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CRAMP Rapid Assessment. Statistical Transformations
In order to determine whether transformations were appropriate, prior to analyses, residual distribution, partial regression plots and coefficient of variation were examined. Data transformations were conducted to satisfy the assumptions of normality, linearity, and homogeneity of variance required for some of the formal statistical tests performed.
To determine the best transformation, histograms and normality plots were generated. Normality was assessed using the Ryan-Joiner test, which is similar to Shapiro-Wilk. Direction and strength of skewness were determined since strong skew can cause leverage problems. Partial regression plots were generated to determine leverage. Since large data sets such as the one this research generated are quite robust against normality violations due to the central limit theorem, data were left in its original form whenever possible. Independent variables that were calculated as percentages and species data containing numerous zero values were transformed.
The transformations used to meet the assumptions of normality and homogeneity of variances included:
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Arcsine square-root, in which variables in percentages were changed to proportions in order to normalize data and obtain a continuous variable. Distributions of proportion data are skewed because they are between 0 and 1 and thus have no tails. Arcsine transformation was used to stretch out the tails on both ends for a more bell-shaped, normal distribution. These are useful in extreme proportions <0.2 or >0.8. Data in degrees was changed to radians.
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Log transformation, in which variables with high positive skewness were log transformed.
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Log (X+1) transformation, in which variables that are counts were log(x+1) transformed to reduce skewness. Variables that contained zero values were also log(x+1) transformed because the log of zero is undefined.
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Square root (X+1/2), in which coral species abundances were square root (X+1/2) transformed since the community ecology matrix is sparse, containing few non-zero values.
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No transformation applied, in which data with a coefficient of variation below 100% were retained in their original form.
Last Update: 04/21/2008
By: Lea Hollingsworth
Hawai‘i Coral Reef Assessment & Monitoring Program
Hawai‘i Institute of Marine Biology
P.O. Box 1346
Kāne‘ohe, HI 96744
808-236-7440 phone
808-236-7443 fax
email: jokiel@hawaii.edu