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  1. 1. Marketing Research Analysis of Variance Lecture 8 Hamendra Dangi [email_address] 9968316938 H.K Dangi, FMS
  2. 2. Session Break Up <ul><li>ANOVA and Its relationship with other technique </li></ul><ul><li>One Way ANOVA </li></ul><ul><li>Practical Exercise </li></ul>H.K Dangi, FMS
  3. 3. Relationship Among Techniques <ul><li>Analysis of variance (ANOVA) is used as a test of means for two or more populations. The null hypothesis, typically, is that all means are equal. </li></ul><ul><li>Analysis of variance must have a dependent variable that is metric (measured using an interval or ratio scale). </li></ul><ul><li>There must also be one or more independent variables that are all categorical (nonmetric). Categorical independent variables are also called factors . </li></ul>H.K Dangi, FMS
  4. 4. Relationship Among Techniques <ul><li>A particular combination of factor levels, or categories, is called a treatment. </li></ul><ul><li>One-way analysis of variance involves only one categorical variable, or a single factor. In one-way analysis of variance, a treatment is the same as a factor level. </li></ul><ul><li>If two or more factors are involved, the analysis is termed n -way analysis of variance . </li></ul><ul><li>If the set of independent variables consists of both categorical and metric variables, the technique is called analysis of covariance (ANCOVA) . In this case, the categorical independent variables are still referred to as factors, whereas the metric-independent variables are referred to as covariates . </li></ul>H.K Dangi, FMS
  5. 5. Relationship Amongst Test, Analysis of Variance, Analysis of Covariance, & Regression H.K Dangi, FMS One Independent One or More Metric Dependent Variable t Test Binary Variable One-Way Analysis of Variance One Factor N-Way Analysis of Variance More than One Factor Analysis of Variance Categorical: Factorial Analysis of Covariance Categorical and Interval Regression Interval Independent Variables
  6. 6. Question for Discussion <ul><li>Identify appropriate test in the following cases </li></ul><ul><li>A comparison of mean salary of IIT and IIM grads </li></ul><ul><li>A study involving age and drinking habits </li></ul><ul><li>A comparative study performance of commerce , Engineering and Others grads in Semester Exam </li></ul><ul><li>Mobile uses in three age group with disposable income </li></ul>H.K Dangi, FMS
  7. 7. One-Way Analysis of Variance <ul><li>Marketing researchers are often interested in examining the differences in the mean values of the dependent variable for several categories of a single independent variable or factor. For example: </li></ul><ul><li>Do the various segments differ in terms of their volume of product consumption? </li></ul><ul><li>Do the brand evaluations of groups exposed to different commercials vary? </li></ul><ul><li>What is the effect of consumers' familiarity with the store (measured as high, medium, and low) on preference for the store? </li></ul>H.K Dangi, FMS
  8. 8. Statistics Associated with One-Way Analysis of Variance <ul><li>eta 2 ( 2 ) . The strength of the effects of X (independent variable or factor) on Y (dependent variable) is measured by eta 2 ( 2 ) . The value of 2 varies between 0 and 1. </li></ul><ul><li>F statistic . The null hypothesis that the category means are equal in the population is tested by an F statistic based on the ratio of mean square related to X and mean square related to error. </li></ul><ul><li>Mean square . This is the sum of squares divided by the appropriate degrees of freedom. </li></ul>H.K Dangi, FMS
  9. 9. Statistics Associated with One-Way Analysis of Variance <ul><li>SS between . Also denoted as SS x , this is the variation in Y related to the variation in the means of the categories of X . This represents variation between the categories of X , or the portion of the sum of squares in Y related to X . </li></ul><ul><li>SS within . Also referred to as SS error , this is the variation in Y due to the variation within each of the categories of X . This variation is not accounted for by X . </li></ul><ul><li>SS y . This is the total variation in Y . </li></ul>H.K Dangi, FMS
  10. 10. Conducting One-Way ANOVA H.K Dangi, FMS Interpret the Results Identify the Dependent and Independent Variables Decompose the Total Variation Measure the Effects Test the Significance Fig. 16.2
  11. 11. Conducting One-Way Analysis of Variance Decompose the Total Variation <ul><li>The total variation in Y , denoted by SS y , can be decomposed into two components: </li></ul><ul><li>SS y = SS between + SS within </li></ul><ul><li>where the subscripts between and within refer to the categories of X . SS between is the variation in Y related to the variation in the means of the categories of X . For this reason, SS between is also denoted as SS x . SS within is the variation in Y related to the variation within each category of X . SS within is not accounted for by X . Therefore it is referred to as SS error . </li></ul>H.K Dangi, FMS
  12. 12. <ul><li>The total variation in Y may be decomposed as: </li></ul><ul><li>SS y = SS x + SS error </li></ul><ul><li>where </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>Y i = individual observation </li></ul><ul><li>j = mean for category j </li></ul><ul><li>= mean over the whole sample, or grand mean </li></ul><ul><li>Y ij = i th observation in the j th category </li></ul>Conducting One-Way Analysis of Variance Decompose the Total Variation H.K Dangi, FMS Y Y S S y = ( Y i - Y ) 2  i = 1 N S S x = n ( Y j - Y ) 2  j = 1 c S S e r r o r =  i n ( Y i j - Y j ) 2  j c
  13. 13. Decomposition of the Total Variation: One-Way ANOVA H.K Dangi, FMS Independent Variable X Total Categories Sample X 1 X 2 X 3 … X c Y 1 Y 1 Y 1 Y 1 Y 1 Y 2 Y 2 Y 2 Y 2 Y 2 : : : : Y n Y n Y n Y n Y N Y 1 Y 2 Y 3 Y c Y Within Category Variation =SS within Between Category Variation = SS between Total Variation =SS y Category Mean Table 16.1
  14. 14. Conducting One-Way Analysis of Variance <ul><li>In analysis of variance, we estimate two measures of variation: within groups ( SS within ) and between groups ( SS between ). Thus, by comparing the Y variance estimates based on between-group and within-group variation, we can test the null hypothesis. </li></ul><ul><li>Measure the Effects </li></ul><ul><li>The strength of the effects of X on Y are measured as follows: </li></ul><ul><li>  </li></ul><ul><li> 2 = SS x / SS y = ( SS y - SS error )/ SS y </li></ul><ul><li>  </li></ul><ul><li>The value of 2 varies between 0 and 1. </li></ul>H.K Dangi, FMS
  15. 15. Conducting One-Way Analysis of Variance Test Significance <ul><li>In one-way analysis of variance, the interest lies in testing the null hypothesis that the category means are equal in the population. </li></ul><ul><li>  </li></ul><ul><li>H 0 : µ 1 = µ 2 = µ 3 = ........... = µ c </li></ul><ul><li>  </li></ul><ul><li>Under the null hypothesis, SS x and SS error come from the same source of variation. In other words, the estimate of the population variance of Y, </li></ul><ul><li>= SS x /( c - 1) </li></ul><ul><li>= Mean square due to X </li></ul><ul><li>= MS x </li></ul><ul><li>or </li></ul><ul><li>= SS error /( N - c ) </li></ul><ul><li>= Mean square due to error </li></ul><ul><li>= MS error </li></ul>H.K Dangi, FMS S y 2 S y 2
  16. 16. Conducting One-Way Analysis of Variance Test Significance <ul><li>The null hypothesis may be tested by the F statistic </li></ul><ul><li>based on the ratio between these two estimates: </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>This statistic follows the F distribution, with ( c - 1) and </li></ul><ul><li>( N - c ) degrees of freedom (df). </li></ul>H.K Dangi, FMS F = S S x / ( c - 1 ) S S e r r o r / ( N - c ) = M S x M S e r r o r
  17. 17. Conducting One-Way Analysis of Variance Interpret the Results <ul><li>If the null hypothesis of equal category means is not rejected, then the independent variable does not have a significant effect on the dependent variable. </li></ul><ul><li>On the other hand, if the null hypothesis is rejected, then the effect of the independent variable is significant. </li></ul><ul><li>A comparison of the category mean values will indicate the nature of the effect of the independent variable. </li></ul>H.K Dangi, FMS
  18. 18. Illustrative Applications of One-Way Analysis of Variance <ul><li>We illustrate the concepts discussed in this chapter using the data presented in Table 16.2. </li></ul><ul><li>The department store is attempting to determine the effect of in-store promotion (X) on sales (Y). For the purpose of illustrating hand calculations, the data of Table 16.2 are transformed in Table 16.3 to show the store sales ( Y ij ) for each level of promotion. </li></ul><ul><li>  </li></ul><ul><li>The null hypothesis is that the category means are equal: </li></ul><ul><li>H 0 : µ 1 = µ 2 = µ 3 . </li></ul>H.K Dangi, FMS
  19. 19. Effect of Promotion and Clientele on Sales H.K Dangi, FMS Table 16.2
  20. 20. Illustrative Applications of One-Way Analysis of Variance <ul><li>EFFECT OF IN-STORE PROMOTION ON SALES </li></ul><ul><li>Store Level of In-store Promotion </li></ul><ul><li>No. High Medium Low </li></ul><ul><li>Normalized Sales </li></ul><ul><li>1 10 8 5 </li></ul><ul><li>2 9 8 7 </li></ul><ul><li>3 10 7 6 </li></ul><ul><li>4 8 9 4 </li></ul><ul><li>5 9 6 5 </li></ul><ul><li>6 8 4 2 </li></ul><ul><li>7 9 5 3 </li></ul><ul><li>8 7 5 2 </li></ul><ul><li>9 7 6 1 </li></ul><ul><li>10 6 4 2 </li></ul><ul><li>  </li></ul><ul><li>Column Totals 83 62 37 </li></ul><ul><li>Category means: j 83/10 62/10 37/10 </li></ul><ul><li> = 8.3 = 6.2 = 3.7 </li></ul><ul><li>Grand mean, = (83 + 62 + 37)/30 = 6.067 </li></ul>H.K Dangi, FMS Table 16.3 Y Y
  21. 21. <ul><li>To test the null hypothesis, the various sums of squares are computed as follows: </li></ul><ul><li>  </li></ul><ul><li>SSy = (10-6.067) 2 + (9-6.067) 2 + (10-6.067) 2 + (8-6.067) 2 + (9-6.067) 2 </li></ul><ul><li>+ (8-6.067) 2 + (9-6.067) 2 + (7-6.067) 2 + (7-6.067) 2 + (6-6.067) 2 </li></ul><ul><li>+ (8-6.067) 2 + (8-6.067) 2 + (7-6.067) 2 + (9-6.067) 2 + (6-6.067) 2 </li></ul><ul><li>(4-6.067) 2 + (5-6.067) 2 + (5-6.067) 2 + (6-6.067) 2 + (4-6.067) 2 </li></ul><ul><li>+ (5-6.067) 2 + (7-6.067) 2 + (6-6.067) 2 + (4-6.067) 2 + (5-6.067) 2 </li></ul><ul><li>+ (2-6.067) 2 + (3-6.067) 2 + (2-6.067) 2 + (1-6.067) 2 + (2-6.067) 2 </li></ul><ul><li>=(3.933) 2 + (2.933) 2 + (3.933) 2 + (1.933) 2 + (2.933) 2 </li></ul><ul><li>+ (1.933) 2 + (2.933) 2 + (0.933) 2 + (0.933) 2 + (-0.067) 2 </li></ul><ul><li>+ (1.933) 2 + (1.933) 2 + (0.933) 2 + (2.933) 2 + (-0.067) 2 </li></ul><ul><li>(-2.067) 2 + (-1.067) 2 + (-1.067) 2 + (-0.067) 2 + (-2.067) 2 </li></ul><ul><li>+ (-1.067) 2 + (0.9333) 2 + (-0.067) 2 + (-2.067) 2 + (-1.067) 2 </li></ul><ul><li>+ (-4.067) 2 + (-3.067) 2 + (-4.067) 2 + (-5.067) 2 + (-4.067) 2 </li></ul><ul><li>= 185.867 </li></ul>Illustrative Applications of One-Way Analysis of Variance H.K Dangi, FMS
  22. 22. <ul><li>SSx = 10(8.3-6.067) 2 + 10(6.2-6.067) 2 + 10(3.7-6.067) 2 </li></ul><ul><li>= 10(2.233) 2 + 10(0.133) 2 + 10(-2.367) 2 </li></ul><ul><li>= 106.067 </li></ul><ul><li>  </li></ul><ul><li>SSerror = (10-8.3) 2 + (9-8.3) 2 + (10-8.3)2 + (8-8.3)2 + (9-8.3)2 </li></ul><ul><li>+ (8-8.3) 2 + (9-8.3)2 + (7-8.3)2 + (7-8.3)2 + (6-8.3)2 </li></ul><ul><li>+ (8-6.2) 2 + (8-6.2)2 + (7-6.2)2 + (9-6.2)2 + (6-6.2)2 </li></ul><ul><li>+ (4-6.2) 2 + (5-6.2)2 + (5-6.2)2 + (6-6.2)2 + (4-6.2)2 </li></ul><ul><li>+ (5-3.7) 2 + (7-3.7)2 + (6-3.7)2 + (4-3.7)2 + (5-3.7)2 </li></ul><ul><li>+ (2-3.7) 2 + (3-3.7)2 + (2-3.7)2 + (1-3.7)2 + (2-3.7)2 </li></ul><ul><li>  </li></ul><ul><li>= (1.7) 2 + (0.7) 2 + (1.7) 2 + (-0.3) 2 + (0.7) 2 </li></ul><ul><li>+ (-0.3) 2 + (0.7) 2 + (-1.3) 2 + (-1.3) 2 + (-2.3) 2 </li></ul><ul><li>+ (1.8) 2 + (1.8) 2 + (0.8) 2 + (2.8) 2 + (-0.2) 2 </li></ul><ul><li>+ (-2.2) 2 + (-1.2) 2 + (-1.2) 2 + (-0.2) 2 + (-2.2) 2 </li></ul><ul><li>+ (1.3) 2 + (3.3) 2 + (2.3) 2 + (0.3) 2 + (1.3) 2 </li></ul><ul><li>+ (-1.7) 2 + (-0.7) 2 + (-1.7) 2 + (-2.7) 2 + (-1.7) 2 </li></ul><ul><li>  = 79.80 </li></ul>Illustrative Applications of One-Way Analysis of Variance H.K Dangi, FMS
  23. 23. <ul><li>It can be verified that </li></ul><ul><li>SSy = SSx + SSerror </li></ul><ul><li>as follows: </li></ul><ul><li>185.867 = 106.067 +79.80 </li></ul><ul><li>The strength of the effects of X on Y are measured as follows: </li></ul><ul><li>2 = SSx / SSy </li></ul><ul><li>= 106.067/185.867 </li></ul><ul><li>= 0.571 </li></ul><ul><li>  In other words, 57.1% of the variation in sales (Y) is accounted for by in-store promotion (X), indicating a modest effect. The null hypothesis may now be tested. </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>  </li></ul><ul><li>= 17.944 </li></ul>Illustrative Applications of One-Way Analysis of Variance H.K Dangi, FMS  F = S S x / ( c - 1 ) S S e r r o r / ( N - c ) = M S X M S e r r o r F = 106.067/(3-1) 79.800/(30-3)
  24. 24. <ul><li>From Table 5 in the Statistical Appendix we see that for 2 and 27 degrees of freedom, the critical value of F is 3.35 for . Because the calculated value of F is greater than the critical value, we reject the null hypothesis. </li></ul><ul><li>We now illustrate the analysis of variance procedure using a computer program. The results of conducting the same analysis by computer are presented in Table 16.4. </li></ul>Illustrative Applications of One-Way Analysis of Variance H.K Dangi, FMS   = 0.05
  25. 25. One-Way ANOVA: Effect of In-store Promotion on Store Sales H.K Dangi, FMS Table 16.4 Cell means Level of Count Mean Promotion High (1) 10 8.300 Medium (2) 10 6.200 Low (3) 10 3.700 TOTAL 30 6.067 Source of Sum of d f Mean F ratio F prob. Variation squares square Between groups 106.067 2 53.033 17.944 0.000 (Promotion) Within groups 79.800 27 2.956 (Error) TOTAL 185.867 29 6.409
  26. 26. <ul><li>A marketing research company wants to test the hypotheses , that in the population there is no difference in the importance attached to shopping by consumer living in the North, Southern ,eastern and western India. A study conducted and analysis of variance is used to analyze the data </li></ul>H.K Dangi, FMS
  27. 27. <ul><li>Source df SS MS F F prob </li></ul><ul><li>Between 3 70.212 23.4 1.12 0.3 </li></ul><ul><li>Within 996 20812.4 20.896 </li></ul><ul><li>Is there sufficient evidence to reject the null hypotheses ? </li></ul><ul><li>What conclusion can be drawn </li></ul><ul><li>What was the total sample size in this study </li></ul><ul><li>MBA(PT) –Nov 2007 </li></ul>H.K Dangi, FMS
  28. 28. <ul><li>No, with p = 0.3, we cannot reject the null hypothesis. </li></ul><ul><li>(b) No difference in shopping behavior exists between the regions. </li></ul><ul><li>C 1000 </li></ul>H.K Dangi, FMS

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