1STAT 13, UCLA, Ivo Dinov Slide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health ScienceszInstructor: Ivo Dinov, Asst. Prof. In Statistics and NeurologyzTeaching Assistants: Janine Miller and Ming ZhengUCLA StatisticsUniversity of California, Los Angeles, Winter 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo Dinov Slide 2Chapter 10: Data on a Continuous VariablezOne-sample issueszTwo independent sampleszMore than 2 sampleszBlocking, stratification and relatedsamplesSTAT 13, UCLA, Ivo DinovSlide 38Comparing two means for independent samplesSuppose we have 2 samples/means/distributions as follows: { } and { }. We’ve seen before that to make inference about we can use a T-test for H0: with And CI( ) =If the 2 samples are independent we use the SE formulawith .This gives a conservative approach for hand calculation of an approximation to the what is known as the Welch procedure, which has a complicated exact formula.)1,1(,1σσσσµµµµNx)2,2(,2σσσσµµµµNx21µµµµµµµµ−−−−021====−−−−µµµµµµµµ21µµµµµµµµ−−−−)(0)(21210xxSExxt−−−−−−−−−−−−====)( 2121xxSEtxx−−−−××××±±±±−−−−2/221/21nsnsSE ++++====)12;11( −−−−−−−−==== nnMindfSTAT 13, UCLA, Ivo DinovSlide 39Means for independent samples –equal or unequal variances?Pooled T-test is used for samples with assumed equal variances. Under data Normal assumptions and equal variances of is exactlyStudent’s t distributed withHere spis called the pooled estimate of the variance, since it pools info from the 2 samples to form a combined estimate of the single variance σ12= σ22=σ2. The book recommends routine use of the Welch unequal variance method.(((()))) (((())))22122)12(21)11(2;2/11/1 where,/0 2121−−−−++++−−−−++++−−−−====++++====−−−−−−−−−−−−nnsnsnpsnnsSExxSExxp)221( −−−−++++==== nndfSTAT 13, UCLA, Ivo DinovSlide 40Comparing two means for independent samples1. How sensitive is the two-sample t-test to non-Normality in the data? (The 2-sample T-tests and CI’s are even more robust than the 1-sample tests, against non-Normality, particularly when the shapes of the 2 distributions are similar and n1=n2=n, even for small n, remember df= n1+n2-2.3. Are there nonparametric alternatives to the two-sample t-test? (Wilcoxon rank-sum-test, Mann-Witney test, equivalent tests, same P-values.)4. What difference is there between the quantities tested and estimated by the two-sample t-procedures and the nonparametric equivalent? (Non-parametric tests are based on ordering, not size, of the data and hence use median, not mean, for the average. The equality of 2 means is tested and CI(µ1~- µ1~).STAT 13, UCLA, Ivo DinovSlide 41One-way ANOVA refers to the situation of having one factor (or categorical variable) which defines group membership – e.g., comparing 4 reading methods, effects of different reading methods on reading comprehension, data: 50 – 13/14 y/o students tested.Hypotheses for the one-way analysis-of-variance F-testNull hypothesis: All of the underlying true means are identical.Alternative: Differences exist between some of the true means.We know how to analyze 1 & 2 sample data.How about if we have than 2 samples –One-way ANOVA, F-test2STAT 13, UCLA, Ivo DinovSlide 42Comparing 4 reading methods, effects of different reading methods on reading comprehension, data: 50 – 13/14 y/o students tested.-Mapping: using diagrams to relate main points in text;-Scanning: reading the intro and skimming for an overview before reading details;-Mapping and Scanning;-Neither.Table below shows increases in test scores, of 4 groups of students taking similar exams twice, w/ & w/o using a reading technique.Research question: Are the results better for students using mapping, scanning or both?Comparing 4 reading methodsSTAT 13, UCLA, Ivo DinovSlide 43TABLE 10.3.1 Increase in Reading Age Both: 0.1 3.2 4.3 -0.5 1.9 3.3 2.5 3.6 0.4 2.3 -1.4 -0.7-0.10.20.40.91.21.41.81.82.43.1Map Only: 1.0 -0.5 1.0 0.6 0.6 1.0 1.0 -1.4 2.2 3.6 3.1 2.6Scan Only: 1.0 3.3 1.4 -0.9 1.0 0.0 0.6Neither: -0.3 -1.3 1.6 -0.4 -0.7 0.6 -1.8 -2.0 -0.7Increase in reading age-2-1012345Scan onlyMap onlyMap and scanNeitherFigure 10.3.1Increases in reading ages with individual 95% CIs.From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.ObservationalstudySTAT 13, UCLA, Ivo DinovSlide 44Increase in reading age-2 -1 0 1 2 3 4Scan onlyMap onlyMap and scanNeitherFigure 10.3.1 Increases in reading ages with individual 95% CIs.From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.One-way Analysis of VarianceAnalysis of Variance for IncreaseSource DF SS MS F PGrp 3 27.06 9.02 4.45 0.008Error 46 93.35 2.03Total 49 120.41Individual 95% CIs For MeanBased on Pooled StDevLevel N Mean StDev ------+---------+---------+---------+MapScan 22 1.459 1.544 (------*-----)MapOnly 12 1.233 1.441 (-------*--------)ScanOnly 7 0.914 1.302 (----------*----------)Neither 9 -0.556 1.135 (--------*---------)------+---------+---------+---------+Pooled StDev = 1.425 -1.0 0.0 1.0 2.0F-statistic P-valueAnova TableFigure 10.3.2 Minitab analysis of variance output for reading agesFrom Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.The F-test indicates thatthere’s real evidence truedifferences exist it does notgive indication of where thedifferences are or how largethey are.STAT 13, UCLA, Ivo DinovSlide 45Computer outputOne-way Analysis of VarianceAnalysis of Variance for IncreaseSource DF SS MS F PGrp 3 27.06 9.02 4.45 0.008Error 46 93.35 2.03Total 49 120.41Individual 95% CIs For MeanBased on Pooled StDevLevel N Mean StDev ------+---------+---------+---------+MapScan 22 1.459 1.544 (------*-----)MapOnly 12 1.233 1.441 (-------*--------)ScanOnly 7 0.914 1.302 (----------*----------)Neither 9 -0.556 1.135 (--------*---------)------+---------+---------+---------+Pooled StDev = 1.425 -1.0 0.0 1.0 2.0F-statistic P-valueAnova TableFigure 10.3.2 Minitab analysis of variance output for reading agesFrom Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.STAT 13, UCLA, Ivo DinovSlide 46Interpreting the P-value from the F-test (The null hypothesis is that all underlying true means are identical.)z A large
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