Math 558 Lecture #2
Initial steps in the planning of an experiment Purpose of the experiment
As providing the interpretation of results is an important final stage in the statistically designed experiments, the statistician should ask the researcher about the purpose of the experiment. The experiment needs to be conducted to satisfy the purpose of the experiment and the results should align with this purpose. In order to have good understanding of the objectives of the experiment, the statistician must ask a series of question to find enough understanding about why the experiment is done? what are the treatments that need to be compared? and more precisely, what is the research question that the researcher wants to answer?The statistician also needs to know the population to which the results will be extended
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Initial steps in the planning of an experiment Replication
Usually an experiment needs to be replicated several times to estimate
the parameters of interest (to detect differences between the effects of
two drugs, to estimate the proportion of population using a particular
phone service). Recall the formula for the variance of the mean of n
observations. It is Var(X ̄ ) = σ2/n. Therefore, increasing n will decrease
the variance of X ̄ which is an unbiased estimator of the population mean
s2/n determines the width of our confidence interval. And narrower the interval, the more precise the
estimate is.
One important thing to keep in mind is that the increase in the replication increases the cost of the experiment. Usually the cost of experiment is one of the biggest constraints that the researcher encounters.
μ. Also recall that the statistic
Initial steps in the planning of an experiment blocking
Another way to reduce Var(X ̄ ) = σ2/n is to reduce σ2 itself. The quantity σ2 also known as experimental error represents the extraneous variations. One way of reducing the experimental errors is to group the experimental units. An experimental unit1 is an object/individual on which a treatment is applied in a single run. An experimental unit can be a plot of a land, a patient in a hospital, a hospital itself, an animal, a body part of the animal etc.These experimental units produce different results every time they are subjected to the same treatment. These differences contribute to the experimental error. Another contribution to the experimental error comes from the differences in outer physical sources.
1more discussion in the next lecture
Initial steps in the planning of an experiment blocking or local control
The expression “outer physical sources” requires some explanation. An outer physical source can be difference in soil fertility, the amount of water a plot gets, sunlight etc. To reduce this source of variation we can group the experimental units into homogeneous(alike) blocks. Although blocking2 helps to decrease the value of σ2, it increases the complexity of the design. It makes the statistical analysis and interpretation of the results more difficult as well.
2details in the coming lectures
Initial steps in the planning of an experiment Randomization
Another important aspect of any experiment that is to reduce the bias is inferences. This means that no experimental unit should be favoured on the others. This also means that all experimental units should be equally likely to receive a treatments. This is done by an experimental tool known as randomization3.
3details in the coming lectures
Three principals of Experimental design by Fisher
Replication Blocking Randomization
https://rss.onlinelibrary.wiley.com/doi/10.1111/
j.1740-9713.2019.01230.x
Rye grass example (Bailey Page 7)
“An experiment was conducted to compare three different varieties of rye-grass in combination with four quantities of nitrogen fertilizer. The response measured was the total weight of dry-matter harvested from each plot. The three varieties of rye-grass were called Cropper, Melle and Melba. The four amounts of fertilizer were 0 kg/ha, 80 kg/ha, 160 kg/ha and 240 kg/ha.” The plan according to which the treatments are applied to the experimental units is given in the next slide.
Rye grass example (Bailey Page 7)
Mella Mella 0 160 240 160 80 0
160 80 80 0 10 80 80 0 160 240 0 240 240 240 0 80 240 160
Notice the absence of pattern in allocation of treatments to the plots. It is called randomization.
Terminology Bailey pg 8
Experimental unit is the smallest unit to which a treatment can be applied.
A treatment is the entire description of what be applied to an experimental unit.
An observational unit is the smallest unit on which the response will be measured.
Examples Euclypts
Different families of euclypts were grown in five-tree plots. The diameter of each tree measured at breast height was the response. Here the families are the treatments.
The plots are the experimental units.
The trees can be the observational units.
Examples Wheat varieties
Different varieties of wheat are grown in plots in a field. Here the varieties are the treatments.
The plots are the experimental units.
The plots can be the observational units.
Examples Rye-grass
Treatments: Cultivar and amount of fertilizer combinations (12 in total). Experimental units: Plots
Observational units: can be plots
Examples Calf feeding
More details. Calves were house in pens with each pen housing ten calves. All the calves in a pen were given the same feed. Each calf was then weighed individually. Treatments: different composition of feeds Experimental Units: Pens
Observational units: calves
Reading for assignment 1
The importance of experimental design in proteomic mass spectrometry experiments: Some cautionary tales
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