[R-sig-ME] fixed effects/log transformations question

[Ben Bolker]

Gus Jespersen <jesper <at> u.washington.edu> writes:

> 
> Dr. Bolker,
> Thanks for the response.  One last question with regard to negative t
> values when using log10 transformed data.  I am assuming the correct
> interpretation of the following output is: if the t value is negative
> and you're using log10 data, to get the fixed effect CI, you must add
> your own negative sign to 10^(est.+1/96*SE), such that the
> backtransformed CI from the output below would be:
> 
> ([1] "95 % REML Confidence interval"
> [1] -0.58261813  0.02578124
> 
> becomes
> 
> -.295       -1.05
> 
> Is this correct,
> 

  No, but the interpretation is a little bit subtle.  Here you
are working with (as far as I can tell) the back-transformed
confidence intervals on the effect of the treatment.
10^{-0.5826,0.02578} is {0.26,1.06} (where did you get 0.295??); 
this says that the lower CI is that the proportional effect of
the treatment is to multiply by 0.26 (a 74% decrease); the upper
CI is a 6% increase (you can subtract 1 from the CI values if you
want to get it in terms of proportional changes).
  If you were using the natural log (log_e) rather than the log10
scale, then you could interpret *small* (near zero) parameters as
being approximately equivalent to proportional changes (without
back-transforming), because exp(x)-1 is approximately x when
x is small ...

  For what it's worth, this isn't an R question, or a mixed-model
question, any more, it's become a general statistical question -- you
might try asking similar questions on http://stats.stackexchange.com
...

> Thanks again for the help
> Gus
> 
> [1] "###############NH4 Results Year Two##################"
> Data: data.sub
> Models:
> Mod.NH4.2.2: log10(NH4Nyeartwo) ~ 1 + (1 | pr)
> Mod.NH4.2.1: log10(NH4Nyeartwo) ~ 1 + sitett + (1 | pr)
>             Df    AIC    BIC   logLik  Chisq Chi Df Pr(>Chisq)
> Mod.NH4.2.2  3 26.427 29.700 -10.2136
> Mod.NH4.2.1  4 25.243 29.607  -8.6216 3.1841      1    0.07436 .
> ---
> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
> Linear mixed model fit by REML
> Formula: log10(NH4Nyeartwo) ~ 1 + sitett + (1 | pr)
>    Data: data.sub
>    AIC   BIC logLik deviance REMLdev
>  30.37 34.73 -11.18    17.24   22.37
> Random effects:
>  Groups   Name        Variance Std.Dev.
>  pr       (Intercept) 0.010942 0.10460
>  Residual             0.130473 0.36121
> Number of obs: 22, groups: pr, 12
> 
> Fixed effects:
>                                                     Estimate Std. Error t value
> (Intercept)                                    0.7305     0.1086   6.729
> sitettToeAdditionsTreatment  -0.2784     0.1552  -1.794
> 
> [1] "95 % REML Confidence interval"
> [1] -0.58261813  0.02578124