Analysis of Variance

Experimental Design u Investigator controls one or more independent variables –Called treatment variables or factors –Contain two or more levels (subcategories) u Observes effect on dependent variable –Response to levels of independent variable u Experimental design: Plan used to test hypotheses

Parametric Test Procedures u Involve population parameters –Example: Population mean u Require interval scale or ratio scale –Whole numbers or fractions –Example: Height in inches: 72, 60.5, 54.7 u Have stringent assumptions Examples: –Normal distribution –Homogeneity of Variance Examples: z - test, t - test

Nonparametric Test Procedures u Statistic does not depend on population distribution u Data may be nominally or ordinally scaled –Examples: Gender [female-male], Birth Order u May involve population parameters such as median u Example: Wilcoxon rank sum test

Advantages of Nonparametric Tests u Used with all scales u Easier to compute –Developed before wide computer use u Make fewer assumptions u Need not involve population parameters u Results may be as exact as parametric procedures © T/Maker Co.

Disadvantages of Nonparametric Tests u May waste information –If data permit using parametric procedures –Example: Converting data from ratio to ordinal scale u Difficult to compute by hand for large samples u Tables not widely available © T/Maker Co.

ANOVA (one-way) One factor, completely randomized design

Completely Randomized Design u Experimental units (subjects) are assigned randomly to treatments –Subjects are assumed homogeneous u One factor or independent variable –two or more treatment levels or classifications u Analyzed by [parametric statistics]: –One-and Two-Way ANOVA

Mini-Case After working for the Jones Graphics Company for one year, you have the choice of being paid by one of three programs: - commission only, - fixed salary, or - combination of the two.

Salary Plans u Commission only? u Fixed salary? u Combination of the two?

Is the average salary under the various plans different?

Assumptions u Homogeneity of Variance u Normality u Additivity u Independence

Homogeneity of Variance Variances associated with each treatment in the experiment are equal.

Normality Each treatment population is normally distributed.

Additivity The effects of the model behave in an additive fashion [e.g. : SST = SSB + SSW]. Non-additivity may be caused by the multiplicative effects existing in the model, exclusion of significant interactions, or by “outliers” - observations that are inconsistent with major responses in the experiment.

Independence Assuming the treatment populations are normally distributed, the errors are not correlated.

u Compares two types of variation to test equality of means u Ratio of variances is comparison basis u If treatment variation is significantly greater than random variation … then means are not equal u Variation measures are obtained by ‘partitioning’ total variation One-Way ANOVA

ANOVA (one-way)

ANOVA Partitions Total Variation Total variation

ANOVA Partitions Total Variation Variation due to treatment Total variation

ANOVA Partitions Total Variation Variation due to treatment Variation due to random sampling Total variation

ANOVA Partitions Total Variation Variation due to treatment Variation due to random sampling Total variation  Sum of squares among  Sum of squares between  Sum of squares model  Among groups variation

ANOVA Partitions Total Variation Variation due to treatment Variation due to random sampling Total variation  Sum of squares within  Sum of squares error  Within groups variation  Sum of squares among  Sum of squares between  Sum of squares model  Among groups variation

Hypothesis H 0 :  1 =  2 =  3 H 1 : Not all means are equal tests: F -ratio = MSB / MSW p-value < 0.05

One-Way ANOVA  H 0 :  1 =  2 =  3 –All population means are equal –No treatment effect u H 1 : Not all means are equal –At least one population mean is different –Treatment effect –  1   2   3 is wrong – is wrong not correct – not correct

StatGraphics Input

StatGraphics Results

Diagnostic Checking u Evaluate hypothesis H 0 :  1 =  2 =  3 H 1 : Not all means equal u F-ratio = {Table value = 3.89} u significance level [p-value] = u Retain null hypothesis [ H 0 ]

ANOVA (two-way) Two factor factorial design

Mini-Case Investigate the effect of decibel output using four different amplifiers and two different popular brand speakers, and the effect of both amplifier and speaker operating jointly.

What effects decibel output? 4 Type of amplifier? 4 Type of speaker? 4 The interaction between amplifier and speaker?

Are the effects of amplifiers, speakers, and interaction significant? [Data in decibel units.]

Hypothesis  Amplifier H 0 :  1 =  2 =  3 =  4 H 1 : Not all means are equal  Speaker H 0 :  1 =  2 H 1 : Not all means are equal u Interaction H 0 : The interaction is not significant H 1 : The interaction is significant

StatGraphics Input

StatGraphics Results

Diagnostics u Amplifier p-value = Reject Null u Speaker p-value = Reject Null u Interaction p-value = Retain Null Thus, based on the data, the type of amplifier and the type of speaker appear to effect the mean decibel output. However, it appears there is no significant interaction between amplifier and speaker mean decibel output.

You and StatGraphics u Specification [Know assumptions underlying various models.] u Estimation [Know mechanics of StatGraphics Plus Win]. u Diagnostic checking

Questions ?

ANOVA

End of Chapter