1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health Sciences!Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology!Teaching Assistants: Sovia Lau, Jason ChengUCLA StatisticsUniversity of California, Los Angeles, Fall 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Chapter 10: Data on a Continuous Variable!One-sample issues!Two independent samples!More than 2 samples!Blocking, stratification and relatedsamplesSTAT 13, UCLA, Ivo DinovSlide 22TABLE 10.1.2 Air Force Head Sizes Data Recruit Cardboard Metal Difference Sign of (mm) (mm) (Card-metal) difference 1 146 145 1 + 2 151 153 -2 - 3 163 161 2 + 4 152 151 1 + 5 151 145 6 + 6 151 150 1 + Measure the head-size of all air force recruits. Using cheaper cardboard or more expensive metal calipers. Are there systematic differences in the two measuring methods? Again, paired comparisons.Flying helmet sizes for NZ Air Force STAT 13, UCLA, Ivo DinovSlide 23TABLE 10.1.2 Air Force Head Sizes Data Recruit Cardboard Metal Difference Sign of (mm) (mm) (Card-metal) difference 1 146 145 1 + 2 151 153 -2 - 3 163 161 2 + 4 152 151 1 + 5 151 145 6 + 6 151 150 1 + 7 149 150 -1 - 8 166 163 3 + 9 149 147 2 + 10 155 154 1 + 11 155 150 5 + 12 156 156 0 013 162 161 1 + 14 150 152 -2 - 15 156 154 2 + 16 158 154 4 + 17 149 147 2 + 18 163 160 3 + Helmet sizes for NZ Air Force – complete tableSTAT 13, UCLA, Ivo DinovSlide 24Head sizes: Does type of caliper make a difference?Differences (Cardboard - Metal)-20246Hypothesized valueFigure 10.1.8 Dot plot of differences in size (with 95% CI).From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.Paired T-Test and Confidence Intervalpaired T for cardboard - metal N Mean StDev SE Meancardboard 18 154.56 5.82 1.37metal 18 152.94 5.54 1.30Difference 18 1.611 2.146 0.50695% CI for mean difference: (0.544, 2.678)T-Test of mean difference=0 (vs not=0): T-Value=3.19 P-Value=0.005Figure 10.1.9 Minitab paired-t output for the size data.From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.H0:µdiff= 0Ha:µdiff!= 0STAT 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(,1σµNx)2,2(,2σµNx21µµ−021=−µµ21µµ−)(0)(21210xxSExxt−−−=)( 2121xxSEtxx−×±−2/221/21nsnsSE +=)12;11( −−= nnMindf2STAT 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-testSTAT 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 -0.7-0.1 0.2 0.4 0.9 1.2 1.4 1.8 1.8 2.4 3.1Map 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 -1 0 1 2 3 4 5Scan 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.ObservationalstudySTAT 13, UCLA, Ivo DinovSlide 44Increase in reading age-2-101234Scan onlyMap onlyMap and scanNeitherFigure 10.3.1Increases in reading ages with individual 95% CIs.From Chance Encounters by C .J. Wild an d G.A.F. Se ber, © Jo hn Wiley & Sons, 20 00.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 49 120.41
View Full Document