MADAM SITI AISYAH ZAKARIA CHAPTER 3 Analysis of Variance (ANOVA) PART 2 =TWO- WAY ANOVA WITHOUT REPLICATION MADAM SITI AISYAH ZAKARIA EQT 271 SEM 1 2016/2017

Two-Way ANOVA without Replication (Randomized Completely Block Design) Known as Randomized Completely Block Design (RBD) For RCBD there is one factor or variable that is of primary interest. However, there are also several other nuisance factor. Nuisance factors are those that may affect the measured result, but not of primary interest. For example, in applying a treatment, nuisance factors might be the specific operator who prepared the treatment, the time of day the experiment was run and the room temperature. So, to control this, the important technique known as blocking.

GOAL RCBD To minimized variability through controlling the effect of factor which is not of interest by bringing experimental units that are similar into a group called a ‘block’ In RCBD, each block is tested against all levels of the primary factor (Treatment) at random order. In RCBD the blocking factor is introduced as an improvement to one-way ANOVA. In this design, experimental units are organized through blocking to reduce the experimental error.

Randomized Block Design, cont. Effects model for RBD: 1. Two Sets of Hypothesis: Treatment Effect: H0: There is no treatments effect on the response variable H1: There is treatments effect on the response variable Block Effect: H0: There is no blocking effect on the response variable H1: There is blocking effect on the response variable

Randomized Block Design, cont. Format for data: Data appear in a table, where location in a specific row and a specific column is important. 2. Test Statistics: i. Sum of squares total (SST) = sum of squared differences between each individual data value (regardless of group membership) minus the grand mean, , across all data... total variation in the data (not variance).

Randomized Block Design, cont. 2. Test Statistics, cont.: ii. Sum of squares treatment (SSTR) = sum of squared differences between each treatment group mean and the grand mean, balanced by sample size... between-treatment-groups variation (not variance). iii. Sum of squares block (SSBL) = sum of squared differences between each block group mean and the grand mean, balanced by sample size... between-block-groups variation (not variance). iv. Sum of squares error (SSE):  SSE = SST – SSTR – SSBL

NOTATION

Randomized Block Design, cont. 2. Test Statistics, cont.: v. Mean square treatment (MSTR) = SSTR/(k – 1), where t is the number of treatment groups. vi. Mean square block (MSBL) = SSBL/(n – 1), where n is the number of block groups. Controls the size of SSE by removing variation that is explained by the blocking categories. vii. Mean square error (MSE):

Randomized Block Design, cont. 2. Test Statistics, cont.: viii. F-Ratios: ( F test) F-Ratio, Treatment = MSTR/MSE, This F-ratio is the test statistic for the hypothesis that the treatment group means are equal. - F-Ratio, Block = MSBL/MSE, This F-ratio is the test statistic for the hypothesis that the block group means are equal.

Randomized Block Design, cont. ix. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F Test p-Value Treatments SSTR k-1 Blocks SSBL n-1 Error SSE (k-1)(n-1) Total SST N-1

Randomized Block Design, cont. 3. Critical Value , F value - Treatment effect - F , k -1,(k-1)(n-1) v1= k -1; v2= (k-1)(n-1) = table (ms 30) - Block effect - F , n -1,(k-1)(n-1) v1= n -1; v2= (k-1)(n-1)= table (ms 30) 4. Rejection Region (Draw diagram) If F-Ratio (F test) > F or p-value <  , reject H0 at the  level

Reject null hypothesis Fail to Reject null hypothesis 4. Conclusion / Decision Reject null hypothesis Fail to Reject null hypothesis Treatment effect Treatment effect Fcalc>Fcv. There is sufficient evidence to support(H1)/to reject (Ho) that there is treatments effect on response variable. Fcalc<Fcv. There is insufficient evidence to support/to reject that there is treatments effect on response variable. Block effect Block effect Fcalc>Fcv. There is sufficient evidence to support/to reject that there is blocking effect on response variable. Fcalc<Fcv. There is insufficient evidence to support/to reject that there is blocking effect on response variable.

Randomized Block Design, cont. EXAMPLE : Crescent Oil Co. Crescent Oil has developed three new blends of gasoline and must decide which blend or blends to produce and distribute. A study of the miles per gallon ratings of the three blends is being conducted to determine if the mean ratings are the same for the three blends. Five automobiles have been tested using each of the three gasoline blends and the miles per gallon ratings are shown on the next slide.

Randomized Block Design, cont. Factor . . . Gasoline blend Treatments . . . Blend X, Blend Y, Blend Z Blocks . . . Automobiles Response variable . . . Miles per gallon

Randomized Block Design, cont. Type of Gasoline (Treatment) Automobile (Block) Block Total Blend X Blend Y Blend Z 1 2 3 4 5 31 30 29 33 26 30 29 31 25 30 29 28 26 91 88 86 93 77 Treatment Total 149 144 142

SOLUTION H1: There is an effect of blends on the miles per gallon 1. Hypothesis: H0: There is no effect of blends on the miles per gallon (The mean ratings are the same for the three blends.) [claim] H1: There is an effect of blends on the miles per gallon (The mean ratings are different for the three blends.)

SOLUTION 2. Test Statistics 1 2 3

SOLUTION 2. Test Statistics – Cont.. 4 5 6 7 8

Randomized Block Design, cont. v. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F p-Value Treatments 5.20 2 2.60 3.82 0.07 Blocks 51.33 4 12.80 *** Error 5.47 8 0.68 Total 62.00 14

Randomized Block Design, cont. 3. F (alfa) value – critical value Treatment : For  = 0.05, F0.05,2,8 = 4.46 (2 d.f. numerator and 8 d.f. denominator) 4. Rejection Region ( Draw picture) Critical Value Approach: Fail to Reject H0 Since F Test < F alfa ; 3.82 < 4.46 p-Value Approach: Fail to Reject H0 Since p-value (0.07) > 0.05

Randomized Block Design, cont. 5. Conclusion: 1. The p-value is greater than .05 (Excel provides a p-value of 0.07). or since F Test > F alfa ; 3.82 < 4.46. Therefore, we fail to reject H0. 2. There is sufficient evidence to reject that there is no effect of blends on the miles per gallon or the mean ratings are the same for the three blends.

CONCLUSION TOPIC CRD RCB To evaluate difference for more than independent groups **One factor Use blocking in order to remove as much as possible variability from random error (MSE). ** lead to better chance in detecting differences among the groups. RCB

EXERCISE A laboratory technician measures the breaking strength of each of kinds of linen thread by means of 4 different instruments and obtains the following results (in ounces). Looking upon the threads as treatments and instruments as blocks, perform the RCBD at the level of significance 0.01.

EXERCISE..CONT.. Measuring instrument I1 I2 I3 I4 Thread 1 20.6 20.7 21.4 Thread 2 24.7 26.5 27.1 24.3 Thread 3 25.2 23.4 21.6 23.9 Thread 4 24.5 21.5 23.6 Thread 5 19.3 22.2

answer

EXERCISE NOTE PAGE 104 (ALL) 2 step 5 IMPORTANT STEP: HYPOTHESIS TESTING TEST STATISTIC – F TEST F (alfa) – VALUE (CRITICAL VALUE) REJECTION REGION CONCLUSION SST SSTR – MSTR SSBL – MSBL SSE = SST – SSTR – SSBL MSE F TEST SSTR= MSTR/MSE F TEST SSBL = MSBL/MSE BUILD ANOVA TABLE