.H reject not do , F If .H reject , > F If o co c FF ≤othersthe from different is means the of one least At ::321a k o H H μμμμ====Chapter 11: Analysis of VarianceExperimental Designa plan and a structure to test hypotheses in which the researcher controls or manipulates one or more variables. Independent VariableTreatment variable 實驗變數- one that the experimenter controls or modifies in the experiment. Classification variable - a characteristic of the experimental subjects that was present prior to the experiment, and is not a resu lt of the experimenter’s manipulations or control.Levels or Classifications - the subcategories of the independent variable used by the researcher in the experimental design.Independent variables are also referred to as factors.Analysis of Variance (ANOVA)變異數分析 – a group of statistical techniques used to analyze experimental designs .ANOVA begins with notion that individual items being studied are all the sameOne-Way ANOVA 單因素變異數分析: Procedural OverviewThe null hypothesis states that the population means for all treatment levels are equalEven if one of the population means is different from the other, the null hypothesis is rejectedTesting the hypothesis is done by portioning the total variance of data int6o the following two variancesVariance resulting from the treatment (columns)Error variance or that portion of the total variance unexplained by the treatmentOne-Way ANOVA: Sums of Squares DefinitionsOne-Way ANOVA: Computational FormulasAssumptions underlie ANOVANormally distributed populationsObservations represent random samplesfrom the populationVariances of the population are equalANOVA is used to determine statistically whether the variance between the treatment level meansis greater than the variances within levels (error variance)ANOVA is computed with the three sums of squaresTotal –Total Sum of Squares (SST); a measure of all variations in the dependent variable Treatment – Sum of Squares Columns (SSC); measures the variations between treatments orcolumns since independent variable levels are present in columnsError –Sum of Squares of Error (SSE); yields the variations within treatments (or columns) Other itemsMSC – Mean Squares of ColumnsMSE - ErrorMST - TotalF value – determined by dividing the treatment variance (MSC) by the error variance (MSE)F value is a ratio of the treatment variance to the error varianceF and t ValuesAnalysis of variance can be used to test hypothesis about the difference in two meansAnalysis of data from two samples by both a t test and an ANOVA shows that the observed Fvalues equals the observed t value squaredF = t2t test of independent samples actually is special case of one way ANOVA when there are only two treatment levelsMultiple Comparison TestsANOVA techniques useful in testing hypothesis about differences of means in multiple groupsAdvantage: Probability of committing a Type I error is controlledMultiple Comparison techniques are used to identify which pairs of means are significantlydifferent given that the ANOVA test reveals overall significanceMultiple comparisons are used when an overall significant difference between groups has been determined using the F value of the analysis of varianceTukey’s honestly significant difference (HSD) test requires equal sample sizesTakes into consideration the number of treatment levels, value of mean square error, and sample size Once HSD is computed, one can examine the absolute value of all differences between pairs of means from treatment levels to determine if it is a significant differenceTukey-Kramer Procedure is used when sample sizes are unequalTukey’s Honestly Significant Difference (HSD) TestRandomized Block Design隨機區集設計Randomized block design - focuses on one independent variable (treatment variable) of interest.Includes a second variable (blocking variable) used to control for confounding or concomitant variables.Variables that are not being controlled by the researcher in the experiment can have an effect on theoutcome of the treatment being studied.Repeated measures design - is a design in which each block level is an individual item or person, and thatperson or item is measured across all treatments.The sum of squares in a completely randomized design isSST = SSC + SSR + SSEIn a randomized block design, the sum of squares isSST = SSC + SSESSR (blocking effects) comes out of the SSESome error in variation in randomized design are due to the blocking effects of the randomized block design.Randomized Block Design Treatment Effects: Procedural OverviewThe observed F value for treatments computed using the randomized block design formula is tested bycomparing it to a table FvalueIf the observed Fvalue is greater than the table value, the null hypothesis is rejected for that alpha value If the F value for blocks is greater than the critical F value, the null hypothesis that all block population means are equal is rejectedFormulas for Computing a Two-Way ANOVA。