# Incomplete block design analysis / design with R

I have a few questions that arose when working on designing the following experiment. The numbers and variable levels are all made up for examples sake.

I want to test the effect of incubation period (15 minutes or 30 minutes) on the outcome of a drug test.

The experimental unit is a well where the drug sample is placed, wells are grouped by 8 into columns. There are 12 columns in a plate. Wells cannot be separated and the smallest unit I can work with is a column (8 wells). I have reason to suspect there will be differences between columns so I want to block them. Here is a picture of a plate, column, and wells: I designed the experiment in the following way. I cannot incubate a column for two different periods since they come as a unit, so I cannot make this a randomized complete block design. I randomly selected 4 columns from a plate, I randomized each column to be incubated for either 15 minutes or 30 minutes. Then I placed 8 samples in each column (all of the samples were spiked with the same concentration).

Here is my sample data I generated.

# Set the block designation
blockCol <- factor(rep(c(1:4), each = 8))
# Setting the treatments (these would be randomized to the blocks)
treatment <- rep(c("15min", "30min"), each = 16)
# Set some made up response data
set.seed(225)
response <- c(rnorm(16, 5, 2), rnorm(16, 7, 2))
# Combine all the data into a data frame
mydata <- data.frame(blockCol, treatment, response)


I have three questions:

1. Using the aov() function, would I analyze this data with

aov1 <- aov(response ~ treament + Error(blockCol), data = mydata)
summary(aov1)


which results in the output

Error: blockCol
Df Sum Sq Mean Sq F value  Pr(>F)
treatment  1  71.88   71.88   107.5 0.00918 **
Residuals  2   1.34    0.67
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Error: Within
Df Sum Sq Mean Sq F value Pr(>F)
Residuals 28  99.14   3.541

2. What is the equivalent lmer analysis using the lme4 package? When I try and run the following, the F-test does not line up with the above aov() analysis.

lmer1 <- lmer(response ~ treatment + (1|blockCol), data = mydata)


anova(lmer1)

Analysis of Variance Table
Df Sum Sq Mean Sq F value
treatment  1 71.881  71.881  21.462

3. Could I design this experiment as a completely randomized design if I only assigned one sample per column?

# Edits

My main concern is with the experimental design, chiefly
1. Is the incomplete block design the most efficient, or can I use a completely randomized design with one sample per column.
2. Are my analyses (in R) partitioning the error term correctly?

• Questions that are only about how to use R are generally off topic here. This strikes me as borderline; you may want to edit it to make the statistical aspects more salient. – gung Apr 11 '17 at 12:15
• This doen't look like a question about how to use R, it is a question about design. – kjetil b halvorsen Apr 11 '17 at 13:56
• I've updated my question with clarification edits at the bottom. I am chiefly interested in creating an effective design and making sure I partition the error terms in the model to match that design. I happen to use R as my program of choice so if anyone is able to give me the answers in terms of R as well, I would appreciate it, but it's not necessary. – Graham Keto Apr 11 '17 at 14:32

## 1 Answer

1.) If the columns really differ systematically, as suspected, you should test as much columns as possible (more than the 4 columns in your example) to estimate/account for this source of variance. Further, you should consider stratifying your samples across (rather than within) columns to avoid that column quality and sample differences become indistinguishable. btw separate incubation runs may also be a random effect to consider.

2.) As discussed in this related post results from aov with random effects and lmer can differ since the underlying algorithms are different. The problem in the simulation above may be that the residual degrees of freedom on which aov bases its F-test is df=2 (there are only two columns, i.e. independent experimental units per treatment). When increasing the number of columns, results from aov and lmer become more similar. I would stick with lmer when dealing with clustered data, since it is more flexible and the aov/strata method is considered a bit old-fashioned.

• I think you misunderstood, all the 8 wells in a column is used, they just always must have the same treatment. This makes this look like a split-plot design. – kjetil b halvorsen Apr 11 '17 at 23:37
• Ah I see: Edit above. – mzunhammer Apr 11 '17 at 23:41