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Some studies have found:Increased attention. [1]Faster reaction times. [1]Decreased risk of liver and heart disease. [1]Improved short-term memory. [2]'Stabilizes' the blood-brain barrier, apparently having numerous positive effects. [3]Reduced risk of Alzheimer's Disease. [4] [5] Some studies have shown this effect to be significant. [6][1] http://en.wikipedia.org/wiki/File:Effects_of_moderate_caffeine_consumption.svg[2] http://www.newscientist.com/article/dn8401-coffees-effects-revealed-in-brain-scans.html[3] http://news.bbc.co.uk/1/hi/7326839.stm[4 Arendash, G. W., Schleif, W., Rezai-Zadeh, K., Jackson, E. K., Zacharia, L. C., Cracchiolo, J. R., Shippy, D., et al. (2006). Caffeine protects Alzheimerʼs mice against cognitive impairment and reduces brain beta-amyloid production. Neuroscience, 142(4), 941-952.[5] Caffeine Reverses Cognitive Impairment and Decreases Brain Amyloid-β Levels in Aged Alzheimer's Disease Mice, doi: 0.3233/JAD-2009-1087[6] Maia, L. and De Mendonça, A. (2002), Does caffeine intake protect from Alzheimer's disease?. European Journal of Neurology, 9: 377–382. doi: 10.1046/j.1468-1331.2002.00421.x

AbstractAlthough previous research has studied power in mediation models, the extent to which the inclusion of a mediator will increase power has not been investigated. First, a study compared analytical power of the mediated effect to the total effect in a single mediator model to identify the situations in which the inclusion of one mediator increased statistical power. Results from the first study indicated that including a mediator increased statistical power in small samples with large coefficients and in large samples with small coefficients, and when coefficients were non-zero and equal across models. Next, a study identified conditions where power was greater for the test of the total mediated effect compared to the test of the total effect in the parallel two mediator model. Results indicated that including two mediators increased power in small samples with large coefficients and in large samples with small coefficients, the same pattern of results found in the first study. Finally, a study assessed analytical power for a sequential (three-path) two mediator model and compared power to detect the three-path mediated effect to power to detect both the test of the total effect and the test of the mediated effect for the single mediator model. Results indicated that the three-path mediated effect had more power than the mediated effect from the single mediator model and the test of the total effect. Practical implications of these results for researchers are then discussed.Go to:IntroductionMost research studies investigate the relation of an independent variable (X) and a dependent variable (Y). The independent variable X represents exposure to a randomized manipulation or an observed variable. An analysis such as linear regression quantifies the relation of X to Y as a regression coefficient and provides a statistical test of whether the coefficient is larger than expected by chance. Prior to such an undertaking, researchers conduct power calculations to assess the probability of observing a statistically significant relation of X to Y given a Type I error rate, sample size, variability, and effect size. Increasing the Type I error rate, increasing the sample size, increasing the effect size, or reducing variability increases power of a statistical test.As recognition of the importance of mediation analysis has increased, researchers have developed advanced statistical and conceptual models to investigate mediated effects (MacKinnon, 2008). In the mediation model, there is a hypothesized chain of relations between X and Y such that X first causes a change in a variable M that then causes a change in Y. This model explains how X affects Y by elaborating the relation to be X to M to Y. The mediator M is selected for change based on theory and prior empirical research. In addition to the strong theoretical and empirical rationale behind including the mediator, including a mediator may also increase the power to detect effects in some circumstances. This article describes the reason for this somewhat counterintuitive result, along with the conditions when this result occurs, and then describes the practical applications of these results for study planning.We present three studies to illustrate when power to detect the mediated effect is greater than power to detect the total effect. Study 1 compares power values for the mediated effect and total effect from a single mediator model to determine when power of the mediated effect is greater than power of the total effect. Study 2 compares power values for the total mediated effect to the total effect from a parallel two mediator model to determine when the test of the parallel two mediator model has more power than the test of the total effect. Study 3 investigates a sequential two mediator model and compares power values for the three-path mediated effect and the total effect to determine when power of the three-path mediated effect is greater than power of the total effect. Study 3 also compares power of the three-path mediated effect to power of the mediated effect from the single mediator model and power of the total effect to see how power increases differentially across models. Results from these studies reveal the nature of how models with one or two mediators increase power over the total effect. Finally, the practical use of these results is described through several research examples.The Single Mediator ModelFigure 1 shows the relation between an independent variable (X) and a dependent variable (Y), and a three variable model including X and Y when a mediator (M) is added to a model.An external file that holds a picture, illustration, etc.Object name is nihms-602858-f0002.jpgFigure 1Path diagrams for the regression and one mediator models. Adapted from MacKinnon, 2008.The following three regression equations are used to investigate mediation for a single mediator using notation in MacKinnon (2008):Y = i1 + cX + ε1 (1)Y = i2 + c’X + bM + ε2 (2)M = i3 + aX + ε3 (3)From Equation 1, i1 is the intercept for the equation, c is the relation between X and Y (also known as the total effect of X on Y), and ε1 is the variability in Y that is not explained by X. In this article the total effect refers to the c coefficient that represents the relation of X to Y, and applies to all subsequent models described. From Equation 2, b is the relation between M and Y adjusting for X, c' is the relation between X and Y adjusting for M (also known as the direct effect of X on Y that is not through M), and ε2 is the variability in Y that is not explained by its relations with X and M. From Equation 3, a is the relation between X and M, and ε3 is the variability in M that is not explained by X. The coefficients i2 and i3 are the intercepts for the equations. The intercepts are not involved in the estimation of mediated effects and could be left out of the equations, but are included here for completeness. Note that both c and c' are coefficients relating X to Y, but c' is a partial effect adjusted for the effects of the mediator. The mediation coefficients from Equations 2 and 3 are estimated using multiple regression or structural equation modeling. A classic example of a single mediator model is the mediation of the relationship between attitudes (X) and behavior (Y) by intentions (M) (Fishbein & Ajzen, 1975).The Parallel Two Mediator ModelThe parallel two mediator model is an extension of the single mediator model, where X is related to Y through a mediator (M1) and also through an additional mediator (M2). The mediators each have their own specific effects within this model (as opposed to transmitting the effect of X to M1 to M2 to Y), hence the use of the term parallel. The parallel two mediator model is depicted in Figure 2.An external file that holds a picture, illustration, etc.Object name is nihms-602858-f0003.jpgFigure 2Path diagram for the parallel two mediator model. Adapted from MacKinnon, 2008.The following equations represent the parallel two mediator model using notation from MacKinnon (2008):Y = i2 + c’X + b1M1 + b2M2 + ε2 (4)M1 = i3 + a1X + ε3 (5)M2 = i4 + a2X + ε4 (6)From Equation 4, c' is the relation between X and Y adjusting for M1 and M2, b1 is the relation between M1 and Y adjusting for M2 and X, b2 is the relation between M2 and Y adjusting for M1 and X, and ε2 is the variability in Y not explained by its relations with X, M1, and M2. From Equation 5, a1 is the relation between M1 and X, and ε3 is the variability in M1 that is not explained by X. From Equation 6, a2 is the relation between M2 and X, and ε4 is the variability in M2 not explained by X. The coefficients i2, i3, and i4 are the intercepts for the equations. An example of a parallel two mediator model is Salthouse's (1984) study looking at how the effect of age (X) on typing efficiency (Y) was mediated by reaction time (M1) and skill (M2).The Sequential Two Mediator ModelIn the sequential two mediator model, also referred to as the three-path mediator model (Taylor, MacKinnon, & Tein, 2008) or micromediational chain (Cook & Campbell, 1979), two mediators (M1 and M2) intervene between X and Y. This model is shown in Figure 3.An external file that holds a picture, illustration, etc.Object name is nihms-602858-f0004.jpgFigure 3Path diagram for the sequential two mediator model. Adapted from MacKinnon, 2008.The sequential two mediator model is represented by Equations 7, 8, and 9, using notation from MacKinnon (2008) and Taylor et al. (2008):Y = i2 + b4X + b3M2 + b6M1 + ε4 (7)M1 = i3 + b1X + ε2 (8)M2 = i4 + b2M1 + b5X + ε3 (9)From Equation 7, b4 is the relation between X and Y adjusting for M1 and M2, b3 is the relation between M2 and Y adjusting for X and M1, b6 is the relation between M1 and Y adjusting for M2 and X, and ε4 is the variability in Y not explained by its relations with X, M1, and M2. From Equation 8, b1 is the relation between M1 and X, and ε2 is the variability in M1 not explained by X. And from Equation 9, b2 is the relation between M2 and M1 adjusting for X, b5 is the relation between M2 and X adjusting for M1, and ε3 is the variability in M2 not explained by its relations with M1 and X. The coefficients i2, i3, and i4 are the intercepts for the equations.The sequential two mediator model contains several different effects of X on Y. In this model, b4 represents the direct effect of X on Y instead of c' as for the other models. The three effects that form the total mediated effect are the three-path mediated effect (b1b2b3), the two-path mediated effect passing through M1 (b1b6), and the two-path mediated effect passing through M2 (b5b3). The total mediated effect of X on Y passing through either mediator is the sum of the three mediated effects, b1b2b3 + b1b6 + b5b3. As in the two previous models, the total mediated effect is equal to the difference between the total and direct effects shown here in Equation 10:b1b2b3 + b1b6 + b5b3 = c − b4 (10)Taylor et al. (2008) describe a sequential two mediator model from Tekleab, Bartol, and Liu (2005) where the effect of pay (X) on turnover (Y) was mediated by perceived distributive justice (M1) and pay raise satisfaction (M2).Statistical PowerThe power of a statistical test is that test's ability to detect an effect when an effect is truly present in the population (Neyman & Pearson, 1933). Power depends on several key parameters involved in hypothesis testing, namely the Type I error rate, sample size, and effect size. The four parameters Type I error, sample size, effect size, and power are interdependent, and given any three, the fourth may be calculated. A power value of .80 is the standard for adequate power of a hypothesis test in the social sciences (Cohen, 1988). The power to detect the total effect of X on Y, c, is the probability of observing a statistically significant t value given the noncentral distribution for the true total effect of X on Y. The probability that both a and b are statistically significant is the product of the probability to detect each effect separately given that power to detect a and b are independent (Tofighi, MacKinnon, & Yoon, 2009). For the sequential two mediator model, the probability that b1, b2, and b3 are statistically significant is the product of the probability to detect each effect separately given that all paths are uncorrelated. The mediated effect ab is the product of two random variables and the total mediated effect a1b1 + a2b2 is the sum of two products of two random variables. The power to detect these effects is not equal to the probability of observing a statistically significant t value given the noncentral distribution (as it would be for one random variable), because the distribution of the product of two random variables is non-normal (Aroian, 1944; Craig, 1936). Appendix A contains additional details on power computations.Tests of MediationThere are different approaches to estimating mediated effects from the regression models (MacKinnon et al., 2002). For the single mediator model, the product of the aˆ and bˆ coefficients (aˆbˆ) is an estimator of the mediated effect. The effect of X on Y after adjustment for M (cˆ′) is an estimator of the direct effect. If a mediated effect is present and cˆ′ is zero then there is complete mediation. If a mediated effect is present and cˆ′ is not zero then there is partial mediation. The mediated effect is also equal to the difference between the cˆ and cˆ′ coefficients, cˆ−cˆ′. As a result, the total effect cˆ breaks down into a direct effect cˆ′ and a mediated effect aˆbˆ. For the multiple regression equations described above, aˆbˆ=cˆ−cˆ′ and therefore cˆ=cˆ′+aˆbˆ. The rationale behind the aˆbˆ mediation measure is that mediation depends on the extent to which the independent variable affects the mediator (coefficient aˆ) and the extent to which the mediator affects the dependent variable (coefficient bˆ). The aˆbˆ measure reflects how much a one unit change in X affects Y indirectly through M, and the cˆ−cˆ′ measure reflects how much M explains the relation between X and Y. The joint significance test of mediation (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002) tests whether the aˆ path is statistically significant and whether the bˆ path is statistically significant, and rejects the hypothesis of no mediation if both paths are statistically significant. In addition to the joint significance test, a commonly implemented test of mediation assesses significance of the mediated effect aˆbˆ by dividing the effect by its multivariate delta standard error based on first derivatives, sab=a2s2b+b2s2a−−−−−−−−−√ (Sobel, 1982).For the parallel two mediator model, the estimator of the mediated effect of M1 is the product of aˆ1 and bˆ1, and the mediated effect of M2 is the product of aˆ2 and bˆ2. Individually, these are the specific mediated effects (aˆ1bˆ1 and aˆ2bˆ2) and together, they are the total mediated effect (aˆ1bˆ1+aˆ2bˆ2). In this model, as cˆ is the total effect and cˆ′ is the direct effect of X on Y adjusting for M1 and M2, aˆ1bˆ1+aˆ2bˆ2=cˆ−cˆ′. As there is no test of joint significance for the total mediated effect aˆ1bˆ1+aˆ2bˆ2, the product of coefficients test assesses significance of the total mediated effect by dividing the total mediated effect by its multivariate delta standard error based on first derivatives, sa1b1+a2b2=s2a1b21+s2b1a21+s2a2b22+s2b2a22+2a1a2sb1b2−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−√ (MacKinnon, 2008).The bootstrap test also assesses significance of the total mediated effect for the parallel two mediator model by bootstrapping confidence intervals around aˆ1bˆ1+aˆ2bˆ2. The bootstrap test generates asymmetric confidence intervals based on the distribution of the mediated effect instead of assuming a normal distribution (MacKinnon et al., 2002; MacKinnon, Lockwood, & Williams, 2004). To obtain a bootstrap estimate of confidence limits for the mediated effect for a sample of size N, the bootstrap procedure takes a bootstrap sample of size N with replacement from the original sample and estimates a mediated effect. This process is repeated a large number of times to create a set of bootstrapped estimates that form a distribution, and that distribution is used to calculate confidence limits for the mediated effect (MacKinnon, 2008). MacKinnon (2008) and MacKinnon et al. (2002) contain additional information on significance tests for these models.For the sequential two mediator model, a joint significance test evaluates the significance of the bˆ1, bˆ2, and bˆ3 coefficients separately (MacKinnon et al., 2002; Taylor et al., 2008). If all paths are statistically significant, mediation is present. For information on other tests of mediation for the sequential two mediator model, see Taylor et al. (2008).Causal assumptions of mediationThis article aims to illustrate when the tests of mediation have more power than the test of the total effect. For the single mediator model, we make several assumptions regarding the causal relations among variables for identification of effects following VanderWeele and Vansteelandt (2009): (1) no confounders of the X to M relation, (2) no confounders of the M to Y relation, (3) no confounders of the X to Y relation, and (4) no effects of X on confounders that then affect the relation of X to Y. Randomization of X satisfies assumptions 1 and 3 which allows for estimators aˆ and cˆ to be treated as causal effects. Randomization of X does not satisfy assumptions 2 and 4 which are required for interpretation of bˆ and cˆ′ as causal effects. Statistical methods that accommodate confounders as well as methods that assess sensitivity may address the violation of these assumptions (Imai, Keele, & Tingley, 2010). Satisfying causal assumptions of mediation becomes more complex with multiple mediators, and is an active area of research (Albert & Nelson, 2011; Avin, Shpitser, & Pearl, 2005; Imai & Yamamoto, 2013; Lange, Rasmussen, & Thygesen, 2013; Robins, 2003).Increased Power Through Inclusion of Additional VariablesThe literature on how including additional variables in statistical analyses can be beneficial is extensive. This idea can be traced back to the foundations of path analysis when it was suggested that researchers should analyze complex or indirect relations among variables, rather than analyzing only direct influences (Wright, 1921; Wright, 1934). R. A. Fisher also summarized the concept: “…When asked in a meeting what can be done in observational studies to clarify the step from association to causation, Sir Ronald Fisher replied: `Make your theories elaborate'” (R. A. Fisher, cited by Cochran, 1965, p. 252). Literature in epidemiology also discussed the idea of making theories elaborate, where researchers proposed that it may be necessary to study the effects of a factor on a disease outcome through multiple channels, requiring more complex theories (Hill, 1965; Yerushalmy & Palmer, 1959). Research on causal inference argues that planned statistical analyses of observational studies should be based on elaborate theories (Rosenbaum, 2010), and that certain definitions of causality are inadequate because they do not address underlying causal mechanisms (Cox, 1992).The increase in power resulting from the inclusion of covariates is particularly well-documented. One of the primary uses of a covariate in analysis of covariance (ANCOVA) is to increase the precision of a randomized experiment by reducing error variability (Cochran, 1957; Cox, 1960; Huck, 1972). Miller and Chapman (2001) agree that the main goal of using ANCOVA should be to increase power to detect effects instead of using it to control for group differences, as is often done in practice. Although theory must determine whether inclusion of a mediator is appropriate and inclusion of a covariate may be data-driven, both serve to increase statistical power to detect effects in some situations.Surrogate endpoints are also frequently used in medical research as a means of increasing power. Researchers may wish to study the effect of X on an outcome Y which is difficult or costly to measure, so they will find an intermediate endpoint Z that is a surrogate for Y such that X affects Z as it would Y, and Z is a predictor of Y (Prentice, 1989, p. 432). For example, the presence of polyps has been used as a surrogate endpoint for the outcome of colon cancer (Freedman & Schatzkin, 1992). Sample size or effect size increases through the use of surrogate endpoints (thereby increasing power).Related to these concepts is the idea of an intensive design: including intermediate points of measurement and using the weighted average of those responses for each subject as an outcome instead of just one response (Kraemer & Thiemann, 1989). The intensive design increases power without requiring an increased sample size over a posttest-only randomized experimental design (Kraemer & Thiemann, 1989), and also increases power over a pretest-posttest design given certain conditions, although this design requires more measurement points (Maxwell, 1998; Venter, Maxwell, & Bolig, 2002).Increased Power Through Inclusion of MediatorsIn light of this research, it follows that including a mediator would increase power in some situations. Cox (1960) showed that including intermediate variables in the relation between X and Y would increase precision. MacKinnon et al. (2002) identified the tests of mediation with the best power and Type I error rates, and found that due to the requirement of a significant X to Y relation, the Baron and Kenny (1986) and Judd and Kenny (1981) causal steps tests for mediation are underpowered. Fritz and MacKinnon (2007) replicated this result, finding that when a and b paths were small and c' was zero, the sample size required to detect the mediated effect at .80 power for the Baron and Kenny test was 20886. Shrout and Bolger (2002) also stated that power to detect mediation is greater than power to detect a total effect of X on Y in some situations, and gave an example of a situation where power to detect the a and b effects would be greater than power to detect the c total effect. This means that some tests of mediation detect significant mediated effects when the relation between X and Y is nonsignificant (Fritz, Cox, & MacKinnon, 2013; MacKinnon, 2008, 394–395; Shrout & Bolger, 2002; Taylor et al., 2008). It follows that including a mediator, or mediators, results in increased power over the total effect in some situations. The purpose of this article is to describe when these situations occur.Go to:Study 1As described above, there is evidence that the single mediator model will have more power to detect effects than the total effect under certain circumstances. Study 1 compared power values of the mediated effect ab and the total effect c for the single mediator model to determine when the mediated effect has more power.MethodAnalytical power for the single mediator model was calculated based on derivations in Appendix A. Both SAS and R programs to compute analytical power for the single mediator model may be downloaded at Research in Prevention Laboratory (MacKinnon). For the single mediator model, population variances and covariances for X, M, and Y were calculated based on different combinations of population path parameters and sample sizes. The covariance matrix for the single mediator model is given in MacKinnon (2008). Sample sizes of 50, 100, 200, 500, 1000, and 5000 were chosen based on those commonly found in the social sciences and in mediation research (Fairchild, MacKinnon, Taborga, & Taylor, 2009; MacKinnon et al., 2002; MacKinnon et al., 2004). Population path parameters were chosen in accordance with those used in prior research on mediation models for the single mediator model (0, .14, .39, and .59) for each of the a, b, and c' paths (Fritz & MacKinnon, 2007; MacKinnon et al., 2002). This 4 × 4 × 4 × 6 factorial design resulted in 384 combinations of path parameters and sample size.The predicted variances and covariances were then used to calculate zero-order and first-order partial correlation effect sizes corresponding to the population a and b paths as found in MacKinnon (2008). The variances and covariances for the single mediator model were also used to calculate the true a and b coefficients and their true standard errors. Power was then calculated in two ways. First the correlations corresponding to a and b coefficients were used to calculate a z test, which was then used to compute analytical power. Then the path coefficients a and b and their true standard errors were used to calculate a t test, which was used to compute analytical power. The z and t tests yielded the same power values. Analytical formulas for power of the joint significance test of mediation for the single mediator model are also found in Wang and Xue (2012). For both methods of computing analytical power, the individual power values for a and b were multiplied to calculate analytical power, analogous to the joint significance test.ResultsSingle mediator model analytical resultsAnalytical power of the test of the total effect and the joint significance test for the mediated effect from the single mediator model where c' is zero or small are shown in Table 1 (for all cases where power of the test of the mediated effect was greater than power of the test of the total effect, the direct effect c' was always zero or small; when c' is equal to zero this indicated full mediation). Only power of the models is considered in this table (combinations where a or b is zero are measures of Type I error rate).Table 1Analytical Power of the Mediated and Total Effects for the Single Mediator Model at c′ = 0 and c′ = . 14N = 50 N = 100 N = 200 N = 500 N = 1000 N = 5000c ′ π c π ab π c π ab π c π ab π c π ab π c π ab π c π aba = .14, b = .14 0 .034 .025 .038 .078 .046 .249 .063 .766 .089 .986 .278 10.14 .189 .025 .341 .078 .598 .249 .939 .766 .999 .986 1 1a = .14, b = .39 0 .054 .117 .072 .269 .106 .499 .204 .875 .361 .993 .949 10.14 .235 .117 .427 .269 .716 .499 .980 .875 1 .993 1 1a = .14, b = .59 0 .070 .153 .104 .279 .168 .499 .354 .875 .612 .993 .999 10.14 .257 .153 .467 .279 .763 .499 .989 .875 1 .993 1 1a = .39, b = .14 0 .056 .117 .077 .269 .115 .499 .225 .875 .400 .993 .969 10.14 .259 .117 .471 .269 .767 .499 .990 .875 1 .993 1 1a = .39, b = .39 0 .161 .551 .285 .928 .509 .999 .883 1 .994 1 1 10.14 .454 .551 .754 .928 .965 .999 1 1 1 1 1 1a = .39, b = .59 0 .271 .720 .492 .963 .789 1 .992 1 1 1 1 10.14 .575 .720 .870 .963 .993 1 1 1 1 1 1 1a = .59, b = .14 0 .081 .153 .124 .279 .208 .499 .445 .875 .732 .993 1 10.14 .323 .153 .577 .279 .866 .499 .998 .875 1 .993 1 1a = .59, b = .39 0 .308 .720 .554 .963 .848 1 .997 1 1 1 1 10.14 .640 .720 .915 .963 .997 1 1 1 1 1 1 1a = .59, b = .59 0 .526 .941 .829 1 .986 1 1 1 1 1 1 10.14 .800 .941 .981 1 1 1 1 1 1 1 1 1Open in a separate windowNote. Analytical power of the total effect is represented here as 'πc', and analytical power calculated using regression coefficients and their standard errors is represented here as 'πab'.Combinations of parameters where the test of the mediated effect had more power than the test of the total effect are underlined. When a and b were small (a = b = .14), the test of the mediated effect had more power than the test of the total effect at larger sample sizes. The test of the total effect had more power than the test of the mediated effect at smaller sample sizes when a and b were small. For example, in Table 1 for the case where a and b were small (a = b = .14), the test of the total effect had more power than the test of the mediated effect for a sample size of 50. The test of the mediated effect had more power than the test of the total effect in every other sample size. However, when a and b were large (a = b = .59), the test of the mediated effect had more power than the test of the total effect at smaller sample sizes. For a = b = .59, the test of the mediated effect had more power than the test of the total effect in sample sizes up to 200. At sample sizes larger than 200, the test of the total effect and the joint significance and product of coefficients tests all had power of approximately one for large a and b. In summary, two patterns of results emerged for the single mediator model. The test of the mediated effect had more power than the test of the total effect when sample size was large and coefficients were small, and when sample size was small and coefficients were large. At large sample sizes with large coefficients, power of both tests approached one. Additionally, the test of the mediated effect had more power than the test of the total effect only in cases where c' was zero or small.The analytical power results were replicated in an empirical simulation study that is not reported here. Results from the empirical simulation are available at Research in Prevention Laboratory (MacKinnon).Comparison of standard errors of total and mediated effectsFor the majority of the combinations of parameters and sample sizes studied, the test of mediation was more powerful than the test of the total effect when c' was equal to zero, and therefore when the total and mediated effects were equal (that is, ab = c). For the test of the mediated effect to be more powerful than the test of the total effect, the t test of significance for ab must be larger than the t test of significance for c. Because the tests of significance are computed by dividing the effects by their standard errors, it follows that if the test of the mediated effect is more powerful than the test of the total effect, the standard error of the total effect must be larger than the standard error of the mediated effect.Table 2 shows a comparison of the analytical standard errors of c and ab for the single mediator model next to power of the tests of c and ab, for combinations where a and b are greater than zero and c' was equal to zero at N = 100 (a subsample of the combinations used in the results above).Table 2Comparison of Power and Standard Errors of ab and c for the Single Mediator Model Where c' = 0 and N = 100a b ab c sc sab π c π ab.14 .14 .020 .020 .102 .020 .038 .078.14 .39 .055 .055 .108 .042 .072 .269.14 .59 .083 .083 .117 .061 .104 .279.39 .14 .055 .055 .102 .042 .077 .269.39 .39 .152 .152 .108 .056 .285 .928.39 .59 .230 .230 .117 .072 .492 .963.59 .14 .083 .083 .102 .062 .124 .279.59 .39 .230 .230 .108 .072 .554 .963.59 .59 .348 .348 .117 .085 .829 1When ab was equal to c and c' was equal to zero, the standard error of c was larger than the standard error of ab. Furthermore, when the standard error of c was larger than the standard error of ab, power of the test of ab was larger than power of the test of c.A further study of situations where the test of the mediated effect had more power than the test of the total effect at smaller sample sizes and smaller coefficients showed that although the standard error of c was always larger than the standard error of ab when ab and c are equal, when a or b approached zero the power of the test of ab became smaller than power of the test of c. This pattern occurred for cases where a or b was very small but not zero. For example at N = 100, when a was equal to .14 and b was equal to .001, c was equal to .0014. The standard error of ab was .01422 and the standard error of c was .10102 so the standard error of ab was smaller than the standard error of c, but power of the test of ab was equal to .00714 and power of the test of c was equal to .02508. Although power of the test of c approached 0.025 for very small coefficients, power of the test of ab decreased to below 0.025.Type I error ratesFor all sample sizes, when a = b = 0 the test of joint significance had Type I error rates of .001. This result is a replication of previous literature on Type I error rates for tests of mediation (MacKinnon et al., 2002).DiscussionA comparison of analytical power of the mediated and total effects shows that the mediated effect has increased power with small parameter sizes and large sample size, and with large parameter sizes and small sample size. Furthermore, the conditions for when power of the test of the mediated effect will be greater than power of the test of the total effect are also found in the comparison of standard errors of c and ab, and therefore in terms of variances and covariances among variables. This means that given an expected covariance matrix among variables based on previous literature, it would be possible to use those variances and covariances to determine if the standard error of c will be larger than the standard error of ab. If it is and ab and c are expected to be equal, the test of mediation will have more power than the test of the total effect.Proximal Versus Distal Mediators: Effects on PowerWhen b is larger than a (that is, M is closer in time or more highly related to Y than to X), M may be considered a distal mediator. When a is larger than b (that is, M is closer in time or more highly related to X than to Y), M may be considered a proximal mediator. According to previous research, the test of ab will be more powerful for models with distal mediators than for models with proximal mediators due to collinearity between X and M (Hoyle & Kenny, 1999; Kenny & Judd, 2013). When collinearity between X and M is high, the standard error of the b path is increased, leading to a less powerful test of significance. Results from Fritz and Mackinnon (2007) support this point, demonstrating that required sample size is larger for conditions where a is larger than b.As the test of ab is more powerful when b is larger than a, it follows that when comparing power of the test of ab to power of the test of c, the gain in power over the test of c would be greater in conditions where b is larger than a. This gain in power is seen in the results from this study. For example, in Table 1 at N = 200, a = .39, b = .59, c' = 0, adding a mediator increased power to detect effects by .211, and at N = 200, a = .59, b = .39, c' = 0, adding a mediator increased power to detect effects by 0.152. The increase in power to detect effects was larger for the condition where b was larger than a.Go to:Study 2Study 1 demonstrated that power to detect the mediated effect was greater than power to detect the total effect for the single mediator model. Study 2 aims to demonstrate that having two mediators also increases statistical power to detect effects, and identifies when the test of the total mediated effect a1b1 + a2b2 will have more power than the test of the total effect c for the parallel two mediator model.MethodDetermination of path parameters for the parallel two mediator modelThe parallel two mediator model covariance matrix is shown in Table 3. Covariance algebra was used to derive formulas for the variance of each variable in the parallel two mediator model, and the covariances between each pair of variables.Table 3Covariance Matrix for the Parallel Two Mediator ModelAn external file that holds a picture, illustration, etc.Object name is nihms-602858-t0001.jpgThe variances and covariances were then used to derive full and partial correlation effect sizes for the parallel two mediator model where ρ is the correlation. Formulas for the effect sizes are as follows:ρXM1=σXM1σ2X−−−√σ2M1−−−√(11)ρM1Y.XM2=ρM1Y.X−ρM1M2.XρM2Y.X1−ρ2M1M2.X−−−−−−−−−√1−ρ2M2Y.X−−−−−−−−√(12)ρXM2=σXM2σ2X−−−√σ2M2−−−√(13)ρM2Y.XM1=ρM2Y.X−ρM1M2.XρM1Y.X1−ρ2M1M2.X−−−−−−−−−√1−ρ2M1Y.X−−−−−−−−√(14)ρXY.M1M2=ρXY.M1−ρXM2.M1ρYM2.M11−ρ2XM2.M1−−−−−−−−−√1−ρ2YM2.M1−−−−−−−−−√(15)Where Equation 11 is the zero-order correlation effect size for a1, Equation 12 is the second-order partial correlation effect size for b1, Equation 13 is the zero-order correlation effect size for a2, Equation 14 is the second-order partial correlation effect size for b2, and Equation 15 is the second-order partial correlation effect size for c'. Appendix B shows how the effect size formulas were produced from the variances and covariances between variables.Regression coefficients for this model were derived based on the above formulas that correspond approximately to Cohen's (1988) small, medium, and large correlation effect sizes. Parameters were identified based on the correlation results that would be as close to Cohen's guidelines for small, medium, and large effect sizes (.1, .3, and .5) as possible. It was determined that for a1, b1, a2, and b2, path coefficients of .101, .314, and .577 corresponded to small, medium, and large effects. For c', path coefficients of .131, .400, and .740 corresponded to small, medium, and large effects. These path coefficients were necessary for generating data to calculate empirical power for the parallel two mediator model.Empirical power for the parallel two mediator modelEmpirical power was calculated by generating data with a known mediated effect and computing the proportion of times the test for mediation was statistically significant. Sample sizes chosen for the parallel two mediator model were 50, 100, 200, 500, and 1000. Parameter values (described above) were 0, .101, .314, and .577 for the a1, a2, b1, and b2 coefficients, and 0, .131, .400, and .577 for the c' coefficient. This 5 × 45 design resulted in 5120 combinations of parameters and sample size, and 500 datasets were simulated for each combination. All variables were simulated to be continuous for both models, and all tests of significance were two-tailed. The product of coefficients test was used to assess significance for the parallel two mediator model. In this study, power was assessed for testing two null hypotheses: H0: c = 0 for the test of the total effect and H0: a1b1 + a2b2 = 0 for the product of coefficients test of significance. For each combination of parameters and sample size, the proportion of times out of that the test was significant out of 500 generated datasets was the empirical power for combinations with non-zero path parameters (the Type I error rate for combinations with path parameters equal to zero).BootstrapBootstrap estimation was used to calculate power for the total mediated effect because there is not a joint significance test for the parallel two mediator model, and the z test using the multivariate delta method standard error for the total mediated effect is known to be inaccurate. Bootstrap estimation was conducted for 10 combinations of parameters and sample size to compare to power of the total effect and total mediated effect. The combinations were randomly selected from the combinations where the total mediated effect was more powerful than the total effect based on the empirical power results. One sample of data was simulated for X, M1, M2, and Y based on the mediation equations, and then bootstrap estimation was conducted for that single dataset by sampling with replacement from that sample of data to get N observations, forming the first bootstrap sample. In each bootstrap sample, the values of a1, b1, a2, b2, and c were generated and saved. This process of randomly sampling with replacement and saving values of the coefficients was repeated 1000 times to create 1000 bootstrap samples for the first simulated data set, and the saved bootstrapped values of a1, b1, a2, b2, and c were used to create confidence intervals for the total mediated effect a1b1 + a2b2 and the total effect c. This process of simulating a dataset and generating a bootstrap confidence interval was repeated 1000 times, simulating 1000 datasets. The means of the number of times the confidence intervals did not include zero from the 1000 simulated datasets were the bootstrapped power values of the total and total mediated effects.ResultsParallel two mediator model empirical resultsTable 4 contains empirical power values for both the product of coefficients test of the total mediated effect for the parallel two mediator model and the test of the total effect where c' = 0 or .131 (corresponding to zero or small effect sizes for c', which were the only values of c' at which the test of the total mediated effect was found to have more power than the test of the total effect). To reduce the amount of information presented, only combinations where a1 = b1 and a2 = b2 are included. Combinations of parameters and sample size where the test of the total mediated effect was more powerful than the test of the total effect are underlined, and combinations of parameters and sample size where the test of the total mediated effect exceeded .80 when the test of the total effect did not are underlined and italicized.

False propaganda against homeopathy by vested interests and others misled by media and also in the name of science has become a modern day fashion. All this coupled with almost no governmental support or funding, no investment by multinational or pharmaceutical companies and no further research in the development of homeopathy, homeopathy still exists as the second largest medical system in the world. Imagine the level of homeopathy if research, funding and government support is the same as to so called modern medicine. Till date, homeopathy research and development was totally based on those doctors of allopathy who started studying homeopathy and quit their lucrative practice and devoted their own life as well as the lives of their dear ones for homeopathic provings. No government and research organisations were ever involved considerably in the development of homeopathy except a few dedicated doctors of homeopathic.You can be one of those devoted doctors like Hahnemann, Dr. J T Kent, Dr. Lippe, Dr. Nash, Dr. Clark, Dr. Swan, Dr. William Boericke, Dr. C Herring, Dr. T F Allen, Dr. W.D. Gentry, Dr. Dewey, Dr. C M Boger and many other stalwarts who sacrificed their lives and went on single handed to help the mankind.Another thing you can do is to bring the reality of homeopathy to counter the false propaganda of media in the name of science by hiding the facts:One such fact is most respected Wikipedia. When Jimmy was contacted by modern day homeopaths about the inaccuracy of information about homeopathy and wanted to update, they were not allowed on the ground that being homeopath and therefore have conflict of interest. All doctors of modern medicine are allowed to update medical information on Wikipedia, not the homeopaths. You can imagine the role of such a media.Another fact is people are touting about the water memory without even knowing an iota of realty. Even some writers and others are touting on Quora as magical or mystic water to water memory. Fact of the matter is being hidden that water memory was discovered by a Noble Prize winner scientist:Dysfunction at Wikipedia on Homeopathic MedicineDana Ullman Evidence Based HomeopathIn April, 2014, I had the happenstance to run into Jimmy Wales, the co-founder of Wikipedia, on the streets of Vancouver. I was there to lecture to a group of medical professionals, while he was attending the TED talks. I expressed my appreciation to him for creating Wikipedia. I also then expressed concern to him about the “unencyclopedic” tone and information in Wikipedia’s article on homeopathy. He then encouraged me to express my concerns in writing, and this is that response.It may surprise and even shock most people to learn that, according to the Washington Post, the two most controversial subjects on Wikipedia in four leading languages (English, French, German, and Spanish) are the articles on “Jesus Christ” and “homeopathy.”Because I know that you want Wikipedia to be the best modern resource of reliable information, my intent in writing is to show you where Wikipedia is falling below your high standards, and in fact, Wikipedia’s article on homeopathy is providing strongly biased, inaccurate information. This strong bias is a symptom of a deeper problem at Wikipedia in select articles on topics that challenge dominant medical and scientific worldviews. After reading the below body of scientific evidence on the subject of homeopathic medicine, I hope that we can engage in a dialogue that will help reduce the amount of misinformation that pervades certain subjects, such as homeopathy.Evidence of the strong bias against homeopathy and against an objective encyclopedic tone is evident throughout the article. I will first focus on the second sentence of the first paragraph of the article and the 6 references which purport to substantiate these claims:Homeopathy i/ˌhoʊmiˈɒpəθi/ (also spelled homoeopathy or homœopathy; from the Greek ὅμοιος hómoios “like-“ and πάθος páthos “suffering”) is a system of alternative medicine created in 1796 by Samuel Hahnemann, based on his doctrine of like cures like, according to which a substance that causes the symptoms of a disease in healthy people will cure similar symptoms in sick people.[1] Homeopathy is a pseudoscience[2][3][4] and its remedies have been found to be no more effective than placebos.[5] [6]References from Wikipedia’s article on “Homeopathy”:Hahnemann, Samuel (1833). The Homœopathic Medical Doctrine, or “Organon of the Healing Art”. Dublin: W.F. Wakeman. pp. iii , 48-49 . “Observation, reflection, and experience have unfolded to me that the best and true method of cure is founded on the principle, similia similibus curentur. To cure in a mild, prompt, safe, and durable manner, it is necessary to choose in each case a medicine that will excite an affection similar (ὅμοιος πάθος) to that against which it is employed.” Translator: Charles H. Devrient, Esq.^ Tuomela R (1987). “Chapter 4: Science, Protoscience, and Pseudoscience”. In Pitt JC, Marcello P. Rational Changes in Science: Essays on Scientific Reasoning. Boston Studies in the Philosophy of Science 98 (Springer). pp. 83-101. doi:10.1007/978-94-009-3779-6_4 . ISBN 978-94-010-8181-8.^ Smith K (2012). “Homeopathy is Unscientific and Unethical”. Bioethics 26 (9): 508-512. doi:10.1111/j.1467-8519.2011.01956.x .^ Baran GR, Kiana MF, Samuel SP (2014). “Chapter 2: Science, Pseudoscience, and Not Science: How Do They Differ?” . Healthcare and Biomedical Technology in the 21st Century (Springer). pp. 19-57. doi:10.1007/978-1-4614-8541-4_2 . ISBN 978-1-4614-8540-7. “within the traditional medical community it is considered to be quackery”^ Shang A, Huwiler-Müntener K, Nartey L, et al. (2005). “Are the clinical effects of homoeopathy placebo effects? Comparative study of placebo-controlled trials of homoeopathy and allopathy”. Lancet 366 (9487): 726-32. doi:10.1016/S0140-6736(05)67177-2 . PMID 16125589 .Evidence Check 2: Homeopathy - Science and Technology Committee , British House of Commons Science and Technology Committee, 22 February 2010, retrieved 2014-04-05Is Homeopathy Really a “Pseudoscience”?Wikipedia asserts that “Pseudoscience is a claim, belief or practice which is presented as scientific, but does not adhere to a valid scientific method, lacks supporting evidence or plausibility, cannot be reliably tested, or otherwise lacks scientific status.”The “editors” at Wikipedia have deemed homeopathy to be a “pseudoscience” even though randomized double-blind and placebo controlled studies that have been published in many of the best medical journals in the world have shown efficacy of homeopathic treatment for many common and serious health problems (below is a partial list of such studies):Chronic obstructive pulmonary disease: Frass, M, Dielacher, C, Linkesch, M, et al.Influence of potassium dichromate on tracheal secretions in critically ill patients, Chest, March, 2005;127:936-941. The journal, Chest, is the official publication of the American College of Chest Physicians.Hayfever: Reilly D, Taylor M, McSharry C, et al., Is homoeopathy a placebo response? controlled trial of homoeopathic potency, with pollen in hayfever as model,” Lancet, October 18, 1986, ii: 881-6.Asthma: Reilly, D, Taylor, M, Beattie, N, et al., “Is Evidence for Homoeopathy Reproducible?” Lancet, December 10, 1994, 344:1601-6.Fibromyalgia: Bell IR, Lewis II DA, Brooks AJ, et al. Improved clinical status in fibromyalgia patients treated with individualized homeopathic remedies versus placebo, Rheumatology. 2004:1111-5. This journal is the official journal of the British Society of Rheumatology.Fibromyalgia: Fisher P, Greenwood A, Huskisson EC, et al., “Effect of Homoeopathic Treatment on Fibrositis (Primary Fibromyalgia),” BMJ, 299(August 5, 1989):365-6.Childhood diarrhea: Jacobs, J, Jimenez, LM, Gloyd, SS, Treatment of Acute Childhood Diarrhea with Homeopathic Medicine: A Randomized Double-blind Controlled Study in Nicaragua, Pediatrics, May, 1994,93,5:719-25.ADD/ADHD: Frei, H, Everts R, von Ammon K, Kaufmann F, Walther D, Hsu-Schmitz SF, Collenberg M, Fuhrer K, Hassink R, Steinlin M, Thurneysen A. Homeopathic treatment of children with attention deficit hyperactivity disorder: a randomised, double blind, placebo controlled crossover trial. Eur J Pediatr., July 27,2005,164:758-767.Jimmy, can you name ONE other system of “pseudoscience” that has a similar body of randomized, double-blind, and placebo-controlled clinical trials published in high-impact medical journals showing efficacy of treatment?It is more than a tad ironic that this first paragraph in the Wikipedia article on homeopathy references only one article that was published in a peer-review medical journal. This one article by Shang, et al. has been thoroughly discredited in an article written by Lüdtke and Rutten that was published in a leading “high impact” journal that specializes in evaluating clinical research. The Shang meta-analysis is highlighted on Wikipedia without reference to any critique of it. The fact that there is no hint of any problems in the Shang review, let alone a reference to the Lüdtke and Rutten article that provided evidence of bias, is itself a cause for concern.The Shang article is also the primary reference used by the widely ridiculed “Evidence Check” reports issued by the Science and Technology Committee of the British House of Commons, which also conveniently omits reference to the severe limitations of this one review of research. Further, the “Evidence Check” was signed off by just three of the 15 members of the original committee, never discussed or endorsed by the whole UK Parliament, and had its recommendations ignored by the UK Department of Health.It should be made clear that the Shang meta-analysis was co-authored by M. Egger who is a well-known skeptic of homeopathy and who wrote to the Lancet that his hypothesis before conducting the review was that homeopathy was only a placebo effect. Readers were never informed of this bias.The meta-analysis by Shang evaluated and compared 110 placebo-controlled trials testing homeopathic medicines with 110 testing conventional drugs, finding 21 homeopathy trials (19%) but only nine (8%) conventional-medicine trials were of “higher quality.” Lüdtke and Rutten found that a positive outcome for homeopathy would have resulted if Shang had simply compared these high quality trials against each other. However, with some clever statistical footwork, Shang chose to limit the high quality trials to only 8 homeopathic and 6 conventional medical trials, a result that led to a “negative” outcome for homeopathy. Lüdtke and Rutten determined this review as biased for its “arbitrarily defined one subset of eight trials” and they deemed the entire review as “falsely negative.”By reducing the number of studies, Shang created convoluted logic that enabled his team to avoid evaluation of ANY of the above high quality studies that were all published in respected medical journals. Further, 7 of 8 homeopathic studies only tested one homeopathic medicine for everyone with the similar disease even though one of the primary tenets of homeopathy requires individualization of treatment. Many other extremely scathing critiques of the Shang research were published in the Lancet shortly after publication, including the exclusion of one high quality homeopathic study due to the questionable assertion that the researchers could not find a study in all of conventional medical research that treated patients with polyarthritis (arthritis that involves five or more joints).Skeptics typically assert that the above high-quality studies published in high-impact medical journals are simply “cherry-picking” the positive studies, and then, they begin cherry-picking studies that had negative results. However, skeptics of homeopathy fail to differentiate good, sound scientific investigations that are respectful of the homeopathic method and those that are not. Just because a study was conducted with a randomized double-blind and placebo controlled method does NOT mean that the study gave the appropriate homeopathic medicine for each patient or even each group of patients. This ignorance is akin to someone saying that antibiotics are ineffective for “infections” without differentiating between bacterial infections, viral infections, and fungal infections. Ironically, skeptics of homeopathy consistently show a very sloppy attitude about scientific investigations.What the Most Comprehensive Review of Homeopathic Research Found...Skeptics commonly assert that various meta-analyses verify that homeopathy doesn’t work and that homeopathic medicines are equivalent to the effects of a placebo. These skeptics typically chose to ignore various meta-analyses that were published in highly respected medical journals and that show positive benefits from homeopathic medicines. Skeptics also ignore the largest and most comprehensive review of research ever conducted...one that was funded by the government of Switzerland.It is useful to know that the Shang/Egger meta-analysis was funded by the same Swiss government’s Complementary Medicine Evaluation Program that also funded a much more detailed and comprehensive review of clinical research, preclinical research (fundamental physio-chemical research, botanical studies, animal studies, and in vitro studies with human cells), epidemiological evidence, and cost-effectiveness studies.This more comprehensive Swiss government-funded report found a particularly strong body of evidence to support the homeopathic treatment of Upper Respiratory Tract Infections and Respiratory Allergies. The report cited 29 studies in “Upper Respiratory Tract Infections/AllergicReactions,” with 24 studies having a positive result in favor of homeopathy. Six out of seven controlled studies that compared homeopathic treatment with conventional medical treatment showed homeopathy to be more effective than conventional medical interventions. When the researchers evaluated only the randomized placebo controlled trials, 12 out of 16 studies showed a positive result in favor of homeopathy.Ironically, the Shang/Egger meta-analysis acknowledged that there have been at least eight clinical trials of patients with acute infections of the upper respiratory tract and that there is “robust evidence that the treatment under investigation works.” And yet, Shang/Egger assert that this limited number of trials is inadequate for evaluating homeopathy, while at the same time they assert that eight other trials provided unquestionable evidence for damning homeopathy (it should be noted that Shang/Egger somehow determined that some of the studies on respiratory infection and allergy were not “high quality,” even though numerous other meta-analyses have unanimously defined three trials by David Reilly as high quality (two were published in the British Medical Journal and one was published in the Lancet).In actual fact, although some meta-analyses have had a “negative” result, there have also been a significant number of meta-analyses that have had positive results, including this partial list:Linde L, Clausius N, Ramirez G, Jonas W, “Are the Clinical Effects of Homoeopathy Placebo Effects? A Meta-analysis of Placebo-Controlled Trials,” Lancet, September 20, 1997, 350:834-843. Although a later review by some of these authors found a reduced significance, the authors never asserted that the significance was no longer present. Further, two of the lead authors of this article provided a very sharp critique of the Shang, et al. review of research (2005). Also, both Linde and Jonas wrote to the Lancet after the Shang/Egger article was published and asserted that the Lancet should be “embarrassed” by their publication of this article and the accompanied editorial (Lancet, 366 December 17, 2005:2081-2).Kleijnen J, Knipschild P ter Riet G. Clinical trials of homoeopathy. BMJ 1991, 302, 316-23. Of the 22 best studies, 15 showed positive results from homeopathic treatment. The researchers concluded, “there is a legitimate case for further evaluation of homeopathy.”Jacobs J, Jonas WB, Jimenez-Perez M, Crothers D, Homeopathy for Childhood Diarrhea: Combined Results and Metaanalysis from Three Randomized, Controlled Clinical Trials, Pediatr Infect Dis J, 2003;22:229-34. This metaanalysis of 242 children showed a highly significant result in the duration of childhood diarrhea (P=0.008).Kassab S, Cummings M, Berkovitz S, van Haselen R, Fisher P. Homeopathic medicines for adverse effects of cancer treatments. Cochrane Database of Systematic Reviews 2009, Issue 2.Taylor, MA, Reilly, D, Llewellyn-Jones, RH, et al., Randomised controlled trial of homoeopathy versus placebo in perennial allergic rhinitis with overview of four trial Series, BMJ, August 19, 2000, 321:471-476. The BMJ published an editorial in the issue in which this study was published asserting, “It may be time to confront the conclusion that homeopathy and placebo differ...... This may be more plausible than the conclusion that their trials have produced serial false positive results” (This week in the BMJ. Homoeopathic dilutions may be better than placebo. BMJ 2000;321:0).Jonas, WB, Linde, Klaus, and Ramirez, Gilbert, “Homeopathy and Rheumatic Disease,” Rheumatic Disease Clinics of North America, February 2000,1:117-123.Is Homeopathy Really “Implausible”?The third paragraph in the Wikipedia article continues to show both strong bias against homeopathy and inaccurate information.Homeopathy lacks biological plausibility[10] and the axioms of homeopathy have been refuted for some time.[11] The postulated mechanisms of action of homeopathic remedies are both scientifically implausible[12][13] and not physically possible.[14] Although some clinical trials produce positive results,[15][16] systematic reviews reveal that this is because of chance, flawed research methods, and reporting bias. Overall there is no evidence of efficacy.[12][17][18][19] Continued homeopathic practice, despite the evidence that it does not work, has been criticized as unethical because it increases the suffering of patients by discouraging the use of real medicine,[20] with the World Health Organisation warning against using homeopathy to try to treat severe diseases such as HIV and malaria.[21] The continued practice, despite a lack of evidence of efficacy, has led to homeopathy being characterized within the scientific and medical communities as nonsense,[22] quackery,[4][23][24] or a sham.[25]Ironically, the article makes reference to articles written by known antagonists to homeopathy (such as E. Ernst and K. Atwood) that have not even been published in peer-review scientific or medical journals. Reference #10 by Ernst was published in “The Skeptical Inquirer,” a magazine that is not listed in Index Medicus or any other respected scientific indexing service, and reference #11 by Atwood wasn’t even published in a magazine but at a website. If and when any person tried to edit the article on homeopathy in any way in which homeopathy is presented in a positive light and makes reference to a “magazine” or a “website,” that person would be laughed off of Wikipedia, and yet, the editors of the homeopathy article allow and even encourage the use of inappropriate skeptical references (according to Wikipedia’s usual standards).In the same way that Wikipedia’s editors have inappropriately deemed homeopathy to be “pseudoscientific,” they have also deemed that homeopathy lacks “plausibility.” The definition of plausibility is: “having an appearance of truth or reason; seemingly worthy of approval or acceptance; credible; believable.”The journal, Langmuir, is the journal of the American Chemical Society, and in 2012, they published an important article that provided a plausible explanation for the actions of homeopathic medicines. First, they verified using three different types of spectroscopy that clearly showed that nanoparticles of six original medicinal agents persisted in solutions even after they were diluted 1:100 six times, thirty times, and even two-hundred times.Avogadro’s number predicts that none of the original medicinal agents would have ANY persisting molecules of the original medicinal substance would remain after 12 dilutions of 1:100. However, the scientists describe reasonable and even predictable factors that lead to the persistence of nanoparticles after their multiple dilutions. The scientists note that the use of double-distilled water in glass vials leads to varying amounts of silica fragments that fall into the water, as much as 6ppm. The vigorous shaking of the glass vial creates bubbles and “nanobubbles” that bring oxygen into the water and that increase substantially the water pressure (William Tiller, PhD, the former head of Stanford’s Department of Material Science, estimated this pressure to be 10,000 atmospheres).Ultimately, this increased water pressure forces whatever medicinal substance is in the double-distilled water into the silica, and every substance will interact with the silica in its own idiosyncratic way. Then, when 90% of the water is dumped out, the silica fragments predictably cling to the glass walls.When skeptics of homeopathy reference Avogadro’s number as “evidence” that homeopathic medicines beyond 24X or 12C have “no remaining molecules left,” they are simply verifying their own ignorance of Avogadro’s number because this widely recognized principle in chemistry does NOT account for the complexities of the silica fragments, the bubbles or nanobubbles, nor the increased water pressure. In fact, any serious scientist or educated individual who asserts that a homeopathic medicine is “beyond Avogadro’s number” has no ground on which they stand. And yet, Avogadro’s number is prominently a part of Wikipedia’s article on homeopathy.Despite the obfuscation throughout Wikipedia’s article on homeopathy, in actual fact, the homeopathic pharmaceutical procedure called “potentization” is a clever, perhaps brilliant, method of creating nanoparticles of whatever substance is originally placed in the glass vial. Even more compelling is the significant and growing body of evidence that nanodoses of medicinal agents have several benefits over crude doses of the same substance, including enhanced bioavailability, adsorptive capacity, intracellular accessibility, increased ability to cross cell membranes and even the blood brain barrier, and of course, a substantial better safety profile.The creation of nanodoses actually increases various characteristics of a substance’s properties. Once a substance has an extremely small size but has larger surface area to volume ratio, the nanodose properties create increased chemical and biological reactivity, electromagnetic, optical, thermal, and quantum effects. Further, the idiosyncratic properties of nanomedicines reduce the required doses by orders of magnitude and predictably reduce toxicity.In light of the above, it is stunning and shocking that Wikipedia’s article on “Nanomedicine” has no mention of homeopathy, which rightly is deemed to be the original nanomedicine and nanopharmacology. At a time in the history of medicine and science in which the field of nanomedicine is becoming increasingly accepted and respected, Wikipedia seems stuck in the 20th century, or perhaps the 18th century. It is not surprising that there is an international and inter-disciplinary journal that focuses on the power of extremely small doses in various biological systems, not just medicine.Given the above, it is no longer accurate to consider homeopathic doses to be “implausible.” Wikipedia’s article on homeopathy asserts otherwise, deeming homeopathy to be “biologically implausible” (cited by a non-peer review magazine, called “The Skeptical Inquirer,” that is not listed in any scientific indexing service), “a sham” (cited at a website!), and running “counter to the laws of chemistry and physics” (what is interesting here is that the article cites an article in the journal, “Homeopathy,” and yet, whenever a positive statement or clinical trial or basic sciences trial is published in this same journal, the Wikipedia editors claim that this journal is not worthy of a citation).Further, just one of the theories of how homeopathic medicines work has been described as the “memory of water.” The Wikipedia article refers to this concept as “erroneous” without any acknowledgement that it is inaccurate to assert such a black-and-white statement. It is more accurate to say that this theory is “controversial” because there is, in fact, evidence of a “memory in water,” as both verified by the above research on nanoparticles remaining in homeopathically potentized water and as evidenced by research conducted by the French virologist Luc Montagnier who discovered the AIDS virus and who won the Nobel Prize for doing so. Dr. Montagnier has not only published research that provides evidence of this “memory of water,” he was interviewed in the prestigious journal, Science, and on July 5, 2014, the French government’s public television station showed an hour-long documentary entitled “We Found the Memory in Water” (“On a retrouvé la mémoire de l’eau”)What is shocking about Wikipedia’s article of homeopathy is that there is NO reference to this Nobel Prize winner or to his interview in one of the most respected scientific journals in the world today or any reference to French government’s documentary on this very subject. Obviously, the people who are editing the homeopathy article have a profound bias.Numerous people have sought to improve Wikipedia’s article on homeopathy, but they have been blocked or prohibited from editing the article. In my case, I was blocked from editing any article to do with homeopathy because I was deemed to have a “conflict of interest” due to the fact that I am a homeopath. Ironically, no medical doctor is prohibited from editing on any medical subject just because she or he is a medical doctor! Further, the bias against homeopathy and against any positive evidence for homeopathy is so strong that the vast majority of the articles from the high impact medical and scientific journals are not referenced or described in the Wikipedia article on homeopathy, while there are numerous low-level references to websites and to non-peer review magazines that populate Wikipedia’s article.I could easily show over a hundred other sentences in Wikipedia’s article that are either errors of fact or that are evidence of bias or spin against homeopathy, but I think that I have adequately and accurately provided you with solid testimony proving serious problems with Wikipedia’s article on homeopathy.I await your reply to this letter which you have requested, and I look forward to collaborating with you to improving the article on homeopathy at Wikipedia as well as in establishing guidelines so that strong bias is minimized throughout your usually excellent website.Pathological SkepticismBrian Josephson, Ph.D., won a Nobel Prize in 1973 when he was only 23 years old and is presently professor emeritus at Cambridge University. Josephson contends that many scientists today suffer from “pathological disbelief” — that is, an unscientific attitude that is typified by the statement “even if it were true I wouldn’t believe it” (Josephson, 1997).Josephson asserts that skeptics of homeopathy suffer from a chronic ignorance of this subject, and he maintains that their criticisms of homeopathy are easily refuted, “The idea that water can have a memory can be readily refuted by any one of a number of easily understood, invalid arguments.”Dr. Luc Montagnier won a Nobel Prize in 2008 for discovering the AIDS virus, and in an interview in Science (Dec. 24, 2010), he similarly expressed real concern about the unscientific atmosphere that presently exists on certain unconventional subjects such as homeopathy, “I am told that some people have reproduced Benveniste’s results (showing effects from homeopathic doses), but they are afraid to publish it because of the intellectual terror from people who don’t understand it.”Montagnier concluded this interview when asked if he is concerned that he is drifting into pseudoscience. He responded adamantly: “No, because it’s not pseudoscience. It’s not quackery. These are real phenomena which deserve further study.”Ultimately, at Wikipedia there is a certain substantial body of editors who embody “pathological skepticism” and who do not allow good evidence from high-quality studies and meta-analyses published in high-impact journals to be included into the body of evidence for homeopathy just because they provide a positive spin to the subject. On the other hand, these same editors allow references to non-peer review sources, such as popular magazine and websites, when the information in these questionably valid sources is offensive to homeopathy. Today, Wikipedia’s article on homeopathy is a classic example of a biased, off-balance, and non-encyclopedic review of the subject.Practical Solutions...Jimmy, I assume that you want your website to be the most reliable resource possible, but it can and will never become one unless you, as the founder of Wikipedia, provide some guidance and guidelines so that information for OR against a subject are fair and accurate. In 2009, at a TED talk, you claimed that Wikipedia’s most important virtue is its objective reporting of information; you asserted, “the biggest and the most important thing (about Wikipedia) is our neutral point-of-view policy.”Larry Sanger, a co-founder of Wikipedia, quit the organization several years ago due to serious concerns about its integrity. He maintained:“In some fields and some topics, there are groups who “squat” on articles and insist on making them reflect their own specific biases. There is no credible mechanism to approve versions of articles...The people with the most influence in the community are the ones who have the most time on their hands—not necessarily the most knowledgeable—and who manipulate Wikipedia’s eminently gameable system.”Ultimately, there are indeed subjects at Wikipedia that will probably remain highly controversial no matter what is or isn’t said, and it makes sense to inform readers about this issue. However, at present, the article on homeopathy strongly suggests that there is no or inconsequential evidence that homeopathic medicines have biological activity and/or clinical efficacy, and this letter clearly dispels that myth. Objective reviews of both basic science research and clinical studies suggest that there are simply too many high quality laboratory and clinical trials that show positive results.One solution to dealing with Wikipedia’s article is to have two separate sections in the article that present the “skeptics’ point of view” and the “homeopaths’ point of view.” Although one could have hoped that the article would have evolved into this multi-view perspective, there are simply too many anti-homeopathy fundamentalists who have squatted on this article and have made it literally impossible to have any positive or even any slightly positive assertions about homeopathy.Because this letter proves that Skeptics are incapable of presenting information on homeopathy with even a modicum of objectivity, perhaps the best solution is to enable both viewpoints to be able to express themselves. Some people claim that debate is the best way to understand complex subjects, and therefore, allowing and even encouraging a multi-perspective viewpoint in articles at Wikipedia may be an important and worthwhile change in your website’s policies.I can provide other specific suggestions for helping Wikipedia create a truly neutral point of view if and when you are open to constructive dialogue.You have now been given strong evidence that Wikipedia is NOT maintaining a “neutral point-of-view” on the subject of homeopathy. My question is to you now is: What do you suggest should be done to rectify this problem?This letter was also signed by:Michael Frass, MD, Professor of Medicine, Medical University of Vienna (Austria)Paolo Bellavite, MD, Professor, Università of Verona (Italy), Department of Pathology and DiagnosticsPaolo Roberti di Sarsina, MD, Observatory and Methods for Health, University of Milano-Bicocca, Italy; Charity for Person Centered Medicine-Moral Entity, Bologna, Italy; Expert for Non-Conventional Medicine (2006-2013), High Council for Health, Ministry of Health, ItalyDr Clare Relton, Senior Research Fellow (Public Health), School of Health & Related Research, University of Sheffield (UK)Stephan Baumgartner, PhD, Institute of Complementary Medicine, University of Bern, Switzerland; Institute of Integrative Medicine, University of Witten-Herdecke, GermanyLex Rutten MD, homeopathic physician, independent researcher.References:Max Ehrenfreund, The Science of Wikipedia Flamewars, Washington Post. July 23, 2013.http://www.washingtonpost.com/blogs/wonkblog/wp/2013/07/23/the-science-of-wikipedia-flamewars/Shang A, Huwiler-Müntener K, Nartey L, et al. (2005). “Are the clinical effects of homoeopathy placebo effects? Comparative study of placebo-controlled trials of homoeopathy and allopathy”. Lancet 366 (9487): 726-32. doi:10.1016/S0140-6736(05)67177-2 . PMID 16125589 . http://www.ncbi.nlm.nih.gov/pubmed/16125589Ludtke R, Rutten ALB. The conclusions on the effectiveness of homeopathy highly depend on the set of analysed trials. Journal of Clinical Epidemiology. October 2008. doi: 10.1016/j.jclinepi.2008.06/015. http://www.ncbi.nlm.nih.gov/pubmed/18834714Dana Ullman. The Disinformation Campaign Against Homeopathy. HuffingtonPost. http://www.huffingtonpost.com/dana-ullman/disinformation-about-homeopathy_b_952967.htmlZoe Mullan, senior editor at The Lancet, acknowledged in the publication’s press release for this article, “Professor Egger stated at the onset that he expected to find that homeopathy had no effect other than that of placebo. His ‘conflict’ was therefore transparent. We saw this as sufficient” EHM News Bureau, 2005). The editors chose not to inform readers of this bias.Bornhöft G, Wolf U, von Ammon K, Righetti M, Maxion-Bergemann S, Baumgartner S, Thurneysen AE, Matthiessen PF. Effectiveness, safety and cost-effectiveness of homeopathy in general practice - summarized health technology assessment. Forschende Komplementärmedizin (2006);13 Suppl 2:19-29. http://www.ncbi.nlm.nih.gov/pubmed/16883077Chikramane PS, Kalita D, Suresh AK, Kane SG, Bellare JR. Why Extreme Dilutions Reach Non-zero Asymptotes: A Nanoparticulate Hypothesis Based on Froth Flotation. Langmuir. 2012 Nov http://www.ncbi.nlm.nih.gov/pubmed/23083226Demangeat, J.-L, Gries, P, Poitevin, B, Droesbeke J.-J, Zahaf, T, Maton, F, Pierart, C, Muller, RN, Low-Field NMR Water Proton Longitudinal Relaxation in Ultrahighly Diluted Aqueous Solutions of Silica-Lactose Prepared in Glass Material for Pharmaceutical Use, Applied Magnetic Resonance, 26, 2004:465-481.Bell IR, Schwartz GE. Adaptive network nanomedicine: an integrated model for homeopathic medicine. Frontiers in Bioscience (Scholar Ed) 2013;5(2):685-708. http://www.ncbi.nlm.nih.gov/pubmed/23277079Bell IR, et al. Advances in integrative nanomedicine for improving infectious disease treatment in public health. Eur J Integr Med (2012), Elsevier: Article Locator. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3685499/Bell IR, Koithan M. A model for homeopathic remedy effects: low dose nanoparticles, allostatic cross-adaptation, and time-dependent sensitization in a complex adaptive system. BMC Complementary and Alternative Medicine 2012;12(1):191. http://www.biomedcentral.com/content/pdf/1472-6882-12-191.pdfRoduner E. Size matters: why nanomaterials are different. Chem. Soc. Rev. 2006;35(7):583-92. http://www.ncbi.nlm.nih.gov/pubmed/16791330Bell IR, MD PhD, Ives JA, Jonas WB. Nonlinear Effects of Nanoparticles: Biological Variability From Hormetic Doses, Small Particle Sizes, and Dynamic Adaptive Interactions. Dose Response. May 2014; 12(2): 202-232.Published online Nov 7, 2013. doi: 10.2203/dose-response.13-025. Bell. PMCID: PMC4036395. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4036395/Bell IR, Sarter B, Koithan M, et al. Integrative Nanomedicine: Treating Cancer with Nanoscale Natural Products. Global Advances in Health and Medicine, January 2014. 36-53. http://tinyurl.com/mqe5p88Armstead AI, Li B. Nanomedicine as an emerging approach against intracellular pathogens. Int J Nanomed. 2011;8(3):188-96.http://www.ncbi.nlm.nih.gov/pubmed?term=(nanomedicine)%20AND%20ARMSTEADUllman D. Let’s have a serious discussion of nanopharmacology and homeopathy. FASEB J December 2006 20:2661; doi:10.1096/fj.06-1205ufmhttp://www.fasebj.org/content/20/14/2661.fullDose-Response: An International Journal. Dose-ResponseMontagnier L, Aissa J, Ferris S, et al, Electromagnetic Signals Are Produced by Aqueous Nanostructures Derived from Bacterial DNA Sequences. Interdiscip Sci Comput Life Sci (2009) 1: 81-90.http://www.springerlink.com/content/0557v31188m3766x/fulltext.pdfEnserink M, Newsmaker Interview: Luc Montagnier, French Nobelist Escapes “Intellectual Terror” to Pursue Radical Ideas in China. Science 24 December 2010: Vol. 330 no. 6012 p. 1732. DOI: 10.1126/science.330.6012.1732emissions, programme tv, infos et jeuxSanger, Larry. Why Citizendium? http://blog.citizendium.org/?p=286Source: Dysfunction at Wikipedia on Homeopathic Medicine