Meaning of the weight argument in glmer and lmer

Question

I have been looking into how to use the weight argument of glmer/lmer to represent "frequency" weights. I was comparing model estimations with an expanded dataset, with the frequency weights inputted directly to the weight argument, and with the weights scaled as in Method A in Carle 2009¹ (i.e. scaled such that their sum matches the number of observations). I have found a puzzling difference in how they affect the results.

Specifically:

• lmer gives identical results for scaled and unscaled weights, but different results with the expanded dataset
• glmer gives identical results with the unscaled weights as with the expanded dataset, but different results with the scaled weights

Could anybody clarify what is the reason for the difference?

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Backgroud

I want to compare performance of two groups on a cognitive psychology task. The groups have been matched by a collaborator according to relevant covariates (e.g. socioeconomic status) and as a result few participants from the control group have a frequency weight of 2, indicating that they should be counted twice in the analyses in order for the groups to be matched on all the covariates. I would like to use multilevel models, and I am trying to see if I can use the frequency weight with the weight argument of lme4 (Note: I am not sure of whether frequency weight is the correct term here; perhaps sampling weight is more appropriate). The dependent variable is binomial, so we'd need to use a GLMM.

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Toy example:

Generate some data

# covariance matrix
sigma = matrix(c(1, 0.7, 0.7, 1), ncol = 2)

# simulate data
set.seed(4321)
df = mvrnorm(n = 100, mu = c(0,0), Sigma = sigma)

# Simulate clustering and calculate weights
df = data.frame(df)
names(df) = c("Y", "X")
df$cluster = rep(1:10, each = 10)
df = within(df, {
  Y = Y + cluster/10
  X = X + cluster/10
  weights = rep(1:2, each = 50)
  scaled_weights = weights*2/3
})

#data set with double weighted observations entered twice
df2 = rbind(df, df[51:100,]) 

Effects of weights in lmer. Here the models with weights gives the same results, regardless of the scaling, and the model with the expanded dataset gives different estimates.

# Models
m.weights = lmer(Y ~ X + (1 | cluster), df, weights = weights) # weights
m.double = lmer(Y ~ X + (1 | cluster), df2) # expanded dataset
m.scaled = lmer(Y ~ X + (1 | cluster), df, weights = scaled_weights) #scaled weights
round(summary(m.weights)$coef,digits=4)
#             Estimate Std. Error t value
# (Intercept)   0.2498     0.0904  2.7645
# X             0.6908     0.0658 10.4928
round(summary(m.scaled)$coef,digits=4)   # SAME AS m.weights
#             Estimate Std. Error t value
# (Intercept)   0.2498     0.0904  2.7645
# X             0.6908     0.0658 10.4928
round(summary(m.double)$coef,digits=4)   # DIFFERENT
#             Estimate Std. Error t value
# (Intercept)   0.2445     0.0842  2.9029
# X             0.6847     0.0538 12.7332

Effects of weights in glmer

# discretize dependent variable in 2 levels
df$dY <- ifelse(df$Y>0.8,1,0)
df2$dY <- ifelse(df2$Y>0.8,1,0)

m2.weights = glmer(dY ~ X + (1 | cluster), df, weights = weights, family =binomial) # weights (unscaled)
m2.double = glmer(dY ~ X + (1 | cluster), df2, family =binomial) # expanded dataset
m2.scaled = glmer(dY ~ X + (1 | cluster), df, weights = scaled_weights, family =binomial) # scaled weights
# Warning message:
# In eval(family$initialize, rho) : non-integer #successes in a binomial glm!

Note that the model with scaled weights give a warning, because in glm(er) the weights should correspond to the number of trials when the dependent variable is the proportion of successes. However as I understand them the weights are taken into account in the log-likelihood as

$$ \log\mathcal{L}(\boldsymbol{\theta}) = \sum_{i = 1}^{n} w_i \log \left[ p(y_i | \boldsymbol{x}_i,\boldsymbol{\theta})\right] $$

where $w_i$ is the weight. If that's correct, each observation $i$ contributes $w_i$ times its log-likelihood to the total log-likelihood, which should be the same as saying that each $i$ represents $w_i$ observations. (Since this is what I would like to express with the weights, I thought I could use this argument for my purposes). Furthermore, based on this I would expect to find that the model with the unscaled weight and that with the expanded dataset to give similar or identical results, and indeed that is what I find - in this case is the model with scaled weights that differs from the other two.

round(summary(m2.weights)$coef,digits=4)
#             Estimate Std. Error z value Pr(>|z|)
# (Intercept)  -1.3028     0.3902 -3.3390    8e-04
# X             1.5741     0.2955  5.3263    0e+00
round(summary(m2.scaled)$coef,digits=4)          # DIFFERENT
#             Estimate Std. Error z value Pr(>|z|)
# (Intercept)  -1.2620     0.4183 -3.0167   0.0026
# X             1.5579     0.3531  4.4114   0.0000
round(summary(m2.double)$coef,digits=4)          # SAME as m.weights
#             Estimate Std. Error z value Pr(>|z|)
# (Intercept)  -1.3028     0.3902 -3.3390    8e-04
# X             1.5741     0.2955  5.3263    0e+00

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¹ _{Carle, A. C. (2009). Fitting multilevel models in complex survey data with design weights: Recommendations. BMC Medical Research Methodology, 9 (1), 1–13.}