Transformation technique and when it is used
- 1. 1 TRANSFORMATION An Assignment on SUBMITTED BY Mehta Kavish Kirtikumar 2nd Sem. Ph. D. (1010121013) BACA, AAU, Anand - 388110 SUBMITTED TO Dr. A. D. Kalola Professor & Head, Dept. of Agril. Statistics, BACA, AAU, Anand - 388110 Ag. Stat 534: Statistical Methods for Crop Protection II
- 2. 2 TRANSFORMATION What is a transformation? It is the technique where the original data are converted into a new scale resulting in a new data that is expected to satisfy the condition of homogeneity or variance. It is a mathematical function that is applied to all the observations of a given variable. Y represents the original variable, Y* is the transformed variable, and f is a mathematical function that is applied to the data . Data transformation is the most appropriate remedial measure for variance heterogeneity where the variance and the mean are functionally related. With this technique, the original data are converted into a new scale resulting in a new data set that is expected to satisfy the condition of 99% homogeneity of variance. It is worth noting that the comparative values between treatments are not changed and comparisons between them remain valid due to application of a common transformation scale to all observations. The appropriate data transformation to be used depends upon the specific type of relationship between the mean and the variance. 𝑌 ∗= 𝑓 (𝑌 )
- 3. 3 WHY TRANSFORMATION IS REQUIRED? It is required when data not following normal distribution. It is required to make the mean and variances independent. It is required when Coefficient of variation is high. It is required to maintain the homogeneity of the test data.
- 4. 4 LOG TRANSFORMATION When the original observation Y is converted to log Y, the conversion is known as log transformation. Although log to any base can be used, log to base 10 is generally easiest. If the observed value is 0, a constant value preferably 1 is added to avoid negative logarithms. When such constant is added, it is added to all the observations. The log transformation is particularly effective in normalizing positively skewed distributions. It is also used to achieve additivity.
- 5. 5 Here, It can be verified that the treatment and replication effects are not additive in case of original observations. However, the treatment and replication effects are additive after log transformation.
- 6. 6 ANGULAR TRANSFORMATION The use of arcsine transformation, also known as inverse transformation (Rao, 1998) or angular transformation (Snedecor and Cochran, 1989) has been open for debate as to the usefulness in analysis of proportion data that tends to be skewed when the distribution is not normal. e.g. data obtained from a count, the data expressed as decimal fractions and percentages. The mechanics of data transformation are greatly facilitated by using a table of the arcsine transformation. where, p is the proportion and Y is the result of the transformation
- 7. 7 Although arcsine transformation is a useful tool in stabilizing variances and normalizing proportional data, there are several reasons why this method can be problematic. The equalization of variance in proportional data when using arcsine transformations requires the numbers of trials to be equal for each data point, while the efficacy of arcsine transformation in normalizing proportional data is dependent on sample size, n, and doesn’t perform well at extreme ends of the distribution (Worton and Hui (2010);Hardy 2002).
- 8. 8 Another argument against arcsine transformation is that it does not confine proportional data between 0 and 1, resulting in the extrapolation of proportional values that aren’t biologically sensible (Hardy 2002). In an example provided by Hardy (2002), the arcsine transformation of the relationship between sex ratio data and distance from a pollutant predicted a sex ratio greater than 1 for males as the distance from the pollutant increased (Hardy, 2002). An alternative to arcsine transformation that is becoming more prevalent in today’s biological analyses is the logistic regression, an analytical method which is designed to deal with proportional data (Jeager, 2008).
- 9. 9 SQUARE ROOT TRANSFORMATION If the original observation Y is converted to a new value by taking its square root, it is known as square root transformation. It is used in case of count data, when some of the observed counts are numerically small, say less than 10, the more appropriate transformation is (Y+0.5)1/2 . The transformation of the type (Y)1/2 +(Y+1)1/2 is also used. The square root transformation is used when the observations follow a Poisson distribution.
- 10. 10 It is useful for small whole no. data: Eg. No. of infested plant in a plot, No. of insects caught in traps, No. of weeds per plot. It is also appropriate for percent data, where range is between 0 - 30% or between 70-100%, but not both. Especially with 0 percent (x+0.5) should be use instead of x, where x is the original data.
- 11. 11 The following rules may be useful in choosing the proper transformation scale for percentage data derived from count data. Rule-1: For percentage data lying within the range 30 to 70 % no transformation is needed. Rule-2: For percentage data lying within the range of either 0 to 20 % or 80 to 100 % but not both, the square root transformation should be used. Rule-3: For percentage data that do not follow the ranges specified in either rule-1 or rule-2, the arc sine transformation should be used.
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