# Mixed effects models

## Generalized Linear Mixed Effects Model: Example Using Seedling Survival Data

What is the response variable?

### exploring the data

Data

``````# read in data file
sep = "\t")

``````##   PLOT SPECIES HEIGHT STATUS LIGHT CONS     CONBA
## 1    1  PSYBRA     52      1 0.205    1   18.0165
## 2    1  PIPGLA     40      1 0.205    1    0.8992
## 3    1  PIPGLA     33      0 0.205    1    0.8992
## 4    1  ANDINE     41      0 0.205    0    0.0000
## 5    1  PREMON     90      1 0.205    6 3633.1027
## 6    1  SLOBER    130      1 0.205    1   67.5457``````

Let's explore the data a bit more.

How many lived & died?

``table(sdata\$STATUS)``
``````##
##    0    1
## 2718 1967``````

Look at ranges and distributions of your predictor variables

``````par(mfrow = c(1, 3))
range(sdata\$CONS)``````
``## [1]   0 243``
``````hist(sdata\$CONS, breaks = 50)

range(sdata\$LIGHT)``````
``## [1] 0.069 0.946``
``````hist(sdata\$LIGHT, breaks = 10)

range(sdata\$HEIGHT)``````
``## [1]  10 600``
``hist(sdata\$HEIGHT, breaks = 50)``

Note that your predictors have very different ranges so, if you want to directly compare the effects or test for interactions, you should rescale them by subtracting the mean and dividing by the standard deviation.

``````sdata\$LIGHT.ADJ <- (sdata\$LIGHT - mean(sdata\$LIGHT))/sd(sdata\$LIGHT)

### start analyzing

Load library to do mixed effects models

``library(lme4)``
``````## Loading required package: lattice

First, run a simple model with only height as a predictor

``````surv.glm <- glm(sdata\$STATUS ~ sdata\$HEIGHT.ADJ, family = binomial)
surv.glmer <- glmer(formula = STATUS ~ HEIGHT.ADJ + (1 | PLOT), data = sdata,
family = binomial)

summary(surv.glm)``````
``````##
## Call:
## glm(formula = sdata\$STATUS ~ sdata\$HEIGHT.ADJ, family = binomial)
##
## Deviance Residuals:
##    Min      1Q  Median      3Q     Max
## -3.072  -0.979  -0.922   1.316   1.465
##
## Coefficients:
##                  Estimate Std. Error z value Pr(>|z|)
## (Intercept)       -0.3049     0.0306   -9.97   <2e-16 ***
## sdata\$HEIGHT.ADJ   0.5943     0.0455   13.05   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6373.9  on 4684  degrees of freedom
## Residual deviance: 6129.9  on 4683  degrees of freedom
## AIC: 6134
##
## Number of Fisher Scoring iterations: 4``````
``summary(surv.glmer)``
``````## Generalized linear mixed model fit by maximum likelihood ['glmerMod']
##  Family: binomial ( logit )
## Formula: STATUS ~ HEIGHT.ADJ + (1 | PLOT)
##    Data: sdata
##
##      AIC      BIC   logLik deviance
##     5649     5669    -2822     5643
##
## Random effects:
##  Groups Name        Variance Std.Dev.
##  PLOT   (Intercept) 0.669    0.818
## Number of obs: 4685, groups: PLOT, 148
##
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)  -0.2335     0.0798   -2.92   0.0034 **
## HEIGHT.ADJ    0.6490     0.0532   12.21   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
##            (Intr)

Is it necessary to include PLOT as a random effect?

Test this using likelihood ratio test, as recommended by Bolker et al. 2009

``````LL1 <- logLik(surv.glm)
LL2 <- logLik(surv.glmer)
D <- as.numeric(-2 * (LL1 - LL2))
pchisq(D, df = 1, lower = FALSE)[1]``````
``## [1] 7.462e-108``

Now compare adding species as a fixed vs. a random effect

``````surv.glm.spp <- glm(STATUS ~ SPECIES + HEIGHT.ADJ, data = sdata, family = binomial)
surv.glmer.spp <- glmer(STATUS ~ HEIGHT.ADJ + (1 | PLOT) + (1 | SPECIES), data = sdata,
family = binomial)

summary(surv.glm.spp)``````
``````##
## Call:
## glm(formula = STATUS ~ SPECIES + HEIGHT.ADJ, family = binomial,
##     data = sdata)
##
## Deviance Residuals:
##    Min      1Q  Median      3Q     Max
## -3.673  -0.859  -0.667   0.956   2.576
##
## Coefficients:
##                Estimate Std. Error z value Pr(>|z|)
## (Intercept)     -0.8855     0.6934   -1.28  0.20155
## SPECIESALCLAT   -1.4111     0.7701   -1.83  0.06691 .
## SPECIESANDINE    1.1706     0.8178    1.43  0.15228
## SPECIESARDGLA   16.8536  1455.3977    0.01  0.99076
## SPECIESARDOBO  -14.3010  1455.3977   -0.01  0.99216
## SPECIESBUCTET   16.6510  1455.3977    0.01  0.99087
## SPECIESBYRSPI    1.0029     0.7258    1.38  0.16701
## SPECIESCALCAL    1.6398     1.1017    1.49  0.13663
## SPECIESCASARB    2.3372     0.7996    2.92  0.00347 **
## SPECIESCASSYL    3.2977     1.2590    2.62  0.00881 **
## SPECIESCECSCH   -3.7088     1.3946   -2.66  0.00783 **
## SPECIESCHIDOM    0.8793     0.8959    0.98  0.32634
## SPECIESCLIERO  -15.0637   998.0202   -0.02  0.98796
## SPECIESCLUROS  -14.2785  1455.3977   -0.01  0.99217
## SPECIESCOMGLA   15.3951  1013.9037    0.02  0.98789
## SPECIESCORBOR    3.2011     1.2659    2.53  0.01145 *
## SPECIESCORSUL  -14.8672   828.8543   -0.02  0.98569
## SPECIESCROPOE    3.0876     1.0291    3.00  0.00270 **
## SPECIESCSSGUI  -14.4136  1455.3977   -0.01  0.99210
## SPECIESDACEXC   -1.7054     0.9144   -1.86  0.06218 .
## SPECIESDENARB   -0.4376     1.4256   -0.31  0.75885
## SPECIESDRYGLA  -14.5262  1455.3977   -0.01  0.99204
## SPECIESEUGDOM    1.7217     0.8083    2.13  0.03316 *
## SPECIESEUGSTA    3.0465     1.2682    2.40  0.01630 *
## SPECIESFAROCC    2.0613     1.3308    1.55  0.12141
## SPECIESGONSPI   16.1289   473.8314    0.03  0.97285
## SPECIESGUAGLA    1.4396     0.9494    1.52  0.12942
## SPECIESGUAGUI    0.1033     0.6966    0.15  0.88215
## SPECIESGUEVAL  -14.4361  1455.3977   -0.01  0.99209
## SPECIESGUTCAR    0.0589     1.3502    0.04  0.96518
## SPECIESHIRRUG    1.1565     0.8638    1.34  0.18061
## SPECIESHOMRAC    1.7935     0.8171    2.19  0.02817 *
## SPECIESINGLAU    1.7515     0.7028    2.49  0.01269 *
## SPECIESINGVER    1.4364     0.8462    1.70  0.08962 .
## SPECIESIXOFER    1.7528     0.7607    2.30  0.02121 *
## SPECIESLAEPRO   -0.9416     1.2929   -0.73  0.46644
## SPECIESLASLAN   16.5868   726.9454    0.02  0.98180
## SPECIESMANBID    1.9517     0.7387    2.64  0.00824 **
## SPECIESMARNOB    1.8252     1.4090    1.30  0.19519
## SPECIESMATDOM   -0.4728     0.8058   -0.59  0.55732
## SPECIESMELHER  -14.7064  1455.3977   -0.01  0.99194
## SPECIESMICIMP   12.4625  1455.3977    0.01  0.99317
## SPECIESMICPRA   15.5709   530.9309    0.03  0.97660
## SPECIESMICRAC   16.2498   830.2057    0.02  0.98438
## SPECIESMYRDEF    1.0401     1.2113    0.86  0.39050
## SPECIESMYRLEP    3.1196     1.2627    2.47  0.01349 *
## SPECIESMYRSPL    2.1950     0.8976    2.45  0.01447 *
## SPECIESOCOLEU    2.9209     0.7717    3.78  0.00015 ***
## SPECIESOCOSIN    1.6466     0.7281    2.26  0.02372 *
## SPECIESORMKRU  -14.4316  1455.3977   -0.01  0.99209
## SPECIESPALRIP   -1.2481     0.7914   -1.58  0.11476
## SPECIESPHYICO  -14.4136  1455.3977   -0.01  0.99210
## SPECIESPIPBLA   16.5834  1455.3977    0.01  0.99091
## SPECIESPIPGLA    0.3960     0.7138    0.55  0.57908
## SPECIESPIPHIS    1.9153     1.3606    1.41  0.15921
## SPECIESPREMON    0.3689     0.6972    0.53  0.59674
## SPECIESPSESPU    2.6686     1.2905    2.07  0.03865 *
## SPECIESPSYBER   -0.1923     0.7078   -0.27  0.78583
## SPECIESPSYBRA    0.4080     0.7284    0.56  0.57543
## SPECIESPSYDEF    2.0402     0.9691    2.11  0.03527 *
## SPECIESPSYMAL    1.6238     0.8066    2.01  0.04411 *
## SPECIESRHEPOR   12.2664  1006.7881    0.01  0.99028
## SPECIESROYBOR   -2.4099     1.0013   -2.41  0.01610 *
## SPECIESSAPLAU   -0.1434     0.7259   -0.20  0.84341
## SPECIESSCHMOR    0.4305     0.7239    0.59  0.55200
## SPECIESSLOBER    1.6228     0.7577    2.14  0.03221 *
## SPECIESSYMMAR   16.6284  1455.3977    0.01  0.99088
## SPECIESSYZJAM    1.9873     0.7743    2.57  0.01027 *
## SPECIESTABHET    1.0586     0.7139    1.48  0.13811
## SPECIESTETBAL    1.7468     0.7963    2.19  0.02826 *
## SPECIESTRIPAL    0.8344     0.7299    1.14  0.25297
## SPECIESZANMAR    0.3912     0.8390    0.47  0.64098
## HEIGHT.ADJ       0.7234     0.0594   12.18  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
##     Null deviance: 6373.9  on 4684  degrees of freedom
## Residual deviance: 5316.4  on 4612  degrees of freedom
## AIC: 5462
##
## Number of Fisher Scoring iterations: 14``````
``summary(surv.glmer.spp)``
``````## Generalized linear mixed model fit by maximum likelihood ['glmerMod']
##  Family: binomial ( logit )
## Formula: STATUS ~ HEIGHT.ADJ + (1 | PLOT) + (1 | SPECIES)
##    Data: sdata
##
##      AIC      BIC   logLik deviance
##     5242     5268    -2617     5234
##
## Random effects:
##  Groups  Name        Variance Std.Dev.
##  PLOT    (Intercept) 0.67     0.819
##  SPECIES (Intercept) 2.06     1.434
## Number of obs: 4685, groups: PLOT, 148; SPECIES, 72
##
## Fixed effects:
##             Estimate Std. Error z value Pr(>|z|)
## (Intercept)   0.0855     0.2150     0.4     0.69
## HEIGHT.ADJ    0.8298     0.0651    12.8   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Correlation of Fixed Effects:
##            (Intr)

## Linear Mixed Effects Model: Example Using Fruit Production Data With Repeated Measures

In this case we have normally-distributed errors.

### exploring the data

Data

The `lmer()` function does not return p values because of uncertainties in calculating the appropriate degrees of freedom, but there are some work arounds.

``````# read in data
sep = "\t")
``````##   TAG DBH FRUIT YEAR
## 1  86 454    45    1
## 2  86 454   662    2
## 3 101 369     6    1
## 4 101 369     7    2
## 5 155 450   700    1
## 6 155 450   225    2``````

The fruit production is log normally distributed, so we can make it normally distributed by log transforming it. Why do we add +1 to each value in this case?

``````fruitdata\$LOGFRUIT <- log(fruitdata\$FRUIT + 1)

par(mfrow = c(1, 2))
hist(fruitdata\$FRUIT)
hist(fruitdata\$LOGFRUIT)``````

Let's plot the relationship we are trying to model

``plot(fruitdata\$DBH, fruitdata\$LOGFRUIT)``

We have 3 years of fruit production data, with the same individuals measured in each year (ie. repeated measures) which we can account for by including individual as a random effect

``````fruit.lmer <- lmer(LOGFRUIT ~ as.factor(YEAR) + DBH + (1 | TAG), data = fruitdata)

summary(fruit.lmer)``````
``````## Linear mixed model fit by REML ['lmerMod']
## Formula: LOGFRUIT ~ as.factor(YEAR) + DBH + (1 | TAG)
##    Data: fruitdata
##
## REML criterion at convergence: 1532
##
## Random effects:
##  Groups   Name        Variance Std.Dev.
##  TAG      (Intercept) 0.825    0.908
##  Residual             1.416    1.190
## Number of obs: 428, groups: TAG, 179
##
## Fixed effects:
##                   Estimate Std. Error t value
## (Intercept)       2.267673   0.311076    7.29
## as.factor(YEAR)2  0.532925   0.143737    3.71
## as.factor(YEAR)3 -0.528386   0.142406   -3.71
## DBH               0.003991   0.000572    6.98
##
## Correlation of Fixed Effects:
##             (Intr) a.(YEAR)2 a.(YEAR)3
## as.f(YEAR)2 -0.180
## as.f(YEAR)3 -0.214  0.446
## DBH         -0.927 -0.020     0.009``````

As with other models we have seen, for factors the estimates are relative to the Intercept, which is the value for the first factor (in this case, YEAR 1)

In this case, we do not test whether we should include individual (TAG) in the model because we know we need to control for repeated measures, so we should keep it in.

#### R 3.0.0 and above

For the latest recommendations see here.

#### < R 3.0.0

Since `lmer()` does not give p values, in older versions of R we can evaluate significance using confidence intervals based on simulations, using two functions: `mcmcsamp()` and `HPDinterval()`.

The function `mcmcsamp()` generates a sample of size n from the posterior distribution of the parameters of our fitted model using Markov Chain Monte Carlo methods ...

``fruit.mcmc <- mcmcsamp(fruit.lmer, n = 1000)   # in reality, may want to use a larger n (eg. 10,000)``

.... to then generate 95% "confidence" intervals, we use HPDinterval with prob = 0.95 for each parameter, it returns the shortest interval with a 95% probability content in the empirical distribution.

`HPDinterval()` is included in two packages (code & lme4), and only the lme4 version seems to be working, so we have to specific lme4

``lme4::HPDinterval(fruit.mcmc, prob = 0.95)  ``

Best yet, there is this function, in the languageR package, which does all of this for you in one step:

``````library('languageR')
pvals.fnc(fruit.lmer, nsim = 1000) ``````

For the latest recommendations see here.

And for previous discussions and far more details than you really want on this issue, see here and related posts.

# Exercises

1. Using the seedling survival data, try adding other fixed effects (eg. light, conspecific neighbors) to the model.

2. How do your results change when you add/remove PLOT as a random effect?