EXPERIMENTAL DESIGNS

AND HYPOTHESIS TESTING

PMA 4570/6228

Lab 5

July 13, 2017

Objectives

Design an experimental layout

Simple calculation and analysis of data

Interpret statistical results

Experimental Designs

Completely Randomized Design (CRD)

Randomized Block Design (RBD) Complete

Incomplete

Latin Square

Factorial

Split plot

Randomization to assign subjects to

treatment groups

Helps to prevent bias amongst treatment groups

Helps to balance the treatment groups with respect

to secondary (unknown) factors

Enables us to attribute any response differences to

the treatment rather than the secondary (unknown)

factors

Completely Randomized Design (CRD)

only 1 primary factor under consideration in

the experiment

test subjects are assigned to treatment levels of

the primary factor at random

assumes secondary factors will not produce a

systematic difference in response

E.g: plant height, petri dish location

CRD (Completely Randomized Design)

Diet A Diet D Diet B Diet D

Diet B Diet B Diet A Diet C

Diet D Diet A Diet C Diet B

Diet A Diet C Diet D Diet C

Example:

Five artificial diets are going to be compared for egg

production of H. axyridis (Coleoptera: Coccinellidae). The

females are randomly selected from the same colony and

randomly assigned to a diet treatment.

Treatment (4): Diet

Response: Egg production

Replicates: 4

Randomized Complete Block Design (RCBD)

AB

CD

BA

DC

AB

DC

DC

BA

Block 1 Block 2Block 3

Block 4

acknowledges potential effect of “secondary” factors, unlike in the CRD

contains blocks that represent the secondary factor, eg. shelves in your incubator, trays of plant, by slope, soil type

treatments in each block are exposed to similar conditions as much as possible

number of blocks represent number of replicates

Balanced Incomplete Block (BIB) Design

Block can only contain a subset of the treatments

Not all treatments are present in every block

BUT each possible pair of trt has to occur together within a block the same number of times to allow for comparison

Not a common design

A C E D

B E D C

E A B D

A B C E

B C D A

Treatment – 5

Replicate – 4

Comparison pairs – 3

Block 1

Block 2

Block 3

Block 4

Block 5

Factorial Design

combines effects of 2 or more factors to understand how

they affect a biological system individually and interactively

2-way factorial or 3-way factorial (more can get very

complicated!)

Treatment = combination of factors used in the experiment

Can be used in a CRD or RCBD

Factorial Design

Example:

Factor 1= Diets A, B, C

Factor 2 = Water in diet (1=10 ml, 2=15 ml)

Diet H2O TRTA 1 A1

A 2 A2

B 1 B1

B 2 B2

C 1 C1

C 2 C2

A 1 A1

A 2 A2

B 1 B1

B 2 B2

C 1 C1

C 2 C2

A1 C2 C1 B2 B1 A2

C1 B2 B1 A2 A1 C2

A2 B1 A1 B2 C2 C1

Rep 1

Rep 2

Rep 3

This is a RCBD experiment

Replicates = blocks

Latin Square

All treatments are replicated equally along the row and

along the column

Also known as row-column design

Could be used in a green house or a field experiment

col 1 col 2 col 3 col 4

row 1 A B C D

row 2 B C D A

row 3 C D A B

row 4 D A B C

Example:

The left light is out in the incubator.

The shelf level (row) AND the

location left to right on each shelf

(col) are sources of variation.

Split Plot

At least two factors have to be present

Whole plot (WP) – assigned to factor 1, generally based

on logistics of experiment (see example)

Subplots (SP) – factor 2, randomly assigned to each WP

Can be a CRD or RCBD

Also evaluates how factors affect your subjects

individually and interactively

Split Plot

Example:

Factor 1 (WP) = temperature – 10, 20°C

Factor 2 (SP) = diet – A, B, C, D

20oC 10oC

C D

B A

D B

A C

B

A

D

C

D

C

B

A

C D

B A

D B

A C

B

A

D

C

D

C

B

A

SP are organized in an RCBD experiment

4 Replicates = 4 blocks in each WP

**Temp is WP because

each incubator can be

set to one temperature

Hypothesis Testing

Purpose of an experiment: test a question/hypothesis

about the effectiveness of a new product/technique

Statistical analysis allow us to determine the

probability (P) that a hypothesis will be true for any

given sample

Null hypothesis (H0) – no difference

E.g. There are no differences in artificial diets for H. axyridis.

Alternative hypothesis (Ha) – there are differences

E.g. At least one of the artificial diets for H. axyridis is different.

(Flint and Gouveia 2001)

Testing hypotheses and levels of significance

p-value: probability that observed variation

among means could occur by chance

P > 0.05: not significant, therefore do not reject H0

P < 0.05: significant, therefore reject H0

Small p-value indicates strong evidence against H0

therefore reject H0

P < 0.0001 highly significant, therefore reject H0

Large p-value “fail to reject ” H0

P = 0.861 not significant, therefore do not reject H0

Testing Common Hypotheses

Comparing 2 treatment means (t-test)

H0: The two treatment means are equal

H1: The two treatment means differ

Comparing 3 or more treatment means (ANOVA)

H0: All of the treatment means are equal

H1: At least one treatment mean differs

T-test and ANOVA give you a p-value

Significant p-value (P ≤ 0.05) treatment differences

Means Separation Tests

0

1

2

3

4

5

6

7

a b c d

Treatment

Av

era

ge

Tukey’s test and LSD (Least Significant Difference)

are common

Only perform if ANOVA is significant (P ≤ 0.05)

Results look like this: Treatment a 5.2 A

Treatment d 4.8 A

Treatment c 4.1 AB

Treatment b 3.0 B

aaab

b

Treatments with similar letters are not sig. different

Simple Linear Regression

Explores and models the relationship between two

variables (x and y)

X = independent or predictor variable

Y = response or dependent variable

Changes in X cause changes in Y

Example: yield loss (y) and pest numbers (x)

Simple Linear Regression

Regression analysis measures the correlation (r) between X and Y

• Correlation coefficient: r

• Measures strength of the linear relationship between x and y

• Ranges between -1 and 1

• Where r ≥ 0.9 means highly positively correlated

• r = – 0.9 negatively correlated

• Coefficient of Determination: r2

• It is the square of correlation coefficient

• Ranges between 0 and 1

• Proportion of total variation in y attributable to variation in x

• Values of r2 >0.65 indicate significant correlation between the factors

r gives you the

direction of the

association

Simple Linear Regression

y = -8.9016x + 1011.2R² = 0.8957

0

100

200

300

400

500

600

700

800

900

1000

0 20 40 60

y = x

R2 = 1

0

5

10

15

20

0 5 10 15 20

# of TSSM

Yie

ld (kg

)

Yie

ld (kg

)

# of Neoseilus californicus

•Values of r2 >0.65 indicate significant correlation between the factors

Homework

Experimental Design and Hypothesis testing

handout

Worth 15 pts

DUE: Tues. July 18 before midnight by email