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I ran a mixed model regression and obtained the following output. Because I log-transformed the concentration variable, I interpreted the coefficient as follows: the mean concentration is reduced $1.5$ percent each year. The years run from 1993-2010.

Can say also state that the mean concentration is reduced by $26\%$ ($0.01509\cdot 17$) over the $17$ year period? Why or why not?

Why do most people generally state $1$ unit increase in $X$ corresponds to a certain number change in $Y$ rather than infer change over the entire period?

baggerTrend <-  lme(Log.Qconc   ~   Yearf,  random=(~Yearf|MineID), Bagger) 

Random  effects:                    
    Formula:    ~Yearf  |   MineID      
    Structure:  General positive-definite,  Log-Cholesky    parametrization 
              StdDev    Corr                
(Intercept) 0.209172851 (Intr)              
Yearf       0.000374953 -0.478              
Residual    0.785367538                 


Fixed effects:  Log.Qconc ~ Year                    
               Value   Std. Error    DF      t-value       pvalue
Intercept   33.71122    6.3901     2762      5.275538       0
Year        -0.01509    0.003193   2762     -4.724678       0
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  • $\begingroup$ If your dependent variable in the analysis is log-transformed concentration, then the coefficient for year is telling you that log-transformed concentration decreases by 0.015 every year, the coefficient is on the same scale as your DV, which is not percentages in this case. $\endgroup$
    – Marius
    Jul 12 '12 at 1:38
  • $\begingroup$ Also, if you did not mean-center Year, and used values like 1993, 1994, etc., then the coefficients may be nonsensical, because the model is fitted for year==0. $\endgroup$
    – Marius
    Jul 12 '12 at 1:40
  • $\begingroup$ @Marius: I interpreted the percent change/decrease based on information from this website: ats.ucla.edu/stat/sas/faq/sas_interpret_log.htm. I am 90% sure but I can be wrong. $\endgroup$
    – Amateur
    Jul 12 '12 at 4:06
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    $\begingroup$ Just had a quick look, and found this: sportsci.org/resource/stats/logtrans.html. Log-transformed coefficients do closely approximate percentages, but only when they are small. $\endgroup$
    – Marius
    Jul 12 '12 at 4:44
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    $\begingroup$ @Amateur so your question is about the interpretation of a regression line and has nothing to do with the mixed model ? $\endgroup$ Jul 12 '12 at 5:03
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I think that would be enough to consider exp coefficient and interpret it this way: the unit change of the regressor x1, creates an increase of exp (b1) of the dependent variable

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  • $\begingroup$ Can you expand on this as it does not seem to add very much to the comments? $\endgroup$
    – mdewey
    Nov 16 '16 at 15:53
  • $\begingroup$ {\displaystyle \log(Y)=a+bX} \log(Y)=a+bX {\displaystyle Y=e^{a}e^{bX}} Y=e^{a}e^{{bX}}) $\endgroup$ Nov 16 '16 at 17:27

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