1、Lecture 3: Some basic statistical concepts, statistics and distributionsParameters and statisticsParametric versus non-parametric statisticsProperties of statisticsSome useful statisticsThe normal distributionThe Students t distributionConfidence intervals for sample statisticsStatistical power and

2、experimental design19991Bio 4118 Applied BiostatisticsParameters, statistics and estimatorsparameters characterize populations (which in general cannot be completely enumerated)statistics (estimators) are estimates of population parameters obtained from a finite sample (e.g., the sample mean is an e

3、stimate of the population mean)The process by which one obtains an estimate of a population parameter from a finite sample is called an estimation procedure.PopulationSample19992Bio 4118 Applied BiostatisticsParametric statistical analysisEstimating model parameters based on a finite sample and infe

4、rring from these estimates the values of the corresponding population parametersTherefore, parametric analysis requires relatively restrictive assumptions about the relationships between the sample and the population, i.e. about the distributions from which samples are drawn and the nature of the dr

5、awing (e.g., normal distributions and random sampling)XYSamplePopulationInferenceX19993Bio 4118 Applied BiostatisticsNon-parametric statistical analysisCalculation of model parameters based on a finite sample, but no inference to corresponding population parametersTherefore, non-parametric analysis

6、requires relatively minimal assumptions about the relationships between the sample and the population (e.g. normal distributions of sampled variables not required)19994Bio 4118 Applied BiostatisticsProperties of statisticsAccuracy: an accurate statistic is one for which its value, averaged over samp

7、les from the same population, is “close” to the true population parameter.Sample PopulationXXLess accuratestatisticMore accuratestatistic19995Bio 4118 Applied BiostatisticsProperties of statisticsConsistency: the more consistent a statistic is, the faster it approaches the true population value as s

8、ample size increases.SamplePopulationXLessconsistentMoreconsistentXSample size (N)19997Bio 4118 Applied BiostatisticsA comparison of some well-known statisticsFrequencyRange19998Bio 4118 Applied BiostatisticsStatistics of dispersion: the rangerange: defined by largest and smallest values in the samp

9、leIt is a simple statistic, but is biased because it consistently underestimates the population (parametric) range.FrequencyPopulation rangeSample range199910Bio 4118 Applied BiostatisticsDispersionThree frequency distributions with identical means and sample sizes but different dispersion patterns1

10、99911Bio 4118 Applied BiostatisticsDispersion statistics: variance, standard deviation and the coefficient of variationVariance: average squared deviation from the meanStandard deviation: square root of the varianceCoefficient of variation: standard deviation divided by the sample mean X 100199912Bi

11、o 4118 Applied BiostatisticsThe standard normal distributionobtained by scaling the distribution by converting observed values to standard normal deviates (Z-scores)resulting distribution has = 0, 2 = 1Probability-3-2-10123ZScaled (Z-transformed)Unscaled199914Bio 4118 Applied BiostatisticsThe standa

12、rd normal distribution68% of the population within 1 of the mean96% within 2 of the meanZProbability-3-2-10123 1 2199915Bio 4118 Applied BiostatisticsConfidence intervals for observations: estimation problemsReplacing and by their sample estimates can lead to serious biases.Simulation: sample standa

13、rd normal population and for each sample, calculate sample mean and variance. Then calculate CI based on sample mean and variance, and see what proportion of the true population fall within the CIs.Average 5%Proportion (%) of the populationoutide 95% CI N =10000100200300400500020406080100Mean = 5% N

14、umber of trials 199917Bio 4118 Applied BiostatisticsConfidence intervals for observations: estimation problemsWhen sample size is large, estimated CIs are very close to true CIs.However, when sample size is small, estimated CIs are far too small.050100150020406080100Proportion (%) of the populationo

15、utide 95% CI N = 5Mean = 23.8% Number of trials 199918Bio 4118 Applied BiostatisticsConfidence intervals for observations: estimation problemsEstimated CIs based on Z-scores approach true CIs as sample size increases, but, for small N, are highly biased (i.e. are smaller than they should be).Sample

16、sizeCIs calculated using Z1010010001000099%99.9%95%90%75%50%305070809095989999.899.9Proportion (%) of population199919Bio 4118 Applied BiostatisticsThe Students t-distributiondistribution of difference between sample mean and population mean divided by the standard error of the meanconverges towards

17、 standard normal distribution when N is largemore peaked and with longer tails at small N-5-4-3-2-10123450.00.10.20.30.4-5-4-3-2-1012345t0.00.10.20.30.4Probabilitydf=2df=1000199920Bio 4118 Applied BiostatisticsConfidence intervals based on t-scoresWhen sample size is small, calculate CIs by replacin

18、g Z with the critical value of the t distribution.This helps, but CIs are still too small when sample sizes are very small.CIs calculated using t10100100010000305070809095989999.899.999%99.9%95%90%75%50%Sample sizeProportion (%) of population199921Bio 4118 Applied BiostatisticsConfidence intervals f

19、or meansinterval that has a certain probability of including the value of the true mean of the populationsmaller than CI for observationsProbabilityorSample meansObservations199922Bio 4118 Applied BiostatisticsConfidence intervals for the medianbased on the binomial distribution b(x) with p = 0.5.Ou

20、t of a sample of n = 10, what is the probability of obtaining only x = 1, 2, n observations below the median?Because b(x) is discrete, confidence intervals wont be exactly at the 1- level.1- a CI: what range of values would we expect the true population median to lie 100(1-a) percent of the time?97.86% CI for the median given by values 1 and 9,89.08% CI for the median given by values 2 and 8012345678910Probability199924Bio 4118 Applied BiostatisticsConfidence interva