Mark Kac gave one of the first results analyzing random polynomial zeros. He considered the case of independent standard normal coefficients and was able to show that the expected number of real zeros for a degree n polynomial is on the order of 2 π log n, as n → ∞. Several years later, Sambandham considered two cases with some dependence assumed among the coefficients. The first case looked at coefficients with an exponentially decaying covariance function, while the second assumed a constant covariance. He showed that the expectation of the number of real zeros for an exponentially decaying covariance matches the independent case, while having a constant covariance reduces the expected number of zeros in half. In this paper we will apply techniques similar to Sambandham's and extend his results to a wider class of covariance functions. Under certain restrictions on the spectral density, we will show that the order of the expected number of real zeros remains the same as in the independent case.

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Many areas of educational research require the analysis of data that have an inherent sequential or temporal ordering. In certain cases, researchers are specifically interested in the transitions between different states-or events-in these sequences, with the goal being to understand the significance of these transitions; one notable example is the study of affect dynamics, which aims to identify important transitions between affective states. Unfortunately, a recent study has revealed a statistical bias with several metrics used to measure and compare these transitions, possibly causing these metrics to return unexpected and inflated values. This issue then causes extra difficulties when interpreting the results of these transition metrics. Building on this previous work, in this study we look in more detail at the specific mechanisms that are responsible for the bias with these metrics. After giving a theoretical explanation for the issue, we present an alternative procedure that attempts to address the problem with the use of marginal models. We then analyze the effectiveness of this procedure, both by running simulations and by applying it to actual student data. The results indicate that the marginal model procedure seemingly compensates for the bias observed in other transition metrics, thus resulting in more accurate estimates of the significance of transitions between states.

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