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):

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(lmerTest)

## 
## Attaching package: 'lmerTest'

## The following object is masked from 'package:lme4':
## 
##     lmer

## The following object is masked from 'package:stats':
## 
##     step

m1 <- lmer(resp2 ~ 1 + elevation + slope + (1|stand))
summary(m1)

## Linear mixed model fit by REML. t-tests use Satterthwaite's method [
## lmerModLmerTest]
## 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         df t value Pr(>|t|)    
## (Intercept) -21.314628   6.602053   3.001313  -3.228   0.0482 *  
## elevation     0.020600   0.004916   3.113482   4.190   0.0230 *  
## slope         0.095105   0.016441  15.868032   5.785 2.88e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##           (Intr) elevtn
## elevation -0.991       
## slope      0.049 -0.148

cat("b0 = ", b0, sep = "")

## b0 = -1

cat("b1 = ", b1, sep = "")

## b1 = 0.005

cat("b2 = ", b2, sep = "")

## b2 = 0.1

Beta^1=0.005 -> a lot smaller than the intial parameter 0.005

Beta^2=0.1 -> pretty close to the initial parameter

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

sim = 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)
}
sim()

## Linear mixed model fit by REML ['lmerModLmerTest']
## 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

sim_result = replicate(n = 1000, expr = sim())

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00350336
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0045514
## (tol = 0.002, component 1)

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00363544
## (tol = 0.002, component 1)

## boundary (singular) fit: see ?isSingular

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00439561
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00239513
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00219495
## (tol = 0.002, component 1)

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00322495
## (tol = 0.002, component 1)

## boundary (singular) fit: see ?isSingular

length(sim_result)

## [1] 1000

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

library(tidyverse)

## -- Attaching packages --------------------------------------------------------------------------------- tidyverse 1.2.1 --

## v ggplot2 3.2.1       v purrr   0.3.2  
## v tibble  2.1.1       v dplyr   0.8.0.1
## v tidyr   0.8.3       v stringr 1.4.0  
## v readr   1.3.1       v forcats 0.4.0

## -- Conflicts ------------------------------------------------------------------------------------ tidyverse_conflicts() --
## x tidyr::expand() masks Matrix::expand()
## x dplyr::filter() masks stats::filter()
## x dplyr::lag()    masks stats::lag()

library(broom)
library(broom.mixed)

## Warning in checkMatrixPackageVersion(): Package version inconsistency detected.
## TMB was built with Matrix version 1.2.17
## Current Matrix version is 1.2.15
## Please re-install 'TMB' from source using install.packages('TMB', type = 'source') or ask CRAN for a binary version of 'TMB' matching CRAN's 'Matrix' package

## 
## Attaching package: 'broom.mixed'

## The following object is masked from 'package:broom':
## 
##     tidyMCMC

variances = sim_result %>% map_dfr(tidy, effects = "ran_pars", scales = "vcov")
variances %>% print(n = 6)

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

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(furrr)

## Loading required package: future

plan(multiprocess)
sim_result_list = c(5, 30, 100) %>% 
  set_names(c("size_5", "size_30", "size_100")) %>% 
  future_map(function(.size) replicate(n = 1000, expr = sim(nstand = .size)))

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00214049
## (tol = 0.002, component 1)

## 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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00253484
## (tol = 0.002, component 1)

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00209146
## (tol = 0.002, component 1)

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00436838
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00278806
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0028052
## (tol = 0.002, component 1)

## 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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00396062
## (tol = 0.002, component 1)

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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00334769
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0267826
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00214099
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0178432
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0171689
## (tol = 0.002, component 1)

sim_result_stand_df = sim_result_list %>% 
  modify_depth(.depth = 2, function(x) tidy(x, effects = "ran_pars", scales = "vcov")) %>% 
  map_dfr(bind_rows, .id = "id") %>% 
  filter(group == "stand")

sim_result_stand_df

## # A tibble: 3,000 x 5
##    id     effect   group term             estimate
##    <chr>  <chr>    <chr> <chr>               <dbl>
##  1 size_5 ran_pars stand var__(Intercept)     4.03
##  2 size_5 ran_pars stand var__(Intercept)     3.62
##  3 size_5 ran_pars stand var__(Intercept)     3.83
##  4 size_5 ran_pars stand var__(Intercept)     1.45
##  5 size_5 ran_pars stand var__(Intercept)     2.35
##  6 size_5 ran_pars stand var__(Intercept)     1.83
##  7 size_5 ran_pars stand var__(Intercept)     2.57
##  8 size_5 ran_pars stand var__(Intercept)     2.78
##  9 size_5 ran_pars stand var__(Intercept)     7.79
## 10 size_5 ran_pars stand var__(Intercept)     5.74
## # ... with 2,990 more rows

sim_result_stand_df %>% 
  mutate(
    id = case_when(
      id == "size_5" ~ "size = 5",
      id == "size_30" ~ "size = 30",
      id == "size_100" ~ "size = 100"
    )
  ) %>% 
  ggplot(aes(x = estimate) ) +
  geom_density(fill = "blue", alpha = .25) +
  facet_wrap(~ id) +
  geom_vline(xintercept = 4) +
  theme_bw()

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

ANLY 505 - Problem Set #2

Simulation of Linear Mixed Models

Nan Li

8/26/2019

Beta^1=0.005 -> a lot smaller than the intial parameter 0.005

Beta^2=0.1 -> pretty close to the initial parameter