Transcript Assumption and Data Transformation

Assumption and Data
Transformation
Assumption of Anova
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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
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
Normally test
 Shapiro-Wilk test
 Lilliefors-Kolmogorov-Smirnov Test
 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 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 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
Homogeneity test of Variance
 Hartley F-max test
 Bartlett’s test
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
 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
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
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
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
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
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