CHAPTER 3 Analysis of Variance (ANOVA) PART 1 MADAM SITI AISYAH ZAKARIA EQT 271 SEM 2 2015/2016

Learning Objectives Describe the relationship between analysis of variance, the design of experiments, and the types of applications to which the experiments are applied. Differentiate one-way and two-way ANOVA techniques. Arrange data into a format that facilitates their analysis by the appropriate ANOVA technique. Use the appropriate methods in testing hypothesis relative to the experimental data.

Key Terms Sum of squares: Treatment Error Block Interaction Total Response variable / dependent variable Factor level /Independent variable treatment, block, interaction Experimental units Replication Within-group variation Between-group variation Completely randomized design Randomized block design Factorial experiment Sum of squares: Treatment Error Block Interaction Total

Key Concepts ANOVA is a statistical inference method used to test the equality of three or more population means. Response variable/dependent variable is the result that your experiment is measuring. A factor / independent is a variable that affects the outcome of the response variable. A treatment is a level of a factor. Experimental units are the objects or individual that experiment uses. Variation between treatment groups captures the effect of the treatment. A : B : C [Between group-variance] Variation within treatment groups represents random error not explained by the experimental treatments. A ONLY [Within group variance]

EXAMPLE: Mazlan studied the effect of three learning skills; peer group discussion, extra exercises, and additional reference books towards student’s score. Define which are response variable, factor, treatment and experimental units. Answer: Response variable – student’s score Factor – Learning skill Treatment - peer group discussion, extra exercises, and additional reference books [level of factor] Experimental units – score from student at random.

EXAMPLE: You are interested to conduct a study in attempt to determine whether or not the means of statistics test score are not the same for the different sleeping times ( 4,6,8 hours). State the response variable, factor, treatment and experimental unit. Answer: Response variable – statistics test scores Factor – sleeping times Treatment – the number of hours of sleep 4, 6, 8 hours [level of factor] Experimental units – students who are selected in random

Analysis of Variance: A Conceptual Overview Assumptions for Analysis of Variance There must be k random samples drawn from each of k population The populations from which the samples are drawn must be normally or approximately normal distributed. The population variances must be equal ; The k samples are independent of each other; that is the subjects in one group are not related to subjects in a second group in any way.

Variability in response variable Comparison of population mean ANOVA Study the total variation in the response variable/ dependent variable, y Total variation The variation of the group means ( )around the grand mean The variation of observations within each group around the group mean VARIATION BETWEEN GROUPS (reflects the effect of the factor levels on the response variable); A: B: C VARIATION WITHIN GROUPS (represents random error from sampling); A Note: large value between group variation higher chance to reject null hypothesis the groups’ population means are significantly different.

F test value approximately equal to 1 F test value = greater than 1 CONCLUSION Fail to Reject H0 Reject H0 No difference in mean Difference in mean Between- group variance estimate approximately equal to the within-group variance Between- group variance estimate will be larger than within-group variance Variability of response variable F test value approximately equal to 1 F test value = greater than 1

One-Way ANOVA (Completely Randomized Design) A completely randomized design (CRD) is often used in scientific experiments with the purpose of comparing treatments, processes, materials or products. Purpose: used when there is only one independent variable or also called as factor which gives influence to the response variable, y Effects model for CRD:

One-Way ANOVA, cont. Example: you would want to compare the effect of three different types of vehicle (small, sedan, multi-purpose-vehicle (MPV) on fuel consumption. Response variable – fuel consumption Factor – three types of vehicle Hypothesis: k= 3 population means of fuel consumption for the three different types of vehicle. H0: µ1 = µ2 = µ3 (all three population means are equal) H1: At least one mean is different from the other. Note: Reject H0 – does not require that ALL means of fuel consumption are significantly differ from each other. It rejected if at least one of the mean is significantly different from the other.

One-Way ANOVA, cont. Format for data: Data appear in separate columns or rows, organized by treatment groups. Sample size of each group may differ. Calculations: 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).

One-Way ANOVA, cont. Calculations, cont.: Sum of squares treatment (SSTR) = sum of squared differences between each group mean and the grand mean, balanced by sample size... between-groups variation

Sum of squares error (SSE) = sum of squared differences between the individual data values and the mean for the group to which each belongs... within-group variation

One-Way ANOVA, cont. Calculations, cont.: Mean square treatment (MSTR) = SSTR/(k– 1), where k is the number of treatment groups... between-groups variance. Mean square error (MSE) = SSE/(N – k), where N is the number of elements sampled and k is the number of treatment groups... within-groups variance. F-Ratio [ F test ] = MSTR/MSE, where numerator degrees of freedom are k – 1 and denominator degrees of freedom are N – k. - F, k-1,N-k refer table ms 30 If F-Ratio > F or p-value <  , reject H0 at the  level.

One-Way ANOVA, cont. Comparing the Variance Estimates: The F Test a Sampling Distribution of MSTR/MSE Sampling Distribution of MSTR/MSE Reject H0 Do Not Reject H0 a MSTR/MSE F, k-1,N-k Critical Value

Characteristics of F-distribution The distribution is not a symmetric distribution F is calculated on ratio of variance so it could not be negative Shape of F-dist is determined by two degree of freedom, d.f df1 = k – 1; df2 = N - k * k = number of group ; * N = Total number of observation

One-Way ANOVA, cont. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F p-Value Treatments SSTR k-1 Error SSE N-k Total SST N-1

5 important step 1. HYPOTHESIS TESTING 2. TEST STATISTIC – F TEST H0: µ1 = µ2 = µ3 (all three population means are equal) H1: At least one mean is different from the other. 2. TEST STATISTIC – F TEST 3. F (alfa) – VALUE (CRITICAL VALUE) 4. REJECTION REGION 5. CONCLUSION/DECISION Reject H0 – There is sufficient evidence to show that at least one population mean is different from the others. Fail to Reject H0 – There is insufficient evidence to show that at least one population mean is different from the others.

EXAMPLE ONE- WAY ANOVA

One-Way ANOVA - An Example 2. AutoShine, Inc. AutoShine, Inc. is considering marketing a long- lasting car wax. Three different waxes (Type 1, Type 2, and Type 3) have been developed. In order to test the durability of these waxes, 5 new cars were waxed with Type 1, 5 with Type 2, and 5 with Type 3. Each car was then repeatedly run through an automatic carwash until the wax coating showed signs of deterioration. The number of times each car went through the carwash before its wax deteriorated is shown on the next slide. AutoShine, Inc. must decide which wax to market. Are the three waxes equally effective? Use

One-Way ANOVA - An Example Factor . . . Car wax Treatments . . . Type I, Type 2, Type 3 Experimental units . . . Cars Response variable . . . Number of washes

One-Way ANOVA - An Example Testing for the Equality of k Population Means: A Completely Randomized Design Wax Type 1 Wax Type 2 Wax Type 3 Observation 1 2 3 4 5 27 30 29 28 31 33 28 31 30 29 28 30 32 31

One-Way ANOVA - An Example Testing for the Equality of k Population Means: A Completely Randomized Design 1. Hypothesis H0: 1=2=3 H1: at least one population mean number of washes for all types is not equal from the other (population mean number of washes for all types are equal) where: 1 = mean number of washes using Type 1 wax 2 = mean number of washes using Type 2 wax 3 = mean number of washes using Type 3 wax

One-Way ANOVA - An Example Testing for the Equality of k Population Means: CRD 2. Test Statistic 1 5 2 6 4 3

One-Way ANOVA - An Example Testing for the Equality of k Population Means: A Completely Randomized Design 2. ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F p-Value Treatments 5.2 2 2.60 0.939 0.42 Error 33.2 12 2.77 Total 38.4 14

One-Way ANOVA - An Example Testing for the Equality of k Population Means: A Completely Randomized Design 3. F value where F0.05,2,12 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom 4. Rejection Rule (Draw picture) Critical Value Approach: Fail to Reject H0 if F < 3.89 p-Value Approach: Fail to Reject H0 if p-value > .05

One-Way ANOVA - An Example Testing for the Equality of k Population Means: A Completely Randomized Design 5. Conclusion/ Decision F test < F alfa (3.89), do not fall in rejection region so we fail to reject H0. 2. There is insufficient evidence to conclude that at least one population mean is different from the other or at least one of the mean number of washes for the three wax types is different from the other. So, this three waxes equally effective. ( same effect)

EXAMPLE 2 2 step: Reject H0 Factor:….. Treatments: ….. Experimental Unit:.. Response variable:… 2 step: 5 IMPORTANT STEP: HYPOTHESIS TESTING TEST STATISTIC – F TEST F (alfa) – VALUE (CRITICAL VALUE) REJECTION REGION CONCLUSION SST SSTR MSTR SSE = SST -SSTR MSE F TEST = MSTR/MSE BUILD ANOVA TABLE Reject H0

answer SSTR = 242.717 SSE = 225.954 SST = 468.671 F TEST = 4.83

EXERCISE NOTE PAGE 33-36 (ALL) Take out all the information from the exercise: 1. Factor:….. 2. Treatments: ….. 3. Experimental Unit:.. 4. Response variable:… 5 IMPORTANT STEP: HYPOTHESIS TESTING TEST STATISTIC – F TEST F (alfa) – VALUE (CRITICAL VALUE) REJECTION REGION CONCLUSION

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