s)_t + \beta_2 * Slope_t + (b_s)_t + \epsilon_t\)

Where:

• \(\beta_0\) is the mean response when both Elevation and Slope are 0
• \(\beta_1\) is the change in mean response for a 1-unit change in elevation. Elevation is measured at the stand level, so all plots in a stand share a single value in elevation.
• \(\beta_2\) is the change in mean response for a 1-unit change in slope. Slope is measured at the plot level, so every plot potentially has a unique value of slope.

Let’s define the parameters:

• the intercept \((\beta_0)\) will be -1
• the coefficient for elevation \((\beta_1)\) will be set to 0.005
• the coefficient for slope \((\beta_2)\) will be set to 0.1

nstand = 5
nplot = 4
b0 = -1
b1 = .005
b2 = .1
sds = 2
sd = 1

Simulate other variables:

set.seed(16)
stand = rep(LETTERS[1:nstand], each = nplot)
standeff = rep( rnorm(nstand, 0, sds), each = nplot)
ploteff = rnorm(nstand*nplot, 0, sd)

Simulate elevation and slope:

elevation = rep( runif(nstand, 1000, 1500), each = nplot)
slope = runif(nstand*nplot, 2, 75)

Simulate response variable:

resp2 = b0 + b1*elevation + b2*slope + standeff + ploteff 

Your tasks (complete each task in its’ own code chunk, make sure to use echo=TRUE so I can see your code):

1. Fit a linear mixed model with the response variable as a function of elevation and slope with stand as a random effect. Are the estimated parameters similar to the intial parameters as we defined them?

library(lme4)

## Loading required package: Matrix

library(Matrix)
fitm1 = lmer(resp2 ~ 1 + elevation + slope + (1|stand))
summary(fitm1)

## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ 1 + elevation + slope + (1 | stand)
## 
## REML criterion at convergence: 82
## 
## Scaled residuals: 
##      Min       1Q   Median       3Q      Max 
## -1.65583 -0.62467 -0.01693  0.53669  1.41736 
## 
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  stand    (Intercept) 1.208    1.099   
##  Residual             1.358    1.165   
## Number of obs: 20, groups:  stand, 5
## 
## Fixed effects:
##               Estimate Std. Error t value
## (Intercept) -21.314628   6.602053  -3.228
## elevation     0.020600   0.004916   4.190
## slope         0.095105   0.016441   5.785
## 
## Correlation of Fixed Effects:
##           (Intr) elevtn
## elevation -0.991       
## slope      0.049 -0.148

2. Create a function for your model and run 1000 simulations of that model.

library(purrr)
newmix_fun = function(nstand = 5, nplot = 4, b0 = -1, b1 = 0.005, b2 = 0.1, sds = 2, sd = 1) {
  stand = rep(LETTERS[1:nstand], each = nplot)
  standeff = rep(rnorm(nstand, 0, sds), each = nplot)
  ploteff = rnorm(nstand * nplot, 0, sd)
  elevation = rep(runif(nstand, 1000, 1500), each = nplot)
  slope = runif(nstand * nplot, 2, 75)
  resp2 = b0 + b1 * elevation + b2 * slope + standeff + ploteff
  dat = data.frame(resp2, elevation, slope, stand)
  lmer(resp2 ~ 1 + elevation + slope + (1|stand), data = dat)
}
newmix_fun()

## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ 1 + elevation + slope + (1 | stand)
##    Data: dat
## REML criterion at convergence: 80.9781
## Random effects:
##  Groups   Name        Std.Dev.
##  stand    (Intercept) 2.3573  
##  Residual             0.9754  
## Number of obs: 20, groups:  stand, 5
## Fixed Effects:
## (Intercept)    elevation        slope  
##   10.584601    -0.005464     0.086839

simul_result <- replicate(n = 1000, expr = newmix_fun())

## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular
## boundary (singular) fit: see ?isSingular

3. Extract the stand and residual variances from this simulation run. Print the first 6 rows of the data.

library(broom)
variances <- simul_result %>% map_dfr(tidy, effects = "ran_pars", scales = "vcov")
variances %>% print(n = 6)

## # A tibble: 2,000 x 3
##   term                     group    estimate
##   <chr>                    <chr>       <dbl>
## 1 var_(Intercept).stand    stand       2.61 
## 2 var_Observation.Residual Residual    1.11 
## 3 var_(Intercept).stand    stand       9.73 
## 4 var_Observation.Residual Residual    1.36 
## 5 var_(Intercept).stand    stand       0.827
## 6 var_Observation.Residual Residual    0.914
## # ... with 1,994 more rows

4. Choose three different sample sizes (your choice) and run 1000 model simulations with each sample size. Create 3 visualizations that compare distributions of the variances for each of the 3 sample sizes. Make sure that the axes are labelled correctly. What do these graphs say about the relationship between sample size and variance?

library(ggplot2)
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

stand_sims = c(100, 200, 300) %>%
  set_names() %>%
  map(~replicate(1000, newmix_fun(nstand = .x)))
stand_vars = stand_sims %>%
  modify_depth(2, ~tidy(.x, effects = "ran_pars", scales = "vcov")) %>%
  map_dfr(bind_rows, .id = "id") %>%
  filter(group == "stand")

ggplot(stand_vars, aes(x = estimate)) +
  geom_density(fill = "black", alpha = "0.25") +
  facet_wrap(~id) +
  geom_vline(xintercept = 4)

5. Plot the coefficients of the estimates of elevation and slope. Hint: the x-axis should have 1000 values. Discuss the graphs.

library(furrr)

## Loading required package: future

simsest <- simul_result %>% 
  future_map(tidy, effects = "fixed") %>% 
  bind_rows()

simsest %>% 
  dplyr::filter(term %in% c("elevation", "slope")) %>% 
  group_by(term) %>% 
  mutate(x = 1 : 1000) %>%
  ungroup() %>% 
  mutate(real_value = ifelse(term == "elevation", 0.005, 0.1)) %>% 
  ggplot(aes(x = x, y = estimate)) +
  geom_line() +
  facet_wrap(~term) +
  geom_hline(aes(yintercept = real_value, color = term), linetype = 4, size = 0.5) +
  theme_bw()

6. Submit a link to this document in R Pubs to your Moodle. This assignment is worth 25 points.