Error in nlme repeated measures - r
I'm trying to run a linear mixed model with repeated measures at 57 different timepoints. But I keep getting the error message:
Error in solve.default(estimates[dimE[1L] - (p:1), dimE[2L] - (p:1), drop = FALSE]) :
system is computationally singular: reciprocal condition number = 7.7782e-18
What does this mean?
My code is such:
model.dataset = data.frame(TimepointM=timepoint,SubjectM=sample,GeneM=gene)
library("nlme")
model = lme(score ~ TimepointM + GeneM,data=model.dataset,random = ~1|SubjectM)
Here's the data:
score = c(2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,6,7,2,-3,11,14,1,7,6,7,2,-3,11,14,1,7,6,2,-3,11,14,1,7,7,2,-3,11,14,1,7,6,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,6,2,-3,11,14,1,7,7,2,-3,11,14,1,7,6,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,6,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,6,7,2,-3,11,14,1,7,6,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7,2,-3,11,14,1,7)
timepoint = c(1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,5,5,5,5,5,5,6,6,6,6,6,6,7,7,7,7,7,7,8,8,8,8,8,8,8,9,9,9,9,9,9,10,10,10,10,10,10,10,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13,13,14,14,14,14,14,14,14,14,15,15,15,15,15,15,16,16,16,16,16,16,16,16,17,17,17,17,17,17,18,18,18,18,18,18,19,19,19,19,19,19,19,20,20,20,20,20,20,21,21,21,21,21,21,24,24,24,24,24,24,24,25,25,25,25,25,25,25,27,27,27,27,27,27,28,28,28,28,28,28,29,29,29,29,29,29,30,30,30,30,30,30,30,31,31,31,31,31,31,31,32,32,32,32,32,32,33,33,33,33,33,33,33,33,34,34,34,34,34,34,34,35,35,35,35,35,35,36,36,36,36,36,36,36,37,37,37,37,37,37,38,38,38,38,38,38,39,39,39,39,39,39,39,40,40,40,40,40,40,40,41,41,41,41,41,41,41,41,42,42,42,42,42,42,42,42,43,43,43,43,43,43,44,44,44,44,44,44,44,45,45,45,45,45,45,46,46,46,46,46,46,47,47,47,47,47,47,48,48,48,48,48,48,49,49,49,49,49,49,49,50,50,50,50,50,50,51,51,51,51,51,51,52,52,52,52,52,52,52,53,53,53,53,53,53,53,54,54,54,54,54,54,55,55,55,55,55,55,56,56,56,56,56,56,57,57,57,57,57,57)
sample = c("S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S13T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S01T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0","S02T0","S03T0","S07T0","S09T0","S10T0","S12T0")
gene =c(24.1215870,-18.8771658,-27.3747309,-41.5740199,26.1561877,-2.7836332,20.8322796,36.5745088,-24.1541743,-11.2362216,4.9042852,7.4230219,155.8663563,16.4465366,-11.7982286,-1.6102783,-35.9559091,27.7909495,-13.9181661,-29.6037658,-68.4297261,-45.0877920,-48.3157529,17.1649982,-26.9084544,19.7358439,-5.8991143,-24.1541743,-23.5960654,13.0780939,-2.7836332,18.6394081,-28.3157487,-49.9186269,-33.7086648,41.6864242,-30.6199654,36.1823804,-36.5745088,-49.9186269,-44.9448864,-4.9042852,-34.3314764,62.3465425,-42.7609951,-11.7982286,-32.2055657,-56.1811080,5.7216661,-17.6296771,4.3857431,-43.6534459,9.6616697,-44.9448864,18.7997599,-12.9902884,109.1064494,7.6750504,-43.6534459,-17.7130611,-25.8433097,5.7216661,-18.5575548,35.2750175,36.1823804,2.3596457,-25.7644526,-55.0574858,15.5302365,-19.4854325,73.3687689,63.1668918,20.8322796,16.5175201,-22.5438960,-28.0905540,15.5302365,7.4230219,39.5062602,107.4657509,36.1823804,-23.5964573,-45.0877920,-43.8212642,4.0869043,-40.8266205,26.3375068,13.1572292,-25.9561030,-40.2569571,-52.8102415,2.4521426,-49.1775202,246.1047731,36.1823804,11.7982286,-35.4261223,-26.9669318,-2.4521426,-38.0429873,38.5656349,9.8679219,16.5175201,8.0513914,-42.6976421,26.9735686,-26.9084544,4.3857431,12.9780515,-32.2055657,-33.7086648,9.8085704,-36.2800196,215.7518511,6.5786146,-9.4385829,-19.3233394,-40.4503978,17.1649982,-7.4230219,14.2536650,-23.5964573,-53.1391834,-52.8102415,22.0692834,-54.7447866,24.1215870,-44.8332688,-24.1541743,-42.6976421,26.9735686,-40.8266205,191.1413737,17.5429723,-70.7893718,-37.0364006,-39.3267756,-4.9042852,-0.9278777,93.5198138,-6.5786146,-24.7762801,-28.9850091,-39.3267756,22.0692834,-50.1053979,14.2536650,23.5964573,-20.9336177,-53.9338637,14.7128556,-39.8987428,4.3857431,-64.8902575,-59.5802966,-33.7086648,22.0692834,2.7836332,46.0503024,-35.3946859,-43.4775137,-53.9338637,30.2430921,-34.3314764,80.3942259,28.5073300,-87.3068919,-24.1541743,-62.9228410,13.0780939,-25.0526990,35.0859447,-24.7762801,-38.6466789,-58.4283523,31.0604729,0.0000000,24.4562563,1.0964358,-27.1359259,-75.6830794,-16.8543324,20.4345217,-11.1345329,74.1390629,18.2282447,-27.3044720,-45.2890768,-46.7707724,15.3258912,-27.9523169,-6.9763039,117.3099418,18.6394081,-21.2368115,-38.6466789,-34.8322870,22.0692834,-48.2496425,6.5786146,-64.8902575,-51.5289052,-80.9007955,23.7040451,-26.9084544,223.1349942,8.7714862,10.6184058,-127.2119846,-31.4614205,0.8173809,-16.7017993,9.8679219,-35.3946859,-54.7494617,-44.9448864,14.7128556,-18.5575548,97.5827836,-166.3550237,-95.0064189,-123.5984376,104.6247509,-121.5519839,33.9895089,-44.8332688,-40.2569571,-56.1811080,51.4949946,0.0000000,-16.9312544,95.9808615,6.5786146,-21.2368115,-9.6616697,-13.4834659,10.6259513,-25.9805767,116.4895926,-1.0964358,-16.5175201,-56.3597400,-44.9448864,13.8954747,-12.9902884,-5.6437515,71.3703842,25.2180227,-41.2938002,-53.1391834,-32.5850426,8.9911895,12.9902884,31.9812582,1.0964358,-70.7893718,-33.8158440,-38.2031534,-15.5302365,-25.0526990,153.4053085,36.1823804,-34.2148630,-41.8672354,-19.1015767,22.8866643,0.9278777,20.8322796,-29.4955716,-43.4775137,-69.6645739,33.5126155,-45.4660092,26.3144585,-33.0350402,24.1541743,-42.6976421,0.0000000,-28.7642099,38.3752520,-7.0789372,-22.5438960,-20.2251989,34.3299964,19.4854325,4.3857431,-61.3507889,-33.8158440,-64.0464631,39.2342816,-28.7642099,183.7582306,-4.3857431,-22.4166344,-28.9850091,-57.3047302,25.3388069,-26.9084544,35.0859447,7.0789372,-33.8158440,-43.8212642,-1.6347617,5.5672664,-35.0859447,-40.1139773,-14.4925046,-12.3598438,21.2519025,-14.8460438,119.7709896,30.7002016,-22.4166344,-46.6980703,-43.8212642,5.7216661,-10.2066551,203.4466124,116.2221917,-83.7674233,-109.4989234,-38.2031534,78.4685632,-56.6005421,21.9287154,-63.7104346,-56.3597400,-4.4944886,25.3388069,-73.3023414,29.6037658,-31.8552173,-46.6980703,-79.7771734,21.2519025,-18.5575548,16.4465366,-27.1359259,-43.4775137,-41.5740199,-11.4433321,-23.1969435,27.4108943,-84.9472461,-53.1391834,-40.4503978,22.8866643,16.7017993)
tl;dr I think your problem is that every individual has exactly the same response value (score) for every time point (i.e. perfect homogeneity within individuals), so the random-effects term completely explains the data; there's nothing left over for the fixed effects. Are you sure you didn't want to use gene as your response variable?? (Discovered after running through a bunch of modeling attempts, by plotting the damn data, something everyone should always do first ...)
## simplifying names etc. slightly
dd <- data.frame(timepoint,sample,gene,score,)
library("nlme")
m0 <- lme(score ~ timepoint + gene, data=dd,
random = ~1|sample)
## reproduces error
As a first check, let's just see if there's something in your fixed-effect model that is singular:
lm(score~timepoint+gene,dd)
##
## Call:
## lm(formula = score ~ timepoint + gene, data = dd)
##
## Coefficients:
## (Intercept) timepoint gene
## 5.414652 -0.004064 -0.024485
No, that works fine.
Let's try it in lme4:
library(lme4)
m1 <- lmer(score ~ timepoint + gene + (1|sample), data=dd)
## Error in fn(x, ...) : Downdated VtV is not positive definite
Let's try scaling & centering the data -- sometimes that helps:
ddsc <- transform(dd,
timepoint=scale(timepoint),
gene=scale(gene))
lme still fails:
m0sc <- lme(score ~ timepoint + gene, data=ddsc,
random = ~1|sample)
lmer works -- sort of!
m1sc <- lmer(score ~ timepoint + gene + (1|sample), data=ddsc)
## Warning message:
## In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, :
## Model is nearly unidentifiable: very large eigenvalue
## - Rescale variables?
The results give coefficients for the parameters that are vanishingly close to zero. (The residual variance is also vanishingly small.)
## m1sc
## Linear mixed model fit by REML ['lmerMod']
## Formula: score ~ timepoint + gene + (1 | sample)
## Data: ddsc
## REML criterion at convergence: -9062.721
## Random effects:
## Groups Name Std.Dev.
## sample (Intercept) 7.838e-01
## Residual 3.344e-07
## Number of obs: 348, groups: sample, 8
## Fixed Effects:
## (Intercept) timepoint gene
## 5.714e+00 -4.194e-16 -1.032e-14
At this point I can only think of a couple of possibilities:
there's something about the experimental design that means the random effects are somehow (?) completely confounded with one or both of the fixed effects
these are simulated data that are artificially constructed to be perfectly balanced ... ?
library(ggplot2); theme_set(theme_bw())
ggplot(dd,aes(timepoint,score,group=sample,colour=gene))+
geom_point(size=4)+
geom_line(colour="red",alpha=0.5)
Aha!
In order for R to solve a matrix, it needs to be computationally invertible. The error you are getting back is telling you that, for computational purposes, your matrix is singular, which means it does not have an inverse.
As this error deals more with the statistical theory side, it's probably better suited for cross-validated. See this link for more information.
Check your data to make sure you do not have perfectly correlated independent variables.
Related
How to incorporate a random effect into a nonlinear mixed effect (nlme) model?
I'd like to build a nonlinear mixed effect model that describes the relationship between two variables, "x" and "y", which vary randomly by a third variable "r" using an exponential rise to a maximum as described by the equation: y = theta(1-exp(-beta*x)).
I've been able to create the nonlinear model for x and y using nls(), but I have not been successful in incorporating a random effect into nlme().
When I build the model using nlme() I end up with an error message: "Error in eval(predvars, data, env) : object 'theta' not found". This error is unexpected to me since the nls() model ran without issue using the same dataframe.
To build the dataset:
x = c(33,35,16,8,31,31,31,23,7,7,7,7,11,11,3,3,6,6,32,32,1,17,17,17,25,40,40,6,6,29,29,13,23,23,44,44,43,43,13,4,6,15,17,22,28,8,11,22,32,6,12,20,27,15,29,29,29,29,29,12,12,16,16,12,12,2,49,49,14,14,14,37,2.87,4.86,7.90,11.95,16.90,16.90,18.90,18.89,22.00,24.08,27.14,30.25,31.22,32.26,7,14,19,31,36,7,14,19,31,36,7,16,16,16,16,16,16,32,32,32,32,32,32,11,11,11,13,13,13,13,13,13,13,13,13,13,13,13,9,9)
y = c(39.61,32.66,27.06,21.74,22.18,38.19,35.02,23.13,9.70,14.20,13.40,15.30,18.80,19.00,3.80,4.90,15.00,14.20,24.90,16.56,1.76,29.29,28.49,18.64,27.10,9.47,14.14,10.27,8.44,26.15,25.43,22.00,19.00,13.00,73.19,67.76,32.34,36.86,8.00,1.57,8.33,16.20,14.69,18.95,20.52,4.92,8.28,15.27,18.37,6.60,10.98,12.56,19.04,5.49,21.00,12.90,17.30,11.40,12.20,15.63,15.22,33.80,17.78,19.33,3.86,8.57,30.40,13.39,11.93,4.55,6.18,12.70,2.71,7.23,5.61,22.74,15.71,16.95,18.31,20.78,17.64,20.00,19.52,24.86,30.06,24.92,4.17,11.02,10.08,14.94,25.98,0.00,3.67,3.67,6.69,11.90,5.06,13.21,10.33,0.00,0.00,6.47,8.38,28.57,25.26,28.67,27.92,33.69,29.61,6.11,7.13,6.93,4.81,15.34,4.90,14.94,8.88,10.24,8.80,10.46,10.48,9.19,9.67,9.40,24.98,50.79)
r = c("A","A","A","A","A","A","A","A","B","B","B","B","B","B","B","B","B","B","C","C","D","E","E","E","F","G","G","H","H","H","H","I","I","I","J","J","J","J","K","L","L","L","L","L","L","L","L","L","L","L","L","L","L","M","N","N","N","N","N","O","P","P","P","P","P","Q","R","R","S","S","S","T","U","U","U","U","U","U","U","U","U","U","U","U","U","U","V","V","V","V","V","V","V","V","V","V","W","X","X","X","X","Y","Y","Z","Z","Z","Z","Z","Z","AA","AA","AA","AB","AB","AB","AB","AB","AB","AB","AB","AB","AB","AB","AB","AC","AC")
df = data.frame(x,y,r)
To build the nonlinear model without "r" as a random effect.
nls_test = nls(y~theta*(1-exp(-beta*x)),
data = df,
start = list(beta = 0.2, theta = 38),
trace = TRUE)
In my model, the only fixed effect is x and the only random effect is r. I've tried building an nlme() model that reflects this, based on the nlme package documentation (https://cran.r-project.org/web/packages/nlme/nlme.pdf),more specifically these lines of code found on page 186 of the documentation linked above.
The nlme() object that I've tried to create with my data is as follows:
nlme_test = nlme(y ~ theta*(1-exp(-beta*x)),
fixed = x~1,
random = r~1,
data = df,
start = c(theta = 38,
beta = 0.2))
And results in the following error.
Error in eval(predvars, data, env) : object 'theta' not found
From what I gather, this is related to 'theta' not being included in the dataframe ("df") used to build the nlme object, but it is unclear to me why this occurs as most examples that I have found for this error are related to the use of the predict() function and missing column or disagreement between column names.
Also, since the nls() model (nls_Test) worked fine using the same start = c(theta = 38, beta = 0.2) and without a 'theta' or 'beta' data column in df, I'm a bit confused as to why I'm receiving this error about column name error.
Does anyone have suggestions or references to help me incorporate the random effect into my nlme model?
Thanks!
Expanding on my (now deleted, because incomplete) comment, I assume this is what you want to do. Please confirm carefully by reading the help page about nlme (i.e. ?nlme::nlme). nlme_test <- nlme(y ~ theta*(1-exp(-beta*x)), fixed = theta + beta ~ 1, random = theta + beta ~ 1, groups = ~ r, data = df, start = c(theta = 38, beta = 0.2)) The fixed and random arguments should not name the variables in your model formula but the regression parameters. This way the function knows which parts of the model are variables (to be found in data) and which parts are parameters. Also, you missed the groups argument in order to specify how the data is clustered. Output: summary(nlme_test) ## Nonlinear mixed-effects model fit by maximum likelihood ## Model: y ~ theta * (1 - exp(-beta * x)) ## Data: df ## AIC BIC logLik ## 887.6224 904.6401 -437.8112 ## ## Random effects: ## Formula: list(theta ~ 1, beta ~ 1) ## Level: r ## Structure: General positive-definite, Log-Cholesky parametrization ## StdDev Corr ## theta 1.145839e+01 theta ## beta 1.061366e-05 0.01 ## Residual 6.215030e+00 ## ## Fixed effects: theta + beta ~ 1 ## Value Std.Error DF t-value p-value ## theta 21.532188 2.8853414 96 7.462614 0e+00 ## beta 0.104404 0.0251567 96 4.150144 1e-04 ## Correlation: ## theta ## beta -0.548 ## ## Standardized Within-Group Residuals: ## Min Q1 Med Q3 Max ## -2.89510795 -0.51882772 -0.09466037 0.34471808 3.66855121 ## ## Number of Observations: 126 ## Number of Groups: 29
lme4: How to specify random slopes while constraining all correlations to 0?
Due to an interesting turn of events, I'm trying use the lme4 package in R to fit a model in which the random slopes are not allowed to correlate with each other or the random intercept. Effectively, I want to estimate the variance parameter for each random slope, but none of the correlations/covariances. From the reading I've done so far, I think what I want is effectively a diagonal variance/covariance structure for the random effects. An answer to a similar question here provides a workaround to specify a model where slopes are correlated with intercepts, but not with each other. I also know the || syntax in lme4 makes slopes that are correlated with each other, but not with the intercepts. Neither of these seems to fully accomplish what I'm looking to do. Borrowing the example from the earlier post, if my model is: m1 <- lmer (Y ~ A + B + (1+A+B|Subject), data=mydata) is there a way to specify the model such that I estimate variance parameters for A and B while constraining all three correlations to 0? I would like to achieve a result that looks something like this: VarCorr(m1) ## Groups Name Std.Dev. Corr ## Subject (Intercept) 1.41450 ## A 1.49374 0.000 ## B 2.47895 0.000 0.000 ## Residual 0.96617 I'd prefer a solution that could achieve this for an arbitrary number of random slopes. For example, if I were to add a random effect for a third variable C, there would be 6 correlation parameters to fix at 0 rather than 3. However, anything that could get me started in the right direction would be extremely helpful. Edit: On asking this question, I misunderstood what the || syntax does in lme4. Struck through the incorrect statement above to avoid misleading anyone in the future.
This is exactly what the double-bar notation does. However, note that the || in lme4 does not work as one might expect for factor variables. It does work 'properly' in glmmTMB, and the afex::mixed() function is a wrapper for [g]lmer which does implement a fully functional version of ||. (I have meant to import this into lme4 for years but just haven't gotten around to it yet ...) simulated example library(lme4) set.seed(101) dd <- data.frame(A = runif(500), B = runif(500), Subject = factor(rep(1:25, 20))) dd$Y <- simulate(~ A + B + (1 + A + B|Subject), newdata = dd, family = gaussian, newparams = list(beta = rep(1,3), theta = rep(1,6), sigma = 1))[[1]] solution summary(m <- lmer (Y ~ A + B + (1+A+B||Subject), data=dd)) The correlations aren't listed because they are structurally absent (internally, the random effects term is expanded to (1|Subject) + (0 + A|Subject) + (0+B|Subject), which is also why the groups are listed as Subject, Subject.1, Subject.2). Random effects: Groups Name Variance Std.Dev. Subject (Intercept) 0.8744 0.9351 Subject.1 A 2.0016 1.4148 Subject.2 B 2.8718 1.6946 Residual 0.9456 0.9724 Number of obs: 500, groups: Subject, 25
Two-level modelling with lme in R
I am interested in estimating a mixed effect model with two random components (I am sorry for the somewhat unprecise notation. I am somewhat new to these kind of models). Finally, I also want also the standard errors of the variances of the random components. That is why I am somewhat boudn to using the package lme. The reason is that I found this description on how to calculate those standard errors and also interesting, the standard error for function of these variances link.
I believe I know how to use the package lmer. I am finally interested in model2. For the model1, both command yield the same estimates. But model2 with lme yields different results than model2 with lmer from the lme4 package. Could you help me to get around how to set up the random components for lme? This would be much appreciated. Thanks. Please find attached my MWE.
Best
Daniel
#### load all packages #####
loadpackage <- function(x){
for( i in x ){
# require returns TRUE invisibly if it was able to load package
if( ! require( i , character.only = TRUE ) ){
# If package was not able to be loaded then re-install
install.packages( i , dependencies = TRUE )
}
# Load package (after installing)
library( i , character.only = TRUE )
}
}
# Then try/install packages...
loadpackage( c("nlme", "msm", "lmeInfo", "lme4"))
alcohol1 <- read.table("https://stats.idre.ucla.edu/stat/r/examples/alda/data/alcohol1_pp.txt", header=T, sep=",")
attach(alcohol1)
id <- as.factor(id)
age <- as.factor(age)
model1.lmer <-lmer(alcuse ~ 1 + peer + (1|id))
summary(model1.lmer)
model2.lmer <-lmer(alcuse ~ 1 + peer + (1|id) + (1|age))
summary(model2.lmer)
model1.lme <- lme(alcuse ~ 1+ peer, data = alcohol1, random = ~ 1 |id, method ="REML")
summary(model1.lme)
model2.lme <- lme(alcuse ~ 1+ peer, data = alcohol1, random = ~ 1 |id + 1|age, method ="REML")
Edit (15/09/2021):
Estimating the model as follows end then returning the estimates via nlme::VarCorr gives me different results. While the estimates seem to be in the ball park, it is as they are switched across components.
model2a.lme <- lme(alcuse ~ 1+ peer, data = alcohol1, random = ~ 1 |id/age, method ="REML")
summary(model2a.lme)
nlme::VarCorr(model2a.lme)
Variance StdDev
id = pdLogChol(1)
(Intercept) 0.38390274 0.6195989
age = pdLogChol(1)
(Intercept) 0.47892113 0.6920413
Residual 0.08282585 0.2877948
EDIT (16/09/2021):
Since Bob pushed me to think more about my model, I want to give some additional information. Please know that the data I use in the MWE do not match my true data. I just used it for illustrative purposes since I can not upload myy true data. I have a household panel with income, demographic informations and parent indicators.
I am interested in intergenerational mobility. Sibling correlations of permanent income are one industry standard. At the very least, contemporanous observations are very bad proxies of permanent income. Due to transitory shocks, i.e., classical measurement error, those estimates are most certainly attenuated. For this reason, we exploit the longitudinal dimension of our data.
For sibling correlations, this amounts to hypothesising that the income process is as follows:
$$Y_{ijt} = \beta X_{ijt} + \epsilon_{ijt}.$$
With Y being income from individual i from family j in year t. X comprises age and survey year indicators to account for life-cycle effects and macroeconmic conditions in survey years. Epsilon is a compund term comprising a random individual and family component as well as a transitory component (measurement error and short lived shocks). It looks as follows:
$$\epsilon_{ijt} = \alpha_i + \gamma_j + \eta_{ijt}.$$
The variance of income is then:
$$\sigma^2_\epsilon = \sigma^2_\alpha + \sigma^2\gamma + \sigma^2\eta.$$
The quantitiy we are interested in is
$$\rho = \frac(\sigma^2\gamma}{\sigma^2_\alpha + \sigma^2\gamma},$$
which reflects the share of shared family (and other characteristics) among siblings of the variation in permanent income.
B.t.w.: The struggle is simply because I want to have a standard errors for all estimates and for \rho.
This is an example of crossed vs nested random effects. (Note that the example you refer to is fitting a different kind of model, a random-slopes model rather than a model with two different grouping variables ...)
If you try with(alcohol1, table(age,id)) you can see that every id is associated with every possible age (14, 15, 16). Or subset(alcohol1, id==1) for example:
id age coa male age_14 alcuse peer cpeer ccoa
1 1 14 1 0 0 1.732051 1.264911 0.2469111 0.549
2 1 15 1 0 1 2.000000 1.264911 0.2469111 0.549
3 1 16 1 0 2 2.000000 1.264911 0.2469111 0.549
There are three possible models you could fit for a model with random effects of age(indexed by i) and id (indexed by j)
crossed ((1|age) + (1|id)): Y_{ij} = beta0 + beta1*peer + eps1_i + eps2_j +epsr_{ij}; alcohol use varies among individuals and, independently, across ages (this model won't work very well because there are only three distinct ages in the data set, more levels are usually needed)
id nested within age ((1|age/id) = (1|age) + (1|age:id)): Y_{ij} = beta0 + beta1*peer + eps1_i + eps2_{ij} + epsr_{ij}; alcohol use varies across ages, and varies across individuals within ages (see note above about number of levels).
age nested within id ((1|id/age) = (1|id) + (1|age:id)): Y_{ij} = beta0 + beta1*peer + eps1_j + eps2_{ij} + epsr_{ij}; alcohol use varies across individuals, and varies across ages within individuals
Here eps1_i, eps2_{ij}, and epsr_{ij} are normal deviates; epsr is the residual error term.
The latter two models actually don't make sense in this case; because there is only one observation per age/id combination, the nested variance (eps2) is completely confounded with the residual variance (epsr). lme doesn't notice this; if you try to fit one of the nested models in lmer it will give an error that
number of levels of each grouping factor must be < number of observations (problems: id:age)
(Although if you try to compute confidence intervals based on model1.lme you'll get an error "cannot get confidence intervals on var-cov components: Non-positive definite approximate variance-covariance", which is a hint that something is wrong.)
You could restate this problem as saying that the residual variation, and the variation among ages within individuals, are jointly unidentifiable (can't be separated from each other, statistically).
The updated answer here shows how to get the standard errors of the variance components from an lmer model, so you shouldn't be stuck with lme (but you should think carefully about which model you're really trying to fit ...)
The GLMM FAQ might also be useful.
More generally, the standard error of
rho = (V_gamma)/(V_alpha + V_gamma)
will be hard to compute accurately, because this is a nonlinear function of the model parameters. You can apply the delta method, but the most reliable approach would be to use parametric bootstrapping: if you have a fitted model m, then something like this should work:
var_ratio <- function(m) {
v <- as.data.frame(sapply(VarCorr(m), as.numeric))
return(v$family/(v$family + v$id))
}
confint(m, method="boot", FUN =var_ratio)
You should specify random effects in lme by using / not + By lmer model2.lmer <-lmer(alcuse ~ 1 + peer + (1|id) + (1|age), data = alcohol1) summary(model2.lmer) Linear mixed model fit by REML ['lmerMod'] Formula: alcuse ~ 1 + peer + (1 | id) + (1 | age) Data: alcohol1 REML criterion at convergence: 651.3 Scaled residuals: Min 1Q Median 3Q Max -2.0228 -0.5310 -0.1329 0.5854 3.1545 Random effects: Groups Name Variance Std.Dev. id (Intercept) 0.08078 0.2842 age (Intercept) 0.30313 0.5506 Residual 0.56175 0.7495 Number of obs: 246, groups: id, 82; age, 82 Fixed effects: Estimate Std. Error t value (Intercept) 0.3039 0.1438 2.113 peer 0.6074 0.1151 5.276 Correlation of Fixed Effects: (Intr) peer -0.814 By lme model2.lme <- lme(alcuse ~ 1+ peer, data = alcohol1, random = ~ 1 |id/age, method ="REML") summary(model2.lme) Linear mixed-effects model fit by REML Data: alcohol1 AIC BIC logLik 661.3109 678.7967 -325.6554 Random effects: Formula: ~1 | id (Intercept) StdDev: 0.4381206 Formula: ~1 | age %in% id (Intercept) Residual StdDev: 0.4381203 0.7494988 Fixed effects: alcuse ~ 1 + peer Value Std.Error DF t-value p-value (Intercept) 0.3038946 0.1438333 164 2.112825 0.0361 peer 0.6073948 0.1151228 80 5.276060 0.0000 Correlation: (Intr) peer -0.814 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -2.0227793 -0.5309669 -0.1329302 0.5853768 3.1544873 Number of Observations: 246 Number of Groups: id age %in% id 82 82
Okay, finally. Just to sketch my confidential data: I have a panel of individuals. The data includes siblings, identified via mnr. income is earnings, wavey survey year, age age factors. female a factor for gender, pid is the factor identifying the individual. m1 <- lmer(income ~ age + wavey + female + (1|pid) + (1 | mnr), data = panel) vv <- vcov(m1, full = TRUE) covvar <- vv[58:60, 58:60] covvar 3 x 3 Matrix of class "dgeMatrix" cov_pid.(Intercept) cov_mnr.(Intercept) residual [1,] 2.6528679 -1.4624588 -0.4077576 [2,] -1.4624588 3.1015001 -0.0597926 [3,] -0.4077576 -0.0597926 1.1634680 mean <- as.data.frame(VarCorr(m1))$vcov mean [1] 17.92341 16.86084 56.77185 deltamethod(~ x2/(x1+x2), mean, covvar, ses =TRUE) [1] 0.04242089 The last scalar should be what I interprete as the shared background of the siblings of permanent income. Thanks to #Ben Bolker who pointed me into this direction.
Correct way to scale for multilevel regression using lmer [R]
Let us say I have data for 300 firms(level 1) in 10 countries (level 2). The level 1 variables are PQ and size. Level 2 variable is the per capita GDP. library(lme4) set.seed(1) PQ <- runif(300,7,21) id <- (1:300) country <- sample(1:10,300,replace=T) size <- sample(1:25000,300,replace=T) GDP <- sample(800:40000,10,replace=T) Country1 <- 1:10 L1data <- as.data.frame(cbind(id,country,PQ,size)) L2data <- as.data.frame(cbind(Country1,GDP)) MLdata <- merge(L1data,L2data, by.x = "country", by.y = "Country1") dummymodel <- lmer(PQ ~ size + GDP + (size|country), data = MLdata, REML = F) When I run this I get warning messages Warning messages: 1: Some predictor variables are on very different scales: consider rescaling 2: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model failed to converge with max|grad| = 1.77081 (tol = 0.002, component 1) 3: In checkConv(attr(opt, "derivs"), opt$par, ctrl = control$checkConv, : Model is nearly unidentifiable: very large eigenvalue - Rescale variables?; Model is nearly unidentifiable: large eigenvalue ratio - Rescale variables? In fact when I run the model with the original data I get an additional warning message: Model failed to converge: degenerate Hessian with 1 negative eigenvalues I guess I need to scale the independent variables to solve this problem. What is the correct way to scale in multilevel regression like this? The question is important as the results of multilevel models are dependent on scaling. Or is there some other problem that I am not able to locate?
tl;dr The models have nearly equivalent goodness of fit; the scaled model is slightly better and more reliable; the fixed effects are nearly equivalent; both models estimate singular random-effects variance-covariance matrices, which makes comparison much harder and means you shouldn't be relying on these models for conclusions about variances in any case ...
The model should have the same meaning after centering and rescaling. As I'll show below, the fixed effects are essentially equivalent. I'm finding it harder to convince myself that the variance-covariance estimates are equivalent, but the model is singular anyway (i.e., there isn't enough information to fit a positive-definite variance-covariance matrix).
Using your example and re-running with scaled parameters:
MLdata <- transform(MLdata,
size_cs=scale(size),
GDP_cs=scale(GDP))
m2 <- lmer(PQ ~ size_cs + GDP_cs + (size_cs|country), data = MLdata,
REML = FALSE)
Comparing the log-likelihoods:
logLik(dummymodel) ## -828.4349
logLik(m2) ## -828.4067
This suggests that the models are pretty close, but the scaled model has done slightly better (although an improvement of 0.03 log-likelihood units is very small).
The fixed effects look different, but I'm going to show that they're equivalent:
fixef(dummymodel)
## (Intercept) size GDP
## 1.345754e+01 3.706393e-05 -5.464550e-06
## fixef(m2)
## (Intercept) size_cs GDP_cs
## 13.86155940 0.27300688 -0.05914308
(If you look at coef(summary(.)) for both models, you'll see that the t-statistics for size and GDP are nearly identical.)
From this question
rescale.coefs <- function(beta,mu,sigma) {
beta2 <- beta ## inherit names etc.
## parameters other than intercept:
beta2[-1] <- sigma[1]*beta[-1]/sigma[-1]
## intercept:
beta2[1] <- sigma[1]*beta[1]+mu[1]-sum(beta2[-1]*mu[-1])
return(beta2)
}
cm <- sapply(MLdata[c("size","GDP")],mean)
csd <- sapply(MLdata[c("size","GDP")],sd)
rescaled <- rescale.coefs(fixef(m2),mu=c(0,cm),sigma=c(1,csd))
all.equal(rescaled,fixef(m2))
cbind(fixef(dummymodel),rescaled)
## rescaled
## (Intercept) 1.345754e+01 1.345833e+01
## size 3.706393e-05 3.713062e-05
## GDP -5.464550e-06 -5.435151e-06
Very similar.
With regard to the variance-covariance matrices:
VarCorr(dummymodel)
## Groups Name Std.Dev. Corr
## country (Intercept) 2.3420e-01
## size 1.5874e-05 -1.000
## Residual 3.8267e+00
VarCorr(m2)
## Groups Name Std.Dev. Corr
## country (Intercept) 0.0000e+00
## size_cs 4.7463e-08 NaN
## Residual 3.8283e+00
Neither variance-covariance matrix is positive definite; the first has the variation in intercept across countries perfectly correlated with variation in slope across countries, while the second assigns zero variance to the intercept across countries. In addition, the among-country variation is very small relative the residual variance in either case. If both matrices were positive definite we could work on finding the transformation that would scale from one case to the other (I think it would just be to multiply each element by (s_i s_j), where s_. is the scaling factor applied to a given predictor). When they're not PD, it becomes tricky.
Error with nlme
For IGF data from nlme library, I'm getting this error message: lme(conc ~ 1, data=IGF, random=~age|Lot) Error in lme.formula(conc ~ 1, data = IGF, random = ~age | Lot) : nlminb problem, convergence error code = 1 message = iteration limit reached without convergence (10) But everything is fine with this code lme(conc ~ age, data=IGF) Linear mixed-effects model fit by REML Data: IGF Log-restricted-likelihood: -297.1831 Fixed: conc ~ age (Intercept) age 5.374974367 -0.002535021 Random effects: Formula: ~age | Lot Structure: General positive-definite StdDev Corr (Intercept) 0.082512196 (Intr) age 0.008092173 -1 Residual 0.820627711 Number of Observations: 237 Number of Groups: 10 As IGF is groupedData, so both codes are identical. I'm confused why the first code produces error. Thanks for your time and help.
I find the other, older answer here unsatisfactory. I distinguish between cases where, statistically, age has no impact and conversely we encounter a computational error. Personally, I have made career mistakes by conflating these two cases. R has signaled the latter and I would like to dive into why that is. The model that OP has specified is a growth model, with random slopes and intercepts. A grand intercept is included but not a grand age slope. One unsavory constraint that is imposed by fitting a random slope without addition of its "grand" term is that you are forcing the random slope to have 0 mean, which is very difficult to optimize. Marginal models indicate age does not have a statistically significant different value from 0 in the model. Furthermore adding age as a fixed effect does not remedy the problem. > lme(conc~ age, random=~age|Lot, data=IGF) Error in lme.formula(conc ~ age, random = ~age | Lot, data = IGF) : nlminb problem, convergence error code = 1 message = iteration limit reached without convergence (10) Here the error is obvious. It might be tempting to set the number of iterations up. lmeControl has many iterative estimands. But even that doesn't work: > fit <- lme(conc~ 1, random=~age|Lot, data=IGF, control = lmeControl(maxIter = 1e8, msMaxIter = 1e8)) Error in lme.formula(conc ~ 1, random = ~age | Lot, data = IGF, control = lmeControl(maxIter = 1e+08, : nlminb problem, convergence error code = 1 message = singular convergence (7) So it's not a precision thing, the optimizer is running out-of-bounds. There must be key differences between the two models you have proposed fitting, and a way to diagnose the error that you have found. One simple approach is specifying a "verbose" fit for the problematic model: > lme(conc~ 1, random=~age|Lot, data=IGF, control = lmeControl(msVerbose = TRUE)) 0: 602.96050: 2.63471 4.78706 141.598 1: 602.85855: 3.09182 4.81754 141.597 2: 602.85312: 3.12199 4.97587 141.598 3: 602.83803: 3.23502 4.93514 141.598 (truncated) 48: 602.76219: 6.22172 4.81029 4211.89 49: 602.76217: 6.26814 4.81000 4425.23 50: 602.76216: 6.31630 4.80997 4638.57 50: 602.76216: 6.31630 4.80997 4638.57 The first term is the REML (I think). The second through fourth terms are the parameters to an object called lmeSt of class lmeStructInt, lmeStruct, and modelStruct. If you use Rstudio's debugger to inspect attributes of this object (the lynchpin of the problem), you'll see it is the random effects component that explodes here. coef(lmeSt) after 50 iterations produces reStruct.Lot1 reStruct.Lot2 reStruct.Lot3 6.316295 4.809975 4638.570586 as seen above and produces > coef(lmeSt, unconstrained = FALSE) reStruct.Lot.var((Intercept)) reStruct.Lot.cov(age,(Intercept)) 306382.7 2567534.6 reStruct.Lot.var(age) 21531399.4 which is the same as the Browse[1]> lmeSt$reStruct$Lot Positive definite matrix structure of class pdLogChol representing (Intercept) age (Intercept) 306382.7 2567535 age 2567534.6 21531399 So it's clear the covariance of the random effects is something that's exploding here for this particular optimizer. PORT routines in nlminb have been criticized for their uninformative errors. The text from David Gay (Bell Labs) is here http://ms.mcmaster.ca/~bolker/misc/port.pdf The PORT documentation suggests our error 7 from using a 1 billion iter max "x may have too many free components. See ยง5.". Rather than fix the algorithm, it behooves us to ask if there are approximate results which should generate similar outcomes. It is, for instance, easy to fit an lmList object to come up with the random intercept and random slope variance: > fit <- lmList(conc ~ age | Lot, data=IGF) > cov(coef(fit)) (Intercept) age (Intercept) 0.13763699 -0.018609973 age -0.01860997 0.003435819 although ideally these would be weighted by their respective precision weights: To use the nlme package I note that unconstrained optimization using BFGS does not produce such an error and gives similar results: > lme(conc ~ 1, data=IGF, random=~age|Lot, control = lmeControl(opt = 'optim')) Linear mixed-effects model fit by REML Data: IGF Log-restricted-likelihood: -292.9675 Fixed: conc ~ 1 (Intercept) 5.333577 Random effects: Formula: ~age | Lot Structure: General positive-definite, Log-Cholesky parametrization StdDev Corr (Intercept) 0.032109976 (Intr) age 0.005647296 -0.698 Residual 0.820819785 Number of Observations: 237 Number of Groups: 10 An alternative syntactical declaration of such a model can be done with the MUCH easier lme4 package: library(lme4) lmer(conc ~ 1 + (age | Lot), data=IGF) which yields: > lmer(conc ~ 1 + (age | Lot), data=IGF) Linear mixed model fit by REML ['lmerMod'] Formula: conc ~ 1 + (age | Lot) Data: IGF REML criterion at convergence: 585.7987 Random effects: Groups Name Std.Dev. Corr Lot (Intercept) 0.056254 age 0.006687 -1.00 Residual 0.820609 Number of obs: 237, groups: Lot, 10 Fixed Effects: (Intercept) 5.331 An attribute of lmer and its optimizer is that random effects correlations which are very close to 1, 0, or -1 are simply set to those values since it simplifies the optimization (and statistical efficiency of the estimation) substantially. Taken together, this does not suggest that age does not have an effect, as was said earlier, and this argument can be supported by the numeric results.
If you plot the data, you can see that there is no effect of age, so it seems strange to be trying to fit a random effect of age in spite of this. No wonder it is not converging. library(nlme) library(ggplot2) dev.new(width=6, height=3) qplot(age, conc, data=IGF) + facet_wrap(~Lot, nrow=2) + geom_smooth(method='lm') I think what you want to do is model a random effect of Lot on the intercept. We can try including age as a fixed effect, but we'll see that it is not significant and can be thrown out: > summary(lme(conc ~ 1 + age, data=IGF, random=~1|Lot)) Linear mixed-effects model fit by REML Data: IGF AIC BIC logLik 604.8711 618.7094 -298.4355 Random effects: Formula: ~1 | Lot (Intercept) Residual StdDev: 0.07153912 0.829998 Fixed effects: conc ~ 1 + age Value Std.Error DF t-value p-value (Intercept) 5.354435 0.10619982 226 50.41849 0.0000 age -0.000817 0.00396984 226 -0.20587 0.8371 Correlation: (Intr) age -0.828 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -5.46774548 -0.43073893 -0.01519143 0.30336310 5.28952876 Number of Observations: 237 Number of Groups: 10