# Using F-test vs. AICc in Mixed Model Selection

I am creating a mixed model in R using the lme4() package. I fit my full model using the lmer() function, with a single random effect.

lme0 = lmer(L50 ~ (1|SITE2) + compost + ZONE + POSITION +
PRED_BIOMASS + CS_DENSITY + RICHNESS + CHLA_RASTER + SST_RASTER, data = gbr)


I am confused on how to proceed with model selection. A stepwise approach, a la dredge(), suggests the following as the 'best' model, one with almost all predictors retained:

head(dredge(lme0))
Fixed term is "(Intercept)"
Global model call: lmer(formula = L50 ~ (1 | SITE2) + compost + ZONE + POSITION +
PRED_BIOMASS + CS_DENSITY + RICHNESS + CHLA_RASTER + SST_RASTER,
data = gbr)
---
Model selection table
(Int) CHL_RAS POS     RIC SST_RAS ZON df   logLik  AICc delta weight
202 -147.40   21.98   +           13.65   +  8 -160.115 340.0  0.00  0.559


But when I apply the lmerTest:step() function, which uses an F test as opposed to an AICc value, the 'best' fit model has only one fixed and one random parameter. I know that the step() function uses maximum liklehihod to fit the random effects, and doesn't seem to compare models on AIC.

s <- lmerTest::step(lme0)

s
Random effects:
Chi.sq Chi.DF elim.num p.value
SITE2  12.96      1     kept   3e-04

Fixed effects:
Sum Sq   Mean Sq NumDF DenDF F.value elim.num Pr(>F)
PRED_BIOMASS    0.0090    0.0090     1 20.59  0.0002        1 0.9898
compost         1.3192    1.3192     1 26.23  0.0259        2 0.8735
CS_DENSITY      7.9498    7.9498     1 35.78  0.1634        3 0.6885
SST_RASTER     27.2554   27.2554     1 25.66  0.5660        4 0.4587
ZONE           89.9116   44.9558     2 19.06  0.9115        5 0.4187
RICHNESS       62.6404   62.6404     1 37.94  1.2645        6 0.2679
CHLA_RASTER   153.7626  153.7626     1 42.82  3.0935        7 0.0858
POSITION     3518.9881 3518.9881     1 22.90 63.8783     kept <1e-07

Least squares means:
POSITION Estimate Standard Error   DF t-value Lower CI Upper CI p-value
POSITION  Mid        1.0   236.56           5.35 23.5   44.24      226      248  <2e-16 ***
POSITION  Outer      2.0   187.55           3.00 21.0   62.43      181      194  <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Differences of LSMEANS:
Estimate Standard Error   DF t-value Lower CI Upper CI p-value
POSITION Mid - Outer     49.0           6.13 22.9    7.99     36.3     61.7  <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Final model:
lme4::lmer(formula = L50 ~ (1 | SITE2) + POSITION, data = gbr,
contrasts = list(POSITION = "contr.SAS"))


Please help me understand what the fundamental difference is between the dredge() and step() functions. Why would these two approaches result in two very different 'best' models?

I'll add that the AICc for step()'s best model is higher than the one proposed by dredge():

lme1 = lmer(L50 ~ (1|SITE2) + POSITION, data = gbr)
AICc(lme1)
[1] 359.5486


• Using F-test automatic backward elimination as the one you suggest regarding lmerTest:step() is a pretty bad idea to use when it comes to model selection. Conceptually AICc is a bit better but only just a bit. Please see this thread on algorithms for automatic model selection. In short, I do not believe the inference produced by F-test backwards elimination because it is very likely to overfit. – usεr11852 Sep 20 '16 at 23:04