Transforming the data Modified from: Gotelli and Allison 2004. Chapter 8; Sokal and Rohlf 2000 Chapter 13

What is a transformation? 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

Most are monotonic: Monotonic functions do not change the rank order of the data, but they do change their relative spacing, and therefore affect the variance and shape of the probability distribution

There are two legitimate reasons to transform your data before analysis The patterns in the transformed data may be easier to understand and communicate than patterns in the raw data. They may be necessary so that the analysis is valid

They are often useful for converting curves into straight lines: The logarithmic function is very useful when two variables are related to each other by multiplicative or exponential functions

Logarithmic (X):

Example: Asi’s growth (50 % each year) weight 1 10.0 2 15.0 3 22.5 4 33.8 5 50.6 6 75.9 7 113.9 8 170.9 9 256.3 10 384.4 11 576.7 12 865.0

Exponential:

Example: Species richness in the Galapagos Islands

Power:

Statistics and transformation Data to be analyzed using analysis of variance must meet to assumptions: The data must be homoscedastic: variances of treatment groups need to be approximately equal The residuals, or deviations from the mean must be normal random variables

Lets look an example A single variate of the simplest type of ANOVA (completely randomized, single classification) decomposes as follows: In this model the components are additive with the error term εij distributed normally

However… We might encounter a situation in which the components are multiplicative in effect, where If we fitted a standard ANOVA model, the observed deviations from the group means would lack normality and homoscedasticity

The logarithmic transformation We can correct this situation by transforming our model into logarithms Wherever the mean is positively correlated with the variance the logarithmic transformation is likely to remedy the situation and make the variance independent of the mean

We would obtain Which is additive and homoscedastic

The square root transformation It is used most frequently with count data. Such distributions are likely to be Poisson distributed rather than normally distributed. In the Poisson distribution the variance is the same as the mean. Transforming the variates to square roots generally makes the variances independents of the means for these type of data. When counts include zero values, it is desirable to code all variates by adding 0.5.

The box-cox transformation Often one do not have a-priori reason for selecting a specific transformation. Box and Cox (1964) developed a procedure for estimating the best transformation to normality within the family of power transformation

The box-cox transformation The value of lambda which maximizes the log-likelihood function: yields the best transformation to normality within the family of transformations s2T is the variance of the transformed values (based on v degrees of freedom). The second term involves the sum of the ln of untransformed values

box-cox in R (for a vector of data Y) >library(MASS) >lamb <- seq(0,2.5,0.5) >boxcox(Y_~1,lamb,plotit=T) >library(car) >transform_Y<-box.cox(Y,lamb) What do you conclude from this plot? Read more in Sokal and Rohlf 2000 page 417

The arcsine transformation Also known as the angular transformation It is especially appropriate to percentages

The arcsine transformation Transformed data It is appropriate only for data expressed as proportions Proportion original data

Since the transformations discussed are NON-LINEAR, confidence limits computed in the transformed scale and changed back to the original scale would be asymmetrical

Evaluating Ecological Responses to Hydrologic Changes in a Payment-for-environmental-services Program on Florida Ranchlands Patrick Bohlen, Elizabeth Boughton, John Fauth, David Jenkins, Pedro Quintana-Ascencio, Sanjay Shukla and Hilary Swain G08K10487

Palaez Ranch Wetland Water Retention

Call: glm(formula = mosqct ~ depth + depth^2, data = pointdata) Deviance Residuals: Min 1Q Median 3Q Max -28.1 -27.4 -25.5 -19.5 6388.3 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 19.40214 42.98439 0.451 0.652 depth 1.11973 4.07803 0.275 0.784 depth^2 -0.03597 0.07907 -0.455 0.649 (Dispersion parameter for gaussian family taken to be 117582.7) Null deviance: 77787101 on 663 degrees of freedom Residual deviance: 77722147 on 661 degrees of freedom AIC: 9641.5 Number of Fisher Scoring iterations: 2

Call: zeroinfl(formula = mosqct ~ depth + depth^2, data = pointdata, dist = "poisson", EM = TRUE) Pearson residuals: Min 1Q Median 3Q Max -6.765e-01 -5.630e-01 -5.316e-01 -4.768e-01 9.393e+05 Count model coefficients (poisson with log link): Estimate Std. Error z value Pr(>|z|) (Intercept) 1.8550639 0.0498470 37.22 <2e-16 *** depth 0.4364981 0.0064139 68.06 <2e-16 *** depth^2 -0.0139134 0.0001914 -72.68 <2e-16 *** Zero-inflation model coefficients (binomial with logit link): (Intercept) 0.6400910 0.3274469 1.955 0.05061 . depth 0.0846763 0.0371673 2.278 0.02271 * depth^2 -0.0027798 0.0009356 -2.971 0.00297 ** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Number of iterations in BFGS optimization: 1 Log-likelihood: -4.728e+04 on 6 Df >