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:
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
What is the equivalent
lmeranalysis using the
lme4package? When I try and run the following, the F-test does not line up with the above
lmer1 <- lmer(response ~ treatment + (1|blockCol), data = mydata)
Analysis of Variance Table Df Sum Sq Mean Sq F value treatment 1 71.881 71.881 21.462
Could I design this experiment as a completely randomized design if I only assigned one sample per column?
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?