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Lecture 11 Parametric hypothesis testing. The logic behind a statistical test. A statistical test is the comparison of the probabilities in favour of a hypothesis H 1 with the respective probabilities of an appropriate null hypothesis H 0 . Type I error. Type II error.

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The logic behind a statistical test



Thelogicbehind a statistical test.

A statistical test isthecomparison of theprobabilitiesinfavour of a hypothesis H1with therespectiveprobabilities of an appropriatenullhypothesis H0.

Type I error

Type II error

Power of a test

Acceptingthewronghypothesis H1istermedtype I error.

Rejectingthe correct hypothesis H1istermedttype II error.

The logic behind a statistical test

Testing simple hypotheses

Karl Pearson threw 24000 times a coin and wanted to see whether in the real world deviations from the expectation of 12000 numbers and 12000 eagles occur. He got 12012 time the numbers.

Does this result deviate from our expectation?

Theexactsolution of thebinomial


The logic behind a statistical test

c2 test

Assume a sum of variances of Z-transformedvariables

Eachvarianceis one. Thustheexpectedvalue of c2is n

Thec2distributionis a group of distributions of variancesindependence on thenumber of elements n.

Observedvalues of c2can be compared to predicted and allow for statisticalhypthesistesting.


Probability of H0

The logic behind a statistical test


9 timesgreen, yellowseed

3 timesgreen, greenseed

3 timesyellow, yellowseed

1 time yellow, greenseed

The Chi2 test has K-1 degrees of freedom.

The logic behind a statistical test

All statisticalprogramsgivetheprobability of thenullhypothesis, H0.

The logic behind a statistical test

  • Advices for applying a χ2-test

  • χ2-tests compare observations and expectations. Total numbers of observations and expectationsmust be equal.

  • The absolute values should not be too small (as a rule the smallest expected value should be larger than 10). At small event numbers the Yates correction should be used.

  • The classification of events must be unequivocal.

  • χ2-tests were found to be quite robust. That means they are conservative and rather favour H0, the hypothesis of no deviation.

  • The applicability of the χ2-test does not depend on the underlying distributions. They need not to be normally of binomial distributed.


The logic behind a statistical test

G-test or log likelihood test

c2relies on absolute differences between observed and expected frequencies. However, it is also possible to take the quotient L = observed / expected as a measure of goodness of fit

G is approximately c2 distributed with k - 1 degrees of freedom

The logic behind a statistical test

A species - area relation is expected to follow a power function of the form S = 10A0..3.

Do the following data points (Area, species number) confirm this expectations:

A1 (1,12), A2 (2,18), A3 (4,14), A4 (8,30), A5 (16,35), A6 (32,38), A7 (64,33), A8 (128,35), A9 (256,56), A10 (512,70)?

We try different tests.

Bothtestsindicatethattheregressionlinedoesn’t fit

Thepatternisbetterseenin a double log plot.

We havesevenpointsabove and 3 pointsbelowtheregressionline.

Isthere a systematicerror?

The logic behind a statistical test

Tests for systematicerrors.


Thec2 test

The logic behind a statistical test

Now we trythebest fit model

theG-testidentifiedeventhebest fit model as havinglargerdeviationsthanexpectedfrom a simplenormal random sample model.

The logic behind a statistical test

Thebest fit model

Kolmogorov-Smirnov test

Observation and expectationcan be compared by a Kolmogorov-Smirnov test.

The test comparesthemaximumcumulativedeviationwiththatexpectedfrom a normaldistribution.

Bothresultsarequalitativelyidentical but differquantitatively.


The logic behind a statistical test

1000 Drosophilaflieswithnormal and curledwings and twoalleles A and B suposed to influence wing form.

Predictednumber of allele A and curledwings

2x2 contingency table

Do flieswith allele havemoreoftencurledwingsthanfielswith allele B?

A contingencytable chi2 test with n rows and m columnshas (n-1) * (m-1) degrees of freedom.

The 2x2 tablehas 1 degree of freedom

The logic behind a statistical test


Abundanceisthetotalnumber of individualsin a population.

Densityrefers to thenumber of individualsin a unit of measurement.


Dominant species





The logic behind a statistical test

Thedistribution of speciesabundancedistributionsacrossvertebrates and invertebrates

Row and columnsumsareidenticaldue to ourclassification. We expectequalentries for eachcell:

3 types of distributions: log-series, powerfunction, lognormal.

We compare 99 suchdistributionsfromallovertheworld.

The logic behind a statistical test

Do vertebrates and invertebratesdifferinabundancedistributions?

Number of log-normalbestfitsonly:

But if we takethewholepattern we get

The logic behind a statistical test

Bivariatecomparisons of means

Student’st-testfor equalsamplesizes and similarvariances

Welch t-testfor unequalvariances and samplesizes


The logic behind a statistical test

In a physiological experiment mean metabolism rates had been measured.

A first treatment gave mean = 100, variance = 45,

a second treatment mean = 120, variance = 55.

In the first case 30 animals in the second case 50 animals had been tested.

Do means and variances differ?


Degrees of freedom

Theprobabilitylevel for thenullhypothesis

The logic behind a statistical test

Thecomparison of variances

Theprobability for thenullhypothesis of no difference, H0.

1-0.287=0.713: probabilitythatthe first variance (50) islargerthanthesecond (30).

One sided test

Degrees of freedom: N-1

0.57  2*0.287

Past givestheprobability for a twosided test that one varianceiseitherlargerorsmallerthanthesecond.

Twosided test

The logic behind a statistical test

Power analysis


In an experiment you estimated two means

Each time youtook 20 replicates.

Was this sample size large enough to confirm differences between both means?

We usethet-distributionwith 19 degrees of freedom.

Youneeded 15 replicates to confirm a differenceatthe 5% errorlevel.

The logic behind a statistical test

From a physiological experiment we want to test whether a certain medicament enhances short time memory.

How many persons should you test (with and without the treatment) to confirm a difference in memory of about 5%?

We don’tknowthevariances and assume a Poisson random sample.


s2 = m

Thet-testcan be used to estimatethenumber of observations to detect a significantsignal for a giveneffectsize.

We don’tknowthedegrees of freedom:

We use a largenumber and get t:

Home work and literature

Home work and literature

  • Refresh:

  • c2 test

  • Mendel rules

  • t-test

  • F-test

  • Contingencytable

  • G-test

  • Prepare to thenextlecture:

  • Coefficient of correlation

  • Maximum, minimum of functions

  • Matrixmultiplication

  • Eigenvalue


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