Linear mixed model for placement of nuclear stress in 10-word turns

Question

I'm trying to model the placement of nuclear stress in 10-word turns in a linear mixed model but am very new to mixed modeling. The model includes these variables:

• STRSS, the binary response variable; the 10-word turns have been selected in such a way that only 1 word carries the nuclear stress
• INFMX, a binary explanatory variable denoting whether a word carries the maximum informativity (i.e., 'surprisal' given the preceding word)
• CLASS, an explanatory variable with three levels: function word, interjection, or content word
• POST, an explanatory variable denoting whether the nuclear stress occurs early in the turn (words 1-3), in mid-turn position (words 4-6), or late in the turn (words 7-10)
• STRCT, an explanatory variable denoting whether the nuclear stress falls on a word inside what is called the turn constructional unit (TCU) or not
• SPKR, a random factor referring to speaker IDs, and
• SEQU, another random factor referring each word to its place in the sequence of exactly 10 words, considered random because only 10-word turns are examined here, not turns of other lengths

Here's some reproducible data:

df <- data.frame(
  SPKR = c(rep("A", 10), rep("B", 10), rep("C", 10)),
  SEQU = rep(1:10, 3),
  STRSS = rep(c(rep("notS", 8), "S", "notS"), 3),
  INFMX = rep(c(rep("notMax", 8), "priorMax", "Max"), 3),
  CLASS = rep(c(rep("fnc", 3), rep("itj", 1), rep("cnt", 6)), 3),
  POST = rep(c(rep("earl", 3), rep("mid", 3), rep("lte", 4)), 3),
  STRCT = rep(c(rep("notTCU", 2), rep("TCU", 6), rep("notTCU", 2)), 3)
)
df
   SPKR SEQU STRSS    INFMX CLASS POST  STRCT
1     A    1  notS   notMax   fnc earl notTCU
2     A    2  notS   notMax   fnc earl notTCU
3     A    3  notS   notMax   fnc earl    TCU
4     A    4  notS   notMax   itj  mid    TCU
5     A    5  notS   notMax   cnt  mid    TCU
6     A    6  notS   notMax   cnt  mid    TCU
7     A    7  notS   notMax   cnt  lte    TCU
8     A    8  notS   notMax   cnt  lte    TCU
9     A    9     S priorMax   cnt  lte notTCU
10    A   10  notS      Max   cnt  lte notTCU
11    B    1  notS   notMax   fnc earl notTCU
12    B    2  notS   notMax   fnc earl notTCU
13    B    3  notS   notMax   fnc earl    TCU
14    B    4  notS   notMax   itj  mid    TCU
15    B    5  notS   notMax   cnt  mid    TCU
16    B    6  notS   notMax   cnt  mid    TCU
17    B    7  notS   notMax   cnt  lte    TCU
18    B    8  notS   notMax   cnt  lte    TCU
19    B    9     S priorMax   cnt  lte notTCU
20    B   10  notS      Max   cnt  lte notTCU
21    C    1  notS   notMax   fnc earl notTCU
22    C    2  notS   notMax   fnc earl notTCU
23    C    3  notS   notMax   fnc earl    TCU
24    C    4  notS   notMax   itj  mid    TCU
25    C    5  notS   notMax   cnt  mid    TCU
26    C    6  notS   notMax   cnt  mid    TCU
27    C    7  notS   notMax   cnt  lte    TCU
28    C    8  notS   notMax   cnt  lte    TCU
29    C    9     S priorMax   cnt  lte notTCU
30    C   10  notS      Max   cnt  lte notTCU

My hypothesis is that a word will carry nuclear stress (i.e., df$STRSS=="S") if

• df$INFMX=="priorMAX", i.e., the word with the greatest informativity immediately follows the word with the nuclear stress
• df$CLASS=="cnt", i.e., the word is a content word
• df$STRCT=="notTCU", i.e., the word lies inside the TCU
• df$POST=="lte", i.e., the word occurs late in the turn

Given that the response variable is binary, I've tried a generalized mixed model so far, using library("mlmRev"):

model1 <- glmer(STRSS ~ (INFMX + CLASS + POST + STRCT)^2 + 
           (1 | SPKR) + (1 | SEQU), data = df, family = binomial(link = "logit"), nAGQ = 1)

The problems I'd appreciate help with are the following:

• Is this the right approach? I.e., is this, at least in principle, the right model?

• The model call produces some unpleasant information--what to make of it?

  fixed-effect model matrix is rank deficient so dropping 19 columns /coefficients
  Warning messages:
  1: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  unable to evaluate scaled gradient
  2: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv,  :
  Hessian is numerically singular: parameters are not uniquely determined
  

• And finally, how to read the output of the model summary?

  summary(model1)
  Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
  Family: binomial  ( logit )
  Formula: STRSS ~ (INFMX + CLASS + POST + STRCT)^2 + (1 | SPKR) + (1 |      SEQU)
    Data: df
  
   AIC      BIC   logLik deviance df.resid 
   18.0     30.6      0.0      0.0       21 
  
  Scaled residuals: 
   Min        1Q    Median        3Q       Max 
  -1.49e-08  1.49e-08  1.49e-08  1.49e-08  1.49e-08 
  
  Random effects:
  Groups Name        Variance Std.Dev.
  SEQU   (Intercept) 0.83102  0.9116  
  SPKR   (Intercept) 0.05073  0.2252  
  Number of obs: 30, groups:  SEQU, 10; SPKR, 3
  
  Fixed effects:
          Estimate Std. Error z value Pr(>|z|)
  (Intercept)    3.972e+01  7.249e+07       0        1
  INFMXnotMax   -4.107e-01  6.711e+07       0        1
  INFMXpriorMax -7.929e+01  5.479e+07       0        1
  CLASSfnc       3.565e-05  4.745e+07       0        1
  CLASSitj       1.581e-06  4.745e+07       0        1
  POSTlte        1.847e-05  3.875e+07       0        1
  STRCTnotTCU   -1.472e-05  4.745e+07       0        1
  
  Correlation of Fixed Effects:
      (Intr) INFMXnM INFMXpM CLASSf CLASSt POSTlt
  INFMXnotMax -0.926                                     
  INFMXprirMx -0.378  0.408                              
  CLASSfnc     0.218 -0.471   0.000                      
  CLASSitj    -0.218  0.000   0.000   0.333              
  POSTlte     -0.535  0.289   0.000   0.408  0.408       
  STRCTnotTCU -0.655  0.707   0.000  -0.667  0.000  0.000
  fit warnings:
  fixed-effect model matrix is rank deficient so dropping 19 columns / coefficients
  convergence code: 0
   unable to evaluate scaled gradient
   Hessian is numerically singular: parameters are not uniquely determined
  
   Warning messages:
   1: In vcov.merMod(object, use.hessian = use.hessian) :
   variance-covariance matrix computed from finite-difference Hessian is
   not positive definite or contains NA values: falling back to var-cov estimated from RX
    2: In vcov.merMod(object, correlation = correlation, sigm = sig) :
    variance-covariance matrix computed from finite-difference Hessian is
    not positive definite or contains NA values: falling back to var-cov estimated from RX
  

Helpful pointers are appreciated all the more!

Answer 1

As the error message suggests, the design matrix of your model is not full-rank, i.e., some of the columns are linearly depended. You can see the rank of the design matrix via

X <- model.matrix(~ (INFMX + CLASS + POST + STRCT)^2, data = df)
qr(X)$rank 

which should be equal to the number of columns of X. You need to simplify the specification of your model to achieve that.

Moreover, it seems to me that the specification of your random effects should be glmer(... + (SEQU | SPKR), ...).

---

EDIT: Solve Complete Separation

In a simplified model in the comments below, i.e.,

fm <- glmer(STRSS ~ INFMX + CLASS + STRCT + (1 | SPKR), data = df,
            family = binomial(), nAGQ = 15)

we have complete separation. This can be checked, for example, by

with(df, table(STRSS, INFMX))

from which we see that we have cells with zero frequency. To solve this problem, you can place a penalty in the fixed-effects coefficients. This option is provided by the GLMMadaptive package. For example,

library("GLMMadaptive")

gm <- mixed_model(STRSS ~ INFMX + CLASS + STRCT, random = ~ 1 | SPKR, 
                  data = df, family = binomial(), nAGQ = 15, penalized = TRUE,
                  initial_values = list(betas = rep(0, 6)))

summary(gm)