Standardizing effect size from linear regression models with logtransformed variables for metaanalysis
 Miguel RodríguezBarranco^{1, 2, 3}Email author,
 Aurelio Tobías^{4},
 Daniel Redondo^{1, 2, 3},
 Elena MolinaPortillo^{1, 2, 3} and
 María José Sánchez^{1, 2, 3}
DOI: 10.1186/s1287401703228
© The Author(s). 2017
Received: 7 September 2016
Accepted: 8 March 2017
Published: 17 March 2017
Abstract
Background
Metaanalysis is very useful to summarize the effect of a treatment or a risk factor for a given disease. Often studies report results based on logtransformed variables in order to achieve the principal assumptions of a linear regression model. If this is the case for some, but not all studies, the effects need to be homogenized.
Methods
We derived a set of formulae to transform absolute changes into relative ones, and vice versa, to allow including all results in a metaanalysis. We applied our procedure to all possible combinations of logtransformed independent or dependent variables. We also evaluated it in a simulation based on two variables either normally or asymmetrically distributed.
Results
In all the scenarios, and based on different change criteria, the effect size estimated by the derived set of formulae was equivalent to the real effect size. To avoid biased estimates of the effect, this procedure should be used with caution in the case of independent variables with asymmetric distributions that significantly differ from the normal distribution. We illustrate an application of this procedure by an application to a metaanalysis on the potential effects on neurodevelopment in children exposed to arsenic and manganese.
Conclusions
The procedure proposed has been shown to be valid and capable of expressing the effect size of a linear regression model based on different change criteria in the variables. Homogenizing the results from different studies beforehand allows them to be combined in a metaanalysis, independently of whether the transformations had been performed on the dependent and/or independent variables.
Keywords
Metaanalysis Systematic review Logtransformation Linear regression Effect size Regression coefficientsBackground
A metaanalysis is a systematic review of the literature that uses statistical methods to combine the results of two or more eligible studies [1]. It is useful because it provides a more accurate effect estimate by identifying clinically important effects, which, because of their size, may not have been detected in the primary studies. Furthermore, with metaanalyses it is possible to obtain a higher level of precision thanks to a larger sample size.
The type of measurement used to calculate effect size depends on the estimators used in the studies included in the metaanalysis [2]. Therefore, one of the possible limitations in a metaanalysis is that published studies report results that were obtained through different analytical approaches and measures of association. When performing a metaanalysis of an effect size estimated with linear regression models, this limitation can be (at least to a certain extent) overcome by using different transformations. Consequently, variables in linear regression models are usually transformed to achieve the principal assumptions of i) linearity of the relationship, ii) independence of the residual values, iii) homoscedasticity (constant variance) of the residuals, and iv) normal distribution of the residuals [3, 4]. Depending on the transformation applied in each case (natural logarithm, base 2 logarithm, base 10, etc.), and whether it is performed on an independent variable, dependent variable or both, the regression coefficient is interpreted differently [3, 4].
Thus, before performing a metaanalysis, some preprocessing procedure to homogenize the magnitude of effect observed in each study is required. This means that recalculating each effect to express it as a change in the dependent variable that corresponds to the same change in the independent variable is required. These changes, depending on the absence or presence of logarithmic transformation, can be expressed in either absolute or relative terms. Recent studies have applied a methodology to standardize the results of linear regression models through the logarithmic transformation of the independent variable in different bases for their inclusion in a metaanalysis [5].
This study aimed to develop a set of formulae to express results from linear regression models with different logtransformations of independent and/or dependent variables as the same effect size to be included in a metaanalysis.
Methods
The estimator \( \widehat{\beta} \) measures the strength of association between X and Y, as this represents the absolute change in the mean of Y for an increase of one unit in X. However, the meaning of \( \widehat{\beta} \) is not as intuitive when variables are transformed.
All possible regression models with all possible combinations of logtransformations for the dependent or independent variables were considered. Thus, the following models were formulated: (i) no transformation (model A), (ii) only the independent variable transformed (model B), (iii) only the dependent variable transformed (model C), and (iv) both the dependent and independent variables transformed (model D) (see Fig. 1). Logtransformations were expressed in a general base a for the dependent variable and in base b for the independent variable. Absolute change in a variable was set as c units and relative change was considered to be a ratio k between values.
Expressions of effect size and the 95% confidence interval estimation for each model and set of change criteria
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  

Absolute change in Y for an absolute change of c units in X  \( \begin{array}{c}\hfill c\cdotp \widehat{\beta}\hfill \\ {}\hfill \mathrm{c}\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \) (5), (6)  \( \begin{array}{c}\hfill { \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}\hfill \\ {}\hfill { \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \) (2), (13), (14)  \( \begin{array}{c}\hfill \left({a}^{c\cdotp \widehat{\beta}}1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{c\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}1\right)\cdotp E\left[ Y\right]\hfill \end{array} \) (4), (22), (23)  \( \begin{array}{c}\hfill \left({a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}1\right)\cdotp E\left[ Y\right]\hfill \end{array} \) (2), (33), (34) 
Absolute change in Y for a relative change of k times in X  \( \begin{array}{c}\hfill \left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}\hfill \\ {}\hfill \left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \) (1), (5), (6)  \( \begin{array}{c}\hfill { \log}_b(k)\cdotp \widehat{\beta}\hfill \\ {}\hfill { \log}_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]\hfill \end{array} \) (13), (14)  \( \begin{array}{c}\hfill \left({a}^{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}1\right)\cdotp E\left[ Y\right]\hfill \end{array} \) (1), (4), (26)  \( \begin{array}{c}\hfill \left({a}^{\log_b(k)\cdotp \widehat{\beta}}1\right)\cdotp E\left[ Y\right]\hfill \\ {}\hfill \left({a}^{\log_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}1\right)\cdotp E\left[ Y\right]\hfill \end{array} \) (4), (29), (30) 
Relative change in Y for an absolute change of c units in X  \( \begin{array}{c}\hfill 1+\frac{c\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{c}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{c}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \) (3), (5), (6)  \( \begin{array}{c}\hfill 1+\frac{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{{ \log}_b\left(1+\frac{c}{E\left[ X\right]}\right)}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \) (2), (17), (19)  \( \begin{array}{c}\hfill {a}^{c\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{c\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \) (22), (23)  \( \begin{array}{c}\hfill {a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\log_b\left(1+\frac{c}{E\left[ X\right]}\right)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \) (2), (29), (30) 
Relative change in Y for a relative change of k times in X  \( \begin{array}{c}\hfill 1+\frac{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \) (1), (3), (10)  \( \begin{array}{c}\hfill 1+\frac{{ \log}_b(k)\cdotp \widehat{\beta}}{E\left[ Y\right]}\hfill \\ {}\hfill \frac{{ \log}_b(k)}{E\left[ Y\right]}\left\{\frac{E\left[ Y\right]}{{ \log}_b(k)}+\widehat{\beta}\pm 1.96\cdotp se\left(\widehat{\beta}\right)\right\}\hfill \end{array} \) (3), (13), (14)  \( \begin{array}{c}\hfill {a}^{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\left( k1\right)\cdotp \mathrm{E}\left[ X\right]\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \) (1), (22), (23)  \( \begin{array}{c}\hfill {a}^{\log_b(k)\cdotp \widehat{\beta}}\hfill \\ {}\hfill {a}^{\log_b(k)\cdotp \left[\widehat{\beta}\pm 1.96\cdotp \mathrm{se}\left(\widehat{\beta}\right)\right]}\hfill \end{array} \) (29), (30) 
 Equivalent absolute change c in X for a relative change k in X.
 Equivalent relative change k in X for an absolute change c in X.
 Equivalent relative change k′ in Y for an absolute change c′ in Y.
 Equivalent absolute change c′ in Y for a relative change k′ in Y.
Thus, with these transformations the formulae in Table 1, based on the combinations of the different models and effect expressions, were obtained (see Additional file 1 for derivations).
Simulation
Simulation results when X and Y are normally distributed
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  

Betahat coefficient and standard error from regression model  \( \widehat{\beta} \) = 0.995 se(\( \widehat{\beta} \)) = 0.054  \( \widehat{\beta} \) = 9.587 se(\( \widehat{\beta} \)) = 0.520  \( \widehat{\beta} \) = 0.020 se(\( \widehat{\beta} \)) = 0.001  \( \widehat{\beta} \) = 0.193 se(\( \widehat{\beta} \)) = 0.011  
Absolute change in Y for an absolute change of c units in X  Effect size  0.995  0.914  1.006  0.928 
95% CI  (0.889–1.101)  (0.817–1.011)  (0.895–1.118)  (0.827–1.029)  
Absolute change in Y for a relative change of k times in X  Effect size  0.995  0.914  1.006  0.928 
95% CI  (0.889–1.101)  (0.817–1.011)  (0.895–1.118)  (0.827–1.029)  
Relative change in Y for an absolute change of c units in X  Effect size  1.0199  1.0183  1.0201  1.0186 
95% CI  (1.0178–1.0220)  (1.0163–1.0202)  (1.0179–1.0224)  (1.0165–1.0206)  
Relative change in Y for a relative change of k times in X  Effect size  1.0199  1.0183  1.0201  1.0186 
95% CI  (1.0178–1.0220)  (1.0163–1.0202)  (1.0179–1.0224)  (1.0165–1.0206) 
Simulation results when X and Y have an asymmetric distribution
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  

Betahat coefficient and standard error from regression model  \( \widehat{\beta} \) = 0.997 se(\( \widehat{\beta} \)) = 0.009  \( \widehat{\beta} \) = 6.071 se(\( \widehat{\beta} \)) = 0.213  \( \widehat{\beta} \) = 0.018 se(\( \widehat{\beta} \)) = 0.0002  \( \widehat{\beta} \) = 0.115 se(\( \widehat{\beta} \)) = 0.003  
Absolute change in Y for an absolute change of c units in X  Effect size  0.997  0.579  0.894  0.551 
95% CI  (0.980–1.014)  (0.539–0.618)  (0.874–0.915)  (0.518–0.584)  
Absolute change in Y for a relative change of k times in X  Effect size  0.997  0.579  0.894  0.551 
95% CI  (0.980–1.014)  (0.539–0.618)  (0.874–0.915)  (0.518–0.584)  
Relative change in Y for an absolute change of c units in X  Effect size  1.0199  1.0116  1.0179  1.0110 
95% CI  (1.0196–1.0203)  (1.0108–1.0124)  (1.0175–1.0183)  (1.0100–1.0117)  
Relative change in Y for a relative change of k times in X  Effect size  1.0199  1.0116  1.0179  1.0110 
95% CI  (1.0196–1.0203)  (1.0108–1.0124)  (1.0175–1.0183)  (1.0100–1.0117) 
Simulation results when Y has an asymmetric distribution
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  

Betahat coefficient and standard error from regression model  \( \widehat{\beta} \) = 0.625 se(\( \widehat{\beta} \)) = 0.254  \( \widehat{\beta} \) = 5.834 se(\( \widehat{\beta} \)) = 2.441  \( \widehat{\beta} \) = 0.011 se(\( \widehat{\beta} \)) = 0.005  \( \widehat{\beta} \) = 0.103 se(\( \widehat{\beta} \)) = 0.044  
Absolute change in Y for an absolute change of c units in X  Effect size  0.625  0.557  0.551  0.493 
95% CI  (0.128–1.122)  (0.101–1.013)  (0.100–1.006)  (0.080–0.909)  
Absolute change in Y for a relative change of k times in X  Effect size  0.625  0.557  0.551  0.493 
95% CI  (0.128–1.122)  (0.101–1.013)  (0.100–1.006)  (0.080–0.909)  
Relative change in Y for an absolute change of c units in X  Effect size  1.0125  1.0111  1.0110  1.0099 
95% CI  (1.0026–1.0224)  (1.0020–1.0203)  (1.0020–1.0201)  (1.0016–1.0182)  
Relative change in Y for a relative change of k times in X  Effect size  1.0125  1.0111  1.0110  1.0099 
95% CI  (1.0026–1.0224)  (1.0020–1.0203)  (1.0020–1.0201)  (1.0016–1.0182) 
Simulation results when X has an asymmetric distribution
Model A \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model B \( \widehat{Y}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  Model C \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp X \)  Model D \( \widehat{{ \log}_a(Y)}=\widehat{\alpha}+\widehat{\beta}\cdotp { \log}_b(X) \)  

Betahat coefficient and standard error from regression model  \( \widehat{\beta} \) = 0.288 se(\( \widehat{\beta} \)) = 0.015  \( \widehat{\beta} \) = 1.517 se(\( \widehat{\beta} \)) = 0.133  \( \widehat{\beta} \) = 0.005 se(\( \widehat{\beta} \)) = 0.0003  \( \widehat{\beta} \) = 0.028 se(\( \widehat{\beta} \)) = 0.003  
Absolute change in Y for an absolute change of c units in X  Effect size  0.288  0.145  0.263  0.133 
95% CI  (0.259–0.317)  (0.120–0.169)  (0.236–0.291)  (0.109–0.156)  
Absolute change in Y for a relative change of k times in X  Effect size  0.288  0.145  0.263  0.133 
95% CI  (0.259–0.317)  (0.120–0.169)  (0.236–0.291)  (0.109–0.156)  
Relative change in Y for an absolute change of c units in X  Effect size  1.0058  1.0029  1.0053  1.0027 
95% CI  (1.0052–1.0063)  (1.0024–1.0034)  (1.0047–1.0058)  (1.0022–1.0031)  
Relative change in Y for a relative change of k times in X  Effect size  1.0058  1.0029  1.0053  1.0027 
95% CI  (1.0052–1.0063)  (1.0024–1.0034)  (1.0047–1.0058)  (1.0022–1.0031) 
For the simulations, the parameters c = 1 and k = 1.1 were fixed to reflect the effect of an absolute change in one unit or a relative change of 10% in the independent variable (equivalent to one unit given that the mean value of X is 10). Tables 2, 3, 4 and 5 show the results of the simulations. The diagonal positions in these tables correspond to the real effect size, which is obtained from the regression coefficient and the standard error of the specific model. The remainder of the values in each row represents the estimated effect size when using the formulae.
Results
In all of the scenarios, the effect size estimated from the formulae, based on different change criteria, was equivalent to the real effect size. In the model without transformation (model A), the variation of a unit in X is associated with a variation of 0.995 units in Y. When the formula to express a variation of X in relative terms was applied (i.e. an increase of 10% in X as equivalent to one unit), the same result (beta = 0.995) was produced. On the other hand, the estimated effect on Y in relative terms was 1.0199, i.e. a variation of 1.99%. Given that the mean value of Y is 50 units, that variation is equivalent to an increase of 0.995 units, which is equal to the real effect observed (Table 1).
For the other models, the result was the same. In model B, the real absolute change was 0.914, whereas the estimated relative change was 1.0183 (1.83% or 0.914 units), while in model C, the real relative change of 1.0201 equaled the estimated absolute change of 1.006 units, and in model D the real relative change of 1.0186 was equivalent to the estimated absolute change of 0.928. This equivalence, based on the different distributions of variables X and Y (Tables 2, 3, 4 and 5), was maintained in all the scenarios contemplated.
The variation in effect size between the various models differed depending on the shape of the distribution of variables. For the relationship between normally distributed variables, the range of variation in the absolute effect was 0.914 to 1.006, and 1.0183 to 1.0201 in the relative effect. When the independent variable only was skewed, the absolute effect varied between 0.133 and 0.288 and the relative effect between 1.0027 and 1.0058, while when the dependent variable only was skewed, the absolute effect varied between 0.493 and 0.625 and the relative effect between 1.0099 and 1.0125, and when both variables were skewed, the absolute effect varied between 0.551 and 0.997 and the relative effect between 1.0110 and 1.0199.
Empirical example
Original regression coefficients and transformed effect size for studies included in the metaanalysis
Author (Year)  Mean of X  Units  Transf. on X  β  SE(β)  θ  SE(θ) 

As in urine  
Hamadani (2011)Girls [7]  μg/L  Ln  −1.40  0.66  −0.57  0.27  
Hamadani (2011)Boys [7]  μg/L  Ln  0.70  0.56  0.28  0.23  
RochaAmador (2007) [8]  μg/gr crea  Ln  −5.72  1.93  −2.32  0.78  
von Ehrenstein (2007) [9]  78  μg/L  None  −0.0007  0.0008  −0.03  0.03 
Wasserman (2007) [10]  μg/gr crea  Ln  −1.78  1.42  −0.72  0.58  
Wasserman (2004) [11]  μg/gr crea  Ln  −2.90  1.71  −1.18  0.69  
As in water  
RochaAmador (2007) [8]  μg/L  Ln  −6.15  1.87  −2.49  0.76  
von Ehrenstein (2007) [9]  147  μg/L  None  −0.0002  0.0004  −0.01  0.03 
Wasserman (2007) [10]  μg/L  Ln  −1.06  0.57  −0.43  0.23  
Wasserman (2004) [11]  μg/L  Ln  −1.64  0.64  −0.66  0.26  
Mn in hair  
Bouchard (2011) [12]  μg/g  log10  −3.30  1.43  −0.58  0.25  
MenezesFilho (2011) [13]  μg/g  log10  −5.78  2.84  −1.02  0.50  
RiojasRodríguez (2010) [14]  6.35  μg/g  None  −0.20  0.11  −0.64  0.36 
Wright (2006) [15]  0.47  μg/g  None  −10.00  5.00  −2.35  1.18 
The results of the metaanalysis suggested that for every 50% increase in arsenic levels (either in urine or in regular drinking water) there could be an approximately 0.5 decrease in the IQ of children aged 5–15 years. Moreover, a 50% increase in manganese levels in hair would be associated with a decrease of 0.7 points in the IQ of children aged 6–13 years [5].
This approach allowed the results from regression models using different formulations to be combined, and, thus obtain a pooled measure of association that included all available results.
Discussion
To establish causality, wellconducted and freeofbias systematic reviews that include a metaanalysis have been proposed as the epidemiological design at the top rank of the evidencebased medicine pyramid [16]. However, the main bias in such design is publication bias and while there are statistical methods that can be used to study the presence of this error, it cannot be controlled [17].
Another problem in metaanalyses is the difficulty of including all the studies dealing with the research topic, either because of a specific transformation performed on the variables of the model or because the effect measurements in said study were not relevant to the research question. When all studies on a specific topic cannot be included, the metaanalysis loses external validity. This difficulty would be solved if it were possible to access the original data (not only the results) that the authors had amassed. However, in almost all cases, accessing this kind of information is practically impossible.
An alternative would be to contact the author of the published study and request the results that were obtained from the original data but which do not appear in the publication. Occasionally this strategy provides a way to access the data required for the study to be included in the metaanalysis. However, such efforts are generally not successful, as positive responses are rare; particularly if the study had been conducted several years beforehand.
On the other hand, there are other initiatives that allow access to anonymized original data obtained in other studies. A relevant example of this is the datasharing policy of the BMJ journals [18]. In fact, after 2013, the publication of the results from any clinical trial on drugs or medical devices requires the authors to make the relevant patientlevel data available (on reasonable request) to other researchers.
In the absence of this type of strategy being consolidated and expanded, there is the urgent need to develop procedures that can be used to standardize results obtained with different methodological approaches so that they can then be validly combined in a metaanalysis. Such procedures would optimize metaanalyses as they would make it possible to include a maximum number of results, even when the analyses carried out were not identical. This would not only increase the statistical potential of the metaanalysis, but would also reduce the risk of any selection bias that might occur if some of the studies identified in the systematic review had to be excluded.
This study proposes a procedure to homogenize the estimated effect sizes with linear regression models that use different transformations of dependent and/or independent variables. The application of these transformations to express all the effect sizes based on the same change criterion enables the results from studies that have built their regression models with different transformations to estimate the effect to be combined. Furthermore, the generalization of the method also allows the effect size to be recalculated, independent of the logarithm base applied in the transformation. Simply reflecting the same change in the independent variable is all that is required.
The simulation results showed that this procedure provided an estimation of the effect that was equal to that obtained with the original model. Moreover, the approximation was not affected by the form of the distribution of the variables. Nevertheless, it is also important to compare the effects of the four models since, from a practical perspective, this procedure will be used to compare the results of different regression models.
As can be observed in the simulation, the effect estimate obtained by using a model without transformations is not the same as that obtained with a model that uses some type of transformation. In other words, if an author presents the result of a model with the logtransformed dependent variable and we then apply the procedure described to recalculate the effect based on a model without transformation, we would not obtain the same result as the author would from their own data in a model without transformation.
This limitation can produce a certain degree of bias in the effect estimate. Based on the simulation results, the size of this bias basically depends on the symmetry of the independent variable (X). When X and Y have a normal distribution, the variation of the effect size in regard to model A is, at most, 8%. When Y has an asymmetric distribution and X a normal distribution, the variation is approximately 10%. However, when the independent variable is asymmetric, the bias can be as high as 50% of the value of the effect estimated with model A.
To apply this model, the standard should be regarded as the most generalized model of all the results, and then the effect should be transformed for those results that use a different model. To apply the proposed formulae featured, an Excel spreadsheet is available as Additional file 2.
Conclusions
In conclusion, the method proposed in this study was shown to be valid and capable of expressing the effect size of a linear regression model consistent with different change criteria in the variables involved. The previous homogenization of the results from different studies allows them to be combined in a metaanalysis, independent of the transformations performed on the dependent and/or independent variables. However, in order to avoid biased effect estimates, this procedure should be used with caution in the case of independent variables with asymmetric distributions that significantly differ from normal ones.
Abbreviations
 As:

Arsenic
 CI:

Confidence interval
 IQ:

Intelligent quotient
 Mn:

Manganese
Declarations
Acknowledgements
The authors would like to thank Begoña Martínez at the Andalusian School of Public Health, for her comments on and suggestions for the manuscript.
Funding
No specific funding.
Availability of data and materials
The datasets used for simulations are available from the corresponding author upon reasonable request.
Authors’ contributions
MRB and AT conceived of the study. MRB developed the method and formulae. DR and EM generated databases, ran the simulations and designed the Excel template. MJS and MRB drafted the original manuscript. All authors contributed to the writing of the final manuscript. All authors have approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
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Authors’ Affiliations
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