ANLY 505 - Simulated Data and Linear Mixed Models

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?

# use this chunk to answer question 1
#install.packages('lme4')
library(lme4)

## Loading required package: Matrix

fit1=lmer(resp2 ~ 1 + elevation + slope + (1|stand))
summary(fit1)

## 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

fit1

## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ 1 + elevation + slope + (1 | stand)
## REML criterion at convergence: 81.9874
## Random effects:
##  Groups   Name        Std.Dev.
##  stand    (Intercept) 1.099   
##  Residual             1.165   
## Number of obs: 20, groups:  stand, 5
## Fixed Effects:
## (Intercept)    elevation        slope  
##   -21.31463      0.02060      0.09511

#The intercept is -21.31 which is far from b0(-1). The elevation is 0.02, also different from b1(0.005). The slop is 0.095 which is similar to b2(0.1). The standard deviation is 1.099 which is different from our preset value 2. The residual is 1.17 which is close to our preset value 1.

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

# use this chunk to answer question 2
library(lme4)
fun1= 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)
  resp = b0 + b1 * elevation + b2 * slope + standeff + ploteff
  dat = data.frame(resp, elevation, slope, stand)
  lmer(resp ~ 1 + elevation + slope + (1|stand), data = dat)
}
set.seed(16)
fun1()

## Linear mixed model fit by REML ['lmerMod']
## Formula: resp ~ 1 + elevation + slope + (1 | stand)
##    Data: dat
## REML criterion at convergence: 81.9874
## Random effects:
##  Groups   Name        Std.Dev.
##  stand    (Intercept) 1.099   
##  Residual             1.165   
## Number of obs: 20, groups:  stand, 5
## Fixed Effects:
## (Intercept)    elevation        slope  
##   -21.31463      0.02060      0.09511

simsmix=replicate(1000,fun1())

## 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
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## 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.

# use this chunk to answer question 3
#install.packages('tidyverse')
library('tidyverse')

## -- Attaching packages --------------------------------------- tidyverse 1.3.0 --

## √ ggplot2 3.3.0     √ purrr   0.3.3
## √ tibble  2.1.3     √ dplyr   0.8.5
## √ tidyr   1.0.2     √ stringr 1.4.0
## √ readr   1.3.1     √ forcats 0.5.0

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

#install.packages('purrr')
library('purrr')
#install.packages('broom')
library('broom')
variances = simsmix %>% map_dfr(tidy, effects = "ran_pars", scales = "vcov")
head(variances,n=6)

## # A tibble: 6 x 3
##   term                     group    estimate
##   <chr>                    <chr>       <dbl>
## 1 var_(Intercept).stand    stand       5.56 
## 2 var_Observation.Residual Residual    0.951
## 3 var_(Intercept).stand    stand       2.61 
## 4 var_Observation.Residual Residual    1.11 
## 5 var_(Intercept).stand    stand       9.73 
## 6 var_Observation.Residual Residual    1.36

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?

# use this chunk to answer question 4
library(ggplot2)
library(dplyr)
stand_sims = c(5, 20, 100) %>%
  set_names() %>%
  map(~replicate(1000, fun1(nstand = .x)))

## 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
## 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

add_prefix = function(string) {
     paste("Number Samples:", string, sep = " ")
}
 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 = "green", alpha = 0.25) +
  facet_wrap(~id, labeller = as_labeller(add_prefix) ) +
  geom_vline(xintercept = 4)

 #We can clearly tell that the size of samples can affect the variance. The more samples we use, the less variance we have.

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

# use this chunk to answer question 5
#install.packages('furrr')
library('furrr')

## Loading required package: future

library('dplyr')
est_e_s <- simsmix %>% future_map(tidy, effects = "fixed")%>%bind_rows()

est_e_s  %>% 
  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_point()+
  facet_wrap(~term) +
  geom_hline(aes(yintercept = real_value, color = term), linetype = 4, size = 0.5)

# the elevation is around 0.005 and slop is around 0.1 which is very close to our preset values. And most of the data points are within a reasonable range.

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

ANLY 505 - Simulated Data and Linear Mixed Models

Week 2

Sizhe Xie

2020-04-04