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)

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

## Loading required package: Matrix

library(lmerTest)

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

## 
## Attaching package: 'lmerTest'

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

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

library(ggplot2)

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

library(future)

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

library(dplyr)

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

## 
## 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

library(bindrcpp)
library(tidyverse)

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

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

## v tibble  2.1.1     v purrr   0.2.5
## v tidyr   0.8.1     v stringr 1.3.1
## v readr   1.1.1     v forcats 0.3.0

## Warning: package 'tibble' 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()

library(broom)

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

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

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

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

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

## 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

> b0 = -1; intercept’s estimated value is -21.3. Significant diff.

> b1 = 0.005; elevation’s coefficient is estimated to be 0.02. Significant diff.

> b2 = 0.1; slope’s coefficient is estimated to be 0.09. Quite close

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

linear_fit <- 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)
}
linear_fit()

## Linear mixed model fit by REML ['lmerModLmerTest']
## Formula: resp2 ~ 1 + elevation + slope + (1 | stand)
##    Data: dat
## REML criterion at convergence: 85.1321
## Random effects:
##  Groups   Name        Std.Dev.
##  stand    (Intercept) 2.992   
##  Residual             1.077   
## Number of obs: 20, groups:  stand, 5
## Fixed Effects:
## (Intercept)    elevation        slope  
##    7.365232    -0.002138     0.102090

simu_R <- replicate(n = 1000, expr = linear_fit())

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control
## $checkConv, : Model failed to converge with max|grad| = 0.00494375 (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| = 0.00426347 (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| = 0.00287986 (tol =
## 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control
## $checkConv, : Model failed to converge with max|grad| = 0.00314794 (tol =
## 0.002, component 1)

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control
## $checkConv, : Model failed to converge with max|grad| = 0.00789635 (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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control
## $checkConv, : Model failed to converge with max|grad| = 0.0162996 (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

## Warning in checkConv(attr(opt, "derivs"), opt$par, ctrl = control
## $checkConv, : Model failed to converge with max|grad| = 0.00307489 (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

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

variance <- simu_R %>% map_dfr(tidy, effects = "ran_pars", scales = "vcov")
variance %>% print(n = 6)

## # A tibble: 2,000 x 4
##   effect   group    term             estimate
##   <chr>    <chr>    <chr>               <dbl>
## 1 ran_pars stand    var__(Intercept)    2.78 
## 2 ran_pars Residual var__Observation    1.20 
## 3 ran_pars stand    var__(Intercept)    2.91 
## 4 ran_pars Residual var__Observation    0.771
## 5 ran_pars stand    var__(Intercept)    1.72 
## 6 ran_pars Residual var__Observation    1.82 
## # ... 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? ``{r, echo=TRUE}

sample_size <- c(10, 15, 20) set_names(c(“10”, “15”, “20”))
map(~replicate(1000, linear_fit(nstand = .x)))

variances <- sample_size modify_depth(.depth = 2, function(x) tidy(x, effects = “ran_pars”, scales = “vcov”)) map_dfr(bind_rows, .id = “id”) filter(group == “stand”)

add_prefix <- function(string) { paste(“sample size:”, string, sep = " “) }

groupmed = variances group_by(id) summarise(mvar = median(estimate))

ggplot(variances, aes(x = estimate)) + geom_density(fill = “purple”, alpha = “0.25”) + facet_wrap(~id, labeller = as_labeller(add_prefix)) + geom_vline(aes(xintercept = 4, linetype = “True Variance”), size=0.5) + geom_vline(data = groupmed, aes(xintercept=mvar,linetype=“Median Variance”), size = 0.5) + theme_bw() + labs(x=“Estimated Variance”)


5. Plot the coefficients of the estimates of elevation and slope.  Hint: the x-axis should have 1000 values.  Discuss the graphs.
``{r,     echo=TRUE}


est <- simu_R
  map(tidy, effects = "fixed") 
  bind_rows()

est 
  dplyr::filter(term %in% c("slope", "elevation")) 
  group_by(term)
  mutate(x = 1 : 1000) 
  ungroup() 
  mutate(real_value = ifelse(term == "elevation", 0.005, 0.1)) 
  ggplot(aes(x = x, y = estimate)) +
  facet_wrap(~term) +
  geom_line() +
  theme_bw() +
  geom_hline(aes(yintercept = real_value, color = term), linetype = 2, size = 1) +
  labs(x="Estimates", y = "Mean Response")

Submit a link to this document in R Pubs to your Moodle.

ANLY 505 - Problem Set #2

Simulation of Linear Mixed Models

Kaiyu Shi

2019-05-06

> b0 = -1; intercept’s estimated value is -21.3. Significant diff.

> b1 = 0.005; elevation’s coefficient is estimated to be 0.02. Significant diff.

> b2 = 0.1; slope’s coefficient is estimated to be 0.09. Quite close