Differentially transcribed genes were determined for each individual treatment group using a General Linear Model (GLM), a negative binomial distribution and a quasi-likelihood test. Ten pairwise comparisons (edgeR, GLM, quasi-likelihood F test) of the experimental groups were analyzed (
Table 1). Instead of correcting the
P values of the differentially transcribed genes with the Bonferroni method for multiple testing error, a different approach was chosen selecting significant genes according to the response of the whole set of CXL and control conditions. For this purpose, a composite null hypothesis, H
0, was created summarizing the five most important comparisons. The condition CXL at 18 mW/cm
2 was excluded in this selection process, as its treatment efficacy is smallest, as shown experimentally
9–11 and clinically
5–8 and, hence, its meaningfulness is lower than the other comparisons.
\(\def\upalpha{\unicode[Times]{x3B1}}\)\(\def\upbeta{\unicode[Times]{x3B2}}\)\(\def\upgamma{\unicode[Times]{x3B3}}\)\(\def\updelta{\unicode[Times]{x3B4}}\)\(\def\upvarepsilon{\unicode[Times]{x3B5}}\)\(\def\upzeta{\unicode[Times]{x3B6}}\)\(\def\upeta{\unicode[Times]{x3B7}}\)\(\def\uptheta{\unicode[Times]{x3B8}}\)\(\def\upiota{\unicode[Times]{x3B9}}\)\(\def\upkappa{\unicode[Times]{x3BA}}\)\(\def\uplambda{\unicode[Times]{x3BB}}\)\(\def\upmu{\unicode[Times]{x3BC}}\)\(\def\upnu{\unicode[Times]{x3BD}}\)\(\def\upxi{\unicode[Times]{x3BE}}\)\(\def\upomicron{\unicode[Times]{x3BF}}\)\(\def\uppi{\unicode[Times]{x3C0}}\)\(\def\uprho{\unicode[Times]{x3C1}}\)\(\def\upsigma{\unicode[Times]{x3C3}}\)\(\def\uptau{\unicode[Times]{x3C4}}\)\(\def\upupsilon{\unicode[Times]{x3C5}}\)\(\def\upphi{\unicode[Times]{x3C6}}\)\(\def\upchi{\unicode[Times]{x3C7}}\)\(\def\uppsy{\unicode[Times]{x3C8}}\)\(\def\upomega{\unicode[Times]{x3C9}}\)\(\def\bialpha{\boldsymbol{\alpha}}\)\(\def\bibeta{\boldsymbol{\beta}}\)\(\def\bigamma{\boldsymbol{\gamma}}\)\(\def\bidelta{\boldsymbol{\delta}}\)\(\def\bivarepsilon{\boldsymbol{\varepsilon}}\)\(\def\bizeta{\boldsymbol{\zeta}}\)\(\def\bieta{\boldsymbol{\eta}}\)\(\def\bitheta{\boldsymbol{\theta}}\)\(\def\biiota{\boldsymbol{\iota}}\)\(\def\bikappa{\boldsymbol{\kappa}}\)\(\def\bilambda{\boldsymbol{\lambda}}\)\(\def\bimu{\boldsymbol{\mu}}\)\(\def\binu{\boldsymbol{\nu}}\)\(\def\bixi{\boldsymbol{\xi}}\)\(\def\biomicron{\boldsymbol{\micron}}\)\(\def\bipi{\boldsymbol{\pi}}\)\(\def\birho{\boldsymbol{\rho}}\)\(\def\bisigma{\boldsymbol{\sigma}}\)\(\def\bitau{\boldsymbol{\tau}}\)\(\def\biupsilon{\boldsymbol{\upsilon}}\)\(\def\biphi{\boldsymbol{\phi}}\)\(\def\bichi{\boldsymbol{\chi}}\)\(\def\bipsy{\boldsymbol{\psy}}\)\(\def\biomega{\boldsymbol{\omega}}\)\(\def\bupalpha{\bf{\alpha}}\)\(\def\bupbeta{\bf{\beta}}\)\(\def\bupgamma{\bf{\gamma}}\)\(\def\bupdelta{\bf{\delta}}\)\(\def\bupvarepsilon{\bf{\varepsilon}}\)\(\def\bupzeta{\bf{\zeta}}\)\(\def\bupeta{\bf{\eta}}\)\(\def\buptheta{\bf{\theta}}\)\(\def\bupiota{\bf{\iota}}\)\(\def\bupkappa{\bf{\kappa}}\)\(\def\buplambda{\bf{\lambda}}\)\(\def\bupmu{\bf{\mu}}\)\(\def\bupnu{\bf{\nu}}\)\(\def\bupxi{\bf{\xi}}\)\(\def\bupomicron{\bf{\micron}}\)\(\def\buppi{\bf{\pi}}\)\(\def\buprho{\bf{\rho}}\)\(\def\bupsigma{\bf{\sigma}}\)\(\def\buptau{\bf{\tau}}\)\(\def\bupupsilon{\bf{\upsilon}}\)\(\def\bupphi{\bf{\phi}}\)\(\def\bupchi{\bf{\chi}}\)\(\def\buppsy{\bf{\psy}}\)\(\def\bupomega{\bf{\omega}}\)\(\def\bGamma{\bf{\Gamma}}\)\(\def\bDelta{\bf{\Delta}}\)\(\def\bTheta{\bf{\Theta}}\)\(\def\bLambda{\bf{\Lambda}}\)\(\def\bXi{\bf{\Xi}}\)\(\def\bPi{\bf{\Pi}}\)\(\def\bSigma{\bf{\Sigma}}\)\(\def\bPhi{\bf{\Phi}}\)\(\def\bPsi{\bf{\Psi}}\)\(\def\bOmega{\bf{\Omega}}\)\begin{equation}\tag{1}{H_{\bf{0}}} = {H_{{\rm{virgin}} = {\rm{3mW}}}}\left| {{H_{{\rm{virgin}} = {\rm{9mW}}}}} \right|{H_{{\rm{ribo}} = {\rm{3mW}}}}\left| {{H_{{\rm{ribo}} = {\rm{9mW}}}}} \right|\, \sim\! {H_{{\rm{virgin}} = {\rm{ribo}}}}\end{equation}
and hence:
\begin{equation}\tag{2}{H_{\bf{1}}} = \, \sim\! {H_0} = \, \sim\! {H_{{\rm{virgin}} = {\rm{3mW}}}}\ \& \, \sim\! {H_{{\rm{virgin}} = {\rm{9mW}}}}\ \& \, \sim\! {H_{{\rm{ribo}} = {\rm{3mW}}}}\ \& \, \sim\! {H_{{\rm{ribo}} = {\rm{9mW}}}}\ \& \, {H_{{\rm{virgin}} = {\rm{ribo}}}}\end{equation}
where
H1 is the composite null hypothesis.
Hx=y represents an individual null hypothesis, that is there is no difference between
x and
y. ∼
Hx=y represent a rejected null hypothesis, that is there is a difference between x and y. Each comparison between CXL (at 3 or 9 mW/cm
2) and control (virgin or riboflavin) is expected to be significant. In contrast, the comparison between the two control conditions is expected not to be significant. A given gene then will be considered significant, if
H1 is true. With a confidence interval of 95%, the probability for a false positive in one comparison is:
\begin{equation}\tag{3}{P_{\bf{i}}} = \left( {{P_{{\rm{virgin}}\, \sim\! = {\rm{3mW}}}}} \right) \cdot \left( {{P_{{\rm{virgin}}\, \sim\! = {\rm{9mW}}}}} \right) \cdot \left( {{P_{{\rm{ribo}}\, \sim\! = {\rm{3mW}}}}} \right) \cdot \left( {{P_{{\rm{ribo}}\, \sim\! = {\rm{9mW}}}}} \right) \cdot \left( {{P_{{\rm{ribo}} = {\rm{virgin}}}}} \right)\end{equation}
The probability of
Pribo=virgin cannot be calculated exactly, as it is the power of the test. However, assuming that the power is 1, we have neglected this term resulting in
Pi ≤ 0.05
4. Applied to the entire set of
n = 9335 analyzed genes, the probability of having at least one false-positive can be calculated:
\begin{equation}\tag{4}{P_{{\bf{cumulative}}}} = {\rm{1}} - {\left( {{\rm{1}} - {P_{\rm{i}}}} \right)^{\rm{n}}} \le 0.0{\rm{567}}\end{equation}
This
P value,
Pcumulative, is comparable to the standard significance level. An alternative correction for multiple testing is the Bonferroni method, which, however, can be applied only to one group at a time. The above-described whole-data-set approach is superior, as it accounts for the reproducibility of the CXL effect before correcting for multiple testing.
Figure 1 illustrates that with Bonferroni correction, lower statistical significance (19 significantly different genes) can be reached than with the whole-data-set approach (297 significantly different genes).