observedexpected

Observedexpected is a descriptive term used in statistics and data analysis to describe the comparison between observed data and the number of events expected under a specified model or null hypothesis. It is not a single method, but a framework in which the fit of a model is assessed by contrasting what was actually observed with what the model predicts.

In practice, for categorical data, each category i has an observed count O_i and an expected count E_i under the null hypothesis. A common summary is the chi-square statistic, X^2 = sum over i of (O_i − E_i)^2 / E_i. The observed-to-expected ratio, O_i / E_i, provides a per-category measure of deviation: values greater than 1 indicate more observations than expected, while values less than 1 indicate fewer. Significant overall deviations lead to small p-values and questions about the model’s adequacy.

Observedexpected comparisons are central in goodness-of-fit tests, contingency table analysis, and disease risk assessments where observed incidence or counts are compared with standardized or theoretically derived expectations. In genetics, expected genotype or allele frequencies under Hardy-Weinberg equilibrium are compared with observed frequencies to detect deviations.

Interpretation requires attention to the scale and assumptions. Large deviations can reflect model misspecification, data quality issues, or true underlying effects. Cautions include the sensitivity to sample size and the requirement that expected counts be sufficiently large for the chi-square approximation to be valid; with small expected counts, exact tests or category grouping may be more appropriate.

Related concepts include the observed-to-expected ratio, standardized incidence or mortality ratios, and residual analysis in regression. Observedexpected comparisons provide a straightforward lens to evaluate how well data align with theoretical expectations.