Assignments

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Assignments

• Assignment 1 handed out October 27,
• due November 3 in class
• Assignment 2 handed out November 22,
• due November 29 in class

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Statistics

• The science of collecting, analyzing,
• presenting and interpreting data

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• Descriptive statistics
• are tabular, graphical and numerical summaries
  of data. The purpose of descriptive statistics is
  to facilitate the presentation and interpretation
  of data
• Inferential statistics
• Inference, in statistics, is the process of
  drawing conclusions about a particular parameter
  of a statistical distribution.

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Characteristics of a statistical problem

• Associated with the problem is a large group
  about which inferences are to be made. This group
  of objects is the population
• There is at least one random variable whose
  behavior is to studied relative to the population
• The population is too large to study in its
  entirety (or techniques used in the study are
  destructive in nature). Conclusions about the
  population must be based on observing only a
  portion or sample of objects drawn from the
  population.

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• State research question
• Formulate null and alternative hypotheses
• Identify population variable and when possible
  its distributions
• Sample data according to chosen sampling
  procedure
• Determine appropriate test statistic
• Calculate appropriate test statistic
• A) Determine critical values for sampling
  distribution and appropriate level of
  significance
• B) Determine P value of the test
  statistic
• Compare the test statistic to critical values.
• Reject or accept null hypothesis
• State conclusion and answer the question in step
  1

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Guidelines for hypothesis testing

• When testing a hypothesis concerning the value of
  some parameter, the statement of equality will
  always be included in H0. In this way H0
  pinpoints a specific numerical value that could
  be the actual value of the parameter.
• Whatever is to be detected or supported is the
  alternative hypothesis (H1).
• Since our research hypothesis is H1, it is hoped
  that the evidence leads us to reject H0 and
  thereby accept H1.

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Random sample

• Random sample of size n from the distribution of
• the random variable X is a collection of n
• Independent random variables, each with the same
• distribution as X
• Random sample is a sample of size n drawn from a
• population of size N in such a way that every
• possible sample of size n has the same
  probability
• of being drawn

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Random Sample?

• Question Do green and red birds of the same
  species occur in the same frequency?
• Sample red and green birds in a forest
• Question What is the size distribution of sugar
  maple in the same forest?
• Sample 100 individuals

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One sample hypothesis

• For example
• We have a population and we assume that it
• has a normal distribution
• We want to know if the population mean is
• smaller or larger than a specific value
• being selected

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Normal distribution
The location on the X-axis depends on the
population mean The shape of the distribution
depends on the population variance These are
the two parameters of the normal distribution
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• We estimate the population mean from which
• we have drawn our sample with the
• Sample mean
• We estimate the population variance from
• which we have drawn our sample with the
• Sample variance

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How good are these estimators?

• Unbiased estimator is centered around the
• right spot of what it is supposed estimate
• Biased estimator

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Unbiased estimator of population variance
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Importance of sample size

• Take many samples of size n from a population
• which is normally distributed then the mean of
• these samples is normally distributed with
  variance

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Standard error of sample mean (mean standard
error)
This estimated with
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Under the assumption that the stated null
hypothesis is true
follows a t-distribution with n-1 degrees of
freedom
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One sample hypothesis test

• Compute the t-ratio.
• Under the assumption that the null hypothesis is
  true
• What is the probability of obtaining this t
• ratio or a more extreme value of the t-ratio?
• If this probability is high- do not reject H0
• If this probability is low-reject H0

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What is considered a high versus a low
probability?

• YOU DECIDE!
• Conventionally, a probability that is 0.05 is
• considered sufficiently low for the null
• hypothesis to be rejected

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Level of significance

• Is under our control and is usually chosen to
• be 0.05, 0.01, or 0.001
• Rejecting an H0 at 0.05, the result is
  significant ()
• Rejecting an H0 at 0.01, the result is highly
  significant ()
• Rejecting an H0 at 0.001, the result is very
  highly significant ()

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• We reject the null hypothesis
• For a two-tailed test
• For a one-tailed test
• or depending on the null hypothesis

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Marine arthropods

• A species of marine arthropods live in seawater
  that
• contains calcium in a concentration of 32
  mmole/kg.
• Question Does members of this species maintain a
• coelomic fluid (extra cellular body fluid) that
  is less
• than that of their environment?

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Coelomic fluid

• assists respiration and circulation by diffusing
  nutrients, and excretion by accumulating wastes
• functions in place of several organ systems in
  higher animals such as mammals
• protects internal organs and also serves as a
  hydrostatic skeleton
• (Just in case you did not know.)

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• Hypothesis
• H0 The calcium concentration of the arthropod is
  the same or higher than the seawater
• H1 The calcium concentration of the arthropod is
  the lower than the seawater

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This is the same as

• H0
• H1

(Remember that the seawater has a concentration
of 32)
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• Thirteen animals are randomly sampled and the
• calcium concentrations in their coelomic fluid
• (extra cellular body fluid) is measured
• 28 27 29 29 30 30 31 30 33 27 30 32 31

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Marine arthropod example
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Calculate t-ratio for experiment
n-113-112 d.f.

• If we look in the t- table we find that
• What is the conclusion?

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• Because prediction in H0 and H1 are written so
  that
• they are mutually exclusive or all inclusive, we
  have
• a situation where one is true and the other is
  false
• 1. When H0 is true, then H1 is false
• -If we accept H0, we have done the right thing
• -If we reject H0, we have made a mistake
• This type of mistake is called Type I error

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Type I error

• Probability of rejecting a true null hypothesis
• Probability of making a type I error
• It is the same as the level of significance

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• 2. When H0 is false, then H1 is true
• -If we accept H0, we have made a mistake
• -If we reject H0, we have done the right thing
• This type of mistake is called Type II error

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Type II error

• Probability of not rejecting a false null
  hypothesis
• Probability of making a type II error

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Statistical power
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Statistical power

• Increases with increasing sample size
• Increases with effect size
• Increases with increasing !

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Fast rotation energy forest

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Basket willow example

• Is waste water influencing the harvest yield for
• a specific variety (clone) of basket willow?
• We choose to measure harvest yield in the
• form of plant height.
• Is this a good indicator of harvest yield?

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Assumptions

• Assume that height of untreated plants has
• a normal distribution with population mean
• Assume that height of treated plants has
• a normal distribution with population mean
• Equal variances

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We set up the hypothesis

• H0 There is no difference between and
• H1 There is a difference between and
• this is the same as
• H0
• H1
• which is the same as
• H0
• H1

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We obtain two random samples from each population
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We obtain two random samples from each population

• We estimate and with and ,
  respectively.
• And we estimate and with and
  respectively.

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Two sample hypothesis test

• Remember that for the one sample hypothesis we
  used
• Here our is
• So that where is
  the
• pooled variance (see my notes)

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has t distribution with degrees of
freedom under the null hypothesis. We reject the
null hypothesis if
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Keep track of the degrees of freedom

• The t-distribution is more spread out than the
  normal distribution. In fact the smaller the
  degrees of freedom the more spread out is the
  t-distribution.

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t-distribution
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Violations of the two-sample t test assumptions

• The two sample t-test assumes that the two
  populations are normally distributed and have
  equal variances!!!
• However, experience has shown that this test
  is rather robust (have high power) even when
  these assumptions are not met.

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Statistical power in two sample hypothesis
testing

• The power improves with increasing sample size
• Also, for a given number of data ( ),
  maximum power is obtained if the sample sizes are
  equal ( )
• If the sample variances are unequal the Type I
  error will tend to be greater than

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Assessing departures from normality

• Graphical assessment of normality
• Check for outliers
• Frequency curve should look normal
• Cumultative frequency curve should be S-shaped
• We will come back to this in a later lecture

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Testing for homoscedasticity (homogeneity among
variances)

• The variance ratio test can be used but
  remember that this test is severely and adversely
  affected by non-normal populations!
• However, understanding this test makes it a
  bit easier to understand the logic behind ANOVA
  (coming lectures)

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Energy forest example

• Question Does treated and untreated plants
• have the same variance for plant height?
• H0
• H1

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Variance ratio tests

• Take the larger of the two sample variances and
  divide it with the smaller e.g.
• if the two samples come from normal populations
  with equal variances this ratio is F distributed
  with and degrees of freedom
  

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The shape of the F-distribution depends on the
degrees of freedom
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Variance ratio test

• So reject the hypothesis of the null hypothesis if

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Manipulation of tuber size distribution in
Solanum tuberosum L

• Breeding goal reduce the size variability
• Let X1 be the tuber size in the year 1954
• Let X2 be the tuber size in the year 2004 (after
  50 generations of breeding)

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Potato example

• Has 50 generations of breeding led to a
• reduced variability in tuber size?
• H0
• H1

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For a specific potato variety in 1954 a random
sample of 30 potatoes had a sample variance of
1367A random sample of 30 potatoes of the
same variety the year 2004 (after 50 years of
extensive breeding)985
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• We use our numbers to calculate the F ratio
• Which is F-distributed with 29 and 29 degrees
• of freedom so that

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• So what do we do when we are unable to tell
• if the two samples originate from populations
• with normal distributions or if there is a
• significant difference between the sample
• variances?
• Well, the problem is that the t-ratio does not
• have t-distribution!

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Nonparametric tests

• These tests do not rely on the normal
  distribution and its parameters

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Important

• If you have a data set where either a
• or a nonparametric test can be applied, then
• go for the parametric. In these situations the
• parametric test is always more powerful than
• the nonparametric (the nonparametric tests
• tend to have a higher Type II error)

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Example from Pollinators entering female
dioecious figs why commit suicide? Patel et
al. 1995 (J Evol Biol)

• In the dioecious fig/pollinator mutualism,
• -female wasps that pollinate figs on female trees
  die without reproducing,
• whereas female wasps that pollinate figs on male
• trees produce offspring.
• Selection should strongly favor wasps
• that avoid female figs and enter only male figs.
• Consequently, fig trees would not be pollinated
  and fig seed
• production would ultimately cease, leading to
• extinction of both wasp and fig.

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• Question Do wasps prefer male figs over
• female figs?
• H0 Equal or larger number of wasps on female
  figs than on male figs
• H1 Fewer wasps on female figs than on male figs

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• In a controlled experiment pollinators in the
  wild (southern India) were presented with a
  choice between male and female figs of the
  species Ficus hispida. This was repeated 3 times
  on 3 different occasions. The data from the first
  experiment is presented on the next slide.

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Results





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Mann-Whitney U Test(Wilcoxon Rank-Sum Test)

• Assumptions
• The variables we are testing are continuous
  random variables
• The samples must be two independent random
  samples, however the samples sizes do not have to
  be equal

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Mann-Whitney U Test

• Pool all observations into one sample
• Observations are ranked from smallest to largest,
  irrespective of which populations each
  observation was sampled from
• Midranks are used for ties values.

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Mann-Whitney U Test

• The test statistic W1 is the sum of the ranks
• from the X1 population (female figs)
• If the sum is too small (too large) then
• this is an indication that the values of the X1
• population tend to be smaller (larger) than
• those of the X2 population (male figs)

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The fig example

• The number of wasps were counted on
• 10 female figs and 9 male figs so we have to
• rank these 19 observations

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Results





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• In our case we will use
• We compare this value to that our critical value

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Conclusion

• Do not reject H0. There is not significantly
  fewer wasps on female figs than on male figs.

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Allergy

• Swedish researchers (Bill Hasselmar et al.)
• claim that children that have pets at home are
• less likely to develop allergies than children
• which have no pets
• Conclusion Buy a furry pet for you child
• Can you see any problems here?