Advanced Statistics - uibk.ac.at

1 Advanced StatisticsAdvanced StatisticsJanette of StatisticsUniversity of InnsbruckAdvanced StatisticsContentsIntroductionBasics/Des criptive StatisticsScales of measurementGraphical exploration of dataDescriptive characteristics for a variableEstimationCharacteristics of an estimatorConfidence intervalStatistical hypothesis testingStatistical testing principleTesting errorsPower analysisWhy multivariate analysis? Advanced StatisticsIntroduction We are pattern-seeking story-telling animals. (Edward Leamer) Statistics does not hand truth to the user on asilver platter. However, Statistics confinesarbitrariness and provides comprehensibleconclusions. Es gibt keine Tatsachen, es gibt nurInterpretationen. (Friedrich Nietzsche) Advanced StatisticsIntroductionPreliminary will learn to apply statistical tools correctly,interpret the findings appropriately and get anidea about the possibilities of analyzingresearch questions employing is not possible and not worthwhile to learnall statistical methods in such a , this course is successful if it enablesyou to improve your knowledge in statisticalmethods on your own.

2 Therefore this coursegives you profound knowledge about somestatistical analyzing tools and shows you thecorrect application of StatisticsIntroductionPreliminary knowing the most sophisticatedanalyzing instruments one may be confrontedwith limits in getting results or findingappropriate interpretations or applying tools inthe given framework. This has to be accepted( If we torture the data long enough, they willconfess. ). aware: Never confuse statistical significancewith biological StatisticsBasics/Descriptive StatisticsScales of measurementScales of Scale. Nominal data are attributes likesex or species, and represent measurement atits weakest level. We can determine if oneobject is different from another, and the onlyformal property of nominal scale data Scale.

3 Some biological variablescannot be measured on a numerical scale, butindividuals can be ranked in relation to oneanother. Two formal properties occur inranking data:equivalenceandgreater StatisticsBasics/Descriptive StatisticsScales of measurementScales of and Ratio Scales. Interval and ratioscales have all the characteristics of the rankingscale, but we know the distances between theclasses. If we have a true zero point, we have aratio scale of StatisticsBasics/Descriptive StatisticsGraphical exploration of dataHistogram 4 3 2 101234050100150200250300 XNormal distributionfrequency (density)0246810121416182005010015020025 0300 YSkewed distributionfrequency (density) Advanced StatisticsBasics/Descriptive StatisticsGraphical exploration of dataBox PlotX 4 3 2 10123 Normal distributionfrequency (density)Y024681012141618 Skewed distributionfrequency (density) Advanced StatisticsBasics/Descriptive StatisticsGraphical exploration of dataQ-Q PlotIMany statistical methods make someassumptions about the distribution of the data( normality).

4 IThe quantile-quantile plot provides a way tovisually investigate such an QQ-plot shows the theoretical quantilesversus the empirical quantiles. If thedistribution assumed (theoretical one) is indeedthe correct one, we should observe a StatisticsBasics/Descriptive StatisticsGraphical exploration of dataQ-Q Plot 2 1012 2 1012 Normal Q Q PlotTheoretical QuantilesSample Quantiles 2 101201020304050 Normal Q Q PlotTheoretical QuantilesSample Quantiles 4 StatisticsBasics/Descriptive StatisticsDescriptive characteristics for a variableSummary StatisticIMean, medianIPercentiles, inter quartile rangeIMinimum, maximum, rangeIStandard deviation, varianceICoefficient of variationIMedian absolute deviation, mean absolutedeviationAdvanced StatisticsEstimationFundamental conceptsPopulations must be defined at the start of anystudy and this definition should include the spatialand temporal limits to the inference.

5 The formalstatistical inference is restricted to these of drawing samples parameters are considered to be fixedbut unknown values (in contrast to the Bayesianapproach). Advanced StatisticsEstimationCharacteristics of an estimatorCharacteristics of an estimatorA good estimator of a population parameter shouldhave the following characteristics:IThe estimator should beunbiased, meaningthat the expected value of the sample statistic(the mean of its probability distribution) shouldequal the should beconsistentso as the sample sizeincreases then the estimator will get closer tothe population should beefficient, meaning it has the lowestvariance among all competing StatisticsEstimationCharacteristics of an estimatorUnbiasedness of sample mean as estimatorfor the population mean12345678910 of samplemean of each samplen = 50 Advanced StatisticsEstimationCharacteristics of an estimatorConsistency of the sample mean asestimator for the population mean12345678910 505n = 1012345678910 505n = 10012345678910 505n = 10.

6 000 Advanced StatisticsEstimationCharacteristics of an estimatorEfficiency of the sample mean and of themedian as an estimator for the populationcentral tendencymeanmedian 5 4 3 2 101234estimatordistribution of the means1,000 samples with n = 100, variabe is normally distributed with population mean zero and standard deviation tenAdvanced StatisticsEstimationConfidence intervalConfidence interval for the populationmeanConsider a population ofNobservations of thevariableX. We take a random sample ofnobservations{x1,x2, ..,xn}from the versus sample mean ( x).IHaving an estimate of a parameter is only thefirst step in estimation. We also need to knowhow precise our estimate is:Standard error of the mean:se x= nIConfidence interval for the population mean:CI(1 ): [ x tdf=n 1,1 se x.]

7 X+tdf=n 1,1 se x] Advanced StatisticsEstimationConfidence interval95% confidence interval for the populationmean12345678910 10 50510n = 1012345678910 10 50510n = 10012345678910 = 10,000 Advanced StatisticsStatistical hypothesis testingStatistical testing principleStatistical tests and scientific hypothesesA statistical test is a confrontation of the real world(observations) to a theory (model) with the aim offalsifying the :H0: = 0 andHa: 6= 0 Real world: x,sAdvanced StatisticsStatistical hypothesis testingStatistical testing principleStatistical tests and scientific hypothesesAs such the statistical test (as a scientific method)fits directly into the philosophy of science describedby the English philosopher Karl Popper (1902 1994)(see The Logic of Scientific Discovery, 1972).

8 Basically the philosophy says that 1) theories cannot be empirically verified but onlyfalsifiedand 2)scientific progress happens by having a theory untilit is falsified. That is, if we observe a phenomenon(data) which under the model (theory) is veryunlikely, then we reject the model (theory). Advanced StatisticsStatistical hypothesis testingStatistical testing principleStatistical tests and scientific hypotheses No amount of experimentation can ever prove me right; asingle experiment can prove me wrong. (Albert Einstein)In other words, experiments can mainly be used forfalsifying a scientific hypothesis never for provingit! When we have a scientific theory, we conduct anexperiment in order to falsify it. Therefore, thestrong conclusion arising from an experiment iswhen a hypothesis is rejected.

9 Accepting (moreprecisely not rejecting) a hypothesis is not a verystrong conclusion (maybe acceptance is simply dueto that the experiment is too small). Advanced StatisticsStatistical hypothesis testingStatistical testing principleExampleSuppose we have a coin, and that our hypothesis isthat the coin is fair, that P(head) = P(tail) =1/2. Suppose we toss a coinn= 25 times andobserve 21 heads. The probability of actuallyobserving these data under the model is P(21 heads,4 tails) = It is a very unlikely (but possible)event to see such data if the model is true. In thisfalsification process we employ the interpretationprinciple of Statistics :Unlikely events do not StatisticsStatistical hypothesis testingStatistical testing principleStatistical tests and scientific hypothesesIf we do not employ this principle we can never sayanything at all on the basis of Statistics (observations): An opponent can always claim thatthe present observations just are an unfortunateoutcome which - no matter how unlikely they are -are StatisticsStatistical hypothesis testingStatistical testing principleStatistical tests and scientific hypothesesIn practice the statistical interpretation principleneeds more structure.

10 IIn a large sample space, all possible outcomeswill have a very small probability, so it will beunlikely to have the data one addition there is also the question abouthow small a probability is needed in order toclassify data as being ofp-value and significance level . Advanced StatisticsStatistical hypothesis testingTesting errorsTwo Types of ErrorsRecall that the following four outcomes are possiblewhen conducting a test:RealityOur DecisionH0 HaH0 Type I Error(Prob = 1 )Prob = HaType II Error Prob = (Prob = 1 )The significance level of any fixed level test is theprobability of a Type I StatisticsStatistical hypothesis testingTesting errorsAcceptable levels of errorsIType I error ( )ITypically = (This convention is due to )IFor more stringent tests = or = or preliminary experiments = II error ( )ITypically unspecified and much less than power=(1 )