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Trying to fit a linear mixed effects model with 2 categorical predictors (group & worker) where worker is a random effect and group a fixed effect. I'm trying to figure out 1) whether I should specify intercept=0 and 2) why these 2 model results seem to give different conclusions about the effect of group.

Model1: tps ~ group + (1 | worker)

Model2: tps ~ group + (1 | worker) + 0

summary(Model1):
Linear mixed model fit by REML ['merModLmerTest']
Formula: tps ~ group + (1 | worker) 
   Data: mydata 

REML criterion at convergence: 3489.872 

Random effects:
 Groups   Name        Variance Std.Dev.
 worker   (Intercept) 1866     43.20   
 Residual             3165     56.26   
Number of obs: 318, groups: worker, 18

Fixed effects:
            Estimate Std. Error     df t value Pr(>|t|)    
(Intercept)    70.15      15.59  11.27   4.501 0.000848 ***
group phone   -20.85      21.75  10.83  -0.959 0.358586    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            (Intr)
group phone -0.717

summary(Model2):
Linear mixed model fit by REML ['merModLmerTest']
Formula: tps ~ group + (1 | worker) + 0 
   Data: mydata 

REML criterion at convergence: 3489.872 

Random effects:
 Groups   Name        Variance Std.Dev.
 worker   (Intercept) 1866     43.20   
 Residual             3165     56.26   
Number of obs: 318, groups: worker, 18

Fixed effects:
               Estimate Std. Error    df t value Pr(>|t|)    
group computer    70.15      15.59 11.27   4.501 0.000848 ***
group phone       49.30      15.17 10.40   3.251 0.008291 ** 
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
            grpcmp
group phone 0.000 

In the first model the 'phone' effect is the same as the difference between the two groups' effects in model 2 (this makes sense because in model1 the 'computer' group is the baseline). In model2, both groups' effects are significant, whereas in model1 only the intercept is significant.

Which is the "right" model for a situation where the group predictor is binary? It must be only one or the other (seems to indicate that model1 is correct, because there the intercept "is the same as" the computer group, right? Model2 allows a "zero" value for group which doesn't make sense). Am I right about this?

And how to interpret the fact that in model1 the intercept is significant but 'phone' is not?

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    $\begingroup$ I don't use R. I assume model2 = intercept through the origin. Why would you do this? I'm sure there are situations where this might make sense, but I don't think they are commonly encountered. I think your reasoning sounds solid in that model1 makes sense. $\endgroup$ – charles Dec 14 '13 at 21:02
  • $\begingroup$ thanks charles. would it also be correct to interpret the fact that in model1, the intercept coefficient is significant but not the "phone group" coefficient to mean that there is no significant difference between the groups, while the intercept coefficient being significant doesn't have a physical interpretation (other than: "yes, you do indeed have data")? $\endgroup$ – andy Dec 14 '13 at 21:44
  • $\begingroup$ Yes. That is how I would interpret it. $\endgroup$ – charles Dec 14 '13 at 21:50
  • $\begingroup$ @charles i think you've answered my question but i can't give you credit unless you post it as an answer... $\endgroup$ – andy Dec 14 '13 at 22:45
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The difference of the outputs from two models dues to the different parameterization of the effect of group.

In the first model, you allowed the intercept, you model would be (disregard the random effect part) y=beta0+beta1*X+e, where X=1 for phone and 0 for computer. So in the output, beta0 should be the effect due to computer (when X=0) and beta0+beta1 is the effect due to phone (when X=1). The difference between these two effects are beta1 (effect due to computer).

While in the second model, the intercept is omitted. So the model will be (also disregard the random effect part) yi=alpha_i+e, where i=1,2 (i=1 for the computer group and 2 for the phone group) Thus the two estimates for the fixed effects are for computer and phone respectively. You can see the difference between the two estimates (i.e. phone-computer) is the beta1 in model 1 (In model one the computer group is treated as reference).

And for the significance part, from model 1, the group effect is not significant (i.e. the effect from phone compared to computer is not significantly different on the outcome.)

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