What are the two purposes for Randomized Block Designs?

• Increase precision of estimates of treatment differences, and power for detecting differences in treatments

• Broaden the basis for conclusions

Exercise 3

- Expected Difference in treatment means

σ - standard deviation in experimental units

Block 1

Block 2

Block 3

Block 4

Why are replicates within a block usually not recommended?

Replicates require larger blocks of experimental units→ more heterogeneity among experimental units in a block

A.

B. use randomized block design and randomize the four treatments to four flowers within each type

What if there is an interaction between block and treatment?

• That means differences in treatments depend on the block!

• Can we draw general conclusions if that is the case?• Treatment MS must be significantly larger than error

(which is equivalent to interaction in this case) if we want to draw general conclusions!

removing treatment differences

response was the activity level of the enzyme EROD in the liver.

Example

Golf Magazine June 2006

(a)Define Objectives

Determine if tee heightaffects golf driving distance

(b) Identify sources of variation

tee height Golfer and ability level brand ball club wind speed repeat swings

c) Choose rule to assign experimental units to treatment factors

Complete Block Design

Blocks will be Golfers (takes into account differences in ability levels and clubs)Treatment Factor tee height, each golfer will hit 5 balls from each tee height in a randomized order

d) Measurements to be made: 1) distance 2) whether or not ball is on fairway “accuracy”

Expected difficulties: miss hit balls may not be representative. Solution 1) use low handicap golfers, 2) warm up 3) use more than 5 balls and don’t count miss hits

What if there is an interaction between block and treatment?

• That means differences in treatments depend on the block!

• Can we draw general conclusions if that is the case?• Treatment MS must be significantly larger than error

(which is equivalent to interaction in this case) if we want to draw general conclusions!

RCB Design nitrogen timing on wheat

Treatment: Timing of Nitrogen application

Blocks: Irrigation gradient

Latin Square

55 Latin-Square laid out in BettgelertForest 1929. Study the effects of exposure On 1)Sitka spruce 2) Norway spruce 3)Japanese larch 4) European larch 5) Pinus contorta

In LSD every treatment occurs in every row and column

Also every row occurs in every column and vise versa

.

Treatment Factor is tire design: types A, B, C, D

Objective to study how different tire designs affect tread life

One Blocking Factor is type of carBecause tires wear at different rates on different type cars

Economy carLuxury CarSUVMuscle Car

Another Blocking factor is Tire Position

Speed of tire wearalso depends on tire position on the car

Left Front Right Front

Left Rear Right Rear

Why not include interactions?

Therefore Use a Latin Square Design

Tire Position

Car Left F Right F Left R Right R

Economy A B C D

Luxury B C D A

Suv C D A B

Muscle D A B C

Ignoring Column Blocks this is a RCB in RowsIgnoring Rows this is a RCB in columns

Example Dairy Cow Experiment

Treatment is Diet:

Response is Milk Yield

1. Row Block is Cow

2. Column Block is period

Calfing------------>

Mil

k Y

ieldDiet 1Diet 2Diet 3

United StatesThe FDA considers two products bioequivalent if the 90% CI of the relative mean Cmax, AUC(0-t) and AUC(0-∞) of the test (e.g. generic formulation) to reference (e.g. innovator brand formulation) should be within 80.00% to 125.00% in the fasting state. Although thereare a few exceptions, generally a bioequivalent comparison of Test toReference formulations also requires administration after an appropriate meal at a specified time before taking the drug, a so-called "fed" or "food-effect" study. A food-effect study requires the same statistical evaluation as the fasting study, described above.

Blocks - Homogeneous Groups of Experimental Units

Reduce variance of experimental error

ijjiij by

RE

ijjiij by ijijjjiijk by RBF

Conclusions apply to population represented by all blocks

Latin Square Design