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# Boundary curves of individual items in the distribution of total depressive symptom scores approximate an exponential pattern in a general population.

^{1}, Kawasaki Y

^{2}, Ide K

^{2}, Akutagawa M

^{2}, Yamada H

^{2}, Furukawa TA

^{3}, Ono Y

^{4}.

### Author information

- 1
- Department of Mental Health, Panasonic Health Center, Tokyo, Japan; Department of Drug Evaluation and Informatics, Graduate School of Pharmaceutical Sciences, University of Shizuoka, Shizuoka, Japan.
- 2
- Department of Drug Evaluation and Informatics, Graduate School of Pharmaceutical Sciences, University of Shizuoka , Shizuoka , Japan.
- 3
- Department of Health Promotion and Human Behavior, Department of Clinical Epidemiology/Graduate School of Medicine/School of Public Health, Kyoto University , Kyoto , Japan.
- 4
- Center for the Development of Cognitive Behavior Therapy Training , Tokyo , Japan.

### Abstract

#### BACKGROUND:

Previously, we proposed a model for ordinal scale scoring in which individual thresholds for each item constitute a distribution by each item. This lead us to hypothesize that the boundary curves of each depressive symptom score in the distribution of total depressive symptom scores follow a common mathematical model, which is expressed as the product of the frequency of the total depressive symptom scores and the probability of the cumulative distribution function of each item threshold. To verify this hypothesis, we investigated the boundary curves of the distribution of total depressive symptom scores in a general population.

#### METHODS:

Data collected from 21,040 subjects who had completed the Center for Epidemiologic Studies Depression Scale (CES-D) questionnaire as part of a national Japanese survey were analyzed. The CES-D consists of 20 items (16 negative items and four positive items). The boundary curves of adjacent item scores in the distribution of total depressive symptom scores for the 16 negative items were analyzed using log-normal scales and curve fitting.

#### RESULTS:

The boundary curves of adjacent item scores for a given symptom approximated a common linear pattern on a log normal scale. Curve fitting showed that an exponential fit had a markedly higher coefficient of determination than either linear or quadratic fits. With negative affect items, the gap between the total score curve and boundary curve continuously increased with increasing total depressive symptom scores on a log-normal scale, whereas the boundary curves of positive affect items, which are not considered manifest variables of the latent trait, did not exhibit such increases in this gap.

#### DISCUSSION:

The results of the present study support the hypothesis that the boundary curves of each depressive symptom score in the distribution of total depressive symptom scores commonly follow the predicted mathematical model, which was verified to approximate an exponential mathematical pattern.

#### KEYWORDS:

Boundary curve; CES-D; Depression; Depressive symptoms; Exponential distribution; Latent trait; Level of measurement; Likert scale; Logistic distribution; Ordinal scale

- PMID:
- 27761346
- PMCID:
- PMC5068372
- DOI:
- 10.7717/peerj.2566

### Conflict of interest statement

The authors declare that they have no competing interests.