# Two way ANOVA and post hoc test

I performed a two way ANOVA over a test set. My two nominal variables are treatment and clone. My outcome variable is area.

> summary(test)
area         clone    treatment
Min.   :0.01418   15A:20   CTRL:30
1st Qu.:0.28725   18B:20   DRUG:30
Median :0.63543   1C :20
Mean   :0.94825
3rd Qu.:1.43322
Max.   :3.27228
area clone treatment
1 0.35096747   15A      CTRL
2 0.31212944   18B      CTRL
3 0.02500775    1C      CTRL
4 0.10188701   15A      CTRL
5 0.53309240   18B      CTRL
6 0.04796319    1C      CTRL


I have fitted the model and only the factor treatment result as significant different.

> model <- aov(data = test,formula = area~treatment*clone)
> summary(model)
Df Sum Sq Mean Sq F value   Pr(>F)
treatment        1 11.277  11.277  19.778 4.38e-05 ***
clone            2  0.456   0.228   0.400    0.673
treatment:clone  2  2.015   1.008   1.767    0.181
Residuals       54 30.791   0.570
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1


When I perform the post-hoc test (Tukey's) to see which comparison is significant I got the following result:

> TukeyHSD(model)
Tukey multiple comparisons of means
95% family-wise confidence level

Fit: aov(formula = area ~ treatment * clone, data = test)

$treatment diff lwr upr p adj DRUG-CTRL 0.8670801 0.4761903 1.25797 4.38e-05$clone
18B-15A 0.1089679 -0.4665064 0.6844423 0.8917976
1C-15A  0.2134299 -0.3620444 0.7889043 0.6465153
1C-18B  0.1044620 -0.4710123 0.6799364 0.9000877

\$treatment:clone
DRUG:15A-CTRL:15A  0.718000960 -0.27971707 1.7157190 0.2897002
CTRL:18B-CTRL:15A  0.212138816 -0.78557921 1.2098568 0.9884101
DRUG:18B-CTRL:15A  0.723797984 -0.27392004 1.7215160 0.2813146
CTRL:1C-CTRL:15A  -0.113359725 -1.11107775 0.8843583 0.9994033
DRUG:1C-CTRL:15A   1.258220555  0.26050253 2.2559386 0.0059201
CTRL:18B-DRUG:15A -0.505862143 -1.50358017 0.4918559 0.6669850
DRUG:18B-DRUG:15A  0.005797025 -0.99192100 1.0035151 1.0000000
CTRL:1C-DRUG:15A  -0.831360685 -1.82907871 0.1663573 0.1539890
DRUG:1C-DRUG:15A   0.540219595 -0.45749843 1.5379376 0.6023064
DRUG:18B-CTRL:18B  0.511659168 -0.48605886 1.5093772 0.6562046
CTRL:1C-CTRL:18B  -0.325498542 -1.32321657 0.6722195 0.9272660
DRUG:1C-CTRL:18B   1.046081739  0.04836371 2.0437998 0.0347021
CTRL:1C-DRUG:18B  -0.837157710 -1.83487574 0.1605603 0.1486268
DRUG:1C-DRUG:18B   0.534422571 -0.46329546 1.5321406 0.6133233
DRUG:1C-CTRL:1C    1.371580280  0.37386225 2.3692983 0.0020981


I know there is not evidence of an interaction between the two nominal variables, but am I allowed to look into those comparisons (like DRUG:1C-CTRL:1C) to pull the ones significantly different or I am just allowed to say that overall CTRL is different from DRUG by blocking for clone?

Here is a depiction of the result where is clear there is no evident interaction between clone and treatment.

• No, if your ANOVA shows no significance for your interaction term then you do not interpret any significant regression terms, the interaction term overall is not significant. – user2974951 Oct 4 '18 at 10:54
• @user2974951 I see, just to confirm. I cannot say that in clone 1C the treatment is significantly different from the control rigth?!. in this case how would you suggest to present the data? I know the way I put them (the boxplots facet by clone) would make raise questions about the comparison within clone. – edoardo pedrini Oct 4 '18 at 12:58
• The only statistically significant results are in the treatments, so you should analyze the treatments (CTRL and DRUG), report their mean etc or draw a plot for the treatments (not separated by clones, since they are not significant). – user2974951 Oct 4 '18 at 13:19