Based on the output of the summary() for the model from Question 1, is there evidence of an association between nox and medv? Choose the correct interpretation.

Question

Based on the output of the summary() for the model from Question 1, is there evidence of an association between nox and medv? Choose the correct interpretation.

BlackTom AI · Accepted Answer

The question asks about interpreting the summary() output for a model and whether there is evidence of an association between nox and medv, with a focus on the significance of an interaction term nox:dis.

First, consider what an interaction term represents: nox:dis tests whether the effect of nox on medv changes depending on the level of dis. If this interaction is statistically significant, it suggests that the relationship between nox and medv is not constant across all values of dis, indicating a contextual association influenced by dis.

Now, evaluating the provided option: 'Yes, there is evidence of an association because the interaction term nox:dis is significant, indicating that the effect of nox on medv depends on dis.'
- This reasoning aligns with the standard interpretation of a significant interaction in a linear model: the effect of one predictor (nox) varies by levels of another predictor (dis), which constitutes an association between nox and medv that depends on dis.
- If the summary() output reports a statistically significant coefficient for nox:dis (e.g., p-value below the chosen alpha level), it supports the claim that the effect of nox on medv changes with dis, implying an interaction-driven association.

Consider potential caveats to avoid overgeneralization:
- A significant interaction does not necessarily imply a main effect is absent; both the main effects and the interaction term can be present and meaningful. One must interpret the main effects in the context of the interaction.
- Statistical significance depends on sample size and variability; a non-significant interaction would lead to claiming no evidence of interaction, even if the effect appears numerically different across groups.
- The interpretation hinges on the model specification: if the model includes other covariates or higher-order terms, the meaning of the interaction term may change, and one should examine diagnostic plots and plots of the interaction to understand the practical impact.

Next, consider what would weaken or falsify this interpretation:
- If nox:dis is non-significant in the summary, the claim of an interaction-based association would be unsupported, and one would conclude that the effect of nox on medv does not significantly depend on dis.
- If there is multicollinearity or model misspecification, the stability and interpretability of the interaction term could be compromised, calling for model re-evaluation.
- If the analysis uses a simplistic or inappropriate scale (e.g., log-transformations or categorization of continuous variables) without justification, the interpretation of the interaction might be misled.

Finally, compile the core takeaway:
- The option correctly identifies that a significant nox:dis interaction would provide evidence that the association between nox and medv varies with dis, which is a precise and standard interpretation of an interaction term in a regression model.
- Absence of an interaction term’s significance would mean the opposite: no evidence that the nox–medv relationship depends on dis.

In summary, relying on a significant nox:dis interaction to claim an association between nox and medv that is contingent on dis is a correct and typical interpretation of interaction effects in regression analysis.

SDS 324E Homework 10

Based on the output of the summary() for the model from Question 1, is there evidence of an association between nox and medv? Choose the correct interpretation.

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