> birch.mod2 <- aov(grn.diam ~ species + Error(location)) > summary(birch.mod2) Error: location Df Sum Sq Mean Sq F value Pr(>F) species 4 1770 442.4 5.049 0.0017 ** Residuals 50 4382 87.6 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Error: Within Df Sum Sq Mean Sq F value Pr(>F) Residuals 1595 3570 2.238 > > library(lmerTest) > > birch.mod3 <- lmer(grn.diam ~ species + (1|species:location)) > summary(birch.mod3) Linear mixed model fit by REML t-tests use Satterthwaite approximations to degrees of freedom [merModLmerTest] Formula: grn.diam ~ species + (1 | species:location) REML criterion at convergence: 6205.4 Scaled residuals: Min 1Q Median 3Q Max -5.2443 -0.6439 0.0147 0.6112 3.5444 Random effects: Groups Name Variance Std.Dev. species:location (Intercept) 2.847 1.687 Residual 2.238 1.496 Number of obs: 1650, groups: species:location, 55 Fixed effects: Estimate Std. Error df t value Pr(>|t|) (Intercept) 23.0338 0.4740 50.0000 48.592 < 2e-16 *** species2 -1.6788 0.8435 50.0000 -1.990 0.052046 . species3 -1.2138 0.8994 50.0000 -1.350 0.183218 species4 -2.3281 0.6152 50.0000 -3.784 0.000413 *** species5 -2.7305 0.6842 50.0000 -3.991 0.000215 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Correlation of Fixed Effects: (Intr) specs2 specs3 specs4 species2 -0.562 species3 -0.527 0.296 species4 -0.771 0.433 0.406 species5 -0.693 0.389 0.365 0.534 > anova(birch.mod3) Analysis of Variance Table of type 3 with Satterthwaite approximation for degrees of freedom Sum Sq Mean Sq NumDF DenDF F.value Pr(>F) species 45.197 11.299 4 50 5.0486 0.001699 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > > if(require(pbkrtest)) + anova(birch.mod3, ddf = "Kenward-Roger") Analysis of Variance Table of type 3 with Kenward-Roger approximation for degrees of freedom Sum Sq Mean Sq NumDF DenDF F.value Pr(>F) species 45.197 11.299 4 50 5.0486 0.001699 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 > step(birch.mod3) Random effects: Chi.sq Chi.DF elim.num p.value species:location 1083.25 1 kept < 1e-07 Fixed effects: Sum Sq Mean Sq NumDF DenDF F.value elim.num Pr(>F) species 45.1973 11.2993 4 50 5.0486 kept 0.0017 Least squares means: species Estimate Standard Error DF t-value Lower CI Upper CI p-value species 1 1 23.034 0.474 50 48.590 22.1 24.0 <2e-16 species 2 2 21.355 0.698 50 30.610 20.0 22.8 <2e-16 species 3 3 21.820 0.764 50 28.550 20.3 23.4 <2e-16 species 4 4 20.706 0.392 50 52.810 19.9 21.5 <2e-16 species 5 5 20.303 0.493 50 41.150 19.3 21.3 <2e-16 species 1 *** species 2 *** species 3 *** species 4 *** species 5 *** --- 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 species 1 - 2 1.7 0.844 50.0 1.99 -0.0155 3.37 0.05 species 1 - 3 1.2 0.899 50.0 1.35 -0.5927 3.02 0.18 species 1 - 4 2.3 0.615 50.0 3.78 1.0924 3.56 4e-04 species 1 - 5 2.7 0.684 50.0 3.99 1.3563 4.10 2e-04 species 2 - 3 -0.5 1.035 50.0 -0.45 -2.5437 1.61 0.66 species 2 - 4 0.6 0.800 50.0 0.81 -0.9584 2.26 0.42 species 2 - 5 1.1 0.855 50.0 1.23 -0.6648 2.77 0.22 species 3 - 4 1.1 0.859 50.0 1.30 -0.6112 2.84 0.20 species 3 - 5 1.5 0.910 50.0 1.67 -0.3106 3.34 0.10 species 4 - 5 0.4 0.630 50.0 0.64 -0.8634 1.67 0.53 species 1 - 2 . species 1 - 3 species 1 - 4 *** species 1 - 5 *** species 2 - 3 species 2 - 4 species 2 - 5 species 3 - 4 species 3 - 5 species 4 - 5 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Final model: lme4::lmer(formula = grn.diam ~ species + (1 | species:location), REML = reml.lmerTest.private, contrasts = l.lmerTest.private.contrast, devFunOnly = devFunOnly.lmerTest.private)