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.
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.
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
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
9 timesgreen, yellowseed
3 timesgreen, greenseed
3 timesyellow, yellowseed
1 time yellow, greenseed
The Chi2 test has K-1 degrees of freedom.
All statisticalprogramsgivetheprobability of thenullhypothesis, H0.
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
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.
Thepatternisbetterseenin a double log plot.
We havesevenpointsabove and 3 pointsbelowtheregressionline.
Isthere a systematicerror?
Tests for systematicerrors.
Now we trythebest fit model
theG-testidentifiedeventhebest fit model as havinglargerdeviationsthanexpectedfrom a simplenormal random sample model.
Thebest fit model
Observation and expectationcan be compared by a Kolmogorov-Smirnov test.
The test comparesthemaximumcumulativedeviationwiththatexpectedfrom a normaldistribution.
Bothresultsarequalitativelyidentical but differquantitatively.
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
Abundanceisthetotalnumber of individualsin a population.
Densityrefers to thenumber of individualsin a unit of measurement.
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.
Do vertebrates and invertebratesdifferinabundancedistributions?
Number of log-normalbestfitsonly:
But if we takethewholepattern we get
Bivariatecomparisons of means
Student’st-testfor equalsamplesizes and similarvariances
Welch t-testfor unequalvariances and samplesizes
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
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
Past givestheprobability for a twosided test that one varianceiseitherlargerorsmallerthanthesecond.
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.
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:
Łomnicki: Statystyka dla biologów