Past exam of the experimental design and methods course of the year 2 of the psychology (experimental) course of the University of Oxford / 2022 / 5

TODO: the supplementary model call and tables are needed to identify the actual analyses. The described categorical predictors are compatible with a multiple linear regression containing a distance factor, a size factor and possibly their interaction; a factorial analysis of variance can represent the same mean structure. The mention of a coefficient table does not establish the exact fitted terms, predictor coding, estimation method or dependence assumptions. No particular fitted model is asserted.

TODO: the model's explained-variance value is absent. For an ordinary least-squares linear regression with an intercept, the coefficient of determination is $R^2=1-S{S}_{\mathrm{residual}}/S{S}_{\mathrm{total}}$. Its value multiplied by one hundred is the corresponding percentage of sample variation explained by that fitted model. Read the actual reported $R^2$ and distinguish it from the adjusted coefficient of determination; neither is a verified prediction-performance estimate on new data. Without the missing summary or sums of squares, no numerical percentage can be recovered.

TODO: the coefficients and their numerical $t$ and $p$ entries cannot be recovered. The coefficient estimate gives a modeled difference relative to the stated reference coding; its standard error quantifies uncertainty, and the usual coefficient test uses $t=\hat{\beta }/\mathrm{SE}(\hat{\beta })$ for a zero-coefficient null hypothesis. The p-value requires the appropriate reference distribution and statistical degrees of freedom. In a conventional R coefficient table an Estimate is usually an unstandardized regression coefficient, not automatically a standardized regression coefficient; the table and model documentation must establish which meaning of $\beta$ is intended. These definitions supply no missing numerical values.

TODO: directions, magnitudes and inferential conclusions require the unavailable coefficient table, coding and model specification. For illustration of interpretation only, let $D$ and $S$ be binary indicators whose reference levels are explicitly defined. If an ordinary linear model includes both factors and their interaction, its conditional mean has the formThen the distance difference is ${\beta }_D$ at the reference size and ${\beta }_D+{\beta }_{DS}$ at the other size; the interaction is the difference between those differences. If the interaction is omitted, the model assumes a common distance difference across sizes. These are conditional model meanings, not a claim that the missing analysis used this model or that either predictor had an observed effect. The uncertainty of a coefficient sum also involves their covariance, not merely their separate p-values.

TODO: no definite follow-up decision can be made without the fitted model, uncertainty and diagnostics. An included interaction can be clarified with simple-effects analysis, fitted cell summaries and confidence intervals on scientifically relevant differences. If the study aims to test whether the predictors combine additively but only an additive model was fitted, an adequately supported, prespecified interaction comparison may be relevant. Respect multiple testing when several comparisons are examined.

Check model fit, influential observations and heteroscedasticity; a response bounded between zero and one thousand may require particular care near its bounds, but that scale alone does not choose a replacement model. If observations are dependent, the analysis must account for that structure. Such checks are possible recommendations; the missing supplementary sheet prevents identifying which problems, if any, actually occurred.

TODO: an empirical Results section requires the missing output and cannot be supplied honestly here. A completed report would name the actual analysis, define the predictor reference levels and outcome units, summarize the relevant cells with sample counts and uncertainty, give overall model-fit information and the actual $R^2$ or other appropriate fit measure, and report each fitted coefficient with its standard error or confidence interval, test statistic and p-value. If an interaction was fitted, interpret conditional differences rather than treating reference-level coefficients as unconditional effects. State relevant regression diagnostics, departures from the planned analysis and any controlled follow-up comparisons. No observed coefficient, significance result or predicted value is inferred from the verbal scenario.