Experimental Design Concepts: Blocking (Chapters 14 & 15)

• Experimental Design Basics. • Completely randomized design revisited.

• Accounting for more than one factor.

• Blocking (and randomized block designs) to remove confounding.

ExpDes-1

Why Experimental Design?

Experimental design

(for an experimental study): the manner in which the experimental units are arranged or grouped, and how the treatments are assigned to them.

To call in the statistician after the experiment is done may be no more than asking him to perform a post-mortem examination: he may be able to say what the experiment died of.

Sir R.A. Fisher (1938)

Fisher compared a dataset to a sample of gold ore. The

finest analysis

could only extract the proportion of gold contained in the ore.

But a good design could yield a sample with more gold!

[ data = ore ; information = gold.] ExpDes-2

Experimental Error

Experimental error is the

variation

in the responses among experimental units (e.u.’s) which are assigned the same treatment, and are observed under the

same

experimental conditions. It is measured by SSE (or MSE).

Ideally, we would like experimental error to be

zero

! This is impossible because of (at least) one or more of the following reasons: 1.

2.

3.

4.

There are inherent differences in the e.u.’s before they receive treatments.

There is variation in the devices that record the measurements.

There is variation in applying or setting the treatments.

There are extraneous factors other than the treatments which affect the response.

ExpDes-3

Experimental Error Control

Control over the magnitude of experimental error can be achieved by: • • • • • Careful choice of e.u.’s.

Homogeneity of e.u.’s decreases MSE.

Taking care with experimental procedures and recording of measurements.

Smaller errors result in smaller MSE.

Blocking of e.u.’s.

Blocking decreases MSE.

Choice of experimental design used.

Designs with more factors generally lead to smaller MSE.

Using

explanatory variables

or

covariates

(variables that are thought to affect the response; related to it).

Models with more variables generally lead to smaller MSE.

ExpDes-4

Review: Completely Randomized Design

(one-factor design)

• CRD in one-way AOV (

test

(pooled variances). • Experimental units are • Treatments are assigned to experimental units at

random

• Each treatment is replicated the same number of times

balanced design

• No accommodation made for (extraneous sources of variation).

• High probability that a large fraction of the experimental units set out at the beginning of the study may be .

appropriate time.

).

generalizes the two-sample t relatively homogeneous

• Experiment will use very

few replicates

.

lost or unavailable for measurement

.

disturbing variables

at the ExpDes-5

Completely Randomized Design

• Experimental Design - Completely randomized design (CRD) • Sampling Design - One-way classification design • • •

Assumptions:

Independent random samples (results of one sample do not effect other samples).

Samples from normal population(s).

Mean and variance for population i are respectively, m i and s 2 .

Model:

y ij

 m  

i

 

ij

E ( y ij )  m   i

AOV model

overall mean random error ~ N(0, s 2 ) effect due to population i

H

0 :

H a

:  1

At

  2

least

 

one

 

t of

 0

the



differ from

0 Requirement for m to be the overall mean:

i t

  1 

i

 0 ExpDes-6

Reference Group Model

Model:

y tj y ij

  m

t

m

t

 

tj

,

i

 

i

 

ij

,

i



t

 1 , 2 ,  ,

t

 1 random error ~ N(0, s 2 ) reference group mean effect due to population i Mean for the last group (i=t) is m t .

Mean for the first group (i=1) is m t Thus,  1 + b 1 is the difference between the mean of the reference group (cell) and the target group mean. Any group can be the reference group.

H

0

H a

:  1   2    : At least one of 

t

 1 the   0  0 This is the model SAS , SPSS and most other packages use.

ExpDes-7

CRD Practical Problems/Limitations

• • • In many situations, the researcher: Does not have sufficient

homogeneous

experimental material in one group (location, batch, etc.) to effectively use the CRD (i.e.

resource constraints

) The study objectives require examining treatments over a broad range of experimental units in order that results can be extended to more situations (i.e.

breadth of study objectives

).

The experimental material must be grouped for administrative or implementation purposes (i.e.

implementation constraints

).

If the researcher knows something about the characteristics of the experimental material, it is often possible to

group

experimental units into

sets of relatively homogenous material

(

blocks

), and then

compare treatment level means within these groups

.

ExpDes-8

Example

A scientist was interested in the use of three chemicals and water on their effectiveness in extracting sulfur from Florida soils. The chemicals of interest are: • Calcium Chloride • Ammonium Acetate • Mono Calcium Phosphate • Water CaCl NH Ca(H H 4 2 2 O 2 OAc PO 4 ) 3 Five soils were chosen for this experiment: • Troup Jackson Co. Paleudults soil • Lakeland • Leon • Chipley • Norfolk Walton Co. Duval Co. Quartzipsamments soil Haplaquads soil Jackson Co. Quartzipsamments soil Alachua Co. Paleudults soil ExpDes-9

Blocking and Control of Extraneous Variation

The main interest in the experiment is the comparison of the four extraction methods.

• The variation imposed on the extraction procedure by the five different soil types represents a source of

extraneous variation

. • Unless

controlled

for in the experiment, this variation has the potential to “swamp” or overwhelm the differences among the extraction procedure.

• High probability of concluding there are no treatment effects when treatment effects are in fact present.

Fair comparisons only occur among extractions within a soil type.

We wish to use the combined experience across soil types to make a stronger statement about the extraction procedures.

ExpDes-10

Graphical View

Note: The pattern of responses to treatments is consistent within a given soil type (a block); but responses vary across soil types.

ExpDes-11

Randomized Block Design (RBD)

Any experimental design in which the randomization of treatments is restricted to groups of experimental units within a predefined block of units assumed to be internally homogeneous is called a

randomized block design

. Blocks of units are created to control known sources of variation in expected (mean) response among experimental units.

There are two

classifications

treatment “effects”.

or

factors

in an RBD: block “effects” and • • Rules for blocking: Carefully examine the situation at hand and identify those factors which are known to affect the proposed response.

Choose one or two of these factors as the basis for creating blocks.

Blocking factors are sometimes referred to as

disturbing factors

.

ExpDes-12

Examples of Typical Blocking Factors

Disturbing Variable

Nutrient gradient Water moisture gradient Slope differences Soil composition Orientation to sun Flow of air Distribution of heat Age Local density Gender Age Socio-demographics

Experimental Unit

Field Plot Location in Greenhouse Tree Person ExpDes-13

Blocking Importance

How blocks are formed is critical to the effectiveness of the analysis

.

• With field plots, blocks are laid out so that they are perpendicular to the maximum direction of change in the • • disturbing factor to be controlled. Wide border (discard) areas are used to overcome interference between neighboring plots (i.e. to maintain independence of responses) within blocks and between blocks.

Time blocks may need discard times between “replications”.

This approach maximizes

within block homogeneity

while simultaneously maximizing

among block heterogeneity

.

ExpDes-14

Blocking Example

T2 T1 T5 T3 T4 T1 T2 T3 T5 T4 T1 T5 T2 T4 T3 T3 T4 T1 T2 T5

Treatment effects confounded with moisture effect!

ExpDes-15

Blocking Example

T1 T5 T3 T4 T3 T2 T5 T4 T3 T1 T2 T5 T4 T1 T5 T2 T2 T4 T1 T3 Block effect now removes moisture effect, fair comparisons among treatments.

ExpDes-16

Advantages and Disadvantages

• • •

Advantages of a Blocked Design

Controls a

single

extraneous source of variation and removes its effect from the estimate of experimental error.

Allows more

flexibility in experimental layout

.

Allows more

flexibility in experimental implementation

and administration.

• •

Disadvantages of a Blocked Design

Generally unsuited when there is a large number of treatments because of possible

loss of within block homogeneity

.

Serious problem with the analysis if a

block factor by treatment interaction effect

actually exists and no replication within blocks has been included. (solution: use replication within blocks when possible).

ExpDes-17

Complete or Incomplete Designs

Can all treatments be accommodated in each block?

Complete Block Design

: Every treatment occurs in each block.

Incomplete Block Design:

Not every treatment occurs in each block.

Complete Incomplete

A B C D A B C B D C D C A A B B D D C A A ExpDes-18

Balance in Designs

Balancing

refers to the specific assignment of treatments to experimental units such that

comparisons of treatment effects are done with equal precision

. This is usually accomplished by equally replicating each treatment.

Balanced Block Design:

The variance of the difference between two treatment means is the same regardless of which two treatments are compared. This usually implies that the overall replication (disregarding which blocks they are in) for the comparison of two treatments is the same for all pairs of treatments.

Partially Balanced Design

: The variance of the difference between two treatments depends on which two treatments are being considered. This usually implies different replication for different treatments.

Unbalanced Designs:

Unequal replication in each block - usually what one ends up with.

ExpDes-19