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1、Will G HopkinsAuckland University of TechnologyAuckland NZQuantitative Data AnalysisSummarizing Data:variables;simple statistics;effect statistics and statistical models;complex models.Generalizing from Sample to Population:precision of estimate,confidence limits,statistical significance,p value,err
2、ors.Reference:Hopkins WG(2002).Quantitative data analysis(Slideshow).Sportscience 6,sportsci.org/jour/0201/Quantitative_analysis.ppt(2046 words)Summarizing DataData are a bunch of values of one or more variables.A variable is something that has different values.Values can be numbers or names,dependi
3、ng on the variable:Numeric,e.g.weightCounting,e.g.number of injuriesOrdinal,petitive level(values are numbers/names)Nominal,e.g.sex(values are namesWhen values are numbers,visualize the distribution of all values in stem and leaf plots or in a frequency histogram.Can also use normal probability plot
4、s to visualize how well the values fit a normal distribution.When values are names,visualize the frequency of each value with a pie chart or a just a list of values and frequencies.A statistic is a number summarizing a bunch of values.Simple or univariate statistics summarize values of one variable.
5、Effect or outcome statistics summarize the relationship between values of two or more variables.Simple statistics for numeric variablesMean:the averageStandard deviation:the typical variationStandard error of the mean:the typical variation in the mean with repeated samplingMultiply by (sample size)t
6、o convert to standard deviation.Use these also for counting and ordinal variables.Use median(middle value or 50th percentile)and quartiles(25th and 75th percentiles)for grossly non-normally distributed data.Summarize these and other simple statistics visually with box and whisker plots.Simple statis
7、tics for nominal variablesFrequencies,proportions,or odds.Can also use these for ordinal variables.Effect statisticsDerived from statistical model(equation)of the form Y(dependent)vs X(predictor or independent).Depend on type of Y and X.Main ones:YXEffect statisticsModel/Testnumericnumericslope,inte
8、rcept,correlation regressionnumericnominalnominalnominalnominalnumericmean differencefrequency difference or ratiofrequency ratio per t test,ANOVA chi-squarecategoricalModel:numeric vs numerice.g.body fat vs sum of skinfoldsModel or test:linear regressionEffect statistics:slope and intercept=paramet
9、erscorrelation coefficient or variance explained(=100correlation2)=measures of goodness of fitOther statistics:typical or standard error of the estimate=residual error=best measure of validity(with criterion variable on the Y axis)sum skinfolds(mm)sum skinfolds(mm)sum skinfolds(mm)body fatbody fatbo
10、dy fat(%BM)(%BM)(%BM)Model:numeric vs nominale.g.strength vs sexModel or test:t test(2 groups)1-way ANOVA(2 groups)Effect statistics:difference between meansexpressed as raw difference,percent difference,or fraction of the root mean square error(Cohens effect-size statistic)variance explained or bet
11、ter(variance explained/100)=measures of goodness of fitOther statistics:root mean square error=average standard deviation of the two groupsfemalefemalemalemalestrengthstrengthsexsexMore on expressing the magnitude of the effectWhat often matters is the difference between means relative to the standa
12、rd deviation:strengthfemalesfemalesmalesmalesTrivial effect:strengthfemalesfemalesmalesmalesVery large effect:Fraction or multiple of a standard deviation is known as the effect-size statistic(or Cohens d).Cohen suggested thresholds for correlations and effect sizes.Hopkins agrees with the threshold
13、s for correlations but suggests others for the effect size:trivialtrivialsmallsmallmoderatemoderatelargelargevery largevery large!0.10.10.30.30.50.50.70.70 00.90.91 1Hopkins:Hopkins:CorrelationsCohen:Cohen:0.10.10.30.30.50.50 00.20.2Hopkins:Hopkins:0.60.61.21.22.02.00 04.04.0Effect Sizes0.20.2Cohen:
14、Cohen:0.50.50.80.80 0For studies of athletic performance,percent differences or changes in the mean are better than Cohen effect sizes.Model:numeric vs nominal (repeated measures)e.g.strength vs trialModel or test:paired t test(2 trials)repeated-measures ANOVA withone within-subject factor(2 trials)
15、Effect statistics:change in mean expressed as raw change,percent change,or fraction of the pre standard deviation Other statistics:within-subject standard deviation(not visible on above plot)=typical error:conveys error of measurementuseful to gauge reliability,individual responses,and magnitude of
16、effects(for measures of athletic performance).prepoststrengthstrengthtrialModel:nominal vs nominale.g.sport vs sexModel or test:chi-squared test or contingency tableEffect statistics:Relative frequencies,expressed as a difference in frequencies,ratio of frequencies(relative risk),or ratio of odds(od
17、ds ratio)Relative risk is appropriate for cross-sectional or prospective designs.risk of having rugby disease for males relative to females is(75/100)/(30/100)=2.5Odds ratio is appropriate for case-control designs.calculated as(75/25)/(30/70)=7.0femalesfemalesmalesmales30%30%75%75%rugby yesrugby yes
18、rugby norugby noModel:nominal vs numerice.g.heart disease vs ageModel or test:categorical modelingEffect statistics:relative risk or odds ratioper unit of the numeric variable(e.g.,2.3 per decade)Model:ordinal or counts vs whateverCan sometimes be analyzed as numeric variables using regression or t
19、testsOtherwise logistic regression or generalized linear modelingComplex modelsMost reducible to t tests,regression,or relative frequencies.Example age(y)age(y)age(y)heartheartheartdiseasediseasedisease(%)(%)(%)0 0 0100100100303030505050707070Model:controlled trial (numeric vs 2 nominals)e.g.strengt
20、h vs trial vs groupModel or test:unpaired t test of change scores(2 trials,2 groups)repeated-measures ANOVA withwithin-and between-subject factors(2 trials or groups)Note:use line diagram,not bar graph,for repeated measures.Effect statistics:difference in change in mean expressed as raw difference,p
21、ercent difference,or fraction of the pre standard deviation Other statistics:standard deviation representing individual responses(derived from within-subject standard deviations in the two groups)prepoststrengthstrengthtrialdrugdrugplaceboplaceboModel:extra predictor variable to control for somethin
22、ge.g.heart disease vs physical activity vs ageCant reduce to anything simpler.Model or test:multiple linear regression or analysis of covariance(ANCOVA)Equivalent to the effect of physical activity with everyone at the same age.Reduction in the effect of physical activity on disease when age is incl
23、uded implies age is at least partly the reason or mechanism for the effect.Same analysis gives the effect of age with everyone at same level of physical activity.Can use special analysis(mixed modeling)to include a mechanism variable in a repeated-measures model.See separate presentation at newstats
24、.org.Problem:some models dont fit uniformly for different subjects That is,between-or within-subject standard deviations differ between some subjects.Equivalently,the residuals are non-uniform(have different standard deviations for different subjects).Determine by examining standard deviations or pl
25、ots of residuals vs predicteds.Non-uniformity makes p values and confidence limits wrong.How to fixUse unpaired t test for groups with unequal variances,orTry taking log of dependent variable before analyzing,orFind some other transformation.As a last resort Use rank transformation:convert dependent
26、 variable to ranks before analyzing(=non-parametric analysissame as Wilcoxon,Kruskal-Wallis and other tests).Generalizing from a Sample to a PopulationYou study a sample to find out about the population.The value of a statistic for a sample is only an estimate of the true(population)value.Express pr
27、ecision or uncertainty in true value using 95%confidence limits.Confidence limits represent likely range of the true value.They do NOT represent a range of values in different subjects.Theres a 5%chance the true value is outside the 95%confidence interval:the Type 0 error rate.Interpret the observed
28、 value and the confidence limits as clinically or practically beneficial,trivial,or harmful.Even better,work out the probability that the effect is clinically or practically beneficial/trivial/harmful.See sportsci.org.Statistical significance is an old-fashioned way of generalizing,based on testing
29、whether the true value could be zero or null.Assume the null hypothesis:that the true value is zero(null).If your observed value falls in a region of extreme values that would occur only 5%of the time,you reject the null hypothesis.That is,you decide that the true value is unlikely to be zero;you ca
30、n state that the result is statistically significant at the 5%level.If the observed value does not fall in the 5%unlikely region,most people mistakenly accept the null hypothesis:they conclude that the true value is zero or null!The p value helps you decide whether your result falls in the unlikely
31、region.If p0.05,your result is in the unlikely region.One meaning of the p value:the probability of a more extreme observed value(positive or negative)when true value is zero.Better meaning of the p value:if you observe a positive effect,1-p/2 is the chance the true value is positive,and p/2 is the
32、chance the true value is negative.Ditto for a negative effect.Example:you observe a 1.5%enhancement of performance(p=0.08).Therefore there is a 96%chance that the true effect is any enhancement and a 4%chance that the true effect is any impairment.This interpretation does not take into account trivi
33、al enhancements and impairments.Therefore,if you must use p values,show exact values,not p0.05.Meta-analysts also need the exact p value(or confidence limits).If the true value is zero,theres a 5%chance of getting statistical significance:the Type I error rate,or rate of false positives or false ala
34、rms.Theres also a chance that the smallest worthwhile true value will produce an observed value that is not statistically significant:the Type II error rate,or rate of false negatives or failed alarms.In the old-fashioned approach to research design,you are supposed to have enough subjects to make a
35、 Type II error rate of 20%:that is,your study is supposed to have a power of 80%to detect the smallest worthwhile effect.If you look at lots of effects in a study,theres an increased chance being wrong about at least one of them.Old-fashioned statisticians like to control this inflation of the Type
36、I error rate within an ANOVA to make sure the increased chance is kept to 5%.This approach is misguided.The standard error of the mean(typical variation in the mean from sample to sample)can convey statistical significance.Non-overlap of the error bars of two groups implies a statistically significa
37、nt difference,but only for groups of equal size(e.g.males vs females).In particular,non-overlap does NOT convey statistical significance in experiments:what-everpostpreHigh reliabilityp=0.003postpreMean SEM in both casespostpreLow reliabilityp=0.2In summaryIf you must use statistical significance,sh
38、ow exact p values.Better still,show confidence limits instead.NEVER show the standard error of the mean!Show the usual between-subject standard deviation to convey the spread between subjects.In population studies,this standard deviation helps convey magnitude of differences or changes in the mean.In interventions,show also the within-subject standard deviation(the typical error)to convey precision of measurement.In athlete studies,this standard deviation helps convey magnitude of differences or changes in mean performance.