Hypothesis Testing and Experimental design.pptx

Hypothesis Testing and Experimental
design
1

• Testing hypothesis involves
• designing a method for collecting information (data) that will allow us to
evaluate the hypothesis, and finally, to accept or reject it.
• Usually, a hypothesis can be formulated based on
• observations of natural phenomena during field (or laboratory) studies.
2
Hypothesis Testing

• These observations include the detection of distinctive patterns (or differences)
in nature.
• For example, you might notice that the number of plants differs on north and
south-facing slopes. You might also notice that each slope receives a different
amount of sunlight during the course of a day.
• It is very important to note that the observed pattern be real and worthy of study.
• Therefore, scientists are careful to measure or quantify their observations.
3
Hypothesis Testing

• In our example, you might count the number of trees along three 20-m paths
(transects) on each slope.
• You could also quantify the number of shrubs in three 1-m2 areas (quadrats) and
measure the amount of sunlight reaching the ground using a light meter.
• In order to tell if the two slopes differ with respect to tree, shrubs or light, you
must show that the differences you see are greater than would be expected by
chance.
• This is what we call the domain of statistical analysis and experimental design.
4
Hypothesis Testing

• The way in which we determine if the hypothesis about an observation is correct is
by performing a hypothesis test.
• We have two types of hypothesis: null and alternative hypothesis
• The null hypothesis (Ho) has the following form:
Ho: The difference between two or more data sets is no greater than that
expected by chance.
HA: The difference between two or more sets of data is so great that it is
unlikely to have occurred due to chance.
5
Hypothesis Testing

• No matter how great the difference between two data sets, there is still a
probability that the difference is due to chance.
• In other words, if there is a 98% probability that there is a difference, then there
is also a 2% probability that there is no difference.
• So, we cannot prove or disprove a hypothesis with 100% certainty but we can
make the decision, based on an overwhelming probability, to accept or reject a
hypothesis.
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Hypothesis Testing

• Experimental design means planning a set of procedures to investigate a
relationship between variables.
• In science, experiments may be designed to measure a pattern in the environment
or to apply a specific treatment to experimental units.
• Experiments designed to measure patterns are termed “mensurative- the
investigator does not apply the treatment”.
• Experiments involving treatments are termed “manipulative-the investigator
keeps all external factors constant and manipulates only the desired treatment
factor(s).
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Experimental Design

• Consider an experiment testing the influence of sunlight on the growth of sugar maple
(Acer saccharum) seedlings.
• An investigator establishes many 2 × 2 m plots in the forest and thins the canopy trees to
expose some plots to low sunlight and others to high sunlight.
• In some plots, the canopy trees are not thinned at all. These plots are termed “controls”
because no treatment was applied.
• Over time, the investigator measures the height of red pine seedlings in each plot.
• Each plot is an experimental unit; light intensity (or thinning) is the treatment; and change
in height over time (growth) is the response variable.
8
Experimental Design

• This experiment is manipulative because the investigator applied thinning to
specific plots.
• If the investigator had merely gone out to a forest and selected plots that had
varying densities of canopy trees (perhaps as a result of tree fall from heavy
winds), the experiment would be considered mensurative.
• Within any population, there exists a certain amount of difference from one
individual object to the next.
• In the previous example, sugar maple seedlings in a given plot would have a
variety of heights (or growth rates).
9
Experimental Design

• This variability can be the result of many factors (e.g., genetics, environmental
influences).
• The ability to distinguish between variability caused by treatment and variability
caused by other sources is a fundamental requirement for a successful study.
• The sources of variability present in any study, other than the treatment, can make
results difficult or impossible to interpret.
• The purpose of experimental design, therefore, is to minimize, or hold constant, all
other sources of variability, except for those that we wish to monitor.
• The features of experimental design that address the problem of variability are sample
size, replication, control, randomization, and interspersion.
10
Experimental Design

 SAMPLE SIZE
• A collection of measurements taken from an experimental unit is termed a sample.
This sample represents a subset of the population being studied.
• In order to estimate the variability within the treatment, it is necessary to have
samples from many experimental units.
• The total number of experimental units sampled is termed the sample size.
• The larger the sample size, the better one is able to account for variability within a
population.
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Experimental Design

 REPLICATION
• To detect differences between treatments, it is necessary to assess the variability
within each treatment.
• To do this you should have multiple experimental units for each treatment (a
replicate).
• A minimum of three replicates is necessary for each experimental treatment.
• In our previous example on the growth response of sugar maple seedlings to
sunlight, the experimental units to be replicated would be the plots of seedlings
exposed to each level of sunlight.
• Replication is the only way to avoid chance events or noise (intrusions).
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Experimental Design

 PSEUDOREPLICATION
• Pseudoreplication describe a statistical error of
using treatment replicates which are in fact
pseudoreplicates (Hurlbert, 1984) - i.e.,
replicates are not independent.
• Three types of pseudoreplication can be
recognized:
1) Simple pseudoreplication,
2) Sacrificial pseudoreplication, and
3) Temporal pseudoreplication
13
Experimental Design
 Yellow and white boxes represent experimental units receiving two
different treatments. Each dot represents sample measurement.

• (I) The simplest and most common type of pseudo-
replication occurs when there is only
one replicate per treatment (see fig.).
• For example, there may be one large burned area and one unburned area.
• If several 1 m² plots are measured within each area, these 1 m² plots are not replicates (they are
subsamples) and they should not be used in a t-test to compare burned vs. unburned areas in
general.
• Because we would not know if we were measuring an area effect or a burn effect.
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Experimental Design
 PSEUDOREPLICATION

PSEUDOREPLICATION
• (II) Sacrificial pseudoreplication occurs when there is a proper, replicated experimental
design but the data for the replicates are pooled together prior to measurement or prior to
statistical analysis.
• This is a simple statistical error and should be a recoverable problem in data analysis
unless the samples were physically pooled before measurement.
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Experimental Design

PSEUDOREPLICATION
• (III) Temporal pseudoreplication is also common in ecological experiments in which a
time series of data are accumulated.
• For example, with burned and unburned plots one might return to sample quadrats each
week/month for four consecutive rounds after the fire. Successive samples over time from
a single experimental unit are clearly not independent samples.
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Experimental Design

 CONTROL
• The control serves as a standard against which the experimental units are
compared. Controls are those treatments receiving no treatment factor.
• In our sugar maple seedling example the amount of sunlight is the treatment, and
therefore we would have one plot that receives no sunlight as a control.
• All other manipulations (i.e., watering, fertilization, etc.) should remain constant
for all experimental units.
• Experimental units assigned to be controls are replicated just as experimental units
receiving a treatment.
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Experimental Design

 INTERSPERSION AND RANDOMIZATION
• The assignment of treatments and controls to experimental units must be such that there is
no interaction between experimental units and therefore, all units are likely, on average, to
receive the same environmental conditions.
• This placement is referred to as interspersion. The way in which interspersion is achieved
is through randomization.
• Ideally, the assignment of treatments and controls is done in a purely random fashion
which allows chance to determine their placement and which experimental units will
receive which treatment.
• For large numbers of replicates this works well since it is not probable that similar
treatments would all end up clumped together.
18
Experimental Design

• For smaller numbers of replicates a
systematic alternating of treatments and
controls is often necessary.
• While desirable, interspersion is often
not possible in mensurative
experiments.
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Experimental Design
 INTERSPERSION AND RANDOMIZATION

 DISCUSSION QUESTIONS:
1. If one were planning to conduct a study investigating the effects of the salting of
highways on stream “health”, what might you measure? What would be your
experimental unit?
2. Write a null and alternative hypothesis for the effect of highway salt on stream health.
3. Identify the pseudoreplication in the situation given below:
In studying the effects of sewage on algal growth in rivers an investigator sets up a
site above and below a sewage treatment plant on a river. Several samples are taken
upstream of the plant and several samples are taken downstream.
4. What are some sources of variability in the experiment on the effects of sunlight on
the growth of sugar maple seedlings example given in the text?
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Experimental Design

 DISCUSSION QUESTIONS:
5. You used a random method of interspersing your experimental units and came
up with the following assignments: AAABBCCCB (A, White, B, Grey, C, Black)
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Experimental Design
 Can you foresee any problem with
this experimental design?
 How might you solve this
problem?

 Completely Randomized Design (CRD)
• This is the simplest and flexible design recommended by many statistical tests.
• Treatments are assigned completely at random (i.e., each experimental units has the same
chances of receiving any treatment.
• Any difference among experimental units receiving the same treatments are considered
experimental error.
• This design is possible when experimental units are very similar (homogeneous)
• Therefore, this design is acceptable for laboratory studies where experimental units are
carefully controlled.
22
Experimental Design
 Types of Experimental Design

 Completely Randomized Design
• In field studies, strict randomization can result in treatments being spatially segregated by
chance, especially if only a few treatment replicates are possible (Hurlbert, 1984).
• Spatial segregation will produce false treatment effects when there are preexisting gradients in
the study area.
• For this reason Hurlbert (1984) recommends against this statistical design in ecological studies
when treatment replicates are few, even though technically this is a perfectly acceptable
statistical design to all professional statisticians.
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Experimental Design

 Completely Randomized Design
24
Experimental Design
Fig. 6 Completely randomised design,
with three treatment levels (white, grey,
black) and four replicates (or
replications) per treatment.

 Completely Randomized Block Design (CRBD)
• There are several ways to control for environmental heterogeneity. Probably the most
popular one in ecology is the Randomised Complete Block Design.
• This is an excellent design for most field experiments because it automatically produces
an interspersion of treatments and thus reduces the effect of chance events on the
results of the experiment.
• Advantage of RCBD is that whole blocks may be lost without compromising the
experiment. E.g., if a bulldozer destroys one set of plots, all is not lost.
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Experimental Design

 Randomized Complete Block Design
• Blocks can be constructed around any known or suspected source of variation.
• Example, a grassland on a SE facing slope could be a block. In this case the
environmental conditions (soil, temperature, rainfall) would be more similar within this
block of grassland than between this site and a nearby SW-facing plot.
• Plots of habitat are the most obvious type of blocks in field ecology but a block could
also be a room in a greenhouse, a lab of aquaria tanks, a day of the week, a group of
animals of similar weight, the measurements taken on instrument X, or the data
collected by technician Y.
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Experimental Design

• A primary distinguishing feature of RCBD from CRD – the presence of blocks of equal
size, each block contains all treatments.
• By putting experimental units that are as similar as possible together in the same group
(generally referred to as a block) and by assigning all treatments into each block
separately and independently, variation among blocks can be measured and removed
from experimental error.
• This process is called blocking. Blocks are relatively uniform internally (as
homogenous as possible).
27
Experimental Design

28
Experimental Design
Fig. 7 The randomised complete block design for three treatment levels.

 Randomized Complete Block
Design
• 3 blocks – eliminate the river gradient
• 8 treatments
• Each block has all treatments
• Within each block – complete
randomization of treatments
29
Experimental Design
A E F
H D D
B F B
G C E
E H H
D A C
C B G
F G A
Blocks
Treatments
Soil moisture

 Factorial Designs
• Deals with several factors of interest at a time.
• For example, the rates of egg production may be measured at three levels of salinity
and two levels of temperatures.
• Thus, the investigator should do all 3 salinities at each of the two temperatures.
• The concept of factorials is the notion that all treatments of one factor should be tried
with all treatments of the other factors.
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Experimental Design

• In an ideal world all factors operate independently.
• For example, salinity will raise or lower egg production, and temperature will
independently change it as well.
• In the real world, factors enhance or
interfere with one another and thus are not
independent in their effects.
• Statisticians say factors interact.
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Experimental Design

• The simplest way to look at and understand
interactions is graphically.
• Fig. 8 illustrates a hypothetical set of data for
this experimental design with and without
interactions.
• When there are no interactions in the data, a
graph will show only sets of parallel lines, as in
Fig. 8(a).
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Experimental Design
Fig. 8 Salinity and temperature effect on egg count

• In this example, level 2 of treatment A (high temperature,
A2) always has a higher mean egg productions than level
1 (low temperature, A1), regardless of what the salinity
is.
• When interaction is present, the lines diverge (Fig. 8(b))
or cross (Fig. 8(c)). In this example, high temperature A2
stimulates more egg laying in the first two low salinities
but lowers egg production in the second two high
salinities relative to low temperature treatment A1.
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Experimental Design

• Another way to understand what
interactions are is to ask a simple question
about each of the factors in the experiment.
• For example, we can ask:
 What effect does temperature have on
number of eggs produced?
 What effect does salinity have on
number of eggs produced? 34
Experimental Design

• When there is no interaction the answer
to these questions is straightforward -
temperature 2 has twice the number of
eggs of temperature 1, or high salinity
increases egg production to 3 times that
of low salinity, for example.
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Experimental Design

• But when there is interaction, these
questions have no simple answer and you
must reply "It all depends ....." and give a
more detailed answer, like:
• For low temperature A1, salinity has no
effect on egg production, but for high
temperature A2 there is a strong effect of
high salinity on egg production.
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Experimental Design

• Interactions produce statistical
headaches but is interesting in ecology.
• There is always a conflict in factorial
designs between the desire for
simplicity with no interactions or for
complexity with interactions that require
ecological understanding.
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Experimental Design

• There is a priority of testing in factorial designs.
• You should first ask whether the interactions are
significant statistically.
• If they are, you should stop there and find out what
is happening.
• It is misleading to present and analyze tests of
significance for the main effects in a factorial when
the interaction term is significant.
• The important thing is to explain the interaction.
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Experimental Design

 Split-Plot/Unit Designs
• When the treatment structure of an experiment involves two or more factors, the best
treatment design is a factorial one.
• The key usage of split-plot designs occurs when two (or more) treatment factors are
being studied and the size of the experimental unit which is appropriate to the first
factor is much larger than that required for the second factor.
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Experimental Design

• Fig. 9 gives a simple experiment from growth
chambers for plants. The two treatment factors
in this simple experiment are CO2 level which
is held in large greenhouses at two levels
(ambient or enhanced CO2) and soil
temperature which is heated to 25°C or 30°C.
• The key element here is that the
experimental units for the two treatments
differ in size.
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Experimental Design

41
Experimental Design
• Greenhouses are held at either ambient or
enhanced CO2.
• Within each greenhouse smaller units
have heated soils.
• This is a simple two-factor design to
investigate CO2 treatment effects and soil
temperature effects.

 Latin Square Designs
• Recall that randomized block designs are useful when there is one source of variation
known before the experiments are carried out.
• In a few cases, scientists can recognize two sources of variation in the experimental units
and wish to correct for both sources in doing an experiment.
• One way to achieve this is to use the latin square design.
• A latin square design is a simple extension of the randomized complete block design
involving blocking in two directions.
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Experimental Design

• For example, assume a field has river on one side
and a road on the other side
• Can have a maximum of 16 plots
• A, B, C, and D are types of treatments
• Each treatment appears only once in row (river
gradient, A) and column ( road gradient, B)
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Experimental Design

• The latin square design is not commonly found in field ecological research but when its
restrictive assumptions can be met it is a very efficient experimental design to apply
particularly in laboratory or common garden experiments.
• Example, if the field in which the trial is conducted has fertility gradient in one direction
and a slope in another; or, if you had a plot of land the fertility of which change in both
directions (North – South and East – West) due to soil or moisture gradients.
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Experimental Design

• By sampling both the control and the
impact sites before and after the
nutrient additions, both temporal and
spatial controls are utilized.
• Green (1979) calls this the BACI
design (Before-After, Control-Impact)
and suggests that it is an optimal
impact design.
45
Experimental Design
 Types of Experimental Design  Before-After-Control-Impact (BACI)

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