Leverage Influential Point and Outlier: Diagnostics (2024)

In this post, a discussion about diagnostics for a Leverage Influential point and outlier will be made. In a regression analysis, certain observations may play a role in influencing the outcomes of the fitted model and its estimates. These observations may be classified as outliers, leverage, and influential points.

Outliers, Leverage, and Influential Observations

• Outliers: An outlier is an extreme observation that is considerably very different from the other observations. An outlier may be due to the recording error and they cannot be explained by the model. However, outlier(s) may contain some important information. An outlier may be in $x$-space, $y$-space, or even in both.
• Leverage: An unusual $x$ value is called a leverage point. The leverage point affects the model summary statistics (such as $R^2$, standard error, etc.), but has little impact on the estimates of the regression coefficients. A leverage point has an unusual predictor value and is different from the bulk of the observations.
• Influence: An unusual $y$ value (and may be an extreme $x$ value), is called an influence point. An influence point has a noticeable impact on the estimated regression coefficients and may change the direction of the slope.

Diagnostics for Outlier Leverage Influential Points

There are some methods to detect/ identify the outlier influential and leverage points

Outliers

Outliers must be treated very carefully. Outliers may be detected by examining the

• Normal Quantile Plots (departer from normality)
• Residual Plots (magnitude of the residuals)
• Scaled residuals (a potential outlier if magnitudes > 3)

Leverage Point

The diagonal elements of the “hat matrix” have an important role in detecting influential observations. $$h_{ii} = x’_i (X’X)^{-1}x_i,$$ where $X$ is matrix of regressors and $x’_i$ is the ith row of the $X$ matrix.

A large diagonal element is an indicator of influential observation as they are remote in $x$-space. Any observation exceeding the average size of the diagonal element of the hat matrix ($\overline{h} = \frac{p}{n}=2h$) is considered as a leverage point, where $p$ is the number of parameters in the model.
It is also useful to observe the studentized residuals in conjunction with $h_{ii}$ (that is, look for large hat diagonal and large residual values).

Note that not all of the leverage points are influential unless they have large residuals. Therefore, observations having large $h_{ii}$ values and large residuals are likely to be R.

Influential Points

• Cook’s Distance: The Cook’s Distance is the Deletion Diagnostic that is used to measure the influence of the $i$th observation by removing it from the regression analysis. It is based on all $n$ points, $\hat{\beta}, and the estimates based on the deletion of the$i$th point,$\hat{\beta}_{(i)}$. • DFBETAS is another Deletion Diagnostic used to measure how the change in each of the$\hat{\beta}j$is due to influential observation. A large value of DFBETAS indicates that the$i$th observation is considerably an influential observation on the$j$th regression coefficient. If$|DFBETAS{j, i} > \frac{2}{\sqrt{n}}$then the$i$th observation warrants further examination. • DFFITS is another deletion diagnostic measure used to measure the deletion influence of the$i$th observation on the predicted or fitted values. DFFITS is the number of standard deviations that the fitted values change if ith observations are removed. If$|DFFITS_i|>\frac{2}{\sqrt{\frac{p}{n}}}$then the$i\$th observation warrants further examination.

Note that the case deletion diagnostics do not provide any information about the overall prediction of the estimation. However, the performance of the model can be measured by using the Generalized Variance (GV) and Covariance Ratio.

In summary, the Outliers, Leverage Points, and Influential Observations are certain data points (observations) that deviate (distant) from the expected patterns. On the other hand, the outliers are extreme values that lie far away from the other data points, while leverage points exert a strong influence on the regression models.