  Items in a test are often used as a basis for making decisions and such tests
are therefore required to have good psychometric properties, like
unidimensionality. In many cases the sum score is used in combination with a
threshold to decide between pass or fail, for instance. Here we consider
whether such a decision function is appropriate, without a latent variable
model, and which properties of a decision function are desirable. We consider
reliability (stability) of the decision function, i.e., does the decision
change upon perturbations, or changes in a fraction of the outcomes of the
items (measurement error). We are concerned with questions of whether the sum
score is the best way to aggregate the items, and if so why. We use ideas from
test theory, social choice theory, graphical models, computer science and
probability theory to answer these questions. We conclude that a weighted sum
score has desirable properties that (i) fit with test theory and is observable
(similar to a condition like conditional association), (ii) has the property
that a decision is stable (reliable), and (iii) satisfies Rousseau's criterion
that the input should match the decision. We use Fourier analysis of Boolean
functions to investigate whether a decision function is stable and to figure
out which (set of) items has proportionally too large an influence on the
decision. To apply these techniques we invoke ideas from graphical models and
use a pseudo-likelihood factorisation of the probability distribution.
