CJDM Lecture Notes # 2 - ANALYSIS OF VARIANCE IN EXPERIMENTS Independent Variable: Presumed cause for the effect. In marketing research, normally these variables will be in marketers’ control. (for example, product, price, promotion, and channel). Dependent Variable: Presumed Effect of the independent variable. Normally these are consumers’ response to marketing programs. (for example, sales, consumer attitude toward a brand, and consumer beliefs about various attributes of the brand). Experiments test the effect of one or more independent variable on one or more dependent variable. In experiments, typically, the independent variables are manipulated and dependent variables are measured. The manipulation can be within groups or between groups. ANALYSIS OF VARIANCE Variance – What is the variance in the following set of numbers? 4, 2, 6, 5, 3 Now, what is the variance in the following set of numbers? 4, 4, 5, 4, 3 Why is the concept of variance important? What is the similarity between the two sets? Concept of Variance in Experimentation Logic behind ANOVA or F test – Experimental Variance is Greater than Error Variance then the independent variable has an effect on the dependent variable. Consider the ice cream flavor experiment discussed earlier. There are three groups. If we want to examine the effect of flavor on consumer preference, we need to conduct an ANOVA (analysis of variance or F test). Step1: Compute Mean for each group Step2: Compute Grand Mean (Mean of Mean mango, Mean vanilla, and Mean Strawberry). Step 3: Compute Total Experimental or Effect Variance (Also called Between- subjects sum of square) = Take Squared Deviation of mean i from the Grand mean for each group, multiply by the number of observations in that group, and then sum it across three groups. Step 4: Compute Total Error Variance (also called Within-group sum of square) = Take the squared deviation of each individual observation from the respective group mean and sum these across the three groups. Step 5: Compute Mean Variance (Mean Square Effect) = Total Effect Variance/ Effect Degree of Freedom. Effect Degree of Freedom = number of levels of the independent variable –1. In our case, because there is only one independent variable at three levels, the effect degree of freedom = number of groups – 1. Step 6: Compute Mean Error Variance (Mean Square Error) = Total Error Variance/Error Degree of Freedom. Error Degree of Freedom = (n-1) * Number of Levels of the Independent Variable. Step 7: Compute F = Mean Square Effect/Mean Square Error Step 8: Compare the Computed F with the table F for the respective level of significance and respective degrees of freedom for the numerator (Effect) and Denominator (Error). Step 9: Apply the rule we talked under t-test for a decision as to whether accept or reject the null hypothesis. We can also do the follow up for each pair. For example mango versus vanilla differences can also be tested by F test (although t test is simpler). Here, the Mean Square Error remains the same as in the three groups F test. However, you need to compute the Mean Square effect fresh by including only the two groups that are relevant (here mango and vanilla). The same applies when you are testing the difference between mango and strawberry. Now, try the following problem at home. Before introducing a new CD player (Brand X), a marketer wanted to compare its performance with that of the leading CD players (Sony and Akai). Five consumers listened to the same song from one of three CD players and responded to the question “How do you rate the overall performance of the CD player?” which was measured on 1 (bad) to 10 (good) scale. The scores are given below. Sony Akai Brand X 5 6 6 4 5 6 4 6 7 5 7 5 6 7 8 What is the independent variable in this problem? What is the dependent variable? Test the null hypothesis that will include all three groups via an F-test. Test the null hypothesis that will include only Sony and Brand X via an F-test.