Alpha= 0.050; t = 2.0796ATopic 19 - ANOVA (III) 19-1 Topic 19 – Estimation in a One-Way ANOVA When Variance is Constant for All Treatments Once we have rejected the null hypothesis that all means are equal, and we have checked the assumptions of the testing procedure, we usually wish to do some specific tests that can elucidate the relationships among the means. These are variously called multiple comparisons tests, contrasts, or tests of linear combinations of means. A priori Hypotheses: hypotheses about population means that are decided during the planning of the experiment and prior to any data analysis. They are the reason for performing the experiment! A posteriori Hypotheses: hypotheses generated as a result of looking at the data after the experiment has been performed. Also called data snooping or data dredging. This is almost ALWAYS inappropriate and to be avoided. The only valid reason for doing so is as an exploratory analysis that will guide future experimentation. Example (a posteriori testing): suppose a 1-way ANOVA is performed and the results are obtained. The analyst looks over the results and decides to test 2 meansTopic 19 - ANOVA (III) 19-2 because they appear to be very different (e.g. the smallest and largest ones). Now, the effect could be due to a real difference in population means or to random occurrence due to sampling that makes them appear different. Investigating only comparisons for which the effect appears large leads to a true confidence level for a conclusion that is lower than the stated confidence level when there is no difference. In other words you are more likely to reject H0: not different. It can be shown that the actual confidence is 60% (!!!!) when 6 levels are used in an experiment and the statistical analysis always includes testing the difference between the largest and smallest means using a stated 95% confidence (note that these means need not be the same treatment means each time).Topic 19 - ANOVA (III) 19-3 There are times when it is possible to do a posteriori testing – BUT the statistical method needs to be modified appropriately to account for the data snooping (see later). 1) Estimation Of A Treatment Mean The population mean for the ith treatment iµ is estimated using the sample mean •=iiyµˆ with a standard error of iinMSEySE =•)( Under our assumptions of normality and random sampling, the (1–α)100% Confidence Interval of the population mean is )(,2•−•±itNiySEtyα where tNt−,2α is the critical value for the upper tail of a t-distribution on N – t df. Hypothesis testing is done using a t-test as is usual for a single population mean.Topic 19 - ANOVA (III) 19-4 2) Estimation Of The Difference Between 2 Means The unbiased estimator of the difference between 2 population means kiDµµ−= is ••−=kiikyyDˆ which has a standard error of ⎟⎠⎞⎜⎝⎛+=kiiknnMSEDSE11)ˆ( assuming the variances are homogeneous. Under our assumptions of normality and random sampling, a (1–α)100% Confidence Interval of the difference of the population means is )ˆ(ˆ,2iktNikDSEtD−±α where tNt−,2α is the critical value for the upper tail of a t-distribution on N – t degrees of freedom. Hypothesis testing is done using the t-test for two independent samples that we reviewed earlier this semester.Topic 19 - ANOVA (III) 19-5 EXAMPLE: Rehabilitation Therapy. A researcher is interested in the relationship between physical fitness in persons prior to knee surgery and the time required in physical therapy after surgery to obtain successful rehabilitation. 24 male subjects with similar knee surgery during the past year were randomly selected from the patient records at the rehabilitation center and the number of days required for successful rehabilitation and prior physical fitness status were recorded. The patients were categorized into one of three levels of fitness. The hypotheses of interest are: 1) the mean time to recovery will differ among the three groups; 2) the above average fitness group will have a shorter recovery period than the below average and average groups and 3) the average group will have a shorter recovery than the below average group. In other words: 1) H0: belowaverageaboveµµµ== HA: at least one mean differs 2) H0: averageaboveµµ= HA: averageaboveµµ< H0: belowaboveµµ= HA: belowaboveµµ< 3) H0: belowaverageµµ=Topic 19 - ANOVA (III) 19-6 HA: belowaverageµµ< (Using Fit Y by X Platform) Oneway Analysis of Recovery Time By Fitness Recovery Time15202530354045above average belowFitness Oneway Anova: Summary of Fit Rsquare 0.660369 Adj Rsquare 0.628024 Root Mean Square Error 4.405084 Mean of Response 31.41667 Observations (or Sum Wgts) 24 Analysis of Variance Source DF Sum of SquaresMean SquareF Ratio Prob > FFitness 2 792.3333 396.167 20.4160 <.0001Error 21 407.5000 19.405 C. Total 23 1199.8333Topic 19 - ANOVA (III) 19-7 Means for Oneway Anova Level n Mean Std Error Lower 95% Upper 95%above 8 24.0000 1.5574 20.761 27.239average 8 32.2500 1.5574 29.011 35.489below 8 38.0000 1.5574 34.761 41.239Std Error uses a pooled estimate of error variance To get pairwise comparisons, use the Fit Model Platform: LSMeans Differences Student's t Alpha= 0.050; t = 2.0796 LSMean[i] By LSMean[j] Mean[i]-Mean[j] Std Err Dif Lower CL Dif Upper CL Dif above average belowabove 0000-8.25 2.20254 -12.83 -3.6696 -142.20254-18.58-9.4196average 8.252.202543.6695612.83040 0 0 0 -5.752.20254-10.33-1.1696below 142.202549.4195618.58045.75 2.20254 1.16956 10.3304 0000Topic 19 - ANOVA (III) 19-8 Level Least Sq Meanbelow A 38.000000average B 32.250000above C 24.000000 Levels not connected by same letter are significantly different 1) Multiple Comparisons The procedures we’ve just seen have 2 IMPORTANT limitations: a. The confidence (1 – α) applies ONLY to a particular estimate (or test) not to the series of estimates (or tests) b. The confidence (1 – α) is appropriate ONLY if the estimate (or test) was not suggested by the data We typically perform multiple tests in order to piece the results together to draw a more complete conclusion. This is sometimes referred to as a “family of statements (or tests)” and it is important to provide some assurance that all of the statements in the family are correct. We call thisTopic 19 - ANOVA (III) 19-9
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