DFFITSi

DFFITSi, often written as DFFITS_i, is a regression diagnostic statistic used to assess the influence of a single observation on the fitted values of a regression model. It measures how much the predicted response for observation i would change if that observation were omitted from the analysis.

Calculation and interpretation: DFFITSi is computed from the difference between the fitted value with the full data and the fitted value with observation i removed, scaled by an estimate of the model’s variability and the observation’s leverage. A common form is DFFITS_i = (ŷ_i − ŷ_i(−i)) / (s_(−i) sqrt(h_ii)), where ŷ_i is the fitted value with all data, ŷ_i(−i) is the fitted value with i removed, h_ii is the i-th diagonal element of the hat matrix (the leverage), and s_(−i) is the standard error from the model without observation i.

Thresholds and usage: Large absolute values indicate influential observations that disproportionately affect fitted values. A typical rule of thumb for identifying influential points is |DFFITS_i| > 2 sqrt(p/n), where p is the number of estimated parameters and n is the sample size. DFFITSi is often examined alongside other influence measures such as Cook’s distance and DFBETAS.

Relation to other diagnostics: DFFITSi complements other regression diagnostics by focusing on the impact of individual observations on predicted values, rather than solely on residuals or parameter estimates. It is most informative when used as part of a broader diagnostic assessment.

Limitations: The measure can be sensitive to data with high leverage and to model specification. Therefore, DFFITSi should be interpreted in context, considering other diagnostics and substantive knowledge about the data and model.