# Handling Skewed Data: A Comparison of Two Popular Methods
## 1 Introduction
However, while the log transformation can decrease skewness, log-transformed data are not guaranteed to satisfy the normality assumption [7]. Thus, the normality of the data should also be checked after transformation. In addition, the use of log transformations can lead to mathematical errors and misinterpretation of results [6; 8]. Similarly, the attitudes of regulatory authorities profoundly influence the trials performed by pharmaceutical companies; Food and Drug Administration (FDA) guidelines state that unnecessary data transformation should be avoided, raising doubts about using transformations. If data transformation is performed, a justification for the optimal data transformation, aside from the interpretation of the estimates of treatment effects based on transformed data, should be given. An industry statistician should not analyze the data using several transformations and choose the transformation that yields  
Figure 1: Simulated data from gamma distribution before and after log transformation. **(a)** The histogram of the sample before the application of log transformation with fitted normal and kernel curves; **(b)** The histogram of the sample after the application of log transformation with fitted normal and kernel curves. the most satisfactory results. Unfortunately, the guideline includes the log transformation with all other kinds of transformation and gives it no special status [9]. An alternative approach is the generalized linear model (GLM), which does not require the normality of data to test for differences between two populations. The GLM is a wide range of models first promoted by [PERSON] and [PERSON] in 1972 and then by [PERSON] and [PERSON] in 1989 [10; 11]. The GLM was presented as a general framework for dealing with a variety of standard statistical models for both normal and non-normal data, like ANOVA, logistic regression, multiple linear regression, log-linear models, and Poisson regression. The GLM can be considered as a flexible generalization of ordinary linear regression, which extends the linear modeling framework to response variables that have non-normal error distributions [12]. It generalizes linear regression by connecting the linear model to the response variable via a link function, and by permitting the magnitude of the variance of each measurement to be a function of its expected value [10]. The GLM consists of:  
1. A linear predictor \[\eta_{i}=\beta_{0}+\beta_{1}x_{1i}+\cdots+\beta_{p}x_{pi}=X\beta,\] (1) where \(\eta_{i},\ i=1,2,\ldots,N\), is a set of independent random variables called response variables, where each \(\eta_{i}\) is a linear function of explanatory variables \(x_{j},\ j=1,\ldots,p\). 2. A link function that defines how \(E(y_{i})=\mu_{i}\) which is the mean or expected value of the outcome \(y_{i}\), depends on the linear predictor, \(g(\mu_{i})=\eta_{i}\), where \(g\) is a monotone, differentiable function. The mean \(\mu\) is thus made a smooth and invertible function of the linear predictor: \[\mu_{i}=\ g^{-1}(\eta_{i}),\] (2)
3. A variance function that defines how the variance, \(Var(y_{i})\), depends on the mean \(Var(y_{i})=\phi V(\mu_{i})\), where the dispersion parameter \(\phi\) is a constant. Replacing the \(\mu_{i}\) in \(V(\mu_{i})\) with \(g^{-1}(\eta_{i})\) also makes the variance a function of the linear predictor. In the GLM, the form of \(E(y_{i})\) and \(Var(y_{i})\) are determined by the distribution of the dependent variable \(y_{i}\) and the link function \(g\). Furthermore, no normality assumption is required [13; 14]. All the major statistical software platforms such as STATA, SAS, R and SPSS include facilities for fitting GLMs to data [15]. Because finding appropriate transformations that simultaneously provide constant variance and approximate normality can be challenging, the GLM becomes a more convenient choice, since the choice of the link function and the random component (which specifies the probability distribution for response variable (\(\gamma\)) are separated.