Happiness Scale Interval Study. Methodological Considerations.
Bottom Line: On the basis of these results, the mean value and variance of the complete distribution can be estimated.An illustration is given in which the method is applied to existing measurement results of 20 surveys in The Netherlands in the period 1990-2008.The results clarify our recommendation to apply the model with a uniform distribution within each of the category intervals, in spite of a better validity of the alternative on the basis of a beta distribution.
Affiliation: Erasmus University, Rotterdam, The Netherlands.
The Happiness Scale Interval Study deals with survey questions on happiness, using verbal response options, such as 'very happy' and 'pretty happy'. The aim is to estimate what degrees of happiness are denoted by such terms in different questions and languages. These degrees are expressed in numerical values on a continuous [0,10] scale, which are then used to compute 'transformed' means and standard deviations. Transforming scores on different questions to the same scale allows to broadening the World Database of Happiness considerably. The central purpose of the Happiness Scale Interval Study is to identify the happiness values at which respondents change their judgment from e.g. 'very happy' to 'pretty happy' or the reverse. This paper deals with the methodological/statistical aspects of this approach. The central question is always how to convert the frequencies at which the different possible responses to the same question given by a sample into information on the happiness distribution in the relevant population. The primary (cl)aim of this approach is to achieve this in a (more) valid way. To this end, a model is introduced that allows for dealing with happiness as a latent continuous random variable, in spite of the fact that it is measured as a discrete one. The [0,10] scale is partitioned in as many contiguous parts as the number of possible ratings in the primary scale sums up to. Any subject with a (self-perceived) happiness in the same subinterval is assumed to select the same response. For the probability density function of this happiness random variable, two options are discussed. The first one postulates a uniform distribution within each of the different subintervals of the [0,10] scale. On the basis of these results, the mean value and variance of the complete distribution can be estimated. The method is described, including the precision of the estimates obtained in this way. The second option assumes the happiness distribution to be described as a beta distribution on the interval [0,10] with two shape parameters (α and β). From their estimates on the basis of the primary information, the mean value and the variance of the happiness distribution in the population can be estimated. An illustration is given in which the method is applied to existing measurement results of 20 surveys in The Netherlands in the period 1990-2008. The results clarify our recommendation to apply the model with a uniform distribution within each of the category intervals, in spite of a better validity of the alternative on the basis of a beta distribution. The reason is that the recommended model allows to construct a confidence interval for the true but unknown population happiness distribution. The paper ends with a listing of actual and potential merits of this approach, which has been described here for verbal happiness questions, but which is also applicable to phenomena which are measured along similar lines.
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Mentions: We will discuss three possible models, which have been represented in Fig. 3. Under the model described in Sect. 3, it is assumed that each respondent with a happiness feeling corresponding to any H-value in the interval (bj−1, bj] will respond as Rj. However, all we know is the number of respondents with Rj, but it is unknown which H-value in the interval (bj−1, bj] belongs to each of them. Therefore, we have to make assumptions on the unknown distribution of H over [0, 10], more precisely, over each of the k intervals ⊂ [0, 10]. The three models differ in these underlying assumptions.
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