Statistics Design of Experiment

Contents An Introduction to Experimental Design Completely Randomized Designs Randomized Block Design Factorial Experiments

An Introduction to Experimental Design Statistical studies can be classified as being either experimental or observational. In an experimental study, one or more factors are controlled so that data can be obtained about how the factors influence the variables of interest. In an observational study, no attempt is made to control the factors.

An Introduction to Experimental Design Cause-and-effect relationships are easier to establish in experimental studies than in observational studies.

An Introduction to Experimental Design A factor is a variable that the experimenter has selected for investigation. A treatment is a level of a factor. Experimental units are the objects of interest in the experiment.

An Introduction to Experimental Design Example: Suppose we want to examine if there is a difference among working methods A, B, and C. 18 workers are randomly chosen. factor is the working methods. treatment is one of A, B, and C. Experimental units are the workers chosen

An Introduction to Experimental Design A completely randomized design is an experimental design in which the treatments are randomly assigned to the experimental units. If the experimental units are heterogeneous, blocking can be used to form homogeneous groups, resulting in a randomized block design.

Completely Randomized Design Recall Simple Random Sampling Finite populations are often defined by lists such as: Organization membership roster Credit card account numbers Inventory product numbers

Completely Randomized Design A simple random sample of size n from a finite population of size N is a sample selected such that each possible sample of size n has the same probability of being selected. Replacing each sampled element before selecting subsequent elements is called sampling with replacement. Sampling without replacement is the procedure used most often.

Completely Randomized Design In large sampling projects, computer-generated random numbers are often used to automate the sample selection process. Excel provides a function for generating random numbers in its worksheets. Infinite populations are often defined by an ongoing process whereby the elements of the population consist of items generated as though the process would operate indefinitely.

Completely Randomized Design A simple random sample from an infinite population is a sample selected such that the following conditions are satisfied. Each element selected comes from the same population. Each element is selected independently.

Completely Randomized Design Random Numbers: the numbers in the table are random, these four-digit numbers are equally likely.

Completely Randomized Design Most experiments have critical error on random sampling. Ex: sampling 8 samples from a production line in one day Wrong method: Get one sample every 3 hours not random!

Completely Randomized Design Ex: sampling 8 samples from a production line Correct method: You can get one sample at each 3 hours interval but not every 3 hours  correct but not a simple random sampling Get 8 samples in 24 hours Maximum population is 24, getting 8 samples two digits 63, 27, 15, 99, 86, 71, 74, 45, 10, 21, 51, … Larger than 24 is discarded So eight samples are collected at: 15, 10, 21, … hour

Completely Randomized Design In Completely Randomized Design, samples are randomly collected by simple random sampling method. Only one factor is concerned in Completely Randomized Design, and k levels in this factor.

Completely Randomized Design Data: Treatment Observations sample sample (level) 1 2 i … nj mean SD 1 x11 x12 x1i … s1 2 x21 x22 x1i … s2 . j xj1 xj2 xji … sj k xk1 xk2 xki … sk

Completely Randomized Design Between-Treatments Estimate of Population Variance The between-samples estimate of σ2 is referred to as the mean square due to treatments (MSTR). denominator is the degrees of freedom associated with SSTR numerator is called the sum of squares due to treatments (SSTR)

Completely Randomized Design Within-Treatments Estimate of Population Variance. The second estimate of σ2, the within-samples estimate, is referred to as the mean square due to error (MSE). denominator is the degrees of freedom associated with SSE numerator is called the sum of squares due to error (SSE)

Completely Randomized Design ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Treatments SSTR k - 1 Error SSE nT - k Total SST nT - 1

Completely Randomized Design--Example Wax Type 3 Type 1 vpe 2 Example: 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.

Completely Randomized Design--Example Wax Type 3 Type 1 vpe 2 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.

Completely Randomized Design--Example Wax Type 3 Type 1 vpe 2 The number of times each car went through the carwash is shown on the next slide. AutoShine, Inc. must decide which wax to market. Are the three waxes equally effective?

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 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 Sample Mean 29.0 30.4 30.0 Sample Variance 2.5 3.3 2.5

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 Hypotheses H0: 1=2=3 Ha: Not all the means are equal where: μ1 = mean number of washes for Type 1 wax μ2 = mean number of washes for Type 2 wax μ3 = mean number of washes for Type 3 wax

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 Mean Square Between Treatments = (29 + 30.4 + 30)/3 = 29.8 Because the sample sizes are all equal: SSTR = 5(29–29.8)2 + 5(30.4–29.8)2 + 5(30–29.8)2 = 5.2 MSTR = 5.2/(3 - 1) = 2.6 Mean Square Error SSE = 4(2.5) + 4(3.3) + 4(2.5) = 33.2 MSE = 33.2/(15 - 3) = 2.77

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 Rejection Rule p-Value Approach: Reject H0 if p-value < .05 Critical Value Approach: Reject H0 if F > 3.89 where F.05 = 3.89 is based on an F distribution with 2 numerator degrees of freedom and 12 denominator degrees of freedom

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 Test Statistic F = MSTR/MSE = 2.6/2.77 = .939 Conclusion The p-value is greater than .10, where F = 2.81. (Excel provides a p-value of .42.) Therefore, we cannot reject H0. There is insufficient evidence to conclude that the mean number of washes for the three wax types are not all the same.

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean F Treatments Error Total 2 14 5.2 33.2 38.4 12 2.60 2.77 .939

Completely Randomized Design--Example Wax Type 3 vpe 2 Type 1 Minitab Outputs One-way ANOVA: Observation versus Type Source DF SS MS F P Type 2 5.20 2.60 0.94 0.418 Error 12 33.20 2.77 Total 14 38.40 S = 1.663 R-Sq = 13.54% R-Sq(adj) = 0.00%

Randomized Block Design In Completely Randomized Experimental Design, the test for a difference among treatment means, we compute a F value by using the ratio A problem can arise whenever differences due to extraneous factors (not considered in the experiment) cause the MSE become large.

Randomized Block Design In previous example, the time after wax is produced is not considered, but it could make differences. To examine the effects on shift in a production line, shift is a factor but gender is not considered. It could also make differences.

Randomized Block Design To isolate the effects of a known effect caused by extraneous factors, one can “block” it by using the Randomized Block Design. After blocking the extraneous effect, the value of MSE becomes smaller and the ability of F test increases.

Randomized Block Design ANOVA Procedure For a randomized block design the sum of squares total (SST) is partitioned into three groups: sum of squares due to treatments, sum of squares due to blocks, and sum of squares due to error. The total degrees of freedom, nT - 1, are partitioned such that k - 1 degrees of freedom go to treatments, b - 1 go to blocks, and (k - 1)(b - 1) go to the error term. SST = SSTR + SSBL + SSE

Randomized Block Design ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Treatments SSTR k - 1 Blocks SSBL b - 1 Error SSE (k – 1)(b – 1) Total SST nT - 1

Randomized Block Design rescent 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.

Randomized Block Design rescent Randomized Block Design 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 rescent Randomized Block Design Type of Gasoline (Treatment) Automobile (Block) Block Means Blend X Blend Y Blend Z 1 2 3 4 5 31 30 29 33 26 30 29 31 25 30 29 28 26 30.333 29.333 28.667 31.000 25.667 Treatment Means 29.8 28.8 28.4

Randomized Block Design rescent Randomized Block Design Mean Square Due to Treatments The overall sample mean is 29. Thus, SSTR = 5[(29.8 - 29)2 + (28.8 - 29)2 + (28.4 - 29)2] = 5.2 MSTR = 5.2/(3 - 1) = 2.6 Mean Square Due to Blocks SSBL = 3[(30.333 - 29)2 + . . . + (25.667 - 29)2] = 51.33 MSBL = 51.33/(5 - 1) = 12.8 SSE = 62 - 5.2 - 51.33 = 5.47 Mean Square Due to Error MSE = 5.47/[(3 - 1)(5 - 1)] = .68

Randomized Block Design rescent ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Treatments 5.20 2 2.60 3.82 Blocks 51.33 4 12.80 Error 5.47 8 .68 Total 62.00 14

Randomized Block Design rescent Randomized Block Design Rejection Rule p-Value Approach: Reject H0 if p-value < .05 Critical Value Approach: Reject H0 if F > 4.46 For  = .05, F.05 = 4.46 (2 d.f. numerator and 8 d.f. denominator)

Randomized Block Design rescent Randomized Block Design Test Statistic F = MSTR/MSE = 2.6/.68 = 3.82 Conclusion The p-value is greater than .05 (where F = 4.46) and less than .10 (where F = 3.11). (Excel provides a p-value of .07). Therefore, we cannot reject H0. There is insufficient evidence to conclude that the miles per gallon ratings differ for the three gasoline blends.

Factorial Experiments In some experiments we want to draw conclusions about more than one variable or factor. Factorial experiments and their corresponding ANOVA computations are valuable designs when simultaneous conclusions about two or more factors are required.

Factorial Experiments The term factorial is used because the experimental conditions include all possible combinations of the factors. For example, for a levels of factor A and b levels of factor B, the experiment will involve collecting data on ab treatment combinations.

Two-Factor Factorial Experiment ANOVA Procedure The ANOVA procedure for the two-factor factorial experiment is similar to the completely randomized experiment and the randomized block experiment We again partition the sum of squares total (SST) into its sources. SST = SSA + SSB + SSAB + SSE

Two-Factor Factorial Experiment SST = SSA + SSB + SSAB + SSE The total degrees of freedom, nT - 1, are partitioned such that (a – 1) d.f go to Factor A, (b – 1) d.f go to Factor B, (a – 1)(b – 1) d.f. go to Interaction, and ab(r – 1) go to Error.

Two-Factor Factorial Experiment Source of Variation Sum of Squares Degrees of Freedom Mean F Factor A Error Total a - 1 nT - 1 SSA SSE SST Factor B SSB b - 1 ab(r – 1) Interaction SSAB (a – 1)(b – 1)

Two-Factor Factorial Experiment Step 1 Compute the total sum of squares Step 2 Compute the sum of squares for factor A Step 3 Compute the sum of squares for factor B

Two-Factor Factorial Experiment Step 4 Compute the sum of squares for interaction Step 5 Compute the sum of squares due to error SSE = SST – SSA – SSB - SSAB

Two-Factor Factorial Experiment Example: State of Ohio Wage Survey A survey was conducted of hourly wages for a sample of workers in two industries at three locations in Ohio. Part of the purpose of the survey was to determine if differences exist in both industry type and location. The sample data are shown on the next slide.

Two-Factor Factorial Experiment Example: State of Ohio Wage Survey Industry Cincinnati Cleveland Columbus I 12.10 11.80 12.90 11.20 12.70 12.00 12.20 II 12.40 12.60 13.00 12.50

Two-Factor Factorial Experiment Factors Factor A: Industry Type (2 levels) Factor B: Location (3 levels) Replications Each experimental condition is repeated 3 times

Two-Factor Factorial Experiment ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Squares F Factor A .50 1 .50 4.19 Factor B 1.12 2 .56 4.69 Interaction .37 2 .19 1.55 Error 1.43 12 .12 Total 3.42 17

Two-Factor Factorial Experiment Conclusions Using the p-Value Approach Industries: p-value = .06 > a = .05 Mean wages do not differ by industry type. Locations: p-value = .03 < a = .05 Mean wages differ by location. Interaction: p-value = .25 > a = .05 Interaction is not significant. (p-values were found using Excel)

Two-Factor Factorial Experiment Conclusions Using the Critical Value Approach Industries: F = 4.19 < Fa = 4.75 Mean wages do not differ by industry type. Locations: F = 4.69 > Fa = 3.89 Mean wages differ by location. Interaction: F = 1.55 < Fa = 3.89 Interaction is not significant.