# [R-sig-ME] anova() and the difference between (x | y) and (1 | y:x) in lme4

Ben Bolker bbolker at gmail.com
Wed Jun 11 16:38:38 CEST 2014

```On 14-06-11 10:21 AM, ONKELINX, Thierry wrote:
> Dear Hans,
>
> I assume that var1 is a factor variable.
>
> The difference is in the distribution of the random effects.
>
> (1|var1:var2) : all random intercept come from the same univariate normal distribution rnorm(mean = 0, sd = sigma)
> (0 + var1|var2): the random intercepts come from a multivariate normal distribution: rmvnorm(mean = 0, sigma = Sigma). Sigma is a positive definite matrix
>
> (0 + var1|var2) is a bit easier to understand because the BLUP's have the same interpretation of those of (1|var1:var2)
>
> The bottom-line is that (var1|var2) and (1|var1:var2) allow the same model fit but (var1|var2) makes less assumptions at the cost of estimation more parameters. (var1|var2) requires n * (n + 1) / 2 parameters, with n = number of levels of var1. (1|var1:var2) requires just 1 parameter.
>
> Best regards,
>
> ir. Thierry Onkelinx
> Instituut voor natuur- en bosonderzoek / Research Institute for Nature and Forest
> team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
> Kliniekstraat 25
> 1070 Anderlecht
> Belgium
> + 32 2 525 02 51
> + 32 54 43 61 85
> Thierry.Onkelinx at inbo.be
> www.inbo.be

Thanks Tierry.

(1) for what it's worth, lme offers an intermediate model (compound
symmetry), which allows for homogeneous but _negative_ within-group
correlation ((1|var1:var2) only allows for non-negative within-group
correlation)
(2) the 'unstructured' (var1|var2) and 'grouped/positive compound
symmetry' models (1|var1:var2) are in principle nested (all
off-diagonals equal to zero, all diagonals identical -> simpler model),
so you should be able to use a likelihood ratio test/ANOVA to test.
I would try scaling&centring your continuous predictors to see if that
makes the eigenvalue warnings go away.

Ben Bolker

>
> To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of.
> ~ Sir Ronald Aylmer Fisher
>
> The plural of anecdote is not data.
> ~ Roger Brinner
>
> The combination of some data and an aching desire for an answer does not ensure that a reasonable answer can be extracted from a given body of data.
> ~ John Tukey
>
> -----Oorspronkelijk bericht-----
> Van: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-bounces at r-project.org] Namens Hans Ekbrand
> Verzonden: woensdag 11 juni 2014 15:58
> Aan: r-sig-mixed-models at r-project.org
> Onderwerp: [R-sig-ME] anova() and the difference between (x | y) and (1 | y:x) in lme4
>
> Dear list,
>
> I have a question about the difference between
>
> y ~ (1 | var2:var1) vs y ~ (var1 | var2).
>
> In reality my model is more complex:
>
> y ~ 1 + var1 + (1 | var2:var1) + var3+ .... + var9
>
> vs
>
> y ~ 1 + var1 + (var1 | var2) + var3+ .... + var9
>
> Following the advice kindly given by Reinhold Kliegl way back ago
> (https://stat.ethz.ch/pipermail/r-sig-mixed-models/2011q2/016545.html)
> I have used the following specification with glmer() in lme4 (version 1.1-7):
>
> fit.flat <- glmer(below.poverty.line ~ 1 + employment.type + (1 | country:employment.type) + gender + age + age.2 + n.adults.minus.n.children + n.children + education + household.type, family = binomial("logit"), data = my.df)
>
> and
>
> fit.hierarchical <- glmer(below.poverty.line ~ 1 + employment.type + (employment.type | country) + gender + age + age.2 + n.adults.minus.n.children + n.children + education + household.type, family = binomial("logit"), data = my.df)
>
> Info on the data:
>
> str(my.df)
> 'data.frame':   93178 obs. of  10 variables:
>  \$ below.poverty.line       : logi  FALSE FALSE FALSE FALSE FALSE FALSE ...
>  \$ employment.type          : Factor w/ 6 levels "Core labour force",..: 1 5 1 1 1 5 5 5 1 1 ...
>  \$ country                  : Factor w/ 22 levels "austria","belgium",..: 1 1 1 1 1 1 1 1 1 1 ...
>  \$ gender                   : Factor w/ 2 levels "female","male": 2 1 2 2 1 1 1 1 2 2 ...
>  \$ age                      : num  22 22 32 56 40 54 42 18 49 20 ...
>  \$ age.2                    : num  3.39e-02 3.39e-02 7.08e-03 2.43e-02 1.71e-05 ...
>  \$ n.adults.minus.n.children: num  3 3 1 5 2 2 3 5 5 5 ...
>  \$ n.children               : num  1 1 2 0 1 0 1 0 0 0 ...
>  \$ education                : Factor w/ 4 levels "primary","lower secondary",..: 2 2 4 2 3 4 2 2 2 3 ...
>  \$ household.type           : Factor w/ 4 levels "couple without children",..: 2 2 3 1 3 4 3 4 1 4 ...
>
> If you want to replicate the analysis - or inspect the data - try this:
>
>
> The total computation time for both models is about one hour on my computer.
>
> My primary question is whether or not anova() is usable to choose between the two models?
>
> Data: my.df
> Models:
> fit.flat: below.poverty.line ~ 1 + employment.type + (1 | country:employment.type) +
> fit.flat:     gender + age + age.2 + n.adults.minus.n.children + n.children +
> fit.flat:     education + household.type
> fit.hierarchical: below.poverty.line ~ 1 + employment.type + (employment.type |
> fit.hierarchical:     country) + gender + age + age.2 + n.adults.minus.n.children +
> fit.hierarchical:     n.children + education + household.type
>                  Df   AIC   BIC logLik deviance  Chisq Chi Df Pr(>Chisq)
> fit.flat         18 38852 39022 -19408    38816
> fit.hierarchical 38 38804 39163 -19364    38728 88.082     20  1.602e-10 ***
> ---
> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> My second question is whether or not I should care about the warnings I get (not entirely sure which one belongs to which model, but the first one should be against fit.hierarchcial).
>
> Warning in checkConv(attr(opt, "derivs"), opt\$par, ctrl = control\$checkConv,  :
>   Model failed to converge with max|grad| = 0.00636715 (tol = 0.001, component 30) Warning in checkConv(attr(opt, "derivs"), opt\$par, ctrl = control\$checkConv,  :
>   Model is nearly unidentifiable: very large eigenvalue
>  - Rescale variables?
> Warning 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?
>
> summary(fit.flat)
> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
>  Family: binomial  ( logit )
> Formula: below.poverty.line ~ 1 + employment.type + (1 | country:employment.type) +
>     gender + age + age.2 + n.adults.minus.n.children + n.children +      education + household.type
>    Data: my.df
>
>      AIC      BIC   logLik deviance df.resid
>  38852.1  39022.1 -19408.1  38816.1    93160
>
> Scaled residuals:
>     Min      1Q  Median      3Q     Max
> -1.5785 -0.2741 -0.1841 -0.1240 14.1696
>
> Random effects:
>  Groups                  Name        Variance Std.Dev.
>  country:employment.type (Intercept) 0.301    0.5486
> Number of obs: 93178, groups:  country:employment.type, 132
>
> Fixed effects:
>                                                      Estimate Std. Error z value Pr(>|z|)
> (Intercept)                                         -2.821530   0.164883 -17.112  < 2e-16 ***
> employment.typeCore self-employed                    1.779764   0.177173  10.045  < 2e-16 ***
> employment.typeInto core labour force                0.873362   0.183095   4.770 1.84e-06 ***
> employment.typeMarginalized peripheral labour force  1.791760   0.185840   9.641  < 2e-16 ***
> employment.typePeripheral labour force               1.036154   0.175026   5.920 3.22e-09 ***
> employment.typePeripheral self-employed              1.699013   0.180444   9.416  < 2e-16 ***
> gendermale                                           0.152666   0.029487   5.177 2.25e-07 ***
> age                                                 -0.008906   0.001537  -5.794 6.86e-09 ***
> age.2                                               -3.647558   1.044310  -3.493 0.000478 ***
> n.adults.minus.n.children                            0.034069   0.010769   3.164 0.001559 **
> n.children                                           0.258188   0.028628   9.019  < 2e-16 ***
> educationlower secondary                            -0.399377   0.051611  -7.738 1.01e-14 ***
> educationupper secondary                            -0.902910   0.049323 -18.306  < 2e-16 ***
> educationpost secondary                             -1.582793   0.056489 -28.019  < 2e-16 ***
> household.typecouple with children                  -0.120115   0.058652  -2.048 0.040568 *
> household.typesingle adult with children             0.514623   0.069323   7.424 1.14e-13 ***
> household.typesingle adult without children          0.195389   0.041295   4.732 2.23e-06 ***
> ---
> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
> summary(fit.hierarchical)
> Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
>  Family: binomial  ( logit )
> Formula: below.poverty.line ~ 1 + employment.type + (employment.type |
>     country) + gender + age + age.2 + n.adults.minus.n.children +      n.children + education + household.type
>    Data: my.df
>
>      AIC      BIC   logLik deviance df.resid
>  38804.0  39162.8 -19364.0  38728.0    93140
>
> Scaled residuals:
>     Min      1Q  Median      3Q     Max
> -1.5710 -0.2728 -0.1835 -0.1243 14.1030
>
> Random effects:
>  Groups  Name                                                Variance Std.Dev. Corr
>  country (Intercept)                                         0.2204   0.4695
>          employment.typeCore self-employed                   0.4457   0.6676   -0.20
>          employment.typeInto core labour force               0.3922   0.6263   -0.35  0.44
>          employment.typeMarginalized peripheral labour force 0.1228   0.3504   -0.63  0.57  0.39
>          employment.typePeripheral labour force              0.1090   0.3301   -0.38  0.24  0.70  0.66
>          employment.typePeripheral self-employed             0.3823   0.6183   -0.32  0.85  0.82  0.66  0.65
> Number of obs: 93178, groups:  country, 22
>
> Fixed effects:
>                                                      Estimate Std. Error z value Pr(>|z|)
> (Intercept)                                         -2.817822   0.153843 -18.316  < 2e-16 ***
> employment.typeCore self-employed                    1.741686   0.156367  11.138  < 2e-16 ***
> employment.typeInto core labour force                0.847817   0.156475   5.418 6.02e-08 ***
> employment.typeMarginalized peripheral labour force  1.771534   0.110705  16.002  < 2e-16 ***
> employment.typePeripheral labour force               1.021857   0.090266  11.321  < 2e-16 ***
> employment.typePeripheral self-employed              1.636287   0.151647  10.790  < 2e-16 ***
> gendermale                                           0.153356   0.029485   5.201 1.98e-07 ***
> age                                                 -0.008938   0.001540  -5.805 6.44e-09 ***
> age.2                                               -3.629799   1.103239  -3.290  0.00100 **
> n.adults.minus.n.children                            0.035107   0.010791   3.253  0.00114 **
> n.children                                           0.257672   0.028595   9.011  < 2e-16 ***
> educationlower secondary                            -0.403009   0.051870  -7.770 7.87e-15 ***
> educationupper secondary                            -0.899745   0.049781 -18.074  < 2e-16 ***
> educationpost secondary                             -1.584911   0.056888 -27.860  < 2e-16 ***
> household.typecouple with children                  -0.117651   0.058688  -2.005  0.04500 *
> household.typesingle adult with children             0.512608   0.069332   7.393 1.43e-13 ***
> household.typesingle adult without children          0.197551   0.041314   4.782 1.74e-06 ***
> ---
> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
>
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