1. Experiment: understand how inputs (explanatory variables) affect outputs (responses) Basic Experimental Design Treatments: the input variables. Typically, discrete factors with a finite number of levels dafs shotgun.df “gun” is a factor with levels: Remington Stevens “range” is a factor with levels:

2. Basic Experimental Design Experimental unit: that to which treatment(s) is(are) applied. In a dataframe, these are probably rows or groups of rows. Block: A group of experimental units that are expected to be more similar to each other (homogeneous) Can increase precision (decrease variance) by collecting together different experimental units that have some kind of commonality dafs anneal.df: Glass RI as a function of temperature and annealing We can block by factors: annealing, square#, replicate# anneal: {pre,post} replicate: {a,b,c} square: {1:150}

3. Basic Experimental Design Replication: same treatment(s) applied to multiple experiments, effectively repeating the experiment. The more the replication, the more the variability of the study can be understood/precisely determined. Blocks can serve as pseudo-replicates Randomization: Random allocation of treatment(s) to experimental units. Effectively makes experiments independent.

4. Basic Designs Completely Randomized Design All experimental units are their own block No block structure. Data model: mean (intercept,  0 ) “treatment effects” errors ~ N(0,  2 ) This is just old N-way ANOVA, Linear regression on a set of (discrete) factors

5. Basic Designs Completely Randomized Design This is a one-way ANOVA. The one “treatment” here is range

6. Basic Designs Randomized Block Design Different experimental units are members of specific blocks Data model is a multidimensional matrix equation and not easy to write down, so lets study a detailed example instead:

7. Basic Designs Randomized Block Design What do we see in the BHH Yield data set? 1 treatment with 4 levels (process) 1 blocking variable with five levels We expect there to be more homogeneity within a blend type 1 run was performed for each process and block level combination It’s easier to read the data in that form, but aov() can’t understand it like that so lets reformat it:

8. Basic Designs So is there evidence for a difference between treatments when we block by blend type? p-value for at least one process being different Look at the treatment means and their difference from the grand mean. Should corroborate the p- value.

9. Basic Designs Randomized Block Designs What if we have two blocking variables? Latin Squares Caution: Blocking variables, just like any other variables, can have dependencies between them. Try to pick “independent” factors to block on Try to do a few replicate runs for each treatment-block Think about using a factorial design instead

10. Basic Designs Full Factorial Design At least one experiment is performed for every combination of every level of every factor. In your experiments, think of every (reasonably) possible variable that can affect the response of interest. Plan experiments that have every level of every factor represented at least once. Caution: You may be doing experiments for your thesis/dissertation for a long time.

11. Basic Designs Full Factorial Design