Assumption and Data Transformation

1. Assumption and Data Transformation

2. Assumption of Anova • The error terms are randomly, independently, and normally distributed • The variance of different samples are homogeneous • Variances and means of different samples are not correlated • The main effects are additive

3. Randomly, independently and Normally distribution • The assumption of normality do not affect the validity of the analysis of variance too seriously • There are test for normality, but it is rather point pointless to apply them unless the number of samples we are dealing with is fairly large • Independence implies that there is no relation between the size of the error terms and the experimental grouping to which the belong • It is important to avoid having all plots receiving a given treatment occupying adjacent positions in the field • The best insurance against seriously violating the first assumption of the analysis of variance is to carry out the randomization appropriate to the particular design

4. Normally test • Shapiro-Wilk test • Lilliefors-Kolmogorov-Smirnov Test • Graphical methods based on residual error (Residual Plotts)

5. Homogeneity of Variance • Unequal variances can have a marked effect on the level of the test, especially if smaller sample sizes are associated with groups having larger variances • Unequal variances will lead to bias conclusion

6. Way to solve the problem of Heterogeneous variances • We can separate the data into groups such that the variances within each group are homogenous • We can use an advance statistic tests rather than analysis of variance • we might be able to transform the data in such a way that they will be homogenous

7. Homogeneity test of Variance • Hartley F-max test • Bartlett’s test • Residual plot for checking the equal variance assumption

8. Independence of Means and Variance • It is a special case and the most common cause of heterogeneity of variance • A positive correlation between means and variances is often encountered when there is a wide range of sample means • Data that often show a relation between variances and means are data based on counts and data consisting of proportion or percentages • Transformation data can frequently solve the problems

9. The Main effects are additive • For each design, there is a mathematical model called a linear additive model. • It means that the value of experimental unit is made up of general mean plus main effects plus an error term • When the effects are not additive, there are multiplicative treatment effect • In the case of multiplication treatment effects, there are again transformation that will change the data to fit the additive model

10. Data Transformation • There are two ways in which the anova assumptions can be violated: 1. Data may consist of measurement on an ordinal or a nominal scale 2. Data may not satisfy at least one of the four requirements • Two options are available to analyze data: 1. It is recommended to use non-parametric data analysis 2. It is recommended to transform the data before analysis

11. Logaritmic Transformation • It is used when the standard deviation of samples are roughly proportional to the means • There is an evidence of multiplicative rather than additive • Data with negative values or zero can not be transformed. It is suggested to add 1 before transformation

12. Square Root Transformation • It is used when we are dealing with counts of rare events • The data tend to follow a Poisson distribution • If there is account less than 10. It is better to add 0.5 to the value

13. Arcus sinus or angular Transformation • It is used when we are dealing with counts expressed as percentages or proportion of the total sample • Such data generally have a binomial distribution • Such data normally show typical characteristics in which the variances are related to the means