[R] Mixed-effecs models- question about syntax to introduce interactions

[Sylvia Anaid Diaz-Palacios]

I would like to as for advice for the use of “lmer” (package ‘lme4’) and writing the proper syntax to best describe my data using a mixed-effects model.

I have just started to use these models, and although I have read some good examples (Extending the Linear Model with R, Faraway 2005; and the R book, Crawley 2007), I am still not sure of the syntax to test my hypothesis.

Thanks in advance for reading me.

Briefly, I describe the data and the situation:

I want to describe the age-specific fecundity of the ith individual from the jth replicate (or line) from the kth strain.

Variables:

Categorical factors:

A[a] = Age (1,2,3…n=8) #Because the fecundity is not linear, I decided to include it in the model as a factor
s[k] =strain (A and B, n=2) # for the moment two, but it’s likely to increase as the work progresses
l[j]   = line (1,2,..n=10)
i[i]   = Ind(1,2… n=50)

(Note: I use capital letters for fixed factors and low case for random effects)

Because the experimental design, the data follows a hierarchical structure: where the ith individual is nested within the jth line, and line within the kth strain

Continuous (response) variable:
Y =Age-specific fecundity (362 observations)

Models:

Because I was (I am still) not sure of how to include all the variables in a single model, I started by splitting up the data and assessing which is the best model for each strain, therefore the “Simplest” model for each strain is:

Linear model.

Y[aij] = A[a] + error[aij]

R code:

m1 <- lm(fecudnity ~ Age)

Reduced mixed-effect model:

Y[aij] = A[a] + l[j] + i[i] + error[aij]

R code:

m2 <- lmer(fecudnity ~ Age + (1 | line/ind), method=”ML”)

And a “Full mixed-effects model” model (looking for interactions between Age and line/ind)

Y[aij] = A[a] + A[a]*l[j] + A[a]*i[i] + error[aij]

R code:

m3 <- lmer(fecudnity ~ Age + (Age | line/ind), method=”ML”)

I have used Likelihood test ratio (LTR) to compare between models, and I have found that for strain A the best model is m3 (X^2 [36 d.f] =164.8, p-value= 4.73e-13), whereas for strain B, the best one is m1 (X^2 [2 d.f] =1.47, p-value= 0.473). Therefore, I interpret these results as follow:

-	The variance between individuals in strain A is large, and it is best described when I include information about the line where the individuals come from. Moreover, there is a significant interaction between age and line/ind. Thus, some individuals have higher fecundity at later ages compared to others.

-	The variance between individuals in strain B is low; therefore the variance between ind/lines and interactions can be ignored. 

These results, on its own, are interesting, but I would like to have a model where I include both strains (and still can make some interpretations)

My first guess is

m4 <- lmer(fecudnity ~ Age + (1 | strain/line/ind), method=”ML”)
m5 <- lmer(fecudnity ~ Age + (Age | strain/line/ind), method=”ML”)

Using LTR, I find that m5 describes better the data (X^2 [105 d.f] = 347.15, p-value  < 2.2 e-16), but I feel like I can not say much of which strain has more individual variance (or perhaps I am wrong and not looking in the right place).

Then I though about using strain as a fixed factor, because I am now interested in the differences between strains

m6 <- lmer(fecudnity ~ Age * Strain +  (Age |strain/line/ind), method=”ML”)

or perhaps include it in the random interaction?

m7 <- lmer(fecudnity ~ Age +  (Age * Strain |strain/line/ind), method=”ML”)

I have to be honest, at this point, I am just not sure of how to write the model to describe the age-specific fecundity and test the hypothesis of whether one strain shows more variance between individuals and lines or not. I hope some one could give some advise.

Thanks in advance!!

Anaid Diaz


-Anaid

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S.Anaid Diaz
Rm 303, Graham Kerr Bldg
Theoretical Ecology Group
Division of Environmental & Evolutionary Biology
University of Glasgow
Glasgow G12 8QQ

tel: 0141 330 2430
fax: 0141 330 5971

http://www.gla.ac.uk/ibls/DEEB/teg/people/diaz.htm