Download Fundamentals of experimentation

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

planned inquiry
 treatment

procedure whose effect will be measured
 factor

class of related treatments
 levels

states of a factor
 variable

measurable characteristic of a plot
 experimental unit (plot)

unit to which a treatment is applied
 replications

experimental units that receive the same
treatment
 sampling unit

part of experimental unit that is measured
 block

group of homogeneous experimental units
 experimental error

variation among experimental units that
are treated alike
Barley Yield Trial
Experiment
Hypothesis
Treatment
Factor
Levels
Variable
Experimental Unit
Replication
Block
Sampling Unit
Error
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
Decision
Accept H0
Reject H0
Reality

H0 is true
1 = 2
Correct
HA is true
1  2

1-
Type II error
Power
Type I error
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
n

SS Y   Yi  Y
i1

2
 Computational formula
– common in older textbooks
uncorrected
sum of squares
correction
factor


  Yi 
n
i1


2
SSY   Yi 
n
i1
n
2
Review of t tests
To test the hypothesis that the mean of a single
population is equal to some value:
Y  0
t
sY
Compare to critical t
for n-1 df for a given 
(0.05 in this graph)
2
s
where s Y 
n
df = 
df = n-1
df = 6
df = 3
Review of t tests
To compare the mean of two populations with equal
variances and equal sample sizes:
Y1  Y 2
t
s Y1  Y2
where
sY1  Y2 
2s2
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:
Y1  Y 2
t
sY1  Y2
where sY1  Y2
1 1
 s    df = (n1-1) + (n2-1)
 n1 n2 
2
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
s (standard deviation)
Y
s2
n
CV (coefficient of variation)
s2
* 100
Y
t ,df
se (standard error of a mean)
L (Confidence Interval for a mean)
2
s
n
sY1Y2
2
(standard error
of a difference
between means)
2s2
n
t ,df
Y  t  ,df
t ,df 2
s2
n
2
LSD (Least Significant Difference between means)
L(Confidence Interval for a difference between means)
t ,df
2s2
n
Y  Y   t
1
2
 ,df
2s2
n