The co-occurrence of self-observed norm-conforming behavior, reduction of zero observations and remaining measurement quality

Abstract

Norm-violating behavior is characterized by clear social norms which prescribe the non-occurrence of that behavior. From the theoretical framework of Allport it is derived that specifically norm-conformation is consistent, while violating norms is expected to be inconsistent and more circumstantial. This is in contrast to test-theoretic approaches of delinquent behavior that assume that various norm-violating responses form a consistent answer pattern that is scalable and reliable. In this study we study the inter-correlations, scalability and reliability of norm-violating responses and their relation with the reduction of zero observations. In concordance with Allport’s view it is expected that different norm-violating self-report items have limited interrelatedness and are limited in scalability and reliability in the norm-violating sub-population. The NLSY98 self-report data show that a large majority of respondents (69 %) conform systematically to all ten different norms, while only nine percent admits more than two different violations. The results show that in subsamples of norm-violating respondents, the correlations between items become closer to zero, dependent on the amount of zero reduction. Furthermore, both Loevinger’s H coefficient of scalability and scale reliability become unsatisfactorily low, when 35 % or more strict norm-conforming subjects are removed.

Appendix

The relationship between mean, variance, covariance and correlation with and without a proportion zero-inflation of $\pi$.

Let there be a zero-inflated variable with $n$ observations, a proportion zero-inflation of $\pi$ and an underlying not inflated distribution with $\tilde{n}$ observations, mean $\tilde{\mu }$ and variance $\tilde{\sigma }^{2}$.

$$\begin{aligned} \mu =\frac{1}{n}\sum y =\frac{1}{n}\sum {\tilde{y}} =\frac{\tilde{n}}{n}\tilde{\mu }=(1-\pi )\tilde{\mu } \end{aligned}$$

Derivation of formula (3)

The variance of a zero-inflated variable $y$ with $n$ observations and mean $\mu$ is:

$$\begin{aligned} \sigma ^{2}&= \frac{1}{n}\sum {{y}^{2}-\mu }^{2} \\ \sigma ^{2}&= \frac{\tilde{n}}{n}\left( \frac{1}{\tilde{n}}\sum {\tilde{y}^{2}} -\frac{\tilde{n}}{n}\tilde{\mu }^{2}\right) \\ \sigma ^{2}&= \frac{\tilde{n}}{n}\left( \frac{1}{\tilde{n}}\sum {\tilde{y}^{2}} -\tilde{\mu }^{2}+\frac{n}{n}\tilde{\mu }^{2}-\frac{\tilde{n}}{n}\tilde{\mu }^{2}\right) \\ \sigma ^{2}&= (1-\pi )\left( \tilde{\sigma }^{2}+\frac{n-\tilde{n}}{n}\tilde{\mu }^{2}\right) =(1-\pi )(\tilde{\sigma }^{2}+\pi \tilde{\mu }^{2}) \end{aligned}$$

and

$$\begin{aligned} {\tilde{\sigma }^{2}} =\frac{\sigma ^{2}}{(1-\pi )}-\frac{\pi }{(1-\pi )^{2}}\tilde{\mu }^{2} \end{aligned}$$

The covariance formula can be derived in similar fashion.

$$\begin{aligned} \sigma _{xy} =(1-\pi )\tilde{\sigma }_{xy} +\pi (1-\pi )\tilde{\mu }_{x} \tilde{\mu }_{y}. \end{aligned}$$

and

$$\begin{aligned} \tilde{\sigma }_{xy} =\frac{\sigma _{xy}}{(1-\pi )}-\frac{\pi }{(1-\pi )^{2}}\tilde{\mu }_{x} \tilde{\mu }_{y}. \end{aligned}$$

The Pearson product moment correlation is

$$\begin{aligned} \tilde{\rho }_{xy} =\frac{\tilde{\sigma }_{xy}}{\tilde{\sigma }_{x} \tilde{\sigma }_{y}}. \end{aligned}$$