How to translate percentage decline into a regression slope? I want to model a negative relationship of count/binary data over 10 years with a known value for the decline rate (in percentage). However, I have problems in calculating the correct slope value for the trend.
Let´s say, I know that the starting value for the response is 50 (intercept value) and that the slope should be a decline of 5% per year.
The slope of the regression line shows the amount of change in the response per 1 unit X. So, in the Gaussian case, I would calculate the 5% of 50 (which is 2.5) and set the slope = 2.5. But this is not true for the poisson or binomial case as we are dealing with log-differences (poisson case).
My question thus is: "How do I translate my percentage decline rate into a slope value for a poisson or binomial model?"
So far I tried this...
    #### POISSON EXAMPLE

## load libraries and source sim.glmm.r from github
library(lme4)
library(visreg)
source("https://raw.githubusercontent.com/pcdjohnson/sim.glmm/master/sim.glmm.R")

## create a data frame with known sampling design
mydata<-expand.grid(SAMPLE=1:20,SUBSAMPLE=1:10, year=1:10) 
mydata$SAMPLE<-factor(mydata$SAMPLE)
mydata$SUBSAMPLE<-factor(mydata$SUBSAMPLE)

## create my slope value
myIntercept <- 50
AnnualPercentageDecline <- 0.05 #(=5%)
mySLOPE <- myIntercept * AnnualPercentageDecline

## simulate data with sim.glmm - here, "year" requires the declining slope
sim.data <- sim.glmm(design.data = mydata, 
              fixed.eff = list(intercept = log(50), year = -(mySLOPE)),
              rand.V = c(SAMPLE = 5, SUBSAMPLE = 1),
              distribution = 'poisson')


## perform the glmer model
mydata.temp<-(glmer(response~ year +(1+year|SAMPLE)+
(1+year|SUBSAMPLE),family="poisson",data=sim.data))

## visualize output
par(mfrow=c(1,3))
visreg(mydata.temp, scale="linear")
par(mfrow=c(1,1))

## END

 A: Linear decline of 2.5 units/year in your Gaussian example is not the same as a decline of 5% per year (as Nick Cox points out). If the response is reduced by 5% each year, then it's being multiplied by 0.95 each year. E.g.:
Linear decline: year1 = 50 - 2.5 * 0 = 50; year10 = 50 - 2.5 * 9 = 27.5
Exponential decline: year1 = 50 * 0.95^0 = 50; year10 = 50 * 0.95^9 = 31.5
NB the intercept in this example isn't the same as the response in year 1 unless year 1 has the value 0, not 1 as in your code.
Do you definitely want to model linear decline and not exponential decline? I don't know a simple way of modelling linear decline in counts. To simulate exponential decline in the Poisson example (which is much easier due to the log link) change
fixed.eff = list(intercept = log(50), year = -(mySLOPE))
to
fixed.eff = list(intercept = log(50), year = log(0.95))
and 
mydata<-expand.grid(SAMPLE=1:20,SUBSAMPLE=1:10, year=1:10)
to 
mydata<-expand.grid(SAMPLE=1:20,SUBSAMPLE=1:10, year=0:9)
Also, 
(1+year|SAMPLE)+(1+year|SUBSAMPLE)
fits random slopes and intercepts, whereas only random intercepts have been simulated in the data. This might be what you want, but to fit the intercepts-only model, change to 
(1|SAMPLE)+(1|SUBSAMPLE)
The same applies to a binomial model with a logit link provided you want to model a 5% annual decline in the odds.
