Same variable as both fixed and random

Dear Joshua,

I agree when the variable is a factor, because then the model with both fixed and random effect becomes unidentifiable. But I disagree when the variable is used as a continuous variable in the fixed effects and makes sense as a factor as well. In that case (provided the variable has enough levels) it makes sense to add the variable as a continuous fixed effect and a random intercept. I'm made a toy example below. The first model has a linear trend along year with additional year-to-year variation. The second one assumes a quadratic effect that is modeled as a linear effect. In the third case the fixed effect uses a quadratic model. The random effects of the first two models capture the discrepancy between the model and the true trend. The random effects in the third model have zero variance because they are not relevant.

Best regards,

Thierry

library(lme4)
library(ggplot2)
set.seed(123546)
n.year <- 10
n.replicate <- 10
intercept <- 1
trend <- 2
quadratic <- -0.5
sd.noise <- 1
sd.year <- 2
test <- expand.grid(year = seq_len(n.year), replicate = seq_len(n.replicate))

test$y.true <- intercept + trend * test$year + rnorm(n.year, sd = sd.year)
test$y <- test$y.true + rnorm(nrow(test), sd = sd.noise)

model <- lmer(y ~ year + (1|year), data = test)
test$fit <- fitted(model)
test$global <- predict(model, re.form = ~0)
ggplot(test, aes(x = year, y = y.true)) + geom_line() + geom_point(aes(y = y)) + geom_line(aes(y = fit), colour = "red") + geom_line(aes(y = global), colour = "blue") + geom_abline(intercept = intercept, slope = trend, colour = "magenta")
plot(ranef(model)$year[, 1])

test$y.true <- intercept + trend * test$year + quadratic * test$year ^ 2
test$y <- test$y.true + rnorm(nrow(test), sd = sd.noise)
model <- lmer(y ~ year + (1|year), data = test)
test$fit <- fitted(model)
test$global <- predict(model, re.form = ~0)
ggplot(test, aes(x = year, y = y.true)) + geom_line() + geom_point(aes(y = y)) + geom_line(aes(y = fit), colour = "red") + geom_line(aes(y = global), colour = "blue") + stat_function(fun = function(x){intercept + trend * x + quadratic * x ^ 2}, colour = "magenta", geom = "line")

plot(ranef(model)$year[, 1])

model <- lmer(y ~ poly(year, 2) + (1|year), data = test)
test$fit <- fitted(model)
test$global <- predict(model, re.form = ~0)
ggplot(test, aes(x = year, y = y.true)) + geom_line() + geom_point(aes(y = y)) + geom_line(aes(y = fit), colour = "red") + geom_line(aes(y = global), colour = "blue") + stat_function(fun = function(x){intercept + trend * x + quadratic * x ^ 2}, colour = "magenta", geom = "line")
plot(ranef(model)$year[, 1])


ir. Thierry Onkelinx
Instituut voor natuur- en bosonderzoek / Research Institute for Nature and Forest
team Biometrie & Kwaliteitszorg / team Biometrics & Quality Assurance
Kliniekstraat 25
1070 Anderlecht
Belgium
+ 32 2 525 02 51
+ 32 54 43 61 85
Thierry.Onkelinx at inbo.be
www.inbo.be

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~ Sir Ronald Aylmer Fisher

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~ Roger Brinner

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-----Oorspronkelijk bericht-----
Van: r-sig-mixed-models-bounces at r-project.org [mailto:r-sig-mixed-models-bounces at r-project.org] Namens Joshua Wiley
Verzonden: donderdag 17 juli 2014 11:24
Aan: Stephen Mayor
CC: r-sig-mixed-models at r-project.org
Onderwerp: Re: [R-sig-ME] Same variable as both fixed and random

Hi Stephen,

In your example, I would recommend not including year as both a fixed and random effect (note that Arrival ~DegreeDays + Year + (1 + Year |
State) --- i.e., allowing the effects of year to differ by state or some such, would be a different scenario).

You will partial the most variability out of the estimates by specifying Year as a fixed effect, however, if it is being treated categorically, this will result in quite a few extra parameters, and you will also not get an estimate of the overall variability in intercept by year.  If those specific ten years are not of interest, and you are controlling for the "key" feature that you expect to change, namely, DegreeDays, then I would suggest:

lmer(Arrival ~ DegreeDays + Longitude + Latitude + (1 | State) + (1 | Year))

A separate issue is how longitude and latitude are included (e.g., depending on the precision of your data, it may be helpful to allow a stronger similarity between nearby locations, although it may not matter much if you only have data at the level of State).

Cheers,

Josh


On Wed, Jul 16, 2014 at 2:47 AM, Stephen Mayor <smayor at neoninc.org> wrote:

Hello,
When should a variable be specified as both a fixed effect AND as a random effect?  If a single variable is defined as both fixed and random, how does one interpret the coefficients?  Is it 'sloppy' practice to include it as both?  Should one be cautious specifying a variable twice, or is it actually more conservative to do so?

I am interested in your thoughts in general, but here is a simplified example if helpful:
I am interested in the date of arrival of house wren, a migratory bird, to each state in the US as a response to year, degree days (climate), latitude, longitude, state.  I primarily want to test if there is a temporal trend in earlier arrival in more recent years, as a result of recent climatic warming.

I specified the model as follows, treating state as a random group.  Because I am interested in testing a linear trend across years, I specified Year as a fixed (and continuous) effect.  But I am unsure as to whether I should ALSO specify it as a random factor, because I am interested in making inferences beyond the 10 years of data that I have and I don't have any specific interest in these 10 years over any other period.

lmer(Arrival ~ DegreeDays + Year + Longitude + Latitude + (1|State) )
OR lmer(Arrival ~ DegreeDays + Year + Longitude + Latitude + (1|State)
+ (1|Year))

Thanks in advance for any help or suggestions, Stephen.


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--
Joshua F. Wiley
Ph.D. Student, UCLA Department of Psychology
http://joshuawiley.com/
Senior Analyst, Elkhart Group Ltd.
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