带有块设计和重复测量的方差分析

Question

I'm attempting to run some statistical analyses on a field trial that was constructed over 2 sites over the same growing season.

At both sites ( Site , levels: HF|NW) the experimental design was a RCBD with 4 (n=4) blocks ( Block , levels: 1|2|3|4 within each Site ). There were 4 treatments - 3 different forms of nitrogen fertiliser and a control (no nitrogen fertiliser) ( Treatment , levels: AN, U, IU, C). During the field trial there were 3 distinct periods that commenced with fertiliser addition and ended with harvesting of the grass. These periods have been given the levels 1|2|3 under the factor N_app .

There are a range of measurements that I would like to test the following null hypothesis H0 on:

Treatment (H0) had no effect on measurement

Two of the measurements I am particularly interested in are: grass yield and ammonia emissions.

Starting with grass yield ( Dry_tonnes_ha ) as shown here, a nice balanced data set

The data can be downloaded in R using the following code:

library(tidyverse)

download.file('https://www.dropbox.com/s/w5ramntwdgpn0e3/HF_NW_grass_yield_data.csv?raw=1', destfile = "HF_NW_grass_yield_data.csv", method = "auto")
raw_data <- read.csv("HF_NW_grass_yield_data.csv", stringsAsFactors = FALSE)

HF_NW_grass <- raw_data %>% mutate_at(vars(Site, N_app, Block, Plot, Treatment), as.factor) %>% 
  mutate(Date = as.Date(Date, format = "%d/%m/%Y"),
         Treatment = factor(Treatment, levels = c("AN", "U", "IU", "C")))

I have had a go at running an ANOVA on this using the following approach:

model_1 <- aov(formula = Dry_tonnes_ha ~ Treatment * N_app + Site/Block, data = HF_NW_grass, projections = TRUE)

I have a few concerns with this.

Firstly, what is the best way to test assumptions? For a simple one-way ANOVA I would use shapiro.test() and bartlett.test() on the dependent variable ( Dry_tonnes_ha ) to assess normality and heterogeneity of variance. Can I use the same approach here?

Secondly, I am concerned that N_app is a repeated measure as the same measurement is taken from the same plot over 3 different periods - what is the best way to build this repeated measures into the model?

Thirdly, I'm not sure of the best way to nest Block within Site . At both sites the levels of Block are 1:4. Do I need to have unique Block levels for each site?

I have another data set for NH3 emissions here . R code to download:

download.file('https://www.dropbox.com/s/0ax16x95m2z3fb5/HF_NW_NH3_emissions.csv?raw=1', destfile = "HF_NW_NH3_emissions.csv", method = "auto")
raw_data_1 <- read.csv("HF_NW_NH3_emissions.csv", stringsAsFactors = FALSE)

HF_NW_NH3 <- raw_data_1 %>% mutate_at(vars(Site, N_app, Block, Plot, Treatment), as.factor) %>% 
  mutate(Treatment = factor(Treatment, levels = c("AN", "U", "IU", "C")))

For this I have all the concerns above with the addition that the data set is unbalanced. At HF for N_app 1 n=3, but for N_app 2 & 3 n=4 At NW n=4 for all N_app levels. At NF measurements were only made on the Treatment levels U and IU At NW measuremnts were made on Treatment levels AN , U and IU

I'm not sure how to deal with this added level of complexity. I am tempted to just analyse as 2 separate site (the fact that the N_app periods are not the same at each site may encourage this approach). Can I use a type iii sum of squares ANOVA here?

It has been suggested to me that a linear mixed modelling approach may be the way forward but I'm not familiar with using these.

I would welcome your thoughts on any of the above. Thanks for your time.

Rory

Answer 1

To answer your first question on the best way of testing assumptions. While your attempt of using another statistical test, implemented in R, is reasonable, I would actually just visualize the distribution and see if the data meet ANOVA assumptions. This approach may seem somewhat subjective, but it does work in most cases.

• independently, identically distributed (iid) data: this is a question that you may already have an answer based on how much you know about your data. It's possible to use a chi-square test to determine independence (or not).
• normally distributed data: use a histogram / QQ plot to check. Based on the distribution, I think it is reasonable to use aov despite the slightly bimodal distribution.

(It appears that log-transformation help further meet normality assumption. This is something you may consider, especially for downstream analyses.)

par(mfrow=c(2,2))
plot(density(HF_NW_grass$Dry_tonnes_ha), col="red", main="Density")
qqnorm(HF_NW_grass$Dry_tonnes_ha, col="red", main="qqplot")
qqline(HF_NW_grass$Dry_tonnes_ha)

DTH_trans <- log10(HF_NW_grass$Dry_tonnes_ha)
plot(density(DTH_trans), col="blue", main="transformed density")
qqnorm(DTH_trans, col="blue", main="transformed density")
qqline(DTH_trans)

Regarding your second question on what the best way to build repeated measures into the model is: Unfortunately, it is difficult to pinpoint such a "best" model, but based on my knowledge (mostly through genomics big data), you may want to use a linear mixed effect model. This can be implemented through the lme4 R package, for example. Since it appears you already know how to construct a linear model in R, you should have no problem with applying lme4 functions.

Your third question regarding whether to nest two variables is tricky. If I were you, I would start with Site and Block as if they were independent factors. However, if you know they are not independent, you should probably nest them.

I think your questions and concerns are quite open-ended. My recommendation is that as long as you have a plausible justification, go ahead and proceed.

Answer 2

I agree with @David C on the use of visual diagnostics. Simple QQ plots should work

# dependent variable.
par(mfrow=c(1,2))
qqnorm(dt[,dry_tonnes_ha]); qqline(dt[,dry_tonnes_ha], probs= c(0.15, 0.85))
qqnorm(log(dt[,dry_tonnes_ha])); qqline(log(dt[,dry_tonnes_ha]), probs= c(0.15, 0.85))

The log transformation looks reasonable to me. You can also see this from the density plot, which is long tailed and somewhat bi-modal

par(mfrow=c(1,1))
plot(density(dt[,dry_tonnes_ha]))

You could alternatively use lineup plots (Buja et al, 2009) if you wish. I'm not sure they're needed in this case. Vignette provided

library(nullabor)
# this may not be the best X variable. I'm not familiar with your data
dt_l <- lineup(null_permute("dry_tonnes_ha"), dt)
qplot(dry_tonnes_ha, treatment, data = dt_l) + facet_wrap(~ .sample)

For the other assumptions, you can just use the standard diagnostic plots from the lm

lm2 <- lm(log(dry_tonnes_ha) ~ treatment * n_app + site/block, data = dt)
plot(lm2)

I don't see anything too troublesome in these plots.