The Scientific Method

Formulation of an H ypothesis

P lanning an experiment to objectively test the hypothesis

Careful observation and collection of D ata from the experiment

I nterpretation of the experimental results

Steps in Experimentation

H Definition of the problem

Statement of objectives

P Selection of treatments

Selection of experimental material

Selection of experimental design

Selection of the unit for observation and the number of replications

Control of the effects of the adjacent units on each other

Consideration of data to be collected

Outlining statistical analysis and summarization of results

D Conducting the experiment

I Analyzing data and interpreting results

Preparation of a complete, readable, and correct report

The Well-Planned Experiment Simplicity

– don’t attempt to do too much– write out the objectives, listed in order of priority

Degree of precision– appropriate design– sufficient replication

Absence of systematic error

Range of validity of conclusions– well-defined reference population– repeat the experiment in time and space– a factorial set of treatments also increases the range

Calculation of degree of uncertainty

Types of variables Continuous

– can take on any value within a range (height, yield, etc.)– measurements are approximate– often normally distributed

Discrete– only certain values are possible (e.g., counts, scores)– not normally distributed, but means may be

Categorical– qualitative; no natural order– often called classification variables– generally interested in frequencies of individuals in each class– binomial and multinomial distributions are common

Terminology

experiment treatment factor levels variable experimental unit (plot) replications

sampling unit block experimental error

planned inquiry

procedure whose effect will be measured

class of related treatments

states of a factor

measurable characteristic of a plot

unit to which a treatment is applied

experimental units that receive the same

treatment

part of experimental unit that is measured

group of homogeneous experimental units

variation among experimental units that

are treated alike

Barley Yield Trial

ExperimentHypothesisTreatmentFactorLevelsVariableExperimental UnitReplicationBlockSampling UnitError

Hypothesis Testing

H0: = ɵ HA: ɵ or H0: 1= 2 HA: 1 2

If the observed (i.e., calculated) test statistic is greater than the critical value, reject H0

If the observed test statistic is less than the critical value, fail to reject H0

The concept of a rejection region (e.g. = 0.05) is not favored by some statisticians

It may be more informative to:– Report the p-value for the observed test statistic– Report confidence intervals for treatment means

Hypothesis testing It is necessary to define a rejection region to determine

the power of a test

Correct

Type I error

Type II error

1 -

Power

Decision

Accept H0 Reject H0

Reality

H0 is true1 = 2

HA is true1 2

Power of the test

Power is greater when– differences among treaments are large– alpha is large– standard errors are small

Review - Corrected Sum of Squares

Definition formula

Computational formula– common in older textbooks

n 2

Y ii 1

SS Y Y

2n

ini 12

Y ii 1

Y

SS Yn

correctionfactor

uncorrectedsum of squares

Review of t tests

To test the hypothesis that the mean of a single population is equal to some value:

0

Y

Yt

s

2

Y

ss

nwhere df = n-1

df = 6

df = 3df = Compare to critical t

for n-1 df for a given (0.05 in this graph)

Review of t tests

To compare the mean of two populations with equal variances and equal sample sizes:

where

1 2

1 2

Y Y

Y Yt

s

1 2

2

Y Y

2ss

n df = 2(n-1)

The pooled s2 should be a weighted average of the two samples

Review of t tests

To compare the mean of two populations with equal variances and unequal sample sizes:

where

1 2

1 2Y Y

Y Yt

s

1 2

2Y Y

1 2

1 1s s

n n

df = (n1-1) + (n2-1)

The pooled s2 should be a weighted average of the two samples

Review of t tests When observations are paired, it may be beneficial

to use a paired t test– for example, feeding rations given to animals from the

same litter

t2 = F in a Completely Randomized Design (CRD) when there are only two treatment levels

Paired t2 = F in a RBD (Randomized Complete Block Design) with two treatment levels

Measures of Variation

2s

n

2s100

Y*

2

df

sY t

n ,

s (standard deviation) CV (coefficient of variation)

se (standard error of a mean)

LSD (Least Significant Difference between means)

L(Confidence Interval for a difference between means)

L (Confidence Interval for a mean)

2s

2

df

2st

n,

n

Y

dft,

dft 2, 2

2

1 2 df

2sY Y t

n ,

(standard error of a difference

between means)

Y Ys 1 2

22s

n

2

dft,