# Interpreting ANOVA, Tukey, Linear Mixed-Effects Results

I am trying to follow the work of a scientist who compared two factors (A and B) each with two levels. The general output from R appears as follows:

Estimate Std. Error z value Pr(>|z|)
A.1 B.1 - A.1 B.2 == 0 0.08015 0.05255 1.525 0.42232
A.2 B.2 - A.1 B.2 == 0 0.17510 0.05255 3.332 0.00481 **
A.2 B.1 - A.1 B.2 == 0 0.27127 0.05259 5.158 < 0.001 ***
A.2 B.2 - A.1 B.1 == 0 0.09496 0.05163 1.839 0.25493
A.2 B.1 - A.1 B.1 == 0 0.19112 0.05171 3.696 0.00133 **
A.2 B.1 - A.2 B.2 == 0 0.09617 0.05171 1.860 0.24556


They also have a barchart with the four treatments and a Tukey letter over each one as follows:

A.1 and B.1 (Tukey letter = A,B)
A.1 and B.2 (Tukey letter = A)
A.2 and B.1 (Tukey letter = C)
A.2 and B.2 (Tukey letter = B)


I am a bit confused on how to interpret these two results, as I wonder if they seem conflicting? Below are two issues I am particularly confused about:

1) In the comparison between treatments A.2,B.1 and A.2,B.2, there are two separate Tukey letters (B and C), which I thought would mean these two treatments are statistically different. However, in the table, there is only a P-value of 0.24556 between these two groups, which indicates they are not statistically different. How should I interpret the difference between these two treatment groups?

2) In the comparison between treatments A.1,B.1 and A.1,B.2, the table indicates they have a P-Value of 0.42232, which indicates they are not statistically different. At the same time, their Tukey letters have some overlap but some non-overlap (A and A,B). The fact that both these Tukey letters have "A" means they are not statistically different; however, does the fact that only Tukey letter has "B" mean there is some intermediate statistical difference?

I do not have reproducible R code that made this plot. But it seems the syntax was somewhat as follows:

dat2 = lme(Results ~ Treatment, data=dat, random =~1|Experiment)
anova(dat2)
summary(glht(dat2, linfct=mcp(Treatment="Tukey")))


Thank you for sharing any advice!

• Pr(>|z|) does not indicate whether there is significant (or insignificant) difference in effects produced by two groups. – Subhash C. Davar Apr 10 '18 at 13:20