商业应用统计英文版ch11

1、Applied Business Statistics, 7th ed.by Ken Black,Chapter 11 Analysis of Variance,Understand the differences between various experimental designs and when to use them. Compute and interpret the results of a one-way ANOVA. Compute and interpret the results of a random block design. Compute and interpr

2、et the results of a two-way ANOVA. Understand and interpret interactions between variables. Know when and how to use multiple comparison techniques.,Learning Objectives,Experimental Design A plan and a structure to test hypotheses in whichthe researcher controls or manipulates one or more variables.

3、 It contains independent and dependent variables.,Introduction to Design of Experiments,In an experimental design, an independent variable may be either a treatment variable or a classification variable. Treatment variable - one that the experimenter controlsor modifies in the experiment. Classifica

4、tion variable - a characteristic of the experimental subjects that was present prior to the experiment, and is not a result of the experimenters manipulations or control. Each independent variable has two or more levels, or classifications Levels or Classifications - the subcategories of the indepen

5、dent variable used by the researcher in the experimental design. Independent variables are also referred to as factors.,Independent Variable,Manipulation of the independent variable depends on the concept being studied Researcher studies the phenomenon being studied under conditions of the aspects o

6、f the variable,Independent Variable,Dependent Variable the response to the different levels of the independent variables. Analysis of Variance (ANOVA) a group of statistical techniques used to analyze experimental designs. ANOVA begins with notion that individual items being studied are all the same

7、,Introduction to Design of Experiments,Three Types of Experimental Designs,Completely Randomized Design subjects are assigned randomly to treatments; single independent variable. Randomized Block Design includes a blocking variable; single independent variable. Factorial Experiments two or more inde

8、pendent variables are explored at the same time; every levelof each factor are studied under every level of all other factors.,The completely randomized design contains onlyone independent variable with two or moretreatment levels. If two treatment levels of the independent variableare present, the

9、design is the same used to test the difference in means of two independent populations presented in chapter 10 which used the t test to analyze the data.,Completely Randomized Design,A technique has been developed that analyzes all the sample means at one time and precludes the buildup of error rate

10、: ANOVA. A completely randomized design is analyzed by one way analysis of variance (One-Way Anova).,Completely Randomized Design,One-Way ANOVA: Procedural Overview,Analysis of Variance,The null hypothesis states that the population means for all treatment levels are equal. Even if one of the popula

11、tion means is different from the other, the null hypothesis is rejected. Testing the hypothesis is done by portioning the total variance of data into the following two variances: Variance resulting from the treatment (columns) Error variance or that portion of the total variance unexplained by the t

12、reatment,One-Way ANOVA: Sums of Squares Definitions,Analysis of Variance,The total sum of square of variation is portioned into the sum of squares of treatment columns and the sum of squares of error. ANOVA compares the relative sizes of the treatment variation and the error variation. The error var

13、iation is unaccounted for variation and can be viewed at the point as variation due to individual differences in the groups. If a significant difference in treatment is present, the treatment variation should be large relative to the error variation.,One-Way ANOVA: Computational Formulas,One-Way ANO

14、VA: Computational Formulas,ANOVA is used to determine statistically whetherthe variance between the treatment level meansis greater than the variances within levels(error variance) Assumptions underlie ANOVA Normally distributed populations Observations represent random samples fromthe population Va

15、riances of the population are equal,One-Way ANOVA: Computational Formulas,ANOVA is computed with the three sums of squares: Total Total Sum of Squares (SST); a measure of all variations in the dependent variable Treatment Sum of Squares Columns (SSC); measuresthe variations between treatments or col

16、umns since independent variable levels are present in columns Error Sum of Squares of Error (SSE); yields the variations within treatments (or columns),One-Way ANOVA: Preliminary Calculations,One-Way ANOVA: Sum of Squares Calculations,One-Way ANOVA: Sum of Squares Calculations,Other items MSC Mean S

17、quares of Columns MSE Mean Squares of Error MST Mean Squares of Total F value determined by dividing the treatment variance (MSC) by the error variance (MSE) F value is a ratio of the treatment variance to the error variance,One-Way ANOVA: Computational Formulas,One-Way ANOVA: Mean Square and F Calc

18、ulations,Source of VariancedfSSMSF Between30.236580.07886010.18 Error200.154920.007746 Total230.39150,Analysis of Variance for Valve Openings,F Table,F distribution table is in Table A7. Associated with every F table are two unique df variables: degrees of freedom in the numerator,and degrees of fre

19、edom in the denominator. Stat computer software packages for computing ANOVA usually give a probability for the F value, which allows hypothesis testing decisions for any values of alpha .,df1,df 2,df2,A Portion of the F Table for = 0.05,One-Way ANOVA: Procedural Summary,Excel Output for the ValveOp

20、ening Example,MINITAB Output for the Valve Opening Example,Copyright 2011 John Wiley & Sons, Inc.,Analysis of variance can be used to test hypothesis about the difference in two means. Analysis of data from two samples by both a t testand an ANOVA shows that the observed F values equals the observed

21、 t value squared. F = t2 t test of independent samples actually is a specialcase of one way ANOVA when there are only two treatment levels.,F and t Values,ANOVA techniques useful in testing hypothesisabout differences of means in multiple groups. Advantage: Probability of committing a Type I error i

22、s controlled. Multiple Comparison techniques are used to identify which pairs of means are significantly different given that the ANOVA test reveals overall significance.,Multiple Comparison Tests,Multiple Comparison Tests,Multiple comparisons are used when an overall significant difference between

23、groups has been determined using the F value of the analysis of variance Tukeys honestly significant difference (HSD) test requires equal sample sizes Takes into consideration the number of treatment levels, value of mean square error, and sample size,Tukeys Honestly Significant Difference (HSD) als

24、o known as the Tukeys T method examines the absolute value of all differences between pairs of means from treatment levels to determine if there is a significant difference. Tukey-Kramer Procedure is used when sample sizes are unequal.,Multiple Comparison Tests,Tukeys Honestly SignificantDifference

25、(HSD) Test,A company has three manufacturing plants, and company officials want to determine whether there is a difference in the average age of workers at the three locations. The following data are the ages of five randomly selected workers at each plant. Perform a one-way ANOVA to determine wheth

26、er there is a significant difference in the mean ages of the workers at the three plants. Use = 0.01 and note that the sample sizes are equal.,Demonstration Example Problem,PLANT (Employee Age) 1 2 3 293225 273324 303124 273425 283026 Group Means28.232.024.8 nj555 C = 3 dfE = N - C = 12MSE = 1.63,Da

27、ta from Demonstration Example,Tukeys HSD test,Since sample sizes are equal, Tukeys HSD testscan be used to compute multiple comparison tests between groups. To compute the HSD, the values of MSE, n andq must be determined,q Values for = 0.01,Tukeys HSD Test for the Employee Age Data,The above values

28、 are larger than HSD(2.88). So, the differences between all pairwise group are significant at 0.5 significance level.,Tukey-Kramer Procedure: The Case of Unequal Sample Sizes,Freighter Example: Means and Sample Sizes for the Four Operators,A metal-manufacturing firm wants to test the tensile strengt

29、h of a given metal under varying conditions of temperature. Suppose that in the design phase, the metal is processed under five different temperature conditions and that random samples of size five are taken under each temperature condition. The data follow.,Freighter Example: Means and Sample Sizes

30、 for the Four Operators,Tukey-Kramer Results forthe Four Operators,Similarity The randomized block design is similar to the completely randomized design in that it focuses on one independent variable (treatment variable) of interest. Difference ：Includes a second variable (blocking variable) used to

31、 control for confounding or concomitant variables. They are Variables that are not being controlled by the researcher in the experiment Can have an effect on the outcome of the treatment being studied.,Randomized Block Design,Randomized Block Design,One way to control for these variables is to inclu

32、de them in the experimental design. The randomized block design has the capability of adding one of these variables into the analysis as a blocking variable. A blocking variable is a variable that the researcher wants to control but is not the treatment variable of interest.,A special case of the ra

33、ndomized block design is the repeated measures design. Repeated measures design - is a design in which each block level is an individual item or person, and that person or item is measured across all treatments.,Randomized Block Design,Randomized Block Design,The sum of squares in a completely rando

34、mizeddesign is SST = SSC + SSE In a randomized block design, the sum of squares is SST = SSC + SSR + SSE SSR (blocking effects) comes out of the SSE Some error in variation in randomized design aredue to the blocking effects of the randomized blockdesign, as shown in the next slide,Randomized Block

35、Design Treatment Effects: Procedural Overview,The observed F value for treatments computed using the randomized block design formula is tested by comparing it to a table F value. If the observed F value is greater than the table value, the null hypothesis is rejected for that alpha value. If the F v

36、alue for blocks is greater than the critical F value, the null hypothesis that all block population means are equal is rejected.,Randomized Block Design TreatmentEffects: Procedural Overview,Randomized Block Design: Computational Formulas,As an example of the application of the randomized block desi

37、gn, consider a tire company that developed a new tire. The company conducted tread-wear tests on the tire to determine whether there is a significant difference in tread wear if the average speed with which the automobile is driven varies. The company set up an experiment in which the independent va

38、riable was speed of automobile. There were three treatment levels.,Randomized Block Design: Tread-Wear Example,Randomized Block Design: Tread-Wear Example,N = 15,Randomized Block Design: Sum of Squares Calculations (Part 1),Randomized Block Design: Sum of Squares Calculations (Part 2),Randomized Blo

39、ck Design: MeanSquare Calculations,Source of Variance SSdfMS F Treatment3.48421.74297.45 Block1.5414 0.3852521.72 Error0.1438 0.017875 Total5.17614,Analysis of Variance for the Tread-Wear Example,Randomized Block Design Treatment Effects: Procedural Summary,Randomized Block Design BlockingEffects: P

40、rocedural Overview,Randomized Block Design BlockingEffects: Procedural Overview,Because the observed value of F for treatment (97.45) is greater than this critical F value, the null hypothesis is rejected. At least one of the population means of the treatment levels is not the same as the others. Th

41、ere is a significant difference in tread wear forcars driven at different speeds The F value for treatment with the blocking was 97.45 and without the blocking was 12.44 By using the random block design, a much largerobserved F value was obtained.,Two-Way ANOVA,Definition Some experiments are design

42、ed so that two or more treatments (independent variables) are explored simultaneously. Such experimental designs are referred to as factorial designs In factorial designs, every level of each treatment is studied under the conditions of every level of all other treatments. Factorial designs can be a

43、rranged such that three, four, or n treatments or independent variables are studied simultaneously in the same experiment.(but not in this text),Two-Way ANOVA: Hypotheses,For factorial designs with two factors (independent variables), a two-way analysis of variance (two-way ANOVA) is used to test hy

44、potheses statistically.,Two-Way ANOVA: Hypotheses,The row effects and the column effects are sometimes referred to as the main effects. Although F values are determined for these main effects, an F value is also computed for interaction effects. Interaction Effect occurs when the effects of one trea

45、tment vary according to the levels of treatment of the other effect.,Formulas for Computing aTwo-Way ANOVA,Example: CEOs Dividend Award Decision,At the end of a financially successful fiscal year, CEOs often must decide whether to award a dividend to stockholders or to make a company investment. One factor in this decision would seem to be whether attractive investment opportunities are available. To determine whether this factor is important, business researchers randomly select 24 CEOs and ask them to rate how important “availability of profitable investment opp