Effect of Copper on Sessile Organism Species Richness

When an intereaction effect exists, then the levels of the two main effects should be combined to form treatment groups (this is the
comb
variable below) and a oneway ANOVA is fit using this to determine which treatment groups differ.  Multiple comparisons confidence intervals are for differences in two means. However, don’t say “the difference in means is between XXX and XXX.” Rather be specific about which group is smaller (Or larger) and by how much. Make note of this in my language below.
 The multiple comparison procedures use a bit of randomness to compute pvalues and confidence intervals. Thus, your values may differ somewhat from what is presented here.
 Interpreting interactions can be difficult. I generally like to describe the effect of the xaxis main effect on the response variable for one of the levels of the other main effect and then describe how this differs for the other level(s) of the other main effect. So, here, I am thinking of how copper level effects species richness for, say, the horizontal orientiation and then how copper effect species richness for the vertical orientation. With an interaction effect the effects of copper on species richness will differ by habitat orientation.
There is not enough information to determine if the individuals (a location) are independent within and among treatments. Independence within treatments would be suspect if the same copper treatment was assigned to adjacent locations or the same habitat orientation locations were adjacent. Independence among treatments would be suspect if different copper treatments were assigned to adjacent locations or different habitat orientations were at adjacent locations. For example, if the researchers found one vertical orientation and then split it into three parts to receive one of the copper treatments, then independence would be lost. There is no evidence either way with the given description so I will continue as if the individuals are independent.
On the original scale (Figure 1) the variances are equal (Levene’s p=0.0750), the residuals are not normal (AndersonDarling p=0.0148) but mostly symmetric without long tails, and there are no significant outliers (p>1). The trialanderror method did not suggest a transformation for species richness that resulted in more normal residuals without sacrificing the other assumptions. Given the symmetry and lack of long tails in the distribution of residuals, it appears that the assumptions for a twoway ANOVA have been largely met on the original scale with no individuals removed.
There is a significant interaction effect (p<0.00005; Table 1). Main effects will not be addressed in the presence of this interaction effect.
Summary plots of these results are in Figure 2.
Mean species richness is most (and significantly) different between the “No Copper and Horizontal Orientation” treatment and the “High Copper and Vertical Orientation Treatment” (Table 2). Specifically, mean species richness was between 21.6 and 27.8 greater at the “No Copper and Horizontal Orientation” treatment than at the “High Copper and Vertical Orientation Treatment” (Table 3).
These results show that species richness generally decreased from locations with no copper to locations with low levels of copper, regardless of whether the location was oriented vertically or horizontally. Species richness further decreased when copper levels were increased from “low” to “high” in verticallyoriented locations but not in horizontallyoriented locations. Thus, copper appears to have negatively affected species richness of sessile organisms, but the effect differed depending on habitat orientation.
Table 1: Analysis of variance table for the twoway ANOVA of untransformed species richness of sessile organisms by copper treatment level and habiat orientation.
Df Sum Sq Mean Sq F value Pr(>F)
Orientation 1 180.3 180.27 32.212 5.619e07
Copper 2 3207.4 1603.72 286.567 < 2.2e16
Orientation:Copper 2 535.8 267.92 47.874 1.096e12
Residuals 54 302.2 5.60
Table 2: Tukey’s multiple comparison results for the twoway ANOVA of untransformed species richness of sessile organisms by copper treatment level and habiat orientation.
Estimate Std. Error t value p value
None:Vertical  None:Horizontal = 0 2.3 1.05795 2.174015 2.668409e01
Low:Horizontal  None:Horizontal = 0 12.2 1.05795 11.531730 0.000000e+00
Low:Vertical  None:Horizontal = 0 9.0 1.05795 8.507014 6.569867e11
High:Horizontal  None:Horizontal = 0 13.4 1.05795 12.665999 0.000000e+00
High:Vertical  None:Horizontal = 0 24.7 1.05795 23.347028 0.000000e+00
Low:Horizontal  None:Vertical = 0 9.9 1.05795 9.357716 2.422285e12
Low:Vertical  None:Vertical = 0 6.7 1.05795 6.333000 8.975828e07
High:Horizontal  None:Vertical = 0 11.1 1.05795 10.491984 2.220446e14
High:Vertical  None:Vertical = 0 22.4 1.05795 21.173013 0.000000e+00
Low:Vertical  Low:Horizontal = 0 3.2 1.05795 3.024716 4.185940e02
High:Horizontal  Low:Horizontal = 0 1.2 1.05795 1.134269 8.648756e01
High:Vertical  Low:Horizontal = 0 12.5 1.05795 11.815298 1.110223e16
High:Horizontal  Low:Vertical = 0 4.4 1.05795 4.158985 1.607308e03
High:Vertical  Low:Vertical = 0 15.7 1.05795 14.840014 0.000000e+00
High:Vertical  High:Horizontal = 0 11.3 1.05795 10.681029 1.443290e15
Table 3: Tukey’s multiple comparison confidence intervals for the twoway ANOVA of untransformed species richness of sessile organisms by copper treatment level and habiat orientation.
Estimate lwr upr
None:Vertical  None:Horizontal 2.3 5.42611464 0.8261146
Low:Horizontal  None:Horizontal 12.2 15.32611464 9.0738854
Low:Vertical  None:Horizontal 9.0 12.12611464 5.8738854
High:Horizontal  None:Horizontal 13.4 16.52611464 10.2738854
High:Vertical  None:Horizontal 24.7 27.82611464 21.5738854
Low:Horizontal  None:Vertical 9.9 13.02611464 6.7738854
Low:Vertical  None:Vertical 6.7 9.82611464 3.5738854
High:Horizontal  None:Vertical 11.1 14.22611464 7.9738854
High:Vertical  None:Vertical 22.4 25.52611464 19.2738854
Low:Vertical  Low:Horizontal 3.2 0.07388536 6.3261146
High:Horizontal  Low:Horizontal 1.2 4.32611464 1.9261146
High:Vertical  Low:Horizontal 12.5 15.62611464 9.3738854
High:Horizontal  Low:Vertical 4.4 7.52611464 1.2738854
High:Vertical  Low:Vertical 15.7 18.82611464 12.5738854
High:Vertical  High:Horizontal 11.3 14.42611464 8.1738854
Figure 1: Histogram of residuals (left) and boxplot of residuals by treatment group (right) for the twoway ANOVA of untransformed species richness of sessile organisms by copper treatment level and habiat orientation.
Figure 2: Interaction plot (mean species richness by treatment group) for the twoway ANOVA of untransformed species richness of sessile organisms by copper treatment level and habiat orientation. Treatment means with different letters are significantly different.
R Appendix.
library(NCStats)
library(multcomp)
df < read.csv("https://raw.githubusercontent.com/droglenc/NCData/master/Sessile.csv")
df$Copper < factor(df$Copper,levels=c("None","Low","High"))
lm1 < lm(Richness~Orientation*Copper,data=df)
transChooser(lm1)
aov1 < anova(lm1)
df$comb < df$Copper:df$Orientation
lm2 < lm(Richness~comb,data=df)
mc < glht(lm2,mcp(comb="Tukey"))
summary(mc)
confint(mc)
fitPlot(lm1,change.order=TRUE,ylab="Species Richness",xlab="Copper Level")
addSigLetters(lm1,change.order=TRUE,lets=c("a","a","b","c","b","d"),pos=c(2,2,2,4,4,4))