What is the null model for a likelihood ratio test of a within-subjects factor? Tissue samples were taken from 4 differention locations and repeatedly measured. This was done identically for 3 animals. The research question was: Are there differences in measurement between the locations?
=> Repeated-measures design, with location as within-subjects factor
m1 <- lmer(meas ~ location + (1 + location | animal))  

I would like to test for a main effect of location by a likelihood ratio test via anova(model, nullmodel) (I am aware of the differing opinions regarding this).
But I am unclear, which, or if any, of the following two is the correct null model:
null.1 <- lmer(meas ~ 1 + (1 | animal), REML=FALSE)
null.2 <- lmer(meas ~ 1 + (1 + location | animal), REML=FALSE)

I am inclinced to think it’s null.1, but I am not entirely sure.
Formulated more generally:
When specifying a maximal random effects structure for a within-subjects factor, what is the null model for that factor for a maximum-likelihood ratio test?
 A: For building models, West, Welsh, and Galecki (2014) propose 2 strategies: step-up and top-down. In the step-up strategy your first step is to find the best "unconditional" model: you start with including all statistically significant as well as theoretically relevant random effects but only the fixed intercept. Then test $H_0: \sigma^2_{random}=0$ using the likelihood test. If the $H_0$ cannot be rejected, remove the random effect from the model. Afterwards, you start including level 1 covariates and remove non-significant ones, and then level-2, and so on.
So in your case, you may want to find the best "unconditional" model by running
null.1_reml <- lmer(meas ~ 1 + (1 | animal), REML=TRUE)
null.2_reml <- lmer(meas ~ 1 + (1 + location | animal), REML=TRUE)

You cannot simply use anova(null.1_reml, null.2_reml) because the the test statistic is distributed as a mixture of $\chi^2_1$ and $\chi^2_2$ distributions with equal weights. So use this instead:
0.5*(1 - pchisq(x, 1)) + 0.5*(1 - pchisq(x, 2))

where x is the difference in the log-likelihood value between the 2 models.
This tests
$
H_0: D=
\left(
\begin{matrix}
  \sigma^2_{int} & 0 \\
  0 & 0
\end{matrix}
\right)
$
$
H_A: D=
\left(
\begin{matrix}
  \sigma^2_{int} & \sigma_{int,location} \\
  \sigma_{int,location} & \sigma^2_{location}
\end{matrix}
\right)
$
where $\sigma_{int,location}$ is the covariance between the intercept and the location effect. If you want to test them individually, you can also fit a model with only the location effect and the intercept without their covariance. After finding the best unconditional model proceed to include fixed effects.
The top-down strategy recommends you first add fixed effects of as many covariates of interest as possible and remove non-significant ones to find the best mean model. Then include random effects.  
Reference
West, B.T., Welch, K.B. and Galecki, A.T. (with Contributions from Brenda W. Gillespie) (2014).  Linear Mixed Models: A Practical Guide using Statistical Software, Second Edition.  Chapman Hall / CRC Press: Boca Raton, FL.
