Assumption and Data Transformation

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

Randomly, independently and Normally distribution The assumption of normality do not affect the validity of the analysis of variance too seriously 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 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 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 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 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

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

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 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 Unequal variances will lead to bias conclusion

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 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 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 we might be able to transform the data in such a way that they will be homogenous

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

Independence of Means and Variance It is a special case and the most common cause of heterogeneity of 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 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 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 Transformation data can frequently solve the problems

The Main effects are additive For each design, there is a mathematical model called a linear additive model. 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 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 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 In the case of multiplication treatment effects, there are again transformation that will change the data to fit the additive model

Data Transformation There are two ways in which the anova assumptions can be violated: 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: 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

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

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

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