# 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. Apr 11, 2017 at 12:15
• This doen't look like a question about how to use R, it is a question about design. Apr 11, 2017 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. Apr 11, 2017 at 14:32