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I am wondering why in my lmer model the summary() only yields t-values rather than z-values, such as here:

model <- lmer(area~Register+(1|subject), data = ev)
summary(model)

Linear mixed model fit by REML 
t-tests use  Satterthwaite approximations to degrees of freedom ['lmerMod']
Formula: area ~ Register + (1 | subject)
   Data: ev

REML criterion at convergence: 166.3

Scaled residuals: 
     Min       1Q   Median       3Q      Max 
-1.60697 -0.46989  0.06558  0.69561  1.41818 

Random effects:
 Groups   Name        Variance Std.Dev.
 subject  (Intercept) 1.185    1.089   
 Residual             1.747    1.322   
Number of obs: 45, groups:  subject, 15

Fixed effects:
                        Estimate Std. Error      df t value Pr(>|t|)    
(Intercept)              11.2827     0.4421 31.6600  25.522  < 2e-16 ***
RegisterIDS               1.2339     0.4826 28.0000   2.557  0.01627 *  
RegisterLombard\nSpeech   1.3630     0.4826 28.0000   2.824  0.00863 ** 
---
Signif. codes:  0 ?**?0.001 ?*?0.01 ??0.05 ??0.1 ??1

Correlation of Fixed Effects:
            (Intr) RgsIDS
RegisterIDS -0.546       
RgstrLmbrdS -0.546  0.500

I am wondering in what circumstances will R report z-values? I saw in some papers that people obtained z-values with binominal dependent values, is this the only circumstance that z-values appear?

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1 Answer 1

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tl;dr lmer (linear mixed models) labels this column as a "t statistic", while glmer (generalized linear mixed models) labels it as a "Z statistic", but they're actually the same number. This mirrors the difference between the way lm and glm report their output.

The "t statistics" reported by lmer (assuming a Gaussian distribution of observations conditional on fixed and random effects) and the "Z statistics" reported by glmer (assuming binomial, Poisson, etc. ... distributions ...) are really the same number, i.e. the point estimate divided by the estimate of its standard error (you can check this via cc <- coef(summary(fitted_model)); cc[,"Estimate"]/cc[,"Std. Error"]). In a standard least-squares fit, we can prove that this quantity is $t$-distributed, and we can derive the corresponding degrees of freedom exactly. We can also do this for some linear mixed models (i.e. balanced designs with single or nested random effects only). For more complex linear mixed models there are various approximations (Satterthwaite, Kenward-Roger; see the pbkrtest or lmerTest packages) for deriving the approximate distribution of $\bar x/\hat \sigma$.

For generalized linear models, the finite-size corrections are less well understood and disseminated; the sampling distribution of $\bar x/\hat \sigma$ for non-Normal responses is not t-distributed even in simple cases (it's also not Z-distributed, except asymptotically). There is some literature on approximate finite-size corrections under the rubric of Bartlett corrections, but the standard practice in GLMs is to report these values as "Z statistics" and assume (when calculating p-values) that people know that they're assuming the sample size is large.

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  • $\begingroup$ Excellent answer! Thanks! One issue now I am concerning is: what is the difference between linear mixed model and generalized linear model? And in what circumstance should I choose glmer? $\endgroup$
    – Ping Tang
    May 28, 2017 at 1:03
  • $\begingroup$ that's a much more general question. Generalized linear (mixed) models handle some kinds of non-Normal data. Typically you want to use GL(M)Ms if your data are of particular discrete types (binary, counts, proportions) that can't easily be transformed to be Normal. $\endgroup$
    – Ben Bolker
    May 28, 2017 at 15:59
  • $\begingroup$ I see, thanks! That is why in some papers I have read people usually put family = “binominal” in glmer and that where I saw they got z values. $\endgroup$
    – Ping Tang
    May 29, 2017 at 2:24
  • $\begingroup$ if my answer resolved your question you are encouraged to click the check-mark to accept it ... $\endgroup$
    – Ben Bolker
    May 29, 2017 at 2:57

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