ANLY 505 - Problem Set #2

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?

# Loading the libraries required
library(lmerTest)

## Warning: package 'lmerTest' was built under R version 3.5.3

## Loading required package: lme4

## Warning: package 'lme4' was built under R version 3.5.3

## Loading required package: Matrix

## 
## Attaching package: 'lmerTest'

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

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

library(lme4)

# Creating a model and running it to view observations

mod1 <- lmer(resp2 ~ 1 + elevation + slope + (1|stand))

summary(mod1)

## 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

# Checking for parameters to crosscheck comparison.

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

## b0 = -1

# Running another sample

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

## b1 = 0.005

# Checking b^2 next

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

## b2 = 0.1

Notice observation clearly shows b^2 being 0.1 near to the original parameters compared to b^1 thats shows 0.005.

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

mod_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)
  sim_standeff = rep(rnorm(nstand, 0, sds), each = nplot)
  sim_ploteff = rnorm(nstand * nplot, 0, sd)
  elevation = rep(runif(nstand, 1000, 1500), each = nplot)
  sim_slope = runif(nstand * nplot, 2, 75)
  sim_resp2 = b0 + b1 * elevation + b2 * sim_slope + sim_standeff + sim_ploteff
  
  simdata = data.frame(sim_resp2, elevation, sim_slope, stand)
  lmer(sim_resp2 ~ 1 + elevation + sim_slope + (1|stand), data =  simdata)
}

# Runnign the model to check simulations.
mod_sim()

## Linear mixed model fit by REML ['lmerModLmerTest']
## Formula: sim_resp2 ~ 1 + elevation + sim_slope + (1 | stand)
##    Data: simdata
## 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    sim_slope  
##   10.584601    -0.005464     0.086839

# Next lets run 1000 simulations

result_1000 = replicate(n = 1000, expr = mod_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

# Lets check the length next

length(result_1000)

## [1] 1000

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

# Loading more libraries for further analysis
library(broom.mixed)

## Warning: package 'broom.mixed' was built under R version 3.5.3

## Warning in checkMatrixPackageVersion(): Package version inconsistency detected.
## TMB was built with Matrix version 1.2.17
## Current Matrix version is 1.2.14
## 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

library(broom)

## Warning: package 'broom' was built under R version 3.5.3

## 
## Attaching package: 'broom'

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

library(tidyverse)

## Warning: package 'tidyverse' was built under R version 3.5.3

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

## v ggplot2 3.2.0     v purrr   0.3.2
## v tibble  2.1.3     v dplyr   0.8.3
## v tidyr   0.8.3     v stringr 1.4.0
## v readr   1.3.1     v forcats 0.4.0

## Warning: package 'ggplot2' was built under R version 3.5.3

## Warning: package 'tibble' was built under R version 3.5.3

## Warning: package 'tidyr' was built under R version 3.5.3

## Warning: package 'readr' was built under R version 3.5.3

## Warning: package 'purrr' was built under R version 3.5.3

## Warning: package 'dplyr' was built under R version 3.5.3

## Warning: package 'stringr' was built under R version 3.5.3

## Warning: package 'forcats' was built under R version 3.5.3

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

# Now, lets print the first 6 rows of the data (n=6)

var = result_1000 %>% map_dfr(tidy, effects = "ran_pars", scales = "vcov")

var %>% 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?

# Installing libraries
library(furrr)

## Warning: package 'furrr' was built under R version 3.5.3

## Loading required package: future

## Warning: package 'future' was built under R version 3.5.3

library(ggplot2)

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

## 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| = 0.00295659
## (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| = 0.0027045
## (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| = 0.00215284
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00395311
## (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| = 0.00417219
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00245866
## (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| = 0.00210622
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.00395452
## (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| = 0.00305796
## (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| = 0.00207284
## (tol = 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl =
## control$checkConv, : Model failed to converge with max|grad| = 0.0146987
## (tol = 0.002, component 1)

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

result_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)   4.92  
##  2 size_5 ran_pars stand var__(Intercept)   0.721 
##  3 size_5 ran_pars stand var__(Intercept)   0.582 
##  4 size_5 ran_pars stand var__(Intercept)   5.68  
##  5 size_5 ran_pars stand var__(Intercept)   1.85  
##  6 size_5 ran_pars stand var__(Intercept)   6.27  
##  7 size_5 ran_pars stand var__(Intercept)   3.56  
##  8 size_5 ran_pars stand var__(Intercept)   7.24  
##  9 size_5 ran_pars stand var__(Intercept)   0.0443
## 10 size_5 ran_pars stand var__(Intercept)   0.888 
## # ... with 2,990 more rows

result_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()

Based on the graphs with the increase in the sample size estimates shows initial value close to 4.

ANLY 505 - Problem Set #2

Simulation of Linear Mixed Models

Bhavesh Vaswani

2019-10-17

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?

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