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  1. 1. ANOVA(Analysis of Variance)<br />Presented byPrasoonJain 2010161RachitAnand 2010169Rajesh Bhatia 2010173Sandeep Kumar Singh 2010201VarunBhargava 2010258<br />
  2. 2. Introduction<br /><ul><li>ANOVA is an abbreviation for the full name of the method: Analysis Of Variance</li></ul> Invented by R.A. Fisher in the 1920’s.<br /><ul><li>ANOVA is used to test the significance of the difference between more than two sample means and to make inferences about whether our samples are drawn from population having same means.</li></li></ul><li>Theoretical Example:Sections At IMT<br />Sec: A Sec: B Sec: C<br />SEC: D<br />
  3. 3. Continued.. <br />ANOVA is comparison of means. Each possible value of a factor or combination of factor is a treatment.<br />The ANOVA is a powerful and common statistical procedure in the social sciences. It can handle a variety of situations.<br />
  4. 4. Why ANOVA instead of multiple t-tests?<br />If you are comparing means between more than two groups, why not just do several two sample t-tests to compare the mean from one group with the mean from each of the other groups?<br />Before ANOVA, this was the only option available to compare means between more than two groups.<br />The problem with the multiple t-tests approach is that as the number of groups increases, the number of two sample t-tests also increases. <br />As the number of tests increases the probability of making a Type I error also increases.<br />
  5. 5. Analysis of Variance : Assumptions<br /><ul><li>We assume independent random sampling from each of the r populations
  6. 6. We assume that the r populations under study:
  7. 7. are normally distributed,
  8. 8. with means mi that may or may not be equal,
  9. 9. but with equal variances, si2.</li></ul>s<br />m1<br />m2<br />m3<br />Population 1<br />Population 2<br />Population 3<br />
  10. 10. ANOVA Hypotheses<br />The Null hypothesis for ANOVA is that the means for all groups are equal:<br />The Alternative hypothesis for ANOVA is that at least two of the means are not equal.<br />The test statistic for ANOVA is the ANOVA <br /> F-statistic.<br />
  11. 11. SST = SSW + SSB<br />This partitioned relationship is also true for the squared differences: <br />The variability between each observation and the overall (or grand) mean is measured by the ‘sum of squares total’ (SST)<br />The variability within groups is measured by the ‘sum of squares within’ (SSW).<br />MSW = SSW/(n-k)<br />The variability between groups is measured by the ‘sum of squares between’ (SSB).<br />MSB = SSB/(k-1)<br />
  12. 12. One & N way ANOVA<br />One way ANOVA<br /> Analysis of variance, so named because they can consider only one independent variable at a time.<br />N way ANOVA<br /> As its name suggests, this is a procedure that allows you to examine the effects of n independent variables concurrently.<br />
  13. 13. ANOVA example: mobility<br />The hypothetical data below represent mobility scores (higher score indicates improved mobility) for 3 groups of patients:<br />Control group did not receive any therapy<br />Treatment group 1 received physical therapy, <br />Treatment group 2 received counseling and physical therapy.<br />Assume that the mobility scores are normally distributed. <br />ControlTrt. 1Trt. 2<br /> 35 38 47<br /> 38 43 53<br />42 45 42<br /> 34 52 45<br /> 28 40 46<br /> 39 46 37<br />. <br />
  14. 14. Example continued..<br />State the Hypotheses<br />Null Hypothesis: m control = m trt 1 = m trt 2<br />Alternative Hypothesis: at least two of the means <br /> (m control , m trt 1 , m trt 2 ) are not equal<br />ANOVA is always a two-sided test<br />ANOVA will identify if at least two means are significantly different but will not identify which two (or more) means are different.<br />Set the significance level<br />a = 0.05.<br />
  15. 15. Example continued..<br />Calculated SD for each group<br />Control: 4.86<br />Trt 1: 4.94<br />Trt 2: 5.33<br />Calculated ratio:<br /> largest SD / smallest SD = 5.33 / 4.86 = 1.1<br />Since the ratio < 2, assume equality of variance between groups<br />
  16. 16. Example continued..<br />Overall mean = average of all 18 observations = 41.7<br />Group means = average of the observations in each group<br />Control mean = 36<br />Treatment 1 mean = 44<br />Treatment 2 mean = 45<br />
  17. 17. Calculate the Within Sum of Squares: SSW<br />Square the difference between each observation and it’s group mean and sum the 18 terms. <br />The SSW for the control group (mean = 36)<br />(35 - 36)2 + (38 – 36)2 + (42 - 36)2 + (34 – 36)2 + (28 – 36)2 + (39 – 36)2 = 118<br />The SSW for the treatment 1 group (mean = 44)<br />(38 - 44)2 + (43 – 44)2 + (45 - 44)2 + (52 – 44)2 + (40 – 44)2 + (46 – 44)2 = 122<br />The SSW for the treatment 2 group (mean = 45)<br />(47 - 45)2 + (53 – 45)2 + (42 - 45)2 + (45 – 45)2 + (46 – 45)2 + (37 – 45)2 = 142<br />SSW = sum of SSW for each group = 118 + 122 + 142 = 382<br />Now MSW = SSW/N-K so, 382/15 = 25.47<br />
  18. 18. Calculate the Between Sum of Squares: SSB<br />The overall mean = 41.7<br />The three group means are:<br />Control: mean = 36<br />Treatment 1: mean = 44<br />Treatment 2: mean = 45<br />For each group square the difference between the group mean and the overall mean and multiply by the group sample size, then sum these 3 terms for the SSB:<br />SSB = 6*(36 - 41.7)2 + 6*(44 – 41.7)2 + 6*(45 - 41.7)2 = 292<br />MSB = SSB/K-1 so MSB = 292/2<br />MSB = 146<br />
  19. 19. Calculate the ANOVA F-statistic<br />The ANOVA F-statistic = MSB/MSW<br />The ANOVA F-statistic will be large when there is more variability between the groups than within the groups. <br />If the variability between groups and within groups is approximately equal the ANOVA F-statistic will be small (close to 1.0)<br />The Null hypothesis of equal means between groups is rejected if the F-statistic is large enough. <br />ANOVA F-statistic for example = 146/25.47 = 5.73<br />
  20. 20. Sampling Distribution of ANOVA F-statistic<br />The sampling distribution of the ANOVA F-statistic is the F-distribution<br />non-negative since all F-statistics are positive. <br />indexed by two degrees of freedom , k-1 and N-k<br />Numerator degree of freedom = number of groups minus 1 (k-1)<br />Denominator degree of freedom = total sample size minus number of groups (N-k)<br />The shape of the F-distribution varies depending on the two degrees of freedom <br />
  21. 21. Find the p-value of the ANOVA F-statistic<br />The p-value of the ANOVA F-statistic is the right tail area greater than the F-statistic under the <br /> F-distribution with (numerator distribution frequency, denominator distribution frequency) <br />The F-statistic for the example data = 5.73<br />The df for the F-distribution are (3-1) = 2 for the numerator and (18-3) = 15 for the denominator.<br />p value of 5.73, 2, 15 = 0.014<br />
  22. 22. Decision about the Hypotheses<br />Since the p-value of 0.014 < the significance level of 0.05, the null hypothesis of equality between all three group means is rejected.<br />We can conclude that that AT LEAST two of the means are significantly different. <br />.<br />
  23. 23. Application of ANOVA<br />ANOVA is designed to detect differences among means from populations subject to different treatments.<br />ANOVA is a joint test<br /> The equality of several population means is tested simultaneously or jointly.<br />ANOVA tests for the equality of several population means by looking at two estimators of the population variance (hence, analysis of variance).<br />
  24. 24. Application of ANOVA in Marketing<br />Product testing, ad copy testing and concept testing are some common applications, though ANOVA Analysis Surveys can be used in retail environments or simulated lab-type environments.<br />We can manipulate certain variables (like promotion, ad copy, display at the point of purchase), and observe changes in other variables (like sales, or consumer preferences, behavior or attitude). The application areas for experiments are wide .<br />
  25. 25. Application of ANOVA in Marketing<br /> Whenever marketing-mix variable (independent variable) is changed, we can determine its effect. Such variables include price, a specific promotion or type of distribution, or specific elements like shelf space, color of packaging etc.<br />
  26. 26. Procedures<br />An experiment can be done with just one independent variable (factor) or with multiple independent variables. <br />ANOVA - The key to success in an ANOVA Analysis Survey is the degree of control on the various independent variable (factors) that are being manipulated during the experiment. <br />
  27. 27. Other ANOVA Procedures<br />One-way ANOVA is Analysis of Variance for one factor<br />More than one factor can be used for a two, three or four-way ANOVA<br />A continuous variable can be added to the model – this is Analysis of Covariance (ANCOVA)<br />Repeated Measures ANOVA can handle replicated measurements on the same observation unit (subject)<br />
  28. 28. Thank You<br />