ANALYSIS OF VARIANCE (ANOVA) BCT 2053 CHAPTER 5

CONTENT 5.1 Introduction to ANOVA 5.2 One-Way ANOVA 5.3 Two-Way ANOVA

5.1 Introduction to ANOVA OBJECTIVES After completing this chapter you should be able to: 1. Explain the purpose of ANOVA 2. Identify the assumptions that underlie the ANOVA technique 3. Describe the ANOVA hypothesis testing procedure

What is ANALYSIS OF VARIANCE (ANOVA) the approach that allows us to use sample data to see if the values of three or more unknown population means are likely to be different Also known as factorial experiments this name is derived from the fact that in order to test for statistical significance between means, we are actually comparing (i.e., analyzing) variances. (so F-distribution will be used)

Example of problems involving ANOVA A manager want to evaluate the performance of three (or more) employees to see if any performance different from others. A marketing executive want to see if there’s a difference in sales productivity in the 5 company region. A teacher wants to see if there’s a difference in student’s performance if he use 3 or more approach to teach.

The Procedural Steps for an ANOVA Test 1.State the Null and Alternative hypothesis 2.Select the level of significance, α 3.Determine the test distribution to use - F test 4.Compute the test statistic 5.Define rejection or critical region – F test > F critical 6.State the decision rule 7.Make the statistical decision - conclusion

5.2 One-Way ANOVA OBJECTIVE After completing this chapter you should be able to: 1. Use the one-way ANOVA technique to determine if there is a significance difference among three or more means

One-Way ANOVA Design Only one classification factor (variable) is considered Factor Treatment 1 2 i (The level of the factor) Response/ outcome/ dependent variable (samples) Replicates (1,… j ) The object to a given treatment

The resulting input grid of factorial experiment where, i = 1, 2, … a is the number of levels being tested. j = 1, 2, … n i is the number of replicates at each level.

Assumptions To use the one-way ANOVA test, the following assumptions must be true The population under study have normal distribution. The samples are drawn randomly, and each sample is independent of the other samples. All the populations from which the samples values are obtained, have the same unknown population variances, that is for k number of populations,

The Null and Alternative hypothesis (All population means are equal) (Not all population means are equal) If Ho is true we have k number of normal populations with If H 1 is true we may have k number of normal populations with Or H 1 : At least one mean is different from others

Source of Variation Degrees of Freedom (Df) Sum of Squares (SS)Mean of Squares (MS)F Value Between sample (Factor Variation)k - 1 Within samples (Error variation) T - k TotalT - 1 The format of a general one-way ANOVA table Reject H o if T = k n

Example 1 The data shows the Math’s test score for 4 groups of student with 3 different methods of study. Test the hypothesis that there is no difference between the Math’s score for 4 groups of student at significance level Score Individually & Group study Group study Individually

Example 2 An experiment was performed to determine whether the annealing temperature of ductile iron affects its tensile strength. Five specimens were annealed at each of four temperatures. The tensile strength (in ksi) was measured for each temperature. The results are presented in the following table. Can you conclude that there are differences among the mean strengths at α = 0.01? Temperature ( o C) Sample Values

Example 3 Three random samples of times (in minutes) that commuters are stuck in traffic are shown below. At α = 0.05, is there a difference in the mean times among the three cities? Eastern ThirdMiddle ThirdWestern Third

Solve one-way ANOVA by EXCEL Excel – key in data (Example 1)

Solve one-way ANOVA by EXCEL Tools – Add Ins – Analysis Toolpak – Data Analysis – ANOVA single factor – enter the data range – set a value for α - ok Reject H 0 if P-value ≤ α or F > F crit P-value < 0.05 so Reject H 0

4.2 Two-Way ANOVA OBJECTIVE After completing this chapter you should be able to: 1. Use the two-way ANOVA technique to determine if there is an effect of interaction between two factors experiment

Two-Way ANOVA Design Two classification factor is considered Factor B j = 12 b Factor A i = 1 k = 1,…n 2 a Example A researcher whishes to test the effects of two different types of plant food and two different types of soil on the growth of certain plant.

Some types of two way ANOVA design B1 B2 B1 B2 B3 B1 B2 A1 A2 A1 A2 A3 A1 A2 A3 A1 A2 A3 A4

Assumptions The standard two-way ANOVA tests are valid under the following conditions: The design must be complete Observations are taken on every possible treatment The design must be balanced The number of replicates is the same for each treatment The number of replicates per treatment, k must be at least 2 Within any treatment, the observations are a simple random sample from a normal population The sample observations are independent of each other (the samples are not matched or paired in any way) The population variance is the same for all treatments.

Null & Alternative Hypothesis H 0 : there is no interaction effect between factor A and factor B. H 1 : there is an interaction effect between factor A and factor B. H 0 : there is no difference in means of factor A. H 1 : there is a difference in means of factor A. H 0 : there is no difference in means of factor B. H 1 : there is a difference in means of factor B. interaction effect Column effect Row effect H0: there is no effect from factor A. H1: there is effect from factor A. H0: there is no effect from factor B. H1: there is effect from factor B. or

Source (Df) Sum of Squares (SS)Mean of Squares (MS) F Value A (row effect) a - 1 B (Column effect) b - 1 Interaction ( interaction effect ) (a-1)(b-1) Errorab(n-1) Totalabn-1 The format of a general two-way ANOVA table Reject if

Procedure for Two-Way ANOVA Yes (Reject Ho) No (Accept Ho) Ho: No interaction between two factors Ho: No effects from the column factor B (the column means are equal) Ho: No effects from the row factor A (the row means are equal)

Example 1 A chemical engineer is studying the effects of various reagents and catalyst on the yield of a certain process. Yield is expressed as a percentage of a theoretical maximum. 2 runs of the process were made for each combination of 3 reagents and 4 catalysts. Catalyst Reagent 123 A B C D a)Construct an ANOVA table. b)Test is there an interaction effect between reagents and catalyst. Use α = c)Do we need to test whether there is an effect that is due to reagents or catalyst? Why? If Yes, test is there an effect from reagents or catalyst.

Example 2 A study was done to determine the effects of two factors on the lather ability of soap. The two factors were type of water and glycerol. The outcome measured was the amount of foam produced in mL. The experiment was repeated 3 times for each combination of factors. The result are presented in the following table.. Water typeGlycerolFoam (mL) De-ionizedAbsent De-ionizedPresent TapAbsent TapPresent Construct an ANOVA table and test is there an interaction effect between factors. Use α = 0.05.

Solve two-way ANOVA by EXCEL Excel – key in data

Solve two-way ANOVA by EXCEL tools – Data Analysis – ANOVA two factor with replication – enter the data range – set a value for α - ok Reject H 0 if P-value ≤ α or F > F crit

Summary The other name for ANOVA is experimental design. ANOVA help researchers to design an experiment properly and analyzed the data it produces in correctly way. Thank You