How can I derive effect sizes in lme4 and describe the magnitude of fixed effects? I have run a mixed effects model with a ln transformed continuous response (seconds) and found a significant effect of the categorical predictor (treatment/control, the only fixed effect in the model). 
I want to:
1- Report an effect size (cohen's d, etc)
2- Describe the magnitude of the effect in terms of mean number of seconds that the treated individuals differed from the control individuals, after accounting for the random effects.
I am not sure how to achieve either of these goals. Thank you very much in advance for any advice you can offer.
My code and results are below.
mod1 = lmer(data=data, ln_duration ~ treatment + (1 | id/date/size) +  
(1 | visitor), na.action=na.exclude)

Linear mixed model fit by REML 
t-tests use  Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: ln_duration ~ treatment + (1 | id/date/size) +  
(1 | visitor)
 Data: data

REML criterion at convergence: 248

Scaled residuals: 
  Min      1Q  Median      3Q     Max 
-2.7323 -0.4963 -0.0206  0.5600  3.8502 

Random effects:
 Groups                             Name        Variance Std.Dev.
 display_size:(date:id)             (Intercept) 0.00000  0.0000  
 date:id                            (Intercept) 0.00000  0.0000  
 visitor                            (Intercept) 0.03574  0.1891  
 id                                 (Intercept) 0.01164  0.1079  
 Residual                                       0.20001  0.4472  
Number of obs: 170, groups:  
size:(date:id), 130; date:id, 128; visitor, 118; id, 58

Fixed effects:
                      Estimate     Std. Error  df t value  Pr(>|t|)    
(Intercept)               -0.17012    0.06334  41.63000   -2.686 0.010348 *      
       treatmenttreatment  0.31172    0.08135  40.27000    3.832 0.000436 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr)
    trtmnt -0.729

 A: The paper suggested by @simone, Brysbaert and Stevens as the title indicates, is focused on 'Power Analysis and Effect Size in Mixed Effects Models', but it includes a calculation of effect size, which is not present in @simone's answer, with a reference to Westfall et al. (2014), for the effect size calculation:
'*First, Westfall et al. (2014) showed how you can calculate the effect size (measured as d) for a design with random participants and random items. The equation is as follows:
d = difference between the means / ( sqrt( var.intercept_part + var.intercept_item + var.slope_part + var.slope_item + var_residual ) )*'
(Sorry about the equation notation)
Basicly meaning:
d = estimate for fixed effect / (sqrt of sum of variances of random effects)
From Westfall et al.: 'We follow there the general procedure specified by Cohen (1988) for simpler designs. This procedure involves first calculating an estimated effect size, analogous to Cohen’s d: the expected mean difference divided by the
expected variation of an individual observation, which in our case
accrues from all the variance components specified above'
Finally, this equation seems to be the same as in Hedges (2007), proposed by @d_williams.
Hope this helps.
A: You can indeed compute an effect size in multilevel models. The one provided is called delta total, where total is the total of the variance components. I generally use it when the co-variate in the model is categorical. It should be close to cohen's d, but I would not call it that. Rather, I would refer to it as an effect size parameter. Computing the interval will be challenging in a frequentist framework, but is easily done using Bayesian methods. Since Bayesian methods provide entire posterior distributions, calculation of delta total is done on the posterior distributions, which readily allows for computing credible intervals via the quantile function in r or some package for obtaining high density intervals. 
This is a simple case, however, and I would recommend reading the paper cited for other ways to compute effect sizes in multilevel models.
$$
\delta_t = \frac{beta_{treat}} {sqrt(sigma_{visitor}^2 + sigma_{date:id}^2 +sigma_{display}^2 + sigma_{resid}^2 )} 
$$
Hedges, L. V. (2007). Effect Sizes in Cluster-Randomized Designs. Journal of Educational and Behavioral Statistics, 32(4), 341–370. 
A: Brysbaert and Stevens recently published a paper on how to compute effect sizes with the lme4 package. 
Code: 
library (lme4)
fit <- lmer(RT ~ prime + (prime|item) + (prime|participant), data = data)
summary(fit)

There is one fixed effect (the effect of prime) and four random effects:
The intercept per participant (capturing the fact that some participants are faster than others).
The intercept per item (capturing the fact that some items are easier than others).
The slope per participant (capturing the possibility that the priming effect is not the same for all participants).
The slope per item (capturing the possibility that the priming effect is not the same for all items).
I have not been able to use this solution as I used the nmle package because it lets me define the autocorrelation structure (Pinheiro & Bates, 2008) more easily, but I thought I will share it anyway. 
