One-Way Analysis of Variance Experimental units(subjects)are assigned randomly to treatments Subjects are assumed homogeneous Only one factor or independent variable With two or more treatment levels Analyzed by one-factor analysis of variance (one-way ANOVA)
One-Way Analysis of Variance Experimental units (subjects) are assigned randomly to treatments Subjects are assumed homogeneous Only one factor or independent variable With two or more treatment levels Analyzed by one-factor analysis of variance (one-way ANOV A )
One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples:Accident rates for 1st,2nd,and 3rd shift Expected mileage for five brands of tires ■Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn
One-Way Analysis of Variance Evaluate the difference among the means of three or more groups Examples: Accident rates for 1st, 2nd, and 3rd shift Expected mileage for five brands of tires Assumptions Populations are normally distributed Populations have equal variances Samples are randomly and independently drawn
xii X~N(4,c2）i=1,2,,a;j=1,2,,n E≈N0,σ2) Level Ar A2 Aa Observations X11 X21 Xal 2 Xaj XInl X2n2 Xana Population 41 42 a mean
Level A1 A2 ··· Aa Observations x11 x21 ··· xa1 x1j x2j ··· xaj x1n1 x2n2 ··· xana Population mean 1 2 a Xi ～N( i, 2 ) xij = i + ij i = 1, 2,···, a；j = 1, 2,···, ni ij ～ N(0, 2 )
Hypotheses of One-Way ANOVA ■H01=2=μ3=…=c All population means are equal i.e.,no treatment effect(no variation in means among groups) H:Not all of the population means are the same At least one population mean is different i.e.,there is a treatment effect Does not mean that all population means are different (some pairs may be the same)
Hypotheses of One-Way ANOVA All population means are equal i.e., no treatment effect (no variation in means among groups) At least one population mean is different i.e., there is a treatment effect Does not mean that all population means are different (some pairs may be the same) 0 μ1 μ 2 μ 3 μ c H : H1 : Not all of the population means are the same
One-Factor ANOVA H0μ1=2=μ3=…=c H,Not all u are the same All Means are the same: The Null Hypothesis is True (No Treatment Effect) 1=2=3
One-Factor ANOVA All Means are the same: The Null Hypothesis is True (No Treatment Effect) 0 μ1 μ 2 μ 3 μ c H : H1 : Not all μi are the same μ1 μ 2 μ 3