Submitted 7 October 2015Accepted 9 December 2015Published 5 January 2016

Corresponding authorShinichiro Tomitaka tomi-takashinichirojppanasoniccom

Academic editorC Robert Cloninger

Additional Information andDeclarations can be found onpage 15

DOI 107717peerj1547

Copyright2016 Tomitaka et al

Distributed underCreative Commons CC-BY 40

OPEN ACCESS

Age-related changes in the distributionsof depressive symptom items in thegeneral population a cross-sectionalstudy using the exponential distributionmodelShinichiro Tomitaka12 Yohei Kawasaki2 Kazuki Ide2Hiroshi Yamada2 Toshiaki A Furukawa3 and Yutaka Ono4

1Department of Mental Health Panasonic Health Center Tokyo Japan2Department of Drug Evaluation and Informatics Graduate School of Pharmaceutical SciencesUniversity of Shizuoka Shizuoka Japan

3Department of Health Promotion and Human Behavior Department of ClinicalEpidemiology Kyoto UniversitGraduate School of MedicineSchool of Public Health Kyoto UniversityKyoto Japan

4Center for the Development of Cognitive Behavior Therapy Training Tokyo Japan

ABSTRACTBackground Previous research has reported inconsistent evidence of the trajectory ofdepressive symptoms across the adult lifespan We investigated how the distributionsof each item score change with age and determined whether the trajectory of depressivesymptoms varied with the scoring methods of the questionnaireMethods We analyzed data collected from 21040 subjects who participated in thenational survey in Japan Depressive symptoms were assessed using the Center forEpidemiologic Studies Depression Scale (CES-D) The CES-D has 20 items each ofwhich is scored in four grades of lsquolsquorarelyrsquorsquo lsquolsquosomersquorsquo lsquolsquomuchrsquorsquo and lsquolsquomost of the timersquorsquoWe used the exponential distribution model which fits the distributions of 16 negativesymptom items of CES-D with the probabilities of lsquolsquosomersquorsquo lsquolsquomuchrsquorsquo lsquolsquomostrsquorsquo andlsquolsquorarelyrsquorsquo expressed as P Pr Pr2 and 1minusPtimes (r2+ r+1)Results The distributions of the responses to 16 negative symptom items followed thecommon exponentialmodel across all age groups Themean of the estimated parameterr of 16 negative items showed a U-shape pattern being high during 12ndash29 yearsremaining low during 30ndash50 years and then increasing again over 60 years Thetrajectory of depressive symptom scores simulating the binary method was differentfrom that of the empirical scores using the Likert methodConclusions Our findings show that the increase in the depressive symptoms scoreduring older age is based on the increase of the parameter r The differences in thescoring method may contribute to the different age-related patterns across the adultlifespan

Subjects Mathematical Biology Epidemiology Psychiatry and Psychology StatisticsKeywords Depressive symptoms CES-D CIS-R Mathematical model Item responseQuestionnaire Depression Exponential distribution Latent trait

How to cite this article Tomitaka et al (2016) Age-related changes in the distributions of depressive symptom items in the general popu-lation a cross-sectional study using the exponential distribution model PeerJ 4e1547 DOI 107717peerj1547

INTRODUCTIONDepression a common mental disorder is one of the leading causes of disability world-wide (Moussavi et al 2007) There has been much interest in understanding the distri-bution of depressive symptoms in the general population given their association withdepression (Blazer amp Kessler 1994 Kroenke et al 2009)

Cross-sectional surveys and longitudinal studies the majority of which made usingthe Center for Epidemiologic Studies Depression Scale (CES-D) have found that thetrajectory of depressive symptoms across the adult lifespan follows a U-shaped patternwith symptoms being high during young adulthood decreasing during middle adulthoodand then increasing again over the age of 70 (Kessler et al 1992 Oh et al 2013 Sutin etal 2013 Tomitaka Kawasaki amp Furukawa 2015a) Conversely some studies particularlycross-sectional surveys using the Revised Clinical Interview Schedule (CIS-R) consistentlyfound that depressive symptoms were high during middle adulthood and decreased overthe age of 55 (Adult psychiatric morbidity in England 2007 Bromley et al 2011)

The reason for the differences observed in age-related manifestations of depressivesymptom between CES-D and CIS-R is unclear It is unlikely that the true effect of agingon depressive symptoms in the general population is inconsistent indicating a possibleinfluence of methodological factors on the results of epidemiological studies

The discrepancy between epidemiological studies could be explained by selection biasHowever this explanation conflicts with the fact that even the large sample surveys in therandomly selected general population consistently show differences between CES-D andCIS-R (Adult psychiatric morbidity in England 2007 Tomitaka Kawasaki amp Furukawa2015a)

Another possible explanation is the difference in choice of items between CES-D andCIS-R The CES-D consists of 16 negative symptom items and 4 positive affect symptomitems (good hopeful happy and enjoyed) that are not included in CIS-R (Radloff 1977)Conversely CIS-R contains 14 negative symptom items (somatic symptom fatigueconcentration sleep irritability worry about physical health depression depressive ideasworry anxiety phobias panic compulsions and obsessions) some of which (eg paniccompulsion and obsession) are not included in CES-D (Lewis et al 1992) However eventhe common items (eg depression depressive ideas sleep anxiety and fatigue) showdifferences in age-related patterns between CES-D and CIS-R (Adult psychiatric morbidityin England 2007 Bromley et al 2011 Cooper et al 2015) Therefore it is difficult toattribute these differences between CES-D and CIS-R to the choice of items it is necessaryto consider why even the same items often differ

CES-D and CIS-R differ in the scoring method used for each item In CES-D eachitem has one question which is scored using a four point Likert method (0-1-2-3) basedon four possible answers lsquolsquorarely or none of the time (rarely)rsquorsquo lsquolsquosome or little of thetime (some)rsquorsquo lsquolsquooccasionally or a moderate amount of time (much)rsquorsquo and lsquolsquomost or allof the time (most)rsquorsquo (Radloff 1977) Conversely each item of CIS-R has two mandatoryquestions to screen the specific symptom (Lewis et al 1992) If responses indicate thespecific symptom four questions are asked and scored using a binary method (0ndash1) The

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 218

four non-mandatory questions contribute a single point to the 0ndash4 scale for each item(except depressive idea which has a 0ndash5 scale) The severity of each symptom appears toweigh more heavily in the CES-D scale than in the CIS-R scale These differences in thescoring method may contribute to the different age-related patterns observed across theadult lifespan In support of this speculation other studies investigating binary methodquestionnaires (eg Bradburn Affect Balance Scale) also reported that negative affectsymptom items were higher during middle adulthood than during old age (Stacey ampGatz 1991 Charles Reynolds amp Gatz 2001) To elucidate the effects of different scoringmethods on each item score it is necessary to understand how the distributions of eachitem score change with age Therefore a method for representing the distribution isnecessary in order to detect age-related changes

Using a variety of multivariate statistical methods eg factor analysis principalcomponents analysis and cluster analysis a lot of population studies on depressivesymptoms have been performed so far A common feature of these multivariate statisticalmethods is that they are focused on the interrelationship of each variable Even thoughthe interrelationship of each variable has been overwhelmingly investigated by manyresearchers to the best of our knowledge little work has been done to elucidate themathematical patterns of the distribution of each depressive symptom Thus in theprevious study we investigated the mathematical patterns of the individual distributionsfor each item of the CES-D

Although the method of the previous study was simple (just a histogram) we found anintriguing phenomenon The distributions of the 16 negative items for CES-D commonlyexhibited exponential patterns between lsquolsquosomersquorsquo and lsquolsquomostrsquorsquo response levels whilelsquolsquorarelyrsquorsquo was not related to the exponential patterns between lsquolsquosomersquorsquo and lsquolsquomostrsquorsquo (Tomi-taka Kawasaki amp Furukawa 2015b) Based on the findings we proposed a mathematicalmodel for the distribution of the 16 negative symptom items which were expressed bytwo parameters P and r where P stands for the probability of lsquolsquosomersquorsquo and r standsfor the equal ratio among lsquolsquosomersquorsquo lsquolsquomuchrsquorsquo and lsquolsquomostrsquorsquo The probabilities of lsquolsquosomersquorsquolsquolsquomuchrsquorsquo lsquolsquomostrsquorsquo and lsquolsquorarelyrsquorsquo are expressed as P Pr Pr2 and 1minusPtimes (r2+ r + 1) Weutilized this mathematical model to quantify the age-related changes in the distribution ofnegative symptom items

The present study used cross-sectional data from a large nationally representativesurvey conducted annually to evaluate the health status of a representative sample fromthe general Japanese population (Ministry of Health Labor and Welfare Statistics andInformation Department 2002) We analyzed more than 20000 CES-D assessmentsperformed in subjects between 12 and 89 years

Here we investigated whether the age-related changes observed in the total CES-D score were mainly due to the 16 negative symptoms After confirming that the 16negative symptoms were responsible for the U-shaped pattern of the total CES-D scoreand the distribution of the 16 negative items followed the mathematical model across allage groups we used the mathematical model to examine how the distribution of thesesymptoms changed according to the age group Finally we tested whether the conversion

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 318

of each item score of CES-D from the standard Likert method to the binary method couldaffect age-related changes in depressive symptoms across adulthood

METHODSThis study used data from the Active Survey of Health and Welfare (ASHW) conductedby the Japanese Ministry of Health Labor and Welfare in 2000 (Ministry of Health Laborand Welfare Statistics and Information Department 2002) ASHW is an annual nationwidesurvey conducted by the Japanese Government to collect data necessary for policy makingand health promotion in compliance with the Statistics Law The legal and ethical approvalof ASWH was given by the Ministry of Health Labor and Welfare Japan In 2000 ASHWexamined depressive symptoms among a representative sample from the general Japanesepopulation To ensure that the sample was adequately representative survey participantswere selected from individuals aged gt12 years living in 300 communities in Japan Thesecommunities were selected from 881851 precincts identified in the 1995 Census using astratified sampling design Verbal informed consent was obtained from all the subjectsThe data and methods used by the survey have been described in detail before (Ministry ofHealth Labor and Welfare Statistics and Information Department 2002)

The questionnaire was returned by 32729 respondents and the response rate wasnot publicized by the Ministry of Health Labor and Welfare and Health However theresponse rates for similar surveys conducted 3 and 4 years earlier were 871 and 896respectively (Kaji et al 2010) Therefore we assumed that the response rate for the presentsurvey was over 80

The present study was secondary analysis of ASWH data The Ministry of HealthLabor and Welfare examined our study and permitted us to analyze data from ASWH incompliance with the Statistics Law The present study was approved by the ethics committeeof Panasonic Health Center (approval number 2014-1) and the procedures were carriedout in accordance with the approved guidelines and regulations

We excluded 1394 respondents as we suspected the validity of their responses (ie thosewho answered lsquolsquorarelyrsquorsquo or lsquolsquomostrsquorsquo for all items regardless of the nature of the item) Atotal of 9588 respondents with missing information on one or more key study variables(ie depressive symptoms or age or sex) were also excluded from the sample An additional50 respondents over 89 years of age were excluded as the number was too small to elucidatethe pattern of distribution The final sample consisted of 20990 respondents between 12and 89 years

MEASURESDepressive symptoms were assessed using the Japanese version of the CES-D (Shima et al1985) This 20-item scale assesses the frequency of a variety of depressive symptoms withinthe previous week (0= rarely or none of the time (less than 1 day) 1= some or little ofthe time (1ndash2 days) 2= occasionally or a moderate amount of time (3ndash4 days) and 3=most or all of the time (5ndash7 days)) yielding a total score of 0ndash60 (Radloff 1977) Higherscores indicate greater psychological distress The 20 items of the CES-D were grouped into

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 418

the following four subscales depressive mood (items 3 6 9 10 14 17 and 18) somaticand retarded activities symptoms (items 1 2 5 7 11 13 and 20) interpersonal relations(items 15 and 19) and positive affect symptoms (items 4 8 12 and 16) The positiveaffect symptom items were reverse-scored The 16 items of depressive mood somatic andretarded activities symptoms and interpersonal relations follow the commonmathematicalmodel while the 4 positive affect symptom items do not (Tomitaka Kawasaki amp Furukawa2015b)

Analysis procedureThe respondents were grouped into the following age groups 12ndash19 20ndash29 30ndash39 40ndash4950ndash59 60ndash69 70ndash79 and 80ndash89 years The final sample consisted of 20990 respondentsbetween the ages of 12ndash89 years (ages 12ndash19 N = 2457 (male n= 1269) ages20ndash29 N = 3748 (male n= 1788) ages 30ndash39 N = 3761 (male n= 1783) ages40ndash49 N = 3629 (male n= 1788) ages 50ndash59 N = 3569 (male n= 1800) ages60ndash69 N = 2253 (male n= 1155) ages 70ndash79 N = 1161 (male n= 517) ages 80ndash89N = 412 (male n= 108))

The first step was to compare the total CES-D score and each item score by age groupand to confirm that the 16 negative symptom items followed a U-shaped pattern acrossadulthood resulting in the U-shaped pattern of the total CES-D score This pattern ofeach negative item was shown to exhibit the lowest score during 30ndash69 years (TomitakaKawasaki amp Furukawa 2015a)

Second in order to confirm that the distribution of the 16 negative items follow themathematical model across all age groups their histograms were evaluated for each agegroup The distributions between lsquolsquosomersquorsquo and lsquolsquomostrsquorsquo were analyzed using a log-normalscale

Third the parameters P and r were estimated from empirical data to quantify the age-related changes in the distribution of the 16 negative items These parameters were assessedfor each item using the mathematical model as follows P = probability of lsquolsquosomersquorsquo r =(probability of lsquolsquomuchrsquorsquo)(probability of lsquolsquosomersquorsquo) + (probability of lsquolsquomostrsquorsquo)(probabilityof lsquolsquomuchrsquorsquo)2 The mean values of the parameters were calculated for each age group

Finally to demonstrate the effect of the scoring methods on age-related changes indepressive symptoms the total CES-D score and 16 negative items scores were comparedbetween the standard Likert method and the binary method We used JMP version 11 forWindows (SAS Institute Inc Cary NC USA) to calculate the descriptive statistics andfrequency distribution curves

RESULTSAge-related changes of each item score and total CES-D score acrossadulthoodThe total scores of 20 items and 16 negative item scores exhibited a similar U-shapedpattern with symptoms being highest during 12ndash29 years decreasing during 30ndash69 yearsand then increasing again during 70ndash89 years whereas the 4 positive items showed aplateau-shaped pattern (Fig 1) The similarity in the pattern of the total score of 20 items

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 518

Figure 1 The relationship between age and the total scores of 20 items 16 negative items score and 4positive items score The total scores of 20 items (blue line) and 16 negative item scores (red line)exhibited a similar U-shaped pattern whereas the 4 positive items (green line) showed a plateau-shapedpattern

and the 16 negative items score suggested that the U-shaped pattern of total CES-D scoreis mainly attributed to the age-related changes of the 16 negative symptom items

Analysis of each item score of CES-D by age group showed that most of the 16 negativeitems in the depressive mood group somatic symptoms and retarded activities groupand interpersonal relations group exhibited U-shaped patterns (Table 1) All 16 negativesymptom items were the highest during 12ndash19 years or 80ndash89 years and 13 negative itemswere the lowest during 30ndash69 years indicating that the U-shaped pattern is mostly common

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 618

Table 1 Mean of each item according to age group in the general Japanese population

Number Age group 12ndash19 20ndash29 30ndash39 40-49 50ndash59 60ndash60 70ndash79 80ndash89Depressed mood

3 Blues 041 040 039 042 039 a033 044 b0636 Depressed 079 079 071 070 059 a047 055 b0799 Failure 074 073 066 063 a061 065 071 b07710 Fearful 027 027 a024 028 028 028 030 b04114 Lonely 033 042 030 029 028 a028 041 b07217 Crying 014 015 011 010 a008 009 012 b01718 Sad 039 041 034 034 a032 032 035 b049

Somatic symptoms and retarded activities1 Bothered a054 061 067 071 065 059 069 b0892 Appetite 041 043 039 037 033 a032 046 b0675 Trouble concentrating 095 072 066 066 058 a052 070 b0867 Effort 105 086 077 075 064 a059 075 b11111 Sleep a046 055 047 049 057 065 077 b08213 Talked a041 047 049 054 049 049 052 b06020 Get going b078 044 035 034 a027 028 038 054

Interpersonal relations15 Unfriendly 026 026 024 026 a025 024 027 b03819 Dislike b031 024 022 023 022 a018 019 030

Positive affects4 Good 158 146 a142 146 167 b174 170 1728 Hopeful 174 152 a147 160 167 168 175 b18712 Happy 163 164 a153 163 172 175 b176 17116 Enjoyed a115 131 128 143 145 146 150 b154

Total 16 negative items 82 78 70 71 66 a63 76 b102Total 4 positive items 61 59 a57 61 65 66 67 b68

Total 20 items 143 137 127 132 131 a129 143 b170

NotesaThe lowest valuebThe highest value The positive symptom items are reverse-scored

to the 16 negative items Only three negative items belonging to the somatic and retardedactivities symptoms subscale (ie lsquolsquobotheredrsquorsquo lsquolsquosleeprsquorsquo and lsquolsquotalkedrsquorsquo) did not show theU-shaped pattern lsquolsquoBotheredrsquorsquo lsquolsquosleeprsquorsquo and lsquolsquotalkedrsquorsquo were the lowest during 12ndash19 yearsand highest during 80ndash89 years

In contrast to the negative items the relationship between age and positive affectsymptom items were mild and differed from each other lsquolsquoGoodrsquorsquo was the highest during60ndash69 years lsquolsquohappyrsquorsquo was the highest during 70ndash79 years and lsquolsquohopefulrsquorsquo and lsquolsquoenjoyedrsquorsquowere the highest during 80ndash89 years

Confirming that the distribution of 16 negative items follow themathematical modelTo confirm that they followed the mathematical model the distributions of the 16 negativeitems were evaluated among each group (12ndash19 20ndash29 30ndash39 40ndash49 50ndash59 60ndash69 70ndash79and 80ndash89 years) (Figs 2Andash2H)

The distribution of the 16 items showed a common pattern across all age groups(Fig 2) The lines for the 16 items seemed to intersect at a single point between lsquolsquorarelyrsquorsquoand lsquolsquosomersquorsquo across all age groups As previously reported all the lines that follow this

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 718

Figure 2 The distributions of 16 negative symptoms among each group (A) 12ndash19 (B) 20ndash29 (C) 30ndash39 (D) 40ndash49 (E) 50ndash59 (F) 60ndash69 (G)70ndash79 and (H) 80ndash89 years Red arrows indicate the probability of lsquolsquomostrsquorsquo and blue arrows indicate the probability of lsquolsquosomersquorsquo Although the distri-butions for each of the 16 items followed a common mathematical pattern across all age groups the graphs of the distributions appeared to changewith age

mathematical model theoretically cross at a single point between lsquolsquorarelyrsquorsquo and lsquolsquosomersquorsquo(Tomitaka Kawasaki amp Furukawa 2015b) Conversely the lines for lsquolsquosomersquorsquo to lsquolsquomostrsquorsquoresponses were regularly skewed towards lsquolsquosomersquorsquo

Although the distributions for each of the 16 items followed the common mathematicalmodel across all age groups the graphs of the distributions appeared to change with ageThis change in the distribution with age was examined by focusing on lsquolsquosomersquorsquo and lsquolsquomostrsquorsquoresponses As the red arrows indicate the frequencies of lsquolsquomostrsquorsquo response for 16 items weredecreasing during 12ndash39 years (Figs 2Andash2C) stable during 40ndash69 years (Figs 2Dndash2F) andthen increasing again during 70ndash89 years (Figs 2G and 2H) Conversely as the blue arrowsindicate the frequencies of lsquolsquosomersquorsquo response for 16 items were slightly increasing during

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 818

12ndash39 years (Figs 2Andash2C) decreasing during 40ndash69 years (Figs 2Dndash2F) and were unclearduring 70ndash89 years (Figs 2G and 2H) It is necessary to quantify the age-related changesin the distributions of 16 negative items in order to evaluate the variations precisely

With a log-normal scale the distribution of the 16 negative symptom items for lsquolsquosomersquorsquoto lsquolsquomostrsquorsquo responses showed a parallel linear pattern across all age groups suggesting thatthey followed an exponential pattern with the same parameter r (Fig 3) Upon examiningby age groups the slopes of the lines for 16 negative items with a log-normal scale wereseen to change according to age (12ndash19 20ndash29 30ndash39 40ndash49 50ndash59 60ndash69 70ndash79 and80ndash89 years ) (Figs 3Andash3H) The slopes were more horizontal during 70ndash89 years (3G and3H) than during 30ndash69 years (Figs 3Cndash3F)

Quantification of age-related changes in the distributions of 16negative itemsThe parameters r and P were estimated from empirical data to quantify the age-relatedchanges in the distributions of 16 negative items Figure 4 shows that the average of theestimated parameter r exhibited a U-shaped pattern being high during 12ndash29 yearsstaying low during 30ndash59 years and then increasing again during 60ndash89 years similar tothe U-shaped pattern of total CES-D score (Fig 1) In contrast the mean of the estimatedparameter P was stable across all age groups as compared with the parameter r The averageof estimated parameter P slightly increased during 12ndash49 years slightly decreased during50ndash79 years and then increased again during 80ndash89 years

Comparison of the Likert scoring and binary scoringFinally to demonstrate the effects of scoring methods in adulthood total CES-D scoreand 16 item CESD scores were compared between the standard Likert (0123) andbinary methods (0-1-1-1) (Fig 5) The total CES-D score and16 item CESD scores inthe binary methods were estimated from empirical data We analyzed the respondentsbetween12ndash79 years as the older participants recruited by other studies were mainly under80 years In the standard Likert method the total CES-D score and total 16ndashitem scoreshowed a U-shaped pattern (Fig 5A) In contrast in the binary method the estimatedtotal CES-D score and total 16-item score showed a downward trajectory with age (Fig5B) The total CES-D score and total 16-item score were higher in the 70ndash79 year groupthan the 30ndash59 year group in the Likert method but lower in the 70ndash79 year group thanthe 30ndash59 year group in the binary method

DISCUSSIONThe aim of the present study was to delineate the age-related changes in the distributionsof each item score and to test whether the trajectory of depressive symptoms varies withthe scoring method

The main findings of this study are that (1) the U-shaped pattern of total CES-D scoreacross adulthood is mainly attributed to the age-related changes of 16 negative symptoms(2) the mathematical model fits the distributions of 16 negative items across the adult spanand (3) the estimated parameter r of 16 negative items showed a U-shaped pattern being

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 918

Figure 3 The distributions of the 16 items for lsquolsquosomersquorsquo to lsquolsquomostrsquorsquo responses by age group using a log-normal scale The distributions of the 16items for lsquolsquosomersquorsquo to lsquolsquomostrsquorsquo responses by age group (A) 12ndash19 (B) 20ndash29 (C) 30ndash39 (D) 40ndash49 (E) 50ndash59 (F) 60ndash69 G) 70ndash79 and (H) 80ndash89years The items in this scale use the same color scheme as in Fig 2 While the distribution of the 16 negative symptom items for lsquolsquosomersquorsquo to lsquolsquomostrsquorsquoresponses showed a parallel linear pattern across all age groups the slopes of the lines for 16 negative items with a log-normal scale were seen tochange according to age

high during 12ndash29 years staying low during 30ndash59 years and then increasing again during60ndash89 years

The trajectory of depressive symptoms across adulthood could be explained by theaforementioned findings Theoretically the expected value of each negative item score isPtimes (3r2+2r+1) with the Likert method (0-1-2-3) and Ptimes (r2+ r+1) with the binarymethod (0-1-1-1) Because expected values include the square of r and the changes ofparameter r across the adult lifespan are more intense than that of parameter P thetrajectory of depressive symptoms are mostly affected by parameter r resulting in thesimilar U-shaped patterns

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 1018

Figure 4 The relationships between age and estimated parameters P and r Red graph indicate the theaverage of the estimated parameter r Blue graph indicate the probability of lsquolsquosomersquorsquo The average of the es-timated parameter r exhibited a U-shaped pattern being high during 12ndash29 years staying low during 30ndash59 years and then increasing again during 60ndash89 years In contrast the average of parameter P slightly in-creased during 12ndash39 years decreased during 40ndash79 years and then slightly increased again during 80ndash89years

Our findings indicate that the increase of depressive symptoms for older age is basedon the increase in parameter r which corresponds to the increase of the probabilitiesof lsquolsquomuchrsquorsquo and lsquolsquomostrsquorsquo suggesting that aging tends to induce more severe depressivesymptoms However this explanation conflicts with the fact that most epidemiologicalevidence suggests that major depressive disorders decline with advancing age (Kessleret al 2010) Generally there is the discrepancy between the fact that rates of clinicaldepression are highest in midlife whereas depressive symptom screening scale scores(using Likert method) are highest in older age (Kessler et al 1992) It remains unclearwhether people are more depressed with aging One possible explanation is that theremay be an age-related change in self-rating for the severity of depressive symptoms Ithas been found that older individuals tend to have problems with retrieving informationthat is highly specific because they are less effective at controlling their retrieval process(Luo amp Craik 2009) Further studies are necessary to clarify the fact that depressivesymptom screening scale scores are highest in older age

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 1118

Figure 5 The relationships between age and the total scores of 20 items and 16 negative items using theLikert and binary methods (A) In the standard Likert method the total CES-D score and total 16ndashitemscore showed a U-shaped pattern (B) In the binary method the total CES-D score and total 16-item scoreshowed a downward trajectory with age

Although the findings of the present paper cannot explain the discrepancy betweenthe rates of clinical depression and depressive symptom screening scale scores they couldexplain the discrepancy between the Likert method and the binary method Using thebinary method the total CES-D and 16-item scores were lower in the 70ndash79 age group thanin the 30ndash59 age group (Fig 5B) This finding could be explained as follows First althoughthe parameter r increases during 60ndash89 years the expected value using the binary methoddoes not reflect the increase in parameter r as much as the Likert method Secondly theparameter P decreases during 50ndash79 years being lowest during 70ndash79 years Thus theCES-D score using binary method exhibits lower scores during 70ndash79 years than during30ndash59 years (Fig 5) Therefore our studies support the hypothesis that the trajectory ofdepressive symptoms across the adult lifespan varies with the scoring method

In addition to the scoring method there is another methodological factor which may beresponsible for the inconsistent evidence of the trajectory of depressive symptoms Whilethe total depressive symptom score follows a U-shaped pattern across adulthood theage-related changes in depressive symptoms seem relatively mild (Sutin et al 2013) Theestimated average change per decade in CES-D total score is less than one point betweenthe ages of 20 and 79 years while the standard deviation of the CES-D scores in communitysurveys is between 5 and 10 points (Kessler et al 1992 Hek et al 2013 Oh et al 2013)The mean total CES-D scores appear to stabilize between the ages of 30 and 69 years

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 1218

in particular As Kessler et al (1992) pointed out a number of studies have worked withsamples having a truncated age range a small number of very old respondents (older than75 years) or a measure of age that combines all respondents over the age of 65 into a singlegroup These truncated age ranges may cause the researcher to overlook the non-linearincrease in depressive symptoms that occurs after the age of 70 (Newmann 1989 Kessleret al 1992)

While 13 of the 16 negative symptom items exhibit U-shaped patterns across the adultlifespan the remaining three negative items belonging to the somatic and retarded activitiessymptoms subscale do not exhibit this pattern lsquolsquoBotheredrsquorsquo lsquolsquosleeprsquorsquo and lsquolsquotalkedrsquorsquo are thelowest during 12ndash19 years instead of 30ndash69 years and the highest during 80ndash89 years Theseresults are congruent with previous reports that some items of the somatic and retardedactivities symptoms subscale exhibit the lowest scores during young adulthood and thehighest scores during older age (Berry Storandt amp Coyne 1984 Hegeman et al 2012)

In comparison with the 16 negative symptom items the age-related changes in fourpositive affect symptom items were mild and differed from each other in the present studyMost of the studies report that the trajectories of positive affect symptom items differ fromthose of negative affect symptoms although the evidence for trajectories of positive affectssymptoms has been somewhat mixed (Kunzmann 2008) While the age-related changes inpositive affect items were mild the total scores of 4 positive item scores increased during30ndash89 years This finding is in line with Carstensenrsquos report that emotional well-beingmildly improves from early adulthood to old age (Carstensen et al 2000)

Recently because of their relative independence positive affect and negative affecthave been commonly recognized as two different phenomena that should be studiedindividually (Lucas Diener amp Suh 1996) A number of cross-cultural comparison studieshave reported that the response patterns for the positive affects items vary accordingto ethnicity or nation (eg skewed plateau-shaped U-shaped and reverse U-shaped)whereas the response patterns for the 16 negative items were generally comparable (IwataRoberts amp Kawakami 1995 Iwata et al 1998) Although the CES-D score is the compositescore of the 20-item scores it could be appropriate to recognize the 16 negative item scoresand the positive scores as different scores (Tomitaka Kawasaki amp Furukawa 2015b)

The present paper indicates that the mathematical model fits the distributions of 16negative items across the adult span The conditions that enable such a commondistributioncan be speculated upon In general self-rating depression scales are used as follows Firsteach subject must determine whether a symptom is present If the level of the symptomdoes not meet the threshold which excludes everyday mild symptoms it is regarded aslsquolsquorarely (less than 1 day)rsquorsquo Next if a depressive symptom that meets the threshold is presentthe duration of the symptom is quantified and divided into lsquolsquosome (1ndash2 days)rsquorsquo lsquolsquomuch (3ndash4days)rsquorsquo and lsquolsquomost (5ndash7 days)rsquorsquo This two-step process increases the possibility that lsquolsquorarelyrsquorsquowill cover the range that does not meet the threshold while each of lsquolsquosomersquorsquo lsquolsquomuchrsquorsquo andlsquolsquomostrsquorsquo will cover the almost fixed ratio of the range that satisfies the threshold If whatwe call the latent trait of depressive symptoms could be exponential distribution whichhas the memoryless property the 16 negative symptom items for CES-D would commonlyexhibit the mathematical distribution that was observed in the present study In general

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 1318

exponential distribution is observed where individual variability and total stability areorganized together such as the BoltzmannndashGibbs law and income distribution (Dragulescuamp Yakovenko 2000) With respect to individual variability and total stability the conditionsthat enable exponential distribution could be present in the distributions of the 16 negativeitems Further consideration regarding this speculation is needed

There are somemethodological advantages in the present investigation First the samplewas a representative of the general Japanese population which reduced selection bias Thelarge sample size (N = 21040) enabled us to elucidate the patterns of the distributions ofdepressive symptom items

Second the mathematical model was used to help understand the age-related changesin the distribution of depressive symptoms Because these distributions are completelydifferent fromnormal distribution the parameters of normal distribution (eg average andstandard deviation) were not sufficient to express the distributions themselves Afterassessing the fit of the mathematical model to the distributions of depressive symptoms weutilized it to quantify the age-related changes in the distributions of depressive symptomsTo the best of our knowledge mathematical modeling is still not well developed in the fieldof psychiatry While depressive symptoms of individuals are difficult to predict the entirepopulation may follow a certain mathematical pattern (Tomitaka Kawasaki amp Furukawa2015a)

This study has some limitations First a standard psychiatric diagnosis with structuredinterview was not performed for the subjects in this study Therefore the study did notencompass a psychiatric diagnosis of depressive symptoms Second although we evaluatedthe fit of the mathematical model to the distributions of depressive symptoms analysisbased on other mathematical models was not performed in this study In general the mostimportant part of model evaluation is testing whether the model fits empirical data betterthan other models However to the best of our knowledge no other mathematical modelsfor the distributions of depressive symptoms have been reported so far Despite theselimitations the present study provides important information regarding the age-relatedchanges in depressive symptoms

CONCLUSIONOur results indicate that the scoring method of depressive symptoms is important inevaluating the age-related changes in depressive symptoms The proposed mathematicalmodel fits the distributions of depressive symptoms across adulthood raising the possibilitythat depressive symptoms in the general population follow a mathematical pattern as awhole

ACKNOWLEDGEMENTSThe authors would like to thank the Active Survey of Health and Welfare project forproviding the data for this study Dr Shinji Sakamoto for helpful advice and Enago(wwwenagojp) for the English language review

Tomitaka et al (2016) PeerJ DOI 107717peerj1547 1418

ADDITIONAL INFORMATION AND DECLARATIONS

FundingThe authors received no funding for this work

Competing InterestsThe authors declare there are no competing interests

Author Contributionsbull Shinichiro Tomitaka conceived and designed the experiments performed theexperiments analyzed the data wrote the paper prepared figures andor tables revieweddrafts of the paperbull Yohei Kawasaki Kazuki Ide Hiroshi Yamada Toshiaki A Furukawa and Yutaka Onoreviewed drafts of the paper

Human EthicsThe following information was supplied relating to ethical approvals (ie approving bodyand any reference numbers)

The present study was approved in 2014 by the ethics committee of Panasonic HealthCenter (approval number 2014-1) The authors assert that all procedures contributingto this work comply with the ethical standards of the relevant national and institutionalcommittees on human experimentation and with the Helsinki Declaration of 1975 asrevised in 2008

Data AvailabilityThe following information was supplied regarding data availability

The present study used data from the Active Survey of Health and Welfare (ASHW)conducted by the Japanese Ministry of Health Labor and Welfare in 2000 however theJapanese Ministry of Health Labor and Welfare does not allow the publication of the data

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