You can certainly do it, i.e. you won't get an error:
data("mtcars")
library(lme4)
## Loading required package: Matrix
lme1 = lmer('mpg ~ 1 + hp + (1|vs)', data = mtcars)
summary(lme1)
## Linear mixed model fit by REML ['lmerMod']
## Formula: mpg ~ 1 + hp + (1 | vs)
## Data: mtcars
##
## REML criterion at convergence: 181.4
##
## Scaled residuals:
## Min 1Q Median 3Q Max
## -1.4603 -0.6050 -0.2401 0.4277 2.1212
##
## Random effects:
## Groups Name Variance Std.Dev.
## vs (Intercept) 1.379 1.174
## Residual 14.577 3.818
## Number of obs: 32, groups: vs, 2
##
## Fixed effects:
## Estimate Std. Error t value
## (Intercept) 29.33120 2.03135 14.439
## hp -0.06254 0.01207 -5.183
##
## Correlation of Fixed Effects:
## (Intr)
## hp -0.849
lm2 = lm('mpg ~ 1 + hp', data = mtcars)
summary(lm2)
##
## Call:
## lm(formula = "mpg ~ 1 + hp", data = mtcars)
##
## Residuals:
## Min 1Q Median 3Q Max
## -5.7121 -2.1122 -0.8854 1.5819 8.2360
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 30.09886 1.63392 18.421 < 2e-16 ***
## hp -0.06823 0.01012 -6.742 1.79e-07 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 3.863 on 30 degrees of freedom
## Multiple R-squared: 0.6024, Adjusted R-squared: 0.5892
## F-statistic: 45.46 on 1 and 30 DF, p-value: 1.788e-07
anova(lme1, lm2)
## refitting model(s) with ML (instead of REML)
## Data: mtcars
## Models:
## lm2: "mpg ~ 1 + hp"
## lme1: mpg ~ 1 + hp + (1 | vs)
## Df AIC BIC logLik deviance Chisq Chi Df Pr(>Chisq)
## lm2 3 181.24 185.64 -87.619 175.24
## lme1 4 183.24 189.10 -87.619 175.24 0 1 1
The AIC that is returned by anova is also commonly used to compare nested models.
However, I would urge you to reconsider why you want to test the effect of the random effect for LAKE in the first place. If the variable was part of your experimental design (as it appears to be the case to me), then some would argue that you must include it in your model no matter the difference its inclusion to the model produces (see Testing the significance of random effects for more discussion). If what you are after is to examine the effect in specific lakes, then perhaps the BLUPs are what you are after:
ranef(lme1, condVar = TRUE)
If instead, you initial question was if some lakes are more conductive for parasitism, then I would advise you to include the variable LAKE as a fixed factor and include an interaction with your other variables.
[...] since I am seeing differences in one of the lakes, I believe this means I would have to treat Lake as a RANDOM effect...
No, this is not a valid reason of why you want to select a variable as a random effect. This issue has been discussed elsewhere, for example: What is the difference between fixed effect, random effect and mixed effect models?
In my opinion, you treat a variable as a random effect if you care about its influence but the specific instantiations are not important. In your case, if it is true that you want to include the effect of LAKE in the model but you wouldn't expect a difference whether the specific lakes in your sample were {Tanganyika, Victoria} or {Caspian, Baikal}, then I would include the variable LAKE as a random factor regardless of if it contributed anything to the model based on any test or metric.
