Dear List,
I have a "Poisson-lognormal" mixed model (structure and output below).  I would like to perform Wald t-tests to obtain P-values for the parameter estimates.  I understand that one way to do this is to use the z-scores provided by summary() as the t-values. My question is what I should use as my df, given the model structure below. The y-data are bird counts recorded during 646 surveys at 44 transect locations.  So each transect was surveyed multiple times...thus, the inclusion of (1|trans), which is essentially the "name" indicating which of the 44 transects were surveyed (e.g., L_01, W_04, etc.).  Data were overdispersed, so a random term representing individual-level variation was included (i.e., (1|obs)).  Should the df for the t-tests be 646 - 1 = 645 or maybe 44 - 1 = 43 ? Is there another way to calculate them? Am I even doing the Wald tests correctly? Thanks in advance for your help.

Mike


library(lme4)
library(arm)
db$obs=1:nrow(db)

b.dens.veg2=glmer(dens~year+site+de+avghtin1+I(avghtin1^2)+avghtin1*site+(1|trans)+(1|obs),data=db,family=poisson)

> summary(b.dens.veg2)
Generalized linear mixed model fit by the Laplace approximation 
Formula: dens ~ year + site + de + avghtin1 + I(avghtin1^2) + avghtin1*site + (1 | trans) + (1 | obs) 
   Data: db 
   AIC   BIC logLik deviance
 697.2 750.9 -336.6    673.2
Random effects:
 Groups Name        Variance Std.Dev.
 obs    (Intercept) 0.094321 0.30712 
 trans  (Intercept) 0.049609 0.22273 
Number of obs: 646, groups: obs, 646; trans, 44

Fixed effects:
               Estimate Std. Error z value Pr(>|z|)    
(Intercept)     0.36736    0.16844   2.181   0.0292 *  
year2           0.17644    0.07472   2.361   0.0182 *  
year3           0.02453    0.07537   0.326   0.7448    
siteP           0.41106    0.22915   1.794   0.0728 .  
siteW           0.31676    0.20634   1.535   0.1248    
deE            -0.53292    0.08145  -6.543 6.02e-11 ***
avghtin1        0.06967    0.18736   0.372   0.7100    
I(avghtin1^2)   0.06706    0.05755   1.165   0.2440    
siteP:avghtin1 -0.28355    0.17117  -1.657   0.0976 .  
siteW:avghtin1  0.31147    0.16166   1.927   0.0540 .  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 

Correlation of Fixed Effects:
            (Intr) year2  year3  siteP  siteW  deE    avght1 I(1^2) stP:v1
year2       -0.161                                                        
year3       -0.145  0.591                                                 
siteP       -0.388 -0.124 -0.134                                          
siteW       -0.612  0.070  0.067  0.492                                   
deE         -0.060 -0.002 -0.003 -0.007 -0.018                            
avghtin1    -0.802 -0.045 -0.080  0.081  0.324 -0.006                     
I(vghtn1^2)  0.437 -0.078 -0.007  0.381  0.085  0.007 -0.759              
sitP:vghtn1  0.287  0.148  0.139 -0.848 -0.467 -0.002 -0.077 -0.512       
sitW:vghtn1  0.542 -0.074 -0.071 -0.433 -0.864  0.018 -0.412 -0.076  0.551

# my attempt at Wald t-tests, assuming 645 df (i.e., number of observations minus 1)

> t=fixef(b.dens.veg2)/se.fixef(b.dens.veg2)
> p=data.frame(p=2*pt(abs(t),df=645,lower.tail=F))
> data.frame(t,p)
                        t            p
(Intercept)     2.1809174 2.955005e-02
year2           2.3613006 1.850732e-02
year3           0.3255303 7.448853e-01
siteP           1.7938740 7.330155e-02
siteW           1.5351187 1.252450e-01
deE            -6.5432216 1.227467e-10
avghtin1        0.3718450 7.101304e-01
I(avghtin1^2)   1.1651422 2.443921e-01
siteP:avghtin1 -1.6565831 9.809013e-02
siteW:avghtin1  1.9266674 5.446011e-02

# my other attempt at Wald t-tests, assuming 43 df (i.e., number of transects minus 1)

> t=fixef(b.dens.veg2)/se.fixef(b.dens.veg2)
> p=data.frame(p=2*pt(abs(t),df=43,lower.tail=F))
> data.frame(t,p)
                        t            p
(Intercept)     2.1809174 3.470626e-02
year2           2.3613006 2.281215e-02
year3           0.3255303 7.463586e-01
siteP           1.7938740 7.986700e-02
siteW           1.5351187 1.320807e-01
deE            -6.5432216 5.923002e-08
avghtin1        0.3718450 7.118344e-01
I(avghtin1^2)   1.1651422 2.503844e-01
siteP:avghtin1 -1.6565831 1.048840e-01
siteW:avghtin1  1.9266674 6.064659e-02
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