ANLY 505 - Problem Set #2
Simulation of Linear Mixed Models
Shaun Irvin
June 28, 2019
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 = 1Simulate 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?
fit2 = lmer(resp2 ~ elevation + slope + (1|stand))
fit2## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ 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 0.09511The estimated parameters for elevation (b1) and slope (b2) are 0.0206 and 0.09511, respectively. These are pretty close to the initial defined parameters (.005 and .1). The large difference is in the intercept parameter (-21.3 vs -1).
- Create a function for your model and run 1000 simulations of that model.
fun2 = function(b0=-1, b1=.005, b2=.1, nstand=5, nplot=4, sds=2, sd=1) {
stand = rep(LETTERS[1:nstand], each = nplot)
standeff = rep(rnorm(nstand, 0, sds), each = nplot) # Stand Effect
ploteff = rnorm(nstand * nplot, 0, sd) # Plot Effect
elevation = rep(runif(nstand, 1000, 1500), each = nplot)
slope = runif(nstand*nplot, 2, 75)
resp2 = b0 + b1 * elevation + b2 * slope + standeff + ploteff
fit2 = lmer(resp2 ~ elevation + slope + (1|stand))
fit2
}
fun2()## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
## 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 0.086839sim_fun2 = replicate(n=1000, fun2(), simplify = F)## 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 ?isSingularlength(sim_fun2)## [1] 1000- Extract the stand and residual variances from this simulation run. Print the first 6 rows of the data.
head(map(sim_fun2, ~summary(.x)), 6)## [[1]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 81.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.64222 -0.50652 -0.07439 0.39168 1.49956
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 2.610 1.616
## Residual 1.106 1.052
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -15.946250 5.626086 -2.834
## elevation 0.015212 0.004263 3.568
## slope 0.121069 0.015653 7.735
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -0.984
## slope -0.058 -0.055
##
## [[2]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 87.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.25126 -0.48517 -0.04175 0.49356 1.54182
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 9.731 3.120
## Residual 1.365 1.168
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -13.92550 13.31884 -1.046
## elevation 0.01556 0.01170 1.330
## slope 0.12435 0.01663 7.479
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -0.993
## slope -0.044 -0.003
##
## [[3]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 76.3
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.9391 -0.3691 -0.1649 0.5172 1.7249
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 0.8269 0.9093
## Residual 0.9138 0.9559
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -12.25232 4.45186 -2.752
## elevation 0.01300 0.00336 3.870
## slope 0.09365 0.01005 9.323
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -0.992
## slope 0.059 -0.137
##
## [[4]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 86.8
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.17685 -0.67552 -0.04607 0.66582 1.92244
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 9.588 3.096
## Residual 1.316 1.147
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -11.92341 21.43384 -0.556
## elevation 0.01345 0.01573 0.856
## slope 0.07667 0.01519 5.048
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -0.997
## slope -0.112 0.080
##
## [[5]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 65.1
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -2.3503 -0.5292 -0.3537 0.7736 1.5564
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 0.000 0.0000
## Residual 0.815 0.9028
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 31.75050 15.80359 2.009
## elevation -0.02351 0.01323 -1.776
## slope 0.12712 0.01061 11.985
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -1.000
## slope 0.072 -0.100
## convergence code: 0
## boundary (singular) fit: see ?isSingular
##
##
## [[6]]
## Linear mixed model fit by REML ['lmerMod']
## Formula: resp2 ~ elevation + slope + (1 | stand)
##
## REML criterion at convergence: 83.7
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.9877 -0.4081 -0.1085 0.6376 1.8509
##
## Random effects:
## Groups Name Variance Std.Dev.
## stand (Intercept) 3.555 1.886
## Residual 1.301 1.141
## Number of obs: 20, groups: stand, 5
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) -29.18993 15.11541 -1.931
## elevation 0.02430 0.01076 2.259
## slope 0.11198 0.01475 7.593
##
## Correlation of Fixed Effects:
## (Intr) elevtn
## elevation -0.998
## slope 0.043 -0.079I was not able to figure out how to extract just the stand and residual variances from the summary, so I have printed out the full summary of the first 6 simulations, which includes the variances. I will continue to try and figure out how to do it.
- 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?
# Creating more variance by adding additional observations
stand_sims = c(5, 200, 100) %>%
set_names() %>%
map(~replicate(1000, fun2(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 ?isSingularstand_vars = stand_sims %>%
modify_depth(2, ~tidy(.x, effects = "ran_pars", scales = "vcov")) %>%
map_dfr(bind_rows, .id = "stand_num") %>%
filter(group == "stand")
head(stand_vars)## # A tibble: 6 x 4
## stand_num term group estimate
## <chr> <chr> <chr> <dbl>
## 1 5 var_(Intercept).stand stand 12.0
## 2 5 var_(Intercept).stand stand 6.84
## 3 5 var_(Intercept).stand stand 2.89
## 4 5 var_(Intercept).stand stand 2.52
## 5 5 var_(Intercept).stand stand 0.966
## 6 5 var_(Intercept).stand stand 2.91ggplot(stand_vars, aes(x = estimate)) +
geom_density(fill = "orange", alpha = .25) +
facet_wrap(~stand_num)
ggplot(stand_vars, aes(x = estimate)) +
geom_histogram()## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.
Plot the coefficients of the estimates of elevation and slope. Hint: the x-axis should have 1000 values. Discuss the graphs.
Submit a link to this document in R Pubs to your Moodle. This assignment is worth 25 points.