In a recent study of the relation between wellbeing and gene expression, Fredrickson et al. (2013) concluded that hedonic wellbeing and eudaimonic wellbeing have “similar affective correlates” (p. 13684) but “markedly divergent gene transcriptional correlates in human immune cells” (p. 13686). Specifically, they asserted that hedonic wellbeing and eudaimonic wellbeing showed “comparably strong positive relationships to total wellbeing and comparably strong inverse relationships to depressive symptoms, and were highly correlated with one another” (p. 13686). In contrast, hedonic wellbeing was positively related to gene expression for the overall set of genes under consideration and for the subset of proinflammatory genes, (marginally) negatively related to gene expression for the subset of genes involved in antibody synthesis, and not related to gene expression for the subset of genes mediating Type I interferon antiviral responses, whereas eudaimonic wellbeing was generally related to the opposite pattern of gene expression.
A number of statistical problems in Fredrickson et al.’s (2013) study cast considerable doubt on the validity of these conclusions. First, an idiosyncratic twofactor scoring rather than the documented and wellvalidated threefactor scoring was used for the items in the instrument assessing wellbeing. Second, the analyses relating hedonic wellbeing and eudaimonic wellbeing to affect did not include the same variables as the analyses relating these two wellbeing variables to gene expression, invalidating any comparison between them. Third, hedonic wellbeing and eudaimonic wellbeing were highly correlated, resulting in untheorized and unrecognized suppression effects that accounted for the supposed differential relations of these two wellbeing variables to gene expression. Fourth, the method of computing p values for the onesample t tests discarded information and violated the assumption of independence for those tests. In this comment, I present the results of my reanalysis of Fredrickson et al.’s (2013) dataset as the basis for a detailed discussion of these statistical problems. I first describe the changes necessary to Fredrickson et al.’s (2013) publicly available dataset (https://www.ncbi.nlm.gov/geo/; accession number GSE45330; March 20, 2013) before it could be reanalyzed.
Changes to the Dataset
Fredrickson et al. (2013) indicated that their sample consisted of 84 healthy adults who completed a 14item instrument assessing wellbeing and provided demographic and healthrelated information. Of these 84 cases (participants), 80 provided blood samples for transcriptome analysis, and when Fredrickson et al. (2013) reported a sample size for an analysis (e.g., Figure 1, p. 13686), it equaled 80. The dataset contains only 79 cases, however. Two of these 79 cases are missing all demographic and healthrelated information, and a third is missing information about alcohol consumption. Fredrickson et al. (2013, Supporting Information, p. 1) indicated that they used SAS 9.3 for their analyses. In a regression analysis, SAS ordinarily deletes any case for which any variable in the analysis has a missing value, unless a value for that variable is imputed. Fredrickson et al. (2013) did not indicate that they imputed values for variables with missing values, so I have omitted these three cases from all of my analyses. Six of the remaining 76 cases are missing responses to one (five cases) or two (one case) of the items in the instrument used to assess wellbeing. Fredrickson et al. (2013) seem to have computed scores (means) for hedonic wellbeing and eudaimonic wellbeing ignoring items with missing values, and I have done the same. Another case has an invalid value (“4”) for race/ethnicity, apparently a mistake made in converting the four race/ethnicity categories shown in Fredrickson et al.’s (2013) Table 1 (p. 13685) to the dichotomous variable nonwhite/white; for this case, I have changed race/ethnicity to nonwhite (“0”). These changes resulted in a dataset with 76, not 79 or 80, usable cases. Fredrickson et al. (2013) standardized the scores for hedonic wellbeing and eudaimonic wellbeing for use in their analyses; I have restandardized these scores across the 76 usable cases. All of my analyses are based on these 76 cases and performed in SAS 9.3.
Gene  Correlation With WellBeing 
p for Meng’s Z  

Hedonic  Eudaimonic  
Type I interferon viral response genes  
IFIT1L  –.25  –.32  .3216 
GBP1  –.02  –.03  .8066 
IFI16  –.19  –.16  .6263 
IFI27  –.08  –.10  .8164 
IFI27L1  .00  .03  .7445 
IFI27L2  –.04  –.14  .1492 
IFI30  –.12  –.17  .3988 
IFI35  –.06  –.07  .9313 
IFI44  .12  .08  .5921 
IFI44L  .09  .06  .6548 
IFI6  .07  .02  .5242 
IFIH1  –.09  –.05  .5403 
IFIT1  .20  .19  .8647 
IFIT2  .24  .19  .5099 
IFIT3  .19  .20  .9833 
IFIT5  –.03  –.05  .7345 
IFITM1  .23  .29  .4272 
IFITM2  .15  .16  .8820 
IFITM3  .00  .03  .7168 
IFITM4P  .11  .04  .3684 
IFITM5  –.05  .01  .4925 
IFNB1  .02  .07  .5192 
IRF2  –.10  –.02  .2957 
IRF7  .12  .10  .8232 
IRF8  –.15  –.17  .8623 
MX1  .13  .17  .5763 
MX2  .02  .03  .9503 
OAS1  –.08  –.06  .7711 
OAS2  –.06  –.07  .8951 
OAS3  .12  .09  .7038 
OASL  .13  .14  .9383 
Mean  .04  .02  .7919 
Antibody synthesis genes  
IGJ  .03  .14  .1480 
IGLL1  –.03  .10  .0925 
IGLL3  .02  .02  .9943 
Mean  .01  .11  .2033 
Proinflammatory genes  
IL1B  .17  .09  .2859 
IL8  .20  .18  .7364 
FOS  .10  .10  .9912 
FOSB  .10  .04  .3891 
FOSL1  .06  –.09  .0537 
FOSL2  .11  .06  .4922 
IL1A  .15  .03  .1272 
IL6  .07  .02  .4521 
JUN  .06  .06  .9892 
JUNB  .02  .02  .9538 
JUND  .07  .07  .9159 
NFKB1  .04  .11  .3663 
NFKB2  .09  .06  .6610 
PTGS1  –.13  –.20  .3741 
PTGS2  .14  .10  .6116 
REL  .06  .03  .6189 
RELA  .17  .18  .9399 
RELB  .10  .04  .3922 
TNF  .18  .09  .2562 
Mean  .14  .09  .4935 
Omnibus mean  .08  .04  .5912 
The results of my independent review of Fredrickson et al.’s (2013) dataset agreed with the results of a similar review reported by Brown, MacDonald, Samanta, Friedman, and Coyne (2014, Supporting Information, pp. 2–4). I note that, in an updated (July 15, 2014) version of the dataset, the invalid race/ethnicity value of “4” appearing in the original version has been changed to “0,” although I can find no explicit mention of this change. Additionally, a more recent article by Fredrickson et al. (2015, Figure 1, p. 5) reported the sample size in the Fredrickson et al. (2013) study to have been 76, indicating that the sample size of 80 reported by Fredrickson et al. (2013) must not have been correct.
The Factor Structure and Scoring for the WellBeing Instrument
Fredrickson et al. (2013) assessed hedonic wellbeing and eudaimonic wellbeing with an instrument that they called the “Short Flourishing Scale” (article, pp. 13684, 13688, Supporting Information, p. 1) that actually is named the “Mental Health Continuum Short Form” (MHC–SF, Keyes, 2009a). (Fredrickson et al., 2013, may have confused the MHC–SF with the “Flourishing Scale” developed by Diener et al., 2010, that is sometimes called the “Short Flourishing Scale” [e.g., http://ogg.osu.edu/media/documents/MB Stream/ShortFlourishingScale.pdf].) The MHC–SF consists of 14 items assessing emotional wellbeing (three items), social wellbeing (five items), and psychological wellbeing (six items) on a sixpoint scale and is an abbreviated version of the original longer instrument (“Mental Health Continuum–Long Form,” Keyes, 2008, available from http://booksite.elsevier.com/9780123745170). The nonorthogonal threefactor structure underlying both the long form and the short form of this instrument has been validated repeatedly (e.g., Gallagher, Lopez, & Preacher, 2009; Jaotombo, 2014; Joshanloo, Rostami, & Nosratabadi, 2006; Joshanloo, Wissing, Khumalo, & Lamers, 2013; Karaś, Cieciuch, & Keyes, 2014; Keyes, 2005, 2009b; Keyes et al., 2008; Lamers, Westerhof, Bohlmeijer, ten Klooster, & Keyes, 2011; Lim, 2014; Petrillo, Capone, Caso, & Keyes, 2015; Robitschek & Keyes, 2009; SalamaYounes, 2011a,b; SalamaYounes & Ismaïl, 2011). Emotional wellbeing reflects the traditional philosophical concept of hedonia; psychological wellbeing and social wellbeing reflect two different aspects of the traditional philosophical concept of eudaimonia.
In his letter to the editor regarding Fredrickson et al.’s (2013) study, Coyne (2013) asserted that the high correlation (r = .79) between hedonic wellbeing and eudaimonic wellbeing indicated that these two variables were interchangeable and so invalidated the results of Fredrickson et al.’s (2013) analyses, which used both wellbeing variables as predictors. In their reply to Coyne, Cole and Fredrickson (2013) stated that they addressed this issue “through exploratory and confirmatory factor analyses of the items measuring hedonic and eudaimonic wellbeing. Consistent with previous observations (e.g., references 6, 14, and 27 from original article), both analyses rejected a single common factor solution in favor of a twofactor solution” (p. E4184).
Cole and Fredrickson’s (2013) reply to Coyne (2013) was misleading because it implied that their two factors represented hedonic wellbeing and eudaimonic wellbeing. Brown et al. (2014) subsequently performed exploratory and confirmatory factor analyses of the 14 items in the MHC–SF for Fredrickson et al.’s (2013) dataset. They confirmed a twofactor solution but did not find that the three emotional wellbeing items loaded on one hedonic factor and the remaining five social wellbeing items and six psychological wellbeing items loaded on a second eudaimonic factor, as should have been the case. Instead, all three emotional wellbeing items, two of the social wellbeing items, and the six psychological wellbeing items loaded on one factor; the remaining three social wellbeing items loaded on a second factor. In reply, Cole and Fredrickson (2014) criticized Brown et al. (2014) for attempting to refactor the MHC–SF “in a sample too small to support reliable factor discovery or item reallocation” (p. E3581). It is the case that Fredrickson et al.’s (2013) sample was too small for “reliable factor discovery,” but it must be noted that Cole and Fredrickson (2013) refactored the MHC–SF for that very same small sample to demonstrate to Coyne (2013) that in their dataset the MHC–SF was best described by two factors, not one.
The results of my independent exploratory and confirmatory factor analyses of Fredrickson et al.’s (2013) dataset using SAS agreed with those of Brown et al. (2014), who used SPSS and AMOS for their factor analyses. Accordingly, I requested a copy of the SAS program that Cole and Fredrickson (2013) used to perform their factor analyses. Examination of that program showed that Cole and Fredrickson (2013) failed to rotate the factor pattern as was needed to interpret the results of the exploratory factor analysis. Had they done so, their results would have been identical to those obtained by Brown et al. (2014) and by me. Demonstrating a twofactor solution for the MHC–SF items for their dataset was necessary but not sufficient for Fredrickson et al. (2013) to have used those two factors as predictors in their regression analyses; they needed to have demonstrated also that the three hedonic items loaded on one factor and the eleven other items on the other factor. They did not so demonstrate.
Cole and Fredrickson (2013) surely must have been aware that it has been shown that the MHC–SF has a threefactor, not a twofactor, structure. Fredrickson et al. (2013) cited the documentation for the MHC–SF (Keyes, 2009a; their article reference 52) which lists several articles providing evidence that both the long form and the short form (and in one instance, an augmented form) of the MHC have a nonorthogonal threefactor structure. Indeed, Fredrickson et al. (2013) themselves cited one of the articles (Lamers et al., 2011; their article reference 38) that validated the threefactor structure. Other articles coauthored by Fredrickson (Catalino & Fredrickson, 2011; Fredrickson & Losada, 2005) also make it clear that she, at least, was familiar with the threefactor structure of the MHC–SF.
Also misleading in the reply to Coyne (2013) were Cole and Fredrickson’s (2013) references for a twofactor structure of the MHC–SF. Both the original article and the reply to Coyne ignored all of the articles listed in the documentation for the MHC–SF (Keyes, 2009a) as validating a threefactor structure and instead stated that the twofactor structure that they found was consistent with references 6, 14, and 27 in their original article. Reference 6 (Friedman, 2012) is a literature review of the relations between aging, wellbeing, and immunity. This review explicitly stated that “most studies of wellbeing have not examined the empirical independence of these conceptually different traditions” (p. 54). No correlations were reported in this review, nor were the results of any factor analyses; brief mention was made of the fact that a few studies have found that hedonic wellbeing and eudaimonic wellbeing are independent or weakly correlated, findings that are not consistent with Fredrickson et al.’s (2013) correlation of .79. Reference 14 (Waterman, 1993) is an empirical study of convergent and divergent validity. The MHC was not used, and no factor analyses were reported. Reference 27 (Keyes, Shmotkin, & Ryff, 2002) is coauthored by Corey L. M. Keyes, the developer of the MHC. This early empirical study included items measuring only emotional wellbeing (called subjective wellbeing) and psychological wellbeing, not social wellbeing. Thus, only two of the three kinds of items now included in the MHC were used in this study, and it is not surprising that a twofactor structure rather than a threefactor structure was validated. These three at best marginally relevant references should not have been cited as evidence for a twofactor structure of the MHC–SF when several studies specifically confirming its threefactor structure were listed in the documentation for the MHC–SF and also otherwise known to Fredrickson and her coauthors.
Surprisingly, in their reply to Brown et al. (2014), Cole and Fredrickson (2014) insisted that their division of the 14 MHC–SF items into those assessing hedonic wellbeing and those assessing eudaimonic wellbeing was based on the developer’s established “scoring,” “allocation,” or “specification” (article, p. E3581, Supporting Information, pp. 16, 22, 27) of those items, asserting that this scoring was based on multiple previous confirmatory factor analyses in studies totaling thousands of participants (Supporting Information, p. 27). It is certainly the case that there have been many previous factor analyses of the MHC involving thousands of participants. But the claim that these studies supported a twofactor hedonic/eudaimonic scoring is simply wrong. All of these studies have validated a nonorthogonal threefactor structure for the MHC. To the best of my knowledge, the developer of the MHC has nowhere combined nor suggested combining the five social wellbeing items and the six psychological wellbeing items into one eudaimonic factor. It is the case that he (Keyes, 2009a) labeled the emotional wellbeing factor as “hedonic” and both the social wellbeing factor and the psychological wellbeing factor as “eudaimonic” in the documentation for the MHC–SF He did so not as an indication of the factor structure or scoring of the MHC–SF, but as a way of relating these factors to the traditional philosophical concepts of hedonia and eudaimonia and of describing how these factors were to be used to categorize mentalhealth status. Brown et al.’s (2014) criticism of Cole and Fredrickson’s (2014) factor analyses is a propos because, as I will show later when I raise the question of whether the unrecognized suppression effects in Fredrickson et al.’s (2013) analyses have theoretical importance or are merely statistical artifacts, analyses using the documented and wellvalidated threefactor scoring of the 14 MHC–SF items do not give the same results as analyses using Fredrickson et al.’s (2013) idiosyncratic twofactor hedonic/eudaimonic scoring.
The Affective Correlates of WellBeing
Fredrickson et al. (2013) reported that symptoms of depression, as assessed by the Center for Epidemiological Studies–Depression (CES–D) instrument (Radloff, 1977), had similarly strong correlations of –.67 and –.66 (p = .8550) with hedonic wellbeing and eudaimonic wellbeing, respectively. They also reported that the CES–D subscale assessing affective symptoms of depression had similarly strong correlations of –.75 and –.71 (p = .3228) with hedonic wellbeing and eudaimonic wellbeing, respectively, and that the CES–D subscale assessing vegetative symptoms of depression had similarly strong correlations of –.45 and –.48 (p = .6297) with hedonic wellbeing and eudaimonic wellbeing, respectively. No information about depression is included in Fredrickson et al.’s (2013) publicly available dataset, so it was not possible to verify these correlations.
Fredrickson et al. (2013) also reported that hedonic wellbeing and eudaimonic wellbeing showed comparably strong positive relations to total wellbeing, although they did not provide any details about this analysis or report the actual correlations. Fredrickson et al. (2013) indicated that they had standardized hedonic wellbeing and eudaimonic wellbeing prior to their use in the analyses. If two standardized variables are summed, the correlation of one variable with the total equals the correlation of the other variable with the total. This is a statistical fact, not an empirical finding. If hedonic wellbeing and eudaimonic wellbeing are standardized before being summed, then the correlation of each with the total equals .94882. If they are not standardized before being summed, then the correlations of hedonic wellbeing and eudaimonic wellbeing with the total equal .94926 and .94838, respectively, neither surprising nor interesting given that the correlation between them equals .80053. (These correlations are computed on the corrected sample of 76 usable cases.)
Although Fredrickson et al. (2013) compared the relations of hedonic wellbeing and eudaimonic wellbeing to depression and total wellbeing to the relations of these two wellbeing variables to gene expression, these comparisons were not valid because the analyses on which the comparisons were based were not the same. The analyses relating hedonic wellbeing and eudaimonic wellbeing to the three depression variables and to total wellbeing included only one or the other of the two wellbeing variables, whereas the analyses relating hedonic wellbeing and eudaimonic wellbeing to gene expression included both. The analyses relating hedonic wellbeing and eudaimonic wellbeing to the three depression variables and to total wellbeing also did not include any covariates, whereas the analyses relating hedonic wellbeing and eudaimonic wellbeing to gene expression did. As will be seen, the inclusion of both of the two wellbeing variables and of the 15 covariates has important consequences for the results of the analyses.
The Differential Relations of Hedonic WellBeing and Eudaimonic WellBeing to Gene Expression
To examine Fredrickson et al.’s (2013) claim that hedonic wellbeing and eudaimonic wellbeing were differentially related to gene expression, I present the results of the following sequence of analyses: (a) two sets of correlational analyses relating wellbeing to gene expression, the first set analogous to the analyses used by Fredrickson et al. (2013) to relate hedonic wellbeing and eudaimonic wellbeing to depression and to total wellbeing, the second set relating hedonic wellbeing and eudaimonic wellbeing to averaged gene expression; (b) three sets of regression analyses relating hedonic wellbeing and eudaimonic wellbeing to averaged gene expression; (c) three sets of regression analyses relating hedonic wellbeing and eudaimonic wellbeing to averaged gene association; and (d) an examination of suppression effects in the regression analyses. Note that these analyses use Fredrickson et al.’s (2013) twofactor scoring of the MHC–SF, not the documented and wellvalidated threefactor scoring, because my intention is to rebut Fredrickson et al.’s (2013) and Cole and Fredrickson’s (2013) claims that the results of their analyses were not affected by the high correlation between hedonic wellbeing and eudaiamonic wellbeing.
Correlational Analyses
To render the analyses relating hedonic wellbeing and eudaimonic wellbeing to gene expression comparable to Fredrickson et al.’s (2013) analyses relating hedonic wellbeing and eudaimonic wellbeing to depression and to total wellbeing, I computed the correlation of each of the two wellbeing variables to gene expression for each of the 53 genes included in Fredrickson et al.’s (2013) study and compared the difference between them using a test for dependent correlations, as had Fredrickson et al. (2013) in testing the difference between the correlation of each of the two wellbeing variables with depression and with total wellbeing. Fredrickson et al. (2013) did not state which of several available tests for the difference between dependent correlations they used; I have used Meng’s Z (Meng, Rosenthal, & Rubin, 1992). The correlations and p values for Meng’s Z from my analysis are shown in Table 1. Examination of the p values in this table shows that the correlation between hedonic wellbeing and gene expression and the correlation between eudaimonic wellbeing and gene expression are not significantly different from one another for any of the 53 genes. Most of the differences between the two correlations are quite small, and for only three pairs of correlations is there a sign reversal. Thus, hedonic wellbeing and eudaimonic wellbeing do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
But Fredrickson et al. (2013, Supporting Information, p. 1) stated that they did not perform any statistical testing
at the level of individual transcriptphenotype [singlegene] associations because the goal of this study is not to discover reliable associations between the expression of individual transcripts and measured levels of hedonic or eudaimonic wellbeing. The goal of this study is to test associations between eudaimonic and hedonic wellbeing and average levels of expression of specific sets of genes selected a priori for analysis here based on their previously observed involvement in the conserved transcriptional response to adversity (CTRA) (e.g., as representative proinflammatory genes, IFNrelated genes, and antibodysynthesis related genes) …. [The] primary focus of this study involves testing potential differences in association of hedonic vs. eudaimonic wellbeing with average expression of one set of 53 genes selected a priori.
Analyzing the relation of wellbeing to the averaged expression of sets of similar genes, rather than to the expression of single genes, is not unreasonable. For example, if one considers a gene to be analogous to an item in a multiitem test, then one can conceive of averaging gene expressions for subsets of similar genes and for the entire set of genes to be similar to the construction of the subscales and the total scale of the test. Cronbach’s alpha equals .91, .82, .91, and .89 for the subset of 31 Type I interferon viral response genes, the subset of 3 antibody synthesis genes, the subset of 19 proinflammatory genes, and the “omnibus” set of all 53 genes, respectively; these values of Cronbach’s alpha are well above the usual .70 “acceptable” criterion. (Following Fredrickson et al., 2013, Supporting Information, p. 2, in computing Cronbach’s alpha, each Type I interferon viral response gene and antibody synthesis gene was weighted by –1 in the omnibus set of genes.)
Fredrickson et al.’s (2013) expectations for the relation of hedonic wellbeing and eudaimonic wellbeing to the averaged expression of the omnibus set of genes and its three subsets are shown in Table 2. The omnibus set of genes and the subset of proinflammatory genes were expected to be upregulated by (have a positive relation to) hedonic wellbeing and downregulated by (have a negative relation to) eudaimonic wellbeing. In contrast, the subset of Type I interferon viral response genes and the subset of antibody synthesis genes were expected to be downregulated by hedonic wellbeing and upregulated by eudaimonic wellbeing. Table 3 presents Fredrickson et al.’s (2013) results. As expected, the omnibus set of genes and the subset of proinflammatory genes had significant positive relations to hedonic wellbeing. The omnibus set of genes also had a significant negative relation to eudaimonic wellbeing, but the negative relation between the subset of proinflammatory genes and eudaimonic wellbeing was not significant. Also as expected, the subset of Type I interferon viral response genes had a significant positive relation to eudaimonic wellbeing; the positive relation between the subset of antibody synthesis genes and eudaimonic wellbeing was marginally significant. The negative relation between the subset of antibody synthesis genes and hedonic wellbeing was marginally significant; the relation between the subset of Type I interferon viral response genes and hedonic wellbeing was positive rather than negative, and not significant. Note that Fredrickson et al.’s (2013) regression analyses included both of the two wellbeing variables and the 15 covariates as predictors of gene expression.
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic  Eudaimonic  
Omnibus  53  positive  negative 
Type I interferon viral response genes  31  negative  positive 
Antibody synthesis genes  3  negative  positive 
Proinflammatory genes  19  positive  negative 
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Effect  p^{a}  Effect  p^{a}  
Omnibus  53  positive large  .0047  negative large  .0045 
Type I interferon viral response genes  31  positive small  _____  positive medium  .0084 
Antibody synthesis genes  3  negative large  .0776  positive large  .0849 
Proinflammatory genes  19  positive large  .0008  negative small  _____ 
Accordingly, to analyze the relation between wellbeing and averaged gene expression, I first averaged gene expressions across the genes in each of the three subsets of specific genes. For the omnibus set of genes, I averaged gene expressions across the genes in the set of all 53 genes, after having first weighted the expression of each Type I interferon viral response gene and antibody synthesis gene by –1. I then computed the correlation of each of the two wellbeing variables with averaged gene expression for each of these four sets of genes and compared the difference between them using Meng’s Z. These statistics are also shown in Table 1 on the lines labeled “Mean.” Examination of the p values for the four pairs of means in this table shows that the correlation between hedonic wellbeing and averaged gene expression and the correlation between eudaimonic wellbeing and averaged gene expression are not significantly different for any of the four sets of genes. For example, for the omnibus set of genes, the correlations between hedonic wellbeing and eudaimonic wellbeing and averaged gene expression equal .08 and .04, respectively (p = .5912). Even if one of these correlations had a negative sign, it would seem a considerable overstatement to claim that hedonic wellbeing and eudaimonic wellbeing have “markedly divergent” relations to gene expression. Again, hedonic wellbeing and eudaimonic wellbeing do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
Regression Analyses for Averaged Gene Expression
Because Fredrickson et al. (2013) used regression instead of correlation in their analyses relating hedonic wellbeing and eudaimonic wellbeing to gene expression, I next performed eight regression analyses, each predicting averaged gene expression from either hedonic wellbeing or eudaimonic wellbeing for one of the four sets of genes. Table 4 shows that both hedonic wellbeing and eudaimonic wellbeing have a positive but nonsignificant relation to averaged gene expression for every set of genes. Thus, as in the correlational analysis of single genes, hedonic wellbeing and eudaimonic wellbeing do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.01560  .5150  0.00743  .7568 
Type I interferon viral response genes  31  0.00722  .7640  0.00319  .8946 
Antibody synthesis genes  3  0.00842  .9076  0.06671  .3567 
Proinflammatory genes  19  0.05664  .2236  0.03645  .4348 
But although Fredrickson et al.’s (2013) analyses relating hedonic wellbeing and eudaimonic wellbeing to depression and to total wellbeing included only one or the other of the two wellbeing variables, their regression analyses relating hedonic wellbeing and eudaimonic wellbeing to gene expression included both. Fredrickson et al. (2013) apparently (and mistakenly) believed that including both wellbeing variables in their regression analyses would result in “a purified index of eudaimonic wellbeing (purged of shared variance with hedonia)” (p. 13687) and “a purified index of hedonic wellbeing (purged of shared variance with eudaimonia)” (p. 13687). I repeated my regression analyses but predicted averaged gene expression from both hedonic wellbeing and eudaimonic wellbeing for each set of genes; the results of the four regression analyses are shown in Table 5. Examination of this table shows that the signs of some of the partial regression coefficients have reversed compared to those of the simple regression coefficients in Table 4, so that, with the exception of those for the subset of Type I interferon viral response genes, the signs in Table 5 are consistent with those expected by Fredrickson et al. (2013). However, none of the partial regression coefficients in Table 5 is significant. Note that, with one exception, the magnitudes of the partial regression coefficients in Table 5 are larger than those of the simple regression coefficients in Table 4, not smaller, as one might expect. Again, hedonic wellbeing and eudaimonic wellbeing do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.02689  .5039  –0.01410  .7258 
Type I interferon viral response genes  31  0.01300  .7476  –0.00722  .8581 
Antibody synthesis genes  3  –0.12523  .2999  0.16696  .1682 
Proinflammatory genes  19  0.07644  .3270  –0.02474  .7504 
Finally, I repeated the regression analyses but predicted averaged gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates. Examination of the results presented in Table 6 shows that the sign of one additional partial regression coefficient (that for eudaimonic wellbeing in the regression predicting averaged gene expression of the subset of Type I interferon viral response genes) has reversed, so that, with one exception, the pattern of signs is now consistent with the pattern expected by Fredrickson et al. (2013) and completely consistent with the pattern that they reported. However, as in Tables 4 and 5, none of the partial regression coefficients in Table 6 is significant. Again, hedonic wellbeing and eudaimonic wellbeing do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.01276  .7667  –0.02717  .5378 
Type I interferon viral response genes  31  0.01661  .6803  0.03074  .4574 
Antibody synthesis genes  3  –0.03899  .7290  0.09016  .4350 
Proinflammatory genes  19  0.05654  .4729  –0.01140  .8874 
Regression Analyses for Averaged Gene Association
Although Fredrickson et al. (2013, Supporting Information) initially stated that the goal of their study was “to test associations between eudaimonic and hedonic wellbeing and average levels of expression of specific genes selected a priori” (p. 1), there was subsequently a subtle and unexplained shift to analyzing “average gene set association” (p. 2) [italics mine] instead of average gene set expression. Fredrickson et al. (2013) did not in fact average gene expressions across the genes in the omnibus set of genes and each of the three subsets of specific genes and then perform four regression analyses, each predicting averaged gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates for one of the four sets of genes.
Instead, they performed 53 separate regression analyses, in which they regressed gene expression for one gene separately on hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates. They then sorted the partial regression coefficients for hedonic wellbeing and for eudaimonic wellbeing from these 53 regression analyses into eight sets: (a) hedonic wellbeing for all 53 genes, (b) hedonic wellbeing for the 31 Type I interferon viral response genes, (c) hedonic wellbeing for the 3 antibody synthesis genes, (d) hedonic wellbeing for the 19 proinflammatory genes, (e) eudaimonic wellbeing for all 53 genes, (f) eudaimonic wellbeing for the 31 Type I interferon viral response genes, (g) eudaimonic wellbeing for the 3 antibody synthesis genes, and (h) eudaimonic wellbeing for the 19 proinflammatory genes. The partial regression coefficients for hedonic wellbeing and for eudaimonic wellbeing for each set of genes were then averaged. (For the set of all 53 genes, the Type I interferon viral response genes and the antibody synthesis genes were weighted by –1 before averaging.)
A twotailed onesample t test was then used to test each mean partial regression coefficient against zero. The standard error for each t test was not computed from the partial regression coefficients in each set; instead, the standard error was estimated by nonparametric bootstrapping (Efron & Tibshirani, 1993, pp. 45–59). These analyses produced the results reported by Fredrickson et al. (2013) that are shown in Table 3. The results of my three sets of reanalyses relating wellbeing to averaged (mean) gene association appear in Tables 7, 8, and 9. The analyses for Table 7, like those for Table 4, used hedonic wellbeing alone or eudaimonic wellbeing alone as a predictor. The analyses for Table 8, like those for Table 5, used both hedonic wellbeing and eudaimonic wellbeing as predictors. The analyses for Table 9, like those for Table 6, used hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates as predictors.
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.01560  .1318  0.00743  .4694 
Type I interferon viral response genes  31  0.00722  .5294  0.00319  .8018 
Antibody synthesis genes  3  0.00842  .5881  0.06671  .2011 
Proinflammatory genes  19  0.05664  .0065  0.03645  .0427 
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.02689  .0150  –0.01410  .1928 
Type I interferon viral response genes  31  0.01300  .0882  –0.00722  .5417 
Antibody synthesis genes  3  –0.12523  .2072  0.16696  .1967 
Proinflammatory genes  19  0.07644  .0005  –0.02474  .0667 
Gene Set  Number of Genes  Relation to WellBeing 


Hedonic 
Eudaimonic 

Coefficient  p  Coefficient  p  
Omnibus  53  0.01276  .0969  –0.02717  .0033 
Type I interferon viral response genes  31  0.01661  .0171  0.03074  .0109 
Antibody synthesis genes  3  –0.03898  .2618  0.09016  .1579 
Proinflammatory genes  19  0.05654  .0001  –0.01140  .0667 
It is not clear why Fredrickson et al. (2013) did not test the difference between the mean partial regression coefficient for hedonic wellbeing and the mean partial regression coefficient for eudaimonic wellbeing for each set of genes as well as testing each mean partial regression coefficient against zero (as shown in Table 3), in analyses analogous to the tests of the difference between the correlations performed for the three depression variables and for total wellbeing. Given that (with one exception) the mean partial regression coefficients had different signs, perhaps they did not consider such a test necessary. Alternatively, this omission may have been an oversight; a more recent article by Fredrickson et al. (2015) did do such an analysis.
Brown et al. (2014) considered Fredrickson et al.’s (2013) analytic procedure to be “unnecessarily complicated” (p. 12708) and suggested the alternative of regressing the predictor variables on the averaged gene expression. Cole and Fredrickson (2014) countered that this alternative “guarantees bias” (p. E3581). This is not so. Although perhaps not intuitively obvious, the averaged (mean) regression coefficients for each set of genes equal the regression coefficients for the averaged (mean) gene expression for each set of genes, as can be seen by comparing the entries in columns 3 and 5 in Tables 7, 8, and 9 (averaged coefficients) to the corresponding entries in columns 3 and 5 in Tables 4, 5, and 6 (averaged gene expressions). Note, though, that the p values associated with the regression coefficients in Tables 7, 8, and 9 are usually lower than the corresponding p values in Tables 4, 5, and 6. Moreover, the p values for the regression coefficients in Table 3 reported by Fredrickson et al. (2013) do not match those in either Table 6 or Table 9, even though these three sets of regression coefficients were obtained from the prediction of gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates. The reasons for these differences will become apparent later when I examine the way that the p values were computed by Fredrickson et al. (2013).
Analysis of Suppression Effects
Why have the signs of some of the regression coefficients in the analyses predicting averaged gene expression (Tables 5 and 6) or averaged gene association (Tables 8 and 9) from both hedonic wellbeing and eudaimonic wellbeing reversed compared to those in the analyses predicting averaged gene expression (Table 4) or averaged gene association (Table 7), respectively, from hedonic wellbeing alone or eudaimonic wellbeing alone, and what implications do these reversals have for the differential relations of hedonic wellbeing and eudaimonic wellbeing to gene expression expected and reported by Fredrickson et al. (2013)?
Fredrickson and her coauthors insisted in their original article (2013) and in their reply (2013) to Coyne (2013) that the differential relations of hedonic wellbeing and eudaimonic wellbeing to gene expression were not due to the high correlation between hedonic wellbeing and eudaimonic wellbeing. Fredrickson and her coauthors were wrong.
Apparently anticipating criticism of their regression analyses that used highly correlated predictors, Fredrickson et al. (2013) substituted total wellbeing (eudaimonic wellbeing plus hedonic wellbeing) and relative eudaimonic predominance (eudaimonic wellbeing minus hedonic wellbeing) for hedonic wellbeing and eudaimonic wellbeing in “ancillary analyses” (p. 13685) to “ensure that differential gene expression estimates were not distorted by the high correlation of hedonic and eudaimonic wellbeing” (p. 13685). They reported that these two new variables were correlated only modestly (r = .21), and that there was no significant relation of total wellbeing to expression of the omnibus set of genes, but a significant negative relation of relative eudaimonic predominance to expression of the omnibus set of genes, stemming primarily from the significant negative relation of relative eudaimonic predominance to expression of the subset of proinflammatory genes.
The logic and the details of these ancillary analyses were not made clear in the original article. In their reply to Coyne (2013), Cole and Fredrickson (2013, p. E4814) expanded the description of these analyses:
Might the correlation between hedonic and eudaimonic wellbeing lead to erroneous conclusions? We addressed this possibility in a reparameterized analysis of gene expression reported in the Results …. If eudaimonic and hedonic scales were truly interchangeable, then only their shared variance (the sum, eudaimonic + hedonic) would systematically correlate with gene expression, and their unique/unshared variance (the difference, eudaimonic – hedonic) would reflect only random measurement error and would not correlate systematically with gene expression. Neither was the case. Gene expression associated only with the difference, and the sum and difference variables were not highly correlated …. Even in the presence of correlated predictor variables and control for confounders, the general linear model parameter estimates we used are well established … to remain unbiased, and their P values remain accurate. [Correlated predictors and multiple covariates increase the sampling variability of point estimates, but their SEs adjust appropriately …. Correlated predictors might potentially lead to conservative falsenegative errors in statistical testing, but they would not induce falsepositive errors.] Thus, there is no reason for concern that the standard multivariate analyses we used led to inaccurate conclusions.
This expanded description is confusing rather than enlightening and contains several inaccurate statements:
 The shared variance between two variables is not their sum; it is the square of their correlation (r^{2}).
 The unique/unshared variance between two variables is not their difference; it is one minus the square of their correlation (1 – r^{2}).
 It is not true that a predictor that has a correlation with the criterion reflecting only measurement error (near zero, presumably) cannot correlate with that criterion in a regression containing two (or more) predictors; a correlation of zero (or nearly zero) can become nonzero and increase the magnitude of the other predictor. This wellknown regression situation is called “traditional suppression.”
 It is not true that correlated predictors cannot induce falsepositive errors. If two predictors are correlated, measurement error in one of them can inflate the falsepositive (Type I) error in tests of the other (Brunner & Austin, 2009; Shear & Zumbo, 2013). This is a moot point in Fredrickson et al.’s (2013) study, however, because, of the 106 possible tests of the coefficients for hedonic wellbeing and eudaimonic wellbeing (53 tests each) in the regression analyses that predicted gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates, only one was significant.
 Traditionally, the “standard multivariate analys[i]s” (p. E4184) for predicting multiple criteria from the same predictors for the same cases is multivariate multiple regression; it is not a set of multiple regression analyses predicting one criterion at a time, like that of Fredrickson et al. (2013). A modern alternative to multivariate multiple regression is mixedeffects linear modeling with the gene expressions treated as repeated measurements, which Brown et al. (2014, Supporting Information, p. 16) suggested. Both of these analytic methods take into account the dependency in the analyses; a set of multiple regression analyses predicting one criterion at a time does not. Moreover, even though each of Fredrickson et al.’s (2013) individual regression analyses might be considered standard in and of itself, their method of computing p values for those analyses cannot, as will be seen.
In their Figure 1B, Fredrickson et al. (2013, p. 13686) provided without explanation what they called a “Tukey meandifference plot,” although it is actually a “Tukey sumdifference plot,” usually called a “BlandAltman plot” (Altman & Bland, 1983) in the medical literature. This plot shows the relation between total wellbeing and relative eudaimonic predominance. Plots like this are often used to determine whether two measures, tests, or instruments can be used interchangeably; the common belief (e.g., Coyne, 2013) that a high correlation between two measures indicates that those measures are interchangeable is not correct. Usually horizontal lines are drawn across a BlandAltman plot at the mean of the measure on the y axis, and at 1.96 (or 2) standard deviations above and below that mean. The two measures are judged interchangeable if all the graphed sumdifference data points fall between the lines at 1.96 standard deviations above and below the mean, called the “lines of agreement.” Fredrickson et al.’s (2013) plot of the relation between total wellbeing and relative eudaimonic prevalence is ambiguous with respect to whether hedonic wellbeing and eudaimonic wellbeing are interchangeable. A few data points fall outside the lines of agreement (which were not drawn), but only a few. But even if one concludes that hedonic wellbeing and eudaimonic wellbeing are not interchangeable, one cannot then conclude that the high correlation between these two predictors cannot be responsible for their differential relations to the criterion, as Fredrickson and her coauthors would have it. And, as will become evident shortly, Fredrickson et al.’s (2013) regression analyses did not in fact create “purified” indices of hedonic wellbeing and eudaimonic wellbeing, as claimed.
A comparison of simple regression coefficients to partial regression coefficients demonstrates that the high correlation between hedonic wellbeing and eudaimonic wellbeing was in fact partly responsible for the differential relations of these two wellbeing variables to gene expression in Fredrickson et al.’s (2013) study. Researchers often seem to believe that in a regression analysis predicting a criterion variable Y from two predictor variables X_{1} and X_{2}, one or the other of two situations must occur: either X_{1} and X_{2} are independent, or X_{1} and X_{2} are redundant:
 Independence occurs when the two predictors are uncorrelated. The partial regression coefficient for each of the two predictors equals its corresponding simple regression coefficient. (For simplicity and generality, I will assume that the predictors and the criterion are standardized. The simple regression coefficient for X_{1} or X_{2} is obtained by regressing Y on either X_{1} alone or X_{2} alone, respectively. The simple regression coefficient is equivalent to the zeroorder correlation between the predictor and the criterion. The partial regression coefficient for X_{1} and for X_{2} is obtained by regressing Y on both X_{1} and X_{2}. Partial regression coefficients are not correlations. In the Fredrickson et al., 2013, study, the predictors hedonic wellbeing and eudaimonic wellbeing were standardized, but the criterion gene expression was not. The relations to be described still apply, but the simple regression coefficient is not equivalent to the correlation.)
 Redundancy occurs when the two predictors are correlated. The magnitude (absolute value) of each partial regression coefficient is less than the magnitude of its corresponding simple regression coefficient, but the sign is unchanged. This is the situation to which Fredrickson et al. (2013) were referring when they claimed that their regression analyses used “purified” indices of hedonic wellbeing and eudaimonic wellbeing.
In fact, three other situations are possible when the two predictors are correlated: cooperative suppression, traditional suppression, and negative suppression (Tzelgov & Henik, 1991).
 Cooperative suppression (also called reciprocal suppression) occurs when the correlation between one predictor and the criterion is positive, the correlation between the other predictor and the criterion is negative, and the correlation between the two predictors is positive (or the scale of one of the variables can be reversed so that the correlations exhibit this pattern). The magnitude of the partial regression coefficient for each of the two predictors is greater than the magnitude of its corresponding simple regression coefficient, but the sign is unchanged.
 Traditional suppression (also called classical suppression) occurs when the two predictors are correlated, and the correlation of one of the predictors with the criterion equals zero (or sometimes, nearly zero). The partial regression coefficient for the predictor having the zero correlation with the criterion is nonzero. The magnitude of the partial regression coefficient for the predictor having the nonzero correlation with the criterion is greater than the magnitude of its corresponding simple regression coefficient; the sign is unchanged.
 Negative suppression (also called net suppression) occurs when the two predictors are correlated, and the partial regression coefficient for one of the predictors has a sign that is reversed from that of its corresponding simple regression coefficient. The partial regression coefficient with the sign that reversed is for the predictor for which the magnitude of the simple regression coefficient is smaller. Negative suppression can occur in one of two ways: (a) the magnitude of the simple regression coefficient for X_{1} is greater than that for X_{2}, and the magnitude of the correlation between the two predictors is greater than the ratio of the magnitude of the simple regression coefficient for X_{2} to the magnitude of the simple regression coefficient for X_{1}; or (b) the magnitude of the simple regression coefficient for X_{2} is greater than that for X_{1}, and the magnitude of the correlation between the two predictors is greater than the ratio of the magnitude of the simple regression coefficient for X_{1} to the magnitude of the simple regression coefficient for X_{2}. The magnitude of the partial regression coefficient with the sign that reversed can be less than, equal to, or greater than the magnitude of its corresponding simple regression coefficient. The magnitude of the partial regression coefficient with the sign that did not reverse is greater than the magnitude of its corresponding simple regression coefficient.
Thus, suppression can cause sign reversals in, and increase the magnitudes of, regression coefficients. The occurrence of traditional suppression or cooperative suppression does not depend on the magnitude of the correlation between the two predictors, but negative suppression is more likely to occur when the correlation between the two predictors is high, as it was in Fredrickson et al.’s (2013) study. The majority of the 53 regression analyses predicting gene expression from hedonic wellbeing and eudaimonic wellbeing were characterized by negative suppression. This suppression resulted in the apparent differential relations of hedonic wellbeing and eudaimonic wellbeing to gene expression.
The regressionanalysis results for the subset of three antibody synthesis genes shown in the middle section of Table 10 provide a small example. For simplicity and clarity, I will compare the results of the regression analyses using hedonic wellbeing alone (column 2) or eudaimonic wellbeing alone (column 3) as predictors to those including both wellbeing variables (columns 4 and 5) as predictors, omitting the 15 covariates. The simple regression coefficients for gene expression predicted from hedonic wellbeing alone and from eudaimonic wellbeing alone are small and with one exception, positive (hedonic wellbeing: 0.02882, –0.01621, and 0.01265 for genes IGJ, IGLL1, and IGLL3, respectively; eudaimonic wellbeing: 0.13347, 0.05429, and 0.01236 for genes IGJ, IGLL1, and IGLL3, respectively). The averaged associations (simple regression coefficients) are therefore 0.00842 and 0.06671 for hedonic wellbeing and eudaimonic wellbeing, respectively, and do not show the differential relations to gene expression expected and reported by Fredrickson et al. (2013).
Gene  WellBeing Coefficients from Regression Analyses 


Simple 
Partial 
Suppression 

Hedonic  Eudaimonic  Hedonic  Eudaimonic  Type^{a}  Keeps Sign, Larger^{b,c}  Reverses Sign, Change^{b,d}  
Type I interferon viral response genes  
IFIT1L  –0.22554  –0.28774  0.01339  –0.29847  N  E  H, < 
GBP1  –0.00648  –0.01402  0.01319  –0.02457  N/T  E  H, > 
IFI16  –0.05918  –0.04820  –0.05735  –0.00228  R  
IFI27  –0.04127  –0.04985  –0.00381  –0.04680  R  
IFI27L1  0.00090  0.00608  –0.01107  0.01494  N/T  E  H, > 
IFI27L2  –0.00669  –0.02652  0.04049  –0.05893  N/T  E  H, > 
IFI30  –0.04097  –0.06427  0.02918  –0.08763  N  E  H, < 
IFI35  –0.01653  –0.01819  –0.00549  –0.01379  R  
IFI44  0.05094  0.03443  0.06509  –0.01768  N  H  E, < 
IFI44L  0.06188  0.03970  0.08381  –0.02740  N  H  E, < 
IFI6  0.02280  0.00773  0.04625  –0.02930  N/T  H  E, > 
IFIH1  –0.02290  –0.01186  –0.03731  0.01801  N  H  E, > 
IFIT1  0.11698  0.10981  0.08096  0.04500  R  
IFIT2  0.12863  0.10301  0.12853  0.00012  R  
IFIT3  0.09687  0.09764  0.05211  0.05592  R  
IFIT5  –0.00534  –0.01015  0.00775  –0.01636  N/T  E  H, > 
IFITM1  0.05282  0.06572  0.00058  0.06525  R  
IFITM2  0.04722  0.05074  0.01838  0.03603  R  
IFITM3  0.00154  0.02236  –0.04554  0.05882  N/T  E  H, > 
IFITM4P  0.00604  0.00236  0.01154  –0.00687  N/T  H  E, > 
IFITM5  –0.00258  0.00031  –0.00788  0.00662  C  
IFNB1  0.00176  0.00527  –0.00683  0.01073  N/T  E  H, > 
IRF2  –0.01620  –0.00357  –0.03714  0.02616  N/T  H  E, > 
IRF7  0.03038  0.02622  0.02617  0.00527  R  
IRF8  –0.04512  –0.04886  –0.01671  –0.03548  R  
MX1  0.04966  0.06555  –0.00784  0.07183  N  E  H, < 
MX2  0.00680  0.00833  0.00037  0.00803  R  
OAS1  –0.03414  –0.02493  –0.03948  0.00668  N  H  E, < 
OAS2  –0.01613  –0.01888  –0.00283  –0.01662  R  
OAS3  0.03921  0.03001  0.04226  –0.00382  N  H  E, < 
OASL  0.04847  0.05055  0.02228  0.03271  R  
Antibody synthesis genes  
IGJ  0.02882  0.13347  –0.21724  0.30737  N  E  H, > 
IGLL1  –0.01621  0.05429  –0.16613  0.18728  C  
IGLL3  0.01265  0.01236  0.00767  0.00623  R  
Proinflammatory genes  
IL1B  0.14056  0.07545  0.22321  –0.10324  N  H  E, > 
IL8  0.31637  0.27763  0.26206  0.06784  R  
FOS  0.08753  0.08685  0.05013  0.04672  R  
FOSB  0.10143  0.03996  0.19333  –0.11481  N  H  E, > 
FOSL1  0.00906  –0.01412  0.05669  –0.05950  C  
FOSL2  0.04684  0.02441  0.07603  –0.03645  N  H  E, > 
IL1A  0.02405  0.00562  0.05444  –0.03796  N/T  H  E, > 
IL6  0.02493  0.00607  0.05588  –0.03866  N/T  H  E, > 
JUN  0.05045  0.05132  0.02609  0.03043  R  
JUNB  0.01093  0.01348  0.00038  0.01318  R  
JUND  0.01012  0.01129  0.00300  0.00890  R  
NFKB1  0.00495  0.01334  –0.01596  0.02612  N/T  E  H, > 
NFKB2  0.01921  0.01213  0.02645  –0.00904  N  H  E, < 
PTGS1  –0.07327  –0.10879  0.03849  –0.13961  N  E  H, < 
PTGS2  0.13299  0.09750  0.15297  –0.02496  N  H  E, < 
REL  0.02267  0.00957  0.04180  –0.02389  N/T  H  E, > 
RELA  0.03548  0.03662  0.01716  0.02288  R  
RELB  0.03214  0.01189  0.06300  –0.03855  N  H  E, > 
TNF  0.07968  0.04242  0.12730  –0.05948  N  H  E, > 
Compare these simple regression coefficients to the partial regression coefficients obtained when gene expression is predicted from both hedonic wellbeing and eudaimonic wellbeing. The regression analysis for gene IGLL3 shows redundancy; the partial regression coefficients for hedonic wellbeing (0.00767) and eudaimonic wellbeing (0.00623) have the same signs (positive) as, and are smaller in magnitude than, their corresponding simple regression coefficients (0.01265 and 0.01236, respectively). The other two regression analyses show suppression effects. The regression analysis for gene IGLL1 shows cooperative suppression; the relation between hedonic wellbeing and gene expression has a different sign (negative) than the relation between eudaimonic wellbeing and gene expression and the relation between hedonic wellbeing and eudaimonic wellbeing (both positive). The partial regression coefficients for both hedonic wellbeing and eudaimonic wellbeing (–0.16613 and 0.18728, respectively) have the same signs as, but are much larger in magnitude than, their corresponding simple regression coefficients (–0.01621 and 0.05429, respectively). The regression analysis for gene IGJ shows negative suppression. The partial regression coefficient for hedonic wellbeing (–0.21724) has a sign reversed from that of its corresponding simple regression coefficient (0.02882) and is much larger in magnitude. The partial regression coefficient for eudaimonic wellbeing (0.30737) has the same sign as its corresponding simple regression coefficient (0.13347) but is much larger in magnitude. Now the averaged associations (partial regression coefficients) are –0.12523 and 0.16696 for hedonic wellbeing and eudaimonic wellbeing, respectively. Although these averaged coefficients are not significant, their different signs are consistent with the expected differential relations of hedonic wellbeing and eudaimonic wellbeing to gene expression. The sign of the relation between hedonic wellbeing and gene expression is now negative, whereas the sign of the relation between eudaimonic wellbeing and gene expression is positive, just as Fredrickson et al. (2013) expected and reported.
The coefficients from all 53 regression analyses predicting gene expression from (a) hedonic wellbeing alone, (b) eudaimonic wellbeing alone, and (c) both hedonic wellbeing and eudaimonic wellbeing are shown in Table 10. Thirtythree of the 53 sets of regression analyses show suppression effects (mostly negative suppression). These effects are responsible for the reversals in regressioncoefficient signs and increases in regressioncoefficient magnitudes that led Fredrickson et al. (2013) to conclude that hedonic wellbeing and eudaimonic wellbeing have differential relations to gene expression. In turn, the negative suppression results from the high correlation between hedonic wellbeing and eudaimonic wellbeing (.79 as reported by Fredrickson et al., 2013; .80 in the corrected sample of 76 usable cases). If the correlation had been low, few of the 53 sets of regression analyses would have shown suppression, and differential relations of hedonic wellbeing and eudaimonic wellbeing to gene expression would not have been evident.
Interestingly, Brown et al. (2014), in their analysis of the relation of each of the 8,191 possible twofactor scorings of the MHC–SF to gene expression, briefly mentioned “a previously reported phenomenon (1), whereby when two highlycorrelated items are incorporated in a regression analysis, their variance is distributed approximately equally, but with appreciate opposite signs, across the two resulting regression coefficients” (Supporting Information, p. 27). Reference 1 is a review of datareduction techniques in electroencephalographic research (Allen, Coan, & Nazarian, 2004). The reported phenomenon appears to be a special case of negative suppression, although it is not called such.
Moreover, the inclusion of the 15 covariates in the regression analysis for each gene creates additional suppression effects; this is the reason for the regression coefficient for eudaimonic wellbeing for the subset of Type I interferon viral response genes having a negative sign in Tables 5 and 8 reversing to a positive sign in Tables 6 and 9, after having reversed from a positive sign in Tables 4 and 7 to a negative sign in Tables 5 and 8.
What is one to make of all these sign reversals? Are the signs of the final regression coefficients (those in Tables 6 and 9) “real,” indicating the direction of the true biological relation between each of the two wellbeing variables and gene expression, or are they merely statistical artifacts, lacking the theoretical importance bestowed on them by Fredrickson et al. (2013)?
Unfortunately, it is difficult to know without more information than is included in the dataset, but a comparison of the results of regression analyses predicting averaged gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates to the results of analogous regression analyses that replace eudaimonic wellbeing with Keyes’s (2009a) social wellbeing and psychological wellbeing is suggestive. Table 11 presents the partial regression coefficients that would have been obtained had Fredrickson et al. (2013) used the documented and wellvalidated threefactor scoring instead of their idiosyncratic twofactor scoring and corrected the aforementioned miscoding of race/ethnicity. (None of these three wellbeing variables, which are highly correlated with one another, has a significant relation with averaged gene expression when used as the sole predictor.) A comparison of the partial regression coefficients in Table 11 to those in Tables 6 and 9 shows that the relation of emotional (hedonic) wellbeing to averaged gene expression has reversed sign for the omnibus set of genes, which should not have happened. Moreover, the sign of the relation of social wellbeing to averaged gene expression is the same as the sign of the relation of eudaimonic wellbeing to averaged gene expression for each of the four sets of genes, but the sign of the relation of psychological wellbeing to averaged gene expression is the reverse of the sign of the relation of eudaimonic wellbeing to averaged gene expression for each of the four sets of genes. That is, the signs of the four coefficients for psychological wellbeing are reversals of the signs of the four coefficients for social wellbeing. If social wellbeing and psychological wellbeing can legitimately be represented by the single construct of eudaimonic wellbeing, as Cole and Fredrickson (2014, article, p. E3581, Supporting Information, pp. 16, 22, 27) insisted in their reply to Brown et al. (2014), then surely their relations to averaged gene expression should have the same direction. But they do not, suggesting that the direction and magnitude of the partial regression coefficients in Fredrickson et al.’s (2013) study are statistical artifacts.
Gene Set  Number of Genes  Coefficients for Relation to WellBeing 


Emotional  Social  Psychological  
Omnibus  53  –0.00907  –0.06839  0.05665 
Type I interferon viral response genes  31  0.04392  0.08317  –0.07207 
Antibody synthesis genes  3  –0.00078  0.14135  –0.07546 
Proinflammatory genes  19  0.04623  –0.03275  0.02851 
Although traditionally suppression effects have been viewed as statistical curiosities, artifacts, or nuisances (Bobko, 2001), they may be of theoretical interest, as, for example, in the relation of guilt and shame to selfreported aggression (Paulhus, Robins, Trzesniewski, & Tracy, 2004; see also Nickerson, 2008). Determination of whether a suppression effect is a statistical artifact or a phenomenon of substantive interest requires strong theoretical reasoning. No such reasoning appeared in Fredrickson et al. (2013); indeed, the existence of the suppression effects was not even recognized. Of course, little or no attention is usually paid to regression coefficients that are not significant, regardless of which of the five situations described above applies to a particular regression analysis.
Computation of P Values
The question of significance for the regression coefficients brings us to the computation of p values. As I noted earlier, the p values for the regression coefficients in Table 9 are different from those reported by Fredrickson et al. (2013), shown in Table 3, and the p values associated with the regression coefficients in Tables 7, 8, and 9 are ususally lower than the p values associated with the regression coefficients in Tables 4, 5, and 6. Indeed, some of the regression coefficients reported in Tables 7, 8, and 9 are significant, when their corresponding and equal regression coefficients reported in Tables 4, 5, and 6 are not.
One reason why the p values in Table 9 are different from those reported by Fredrickson et al. (2013) is the uncorrected value (“4”) for race/ethnicity for one case in Fredrickson et al.’s (2013) dataset. As noted by Brown et al. (2014, Supporting Information, p. 24), this seemingly small error had a surprisingly large effect on the results (especially those for hedonic wellbeing) obtained by Fredrickson et al. (2013). For example, had this error not been corrected, the averaged partial regression coefficient for hedonic wellbeing for the omnibus set of genes shown in Table 9 would equal 0.02801 instead of the correct 0.01276; for the subset of antibody synthesis genes that coefficient would equal –0.07938 instead of the correct –0.03898.
A second reason is that Fredrickson et al. (2013) did not use the standard error computed across the regression coefficients for each set of genes in computing the onesample t test for that set of genes, as I have done. Instead, they bootstrapped the standard error “to estimate a stable SE, as recommended” (Supporting Information, p. 2). Bootstrapping is useful when the theoretical distribution of a statistic is mathematically complicated or unknown; if an assumption, such as the normality assumption, of a statistical test is violated (as it is in Fredrickson et al.’s, 2013, onesample t tests); if the number of observations is too small for valid inference from a statistical test (also the case in some of Fredrickson et al.’s, 2013, onesample t tests); and so on. Each run of a bootstrapping algorithm would yield a different standard error, so it is unlikely that a p value for a t value based on a computed standard error would exactly equal one based on a bootstrapped standard error.
As noted earlier, some of the regression coefficients reported in Tables 7, 8, and 9 are significant, when their corresponding and equal regression coefficients reported in Tables 4, 5, and 6 are not. For example, in Table 6, the coefficient for eudaimonic wellbeing in the regression analysis predicting gene expression averaged across all 53 genes from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates for 76 cases equals –0.02717. The standard error for this coefficient equals 0.04383, t = –0.62, p = .5378, n = 76, not significant. In Table 9, the average of the partial regression coefficients for eudaimonic wellbeing from the 53 regression analyses predicting gene expression from hedonic wellbeing, eudaimonic wellbeing, and the 15 covariates also equals –0.02717. But the standard error for this averaged coefficient equals 0.00882, t = –3.08, p = .0033, n = 53, significant.
The reason why some of the regression coefficients reported in Tables 7, 8, and 9 are significant, when their corresponding and equal regression coefficients reported in Tables 4, 5, and 6 are not, is that the standard errors for the statistical tests reported in Tables 7, 8, and 9 are computed differently from the standard errors for the statistical tests reported in Tables 4, 5, and 6. For the latter tests, the standard errors are based on the variation in the criterion variable and the predictor variables, and the covariation between these variables, across the 76 cases for each set of genes. For the former tests, the standard errors are based on the variation in regression coefficients across the genes (n = 53, 31, 3, or 19) for each set of genes. (These are very small sample sizes for statistical tests, especially n = 3.) Which sets of p values are correct, those in Tables 4, 5, and 6, or those in Tables 7, 8, and 9?
The p values in Tables 4, 5, and 6—not those in Tables 7, 8, and 9—are correct, for (at least) two reasons. First, Fredrickson et al.’s (2013) method of computing p values treats two regression coefficients of the same value as equivalent. But one of those regression coefficients may have a large standard error and a wide confidence interval, whereas the other may have a small standard error and a narrow confidence interval. Moreover, one regression coefficient may have a confidence interval that includes zero and so is not significant, whereas the other may have a confidence interval that does not include zero and so is significant. That is, Fredrickson et al.’s (2013) method of computing p values discarded valuable information about the accuracy of the regression coefficients and left ambiguous the direction of the relation between the criterion and the predictor (Kelley & Maxwell, 2003). Unfortunately, the partial regression coefficients for hedonic wellbeing and eudaimonic wellbeing in Fredrickson et al.’s (2013) study all have wide confidence intervals, probably as a result of the effects of the wellbeing variables on gene expression being small, the wellbeing variables being highly correlated, and the use of a sample size of 76 that is much too small for the large number (17) of predictors. Thus, the partial regression coefficients for hedonic wellbeing and eudaimonic wellbeing are unlikely to be accurate. Moreover, all but one of the 106 partial regression coefficients have confidence intervals that include zero, so that the directions of the relations between the criterion and the predictors of interest are not certain. This is so for both the partial regression coefficients from the analyses predicting individual gene expression and for the partial regression coefficients from the analyses predicting averaged gene expression.
Second, the observations in Fredrickson et al.’s (2013) onesample t tests—the partial regression coefficients—were not independent, which invalidates these tests. The onesample t test assumes that observations are randomly and independently drawn from a normally distributed population. The observations in Fredrickson et al.’s (2013) t tests did not satisfy this requirement. These observations were the 106 partial regression coefficients that had been sorted into eight sets. These coefficients were not independent because they were computed from 53 regression analyses that used the same cases and the same predictor variables; only the criterion variable (expression for a particular gene) varied. Moreover, the sets of genes were specifically selected a priori on the basis of their similarity in biological functioning, so they were not independent, either.
The onesample t test is robust to moderate violations of the normality assumption, but it is not robust to violations of the independence assumption. The nonindependence of observations both increases the probability of falsepositive (Type I) errors and decreases the probability of falsenegative (Type II) errors. As the number of observations that are pairwisecorrelated increases and as the magnitude of the correlation between the observations increases, the probability of a falsepositive error increases rapidly. For example, for 40 observations having a (population) intercorrelation of .30, an apparent p value of .05 is actually about .347 (Zimmerman, Williams, & Zumbo, 1992, Table 2, p. 1017).
Brown et al. (2014, Supporting Information, p. 16) remarked that they had been unable to locate any other studies using the analytic method of Fredrickson et al. (2013). Cole and Fredrickson (2014) replied that “pooling gene associations [for use in a onesample t test] is actually commonplace and represents an elementary statistical sum of random variables” (p. 3581), but did not provide any examples of such pooling. Cole and Fredrickson (2014) have failed to distinguish between a sum of independent random variables and a sum of correlated (dependent) random variables. For independent random variables, the mean is the sum of the means, and the variance is the sum of the variances. For correlated random variables, the mean is also the sum of the means, but it does not generally converge to the population mean, as the Law of Large Numbers states for independent random variables. Also, for correlated random variables, the variance is not the sum of the variances but takes the correlation into account. The variance for correlated random variables is smaller than the variance for independent random variables. Assuming that correlated random variables have the same variance as independent random variables, as Fredrickson et al. (2013) seem to have done, introduces bias in any computation using the variance (or its square root, the standard deviation).
Two recent studies (Knight et al., 2016; Vedhara et al., 2015) coauthored by one of the coauthors of the Fredrickson et al. (2013) study (Steven W. Cole) used the same analytic method as Fredrickson et al. (2013) but altered the bootstrapping method for generating standard errors for the onesample t tests “to account for potential correlation among residuals across genes” (Knight et al., 2016, p. 71). This change to the analysis is a tacit admission that the computation of p values in Fredrickson et al.’s (2013) study was not correct, but no corrigendum to, or retraction of, that article has appeared.
Conclusions
As I have demonstrated, the statistical problems in Fredrickson et al.’s (2013) study cast considerable doubt on the validity of their conclusions that hedonic wellbeing and eudaimonic wellbeing have “similar affective correlates” but “markedly divergent gene transcriptional correlates in human immune cells”. Generally, I agree with Coyne (2013) and with Brown et al. (2014) that Fredrickson et al.’s (2013) study is seriously flawed, although some of the specific points made by Coyne (2013) and Brown et al. (2014) were not quite on the mark. For example, Coyne (2013) was correct in asserting that the high correlation between hedonic wellbeing and eudaimonic wellbeing in Fredrickson et al.’s (2013) study was problematical, but not correct in stating that hedonic wellbeing and eudaimonic wellbeing were correlated to the degree that their respective internal consistencies allowed. (The maximum correlation between two variables equals the square root of the product of their reliabilities. Assuming that Cronbach’s alpha is being used as the measure of reliability, the maximum correlation equals the square root of the product of .93 [for hedonic wellbeing] and .92 [for eudaimonic wellbeing], which equals .92, not .79.) Brown et al. (2014) were correct in asserting that the MHC–SF has a validated threefactor structure and scoring. But in analyzing the relation of each of the 8,191 possible twofactor scorings of the MHC–SF to gene expression, they, like Fredrickson et al. (2013), did not realize that their onesample t tests violated the assumption of independence. Nonetheless, the letter to the editor of Coyne (2013), the reanalysis of Brown et al. (2014), and my review of the statistical problems in Fredrickson et al. (2013)’s analyses taken together strongly suggest that there is no good evidence for Fredrickson et al.’s (2013) claim that hedonic wellbeing and eudaimonic wellbeing are differentially related to gene expression or for their recommendation that, as regards disease prevention, hedonia is not the way to live (p. 13688).
Data Accessibility Statement
The dataset used in Fredrickson et al.’s (2013) article is available from the National Center for Biotechnology Information Gene Expression Omnibus data repository (https://www.ncbi.nlm.gov/geo/; accession number GSE45330; March 20, 2013).