### Introduction

### Distribution of each variable and relationship between variables

### Non-linear transformations

^{st}and 3

^{rd}quartiles of the smoking period when the original data have violated the normality assumption. If we compare the smoking periods between two groups and they require square transformation to keep the normality assumption, it is hard to interpret the clinical meaning of the mean difference of squared data. The difference between each squared value is not the same as the squared difference between the two original scaled values. In this situation, non-parametric analysis makes it easier to interpret the results.

### Logarithmic transformation

^{1)}.

*e*

^{3.00}= 20.09, respectively (Fig. 1). The mean value of 20.09, which is known as the geometric mean, is the back-transformed mean value from transformed data. The geometric mean is less affected by the very large values of original data than the corresponding arithmetic mean, which comes from a skewed distribution. SD should also be considered for back-transformation from estimated values. However, for the moment of back-transformation, the meaning of ‘standard’ deviation loses its additive meaning because such data are not normally distributed [5]. Its interpretation does not make sense after back-transformation. Hence, the CI is usually reported for this situation [4,6]. A back-transformed CI allows better understanding on the original scaled data. For example, the mean and SD are 3.00 and 0.201 for natural logarithmic transformed data with a sample size of 100 and its 95% CI is from 2.96 to 3.03. When back-transformation is performed with an exponential function, it changes into from 19.31 to 20.89. Considering the geometric mean is 20.09, a back-transformed 95% CI does not have symmetric placement from the geometric mean value. We sometimes use a variable with the transformed form by default. Back-transformation is essential for statistical analysis and should be returned to its original scale when reporting the results. The one example is the antibody titer. If one patient with myasthenia gravis tested positive for anticholinesterase with an antibody titer 1:32, it means that the number of dilutions should be repeated five times until the last seropositive results (2

^{5}= 32). Antibody titer itself has a characteristic of the powered value of dilution numbers, which is always is reported as 1:2

^{dilution numbers}. Hence, the geometric mean should be presented as 2

^{mean dilution number}, not the mean of titers.

*X*

_{1}-

*X*

_{2}, and back-transformation results in

*e*

^{0.5}= 1.65, mean from one sample has 65% higher value compared to the other mean. This should not be interpreted as 165%, and we should consider the difference, not a simple ratio. The CI of the mean difference also can be interpreted in a similar way. If the estimated 95% CI of the above sample is 0.4–0.6, the back-transformed range is 1.49–1.83, its interpretation is ‘mean from one sample has a 65% higher value with a 95% CI ranging from 49% to 83% compared to the other mean.’ Reporting statistics can be estimated using logarithmically transformed data. When reporting this, the information regarding transformation should be accompanied. Corresponding effect sizes and P values can also be reported as estimated. This interpretation approach can be applied to the statistical method of mean comparison.

*e*

^{0.1}= 1.105, means ‘for a one-unit increase in the independent variable, dependent variable increases by 10.5%.’ Similar to the explanation of the mean difference, it should be noted that the interpreted value is not 110.5%. If the dependent variable is a common logarithmic transformed variable, a one-unit change in the independent variable is the same as a tenfold increment in the original matric variable. That is, a tenfold increment in independent variable produces dependent variable changes by the estimated regression coefficient. Description with a 1% increment of the independent variable is also possible. For the convenience, common logarithmic transformation is better for independent variable transformation. If natural logarithmic transformation is applied, interpretation is not easy with e as the base of the natural logarithm; the approximated value is 2.71828. If both dependent and independent variables are transformed with logarithmic transformation, we can interpret the result as a percentile increment of the independent variable produces a percentile change in the dependent variable following the rule explained above. These interpretation rules can be applied to the other kind of general linear modeling method including ANCOVA and MANOVA.

### Power transformation and Box-Cox transformation

*λ*, this performs various types of non-linear transformation. For example,

*λ*= -1 produces a reciprocal transformation,

*λ*= 2 a square transformation, and

*λ*= 0.5 a square root transformation. Because it contains a constant, a somewhat linear transformation is also applied as we already know the linear transformation hardly effects the estimated statistics. However, we should be cautious as such transformation could affect the statistical results, as described in the previous section. If

*λ*= 0, the Box-Cox transformation is same as logarithmic transformation. Then, how can we determine the value of

*λ*and how the Box-Cox transformation stabilizes the variance? We consider this using linear regression. Linear regression requires several assumptions including homoscedasticity, which means all observed values are equally scattered from the estimated regression line. Several residual diagnostics provide about homoscedasticity. When the homoscedasticity is violated, the Box-Cox transformation could stabilize the variance of residuals. Using

*y*instead of

^{λ}*y*in the linear regression model, for example,

*y*=

^{λ}*αx*+

*β*+

*ε*(

*α*: regression coefficient,

*β*: constant,

*ε*: error), several statistical software programs

^{2)}find best estimated values of

*λ*and its 95% CI based on the maximal likelihood method. Using this result, we can estimate the linear regression model with homoscedasticity. Although the Box-Cox transformation is an excellent tool to obtain the best results of linear regression, it also has a serious problem of result interpretation. Back-transformation for this is not as simple as other non-linear transformations because it includes the error term, which is essential for the linear regression. There are several proposed back-transformation methods from the Box-Cox transformation [10,11], which require complex statistical process. If we try to interpret as the transformed variable itself, we should also consider the transformed unit, which could lose its real meaning after transformation. Only when the other measures for stabilizing variance (homoscedasticity) have failed, should the Box-Cox transformation be considered.