Influential Observations, High Leverage Points, and Outliers in Linear Regression Samprit Chatterjee and Ali S. Hadi Abstract. Influence: An observation is said to be influential if removing the observation substantially changes the estimate of coefficients. However, rather than calling them x- or y-unusual observations, they are categorized as outlier, leverage, and influential points according to their impact on the regression model. Sample data: The points marked in red and blue are clearly not like the main cloud of the data points, even though their xand ycoordinates are quite typical of the data as a whole: the xcoordinates of those points aren’t related to the ycoordinates in the right way, they break a pattern. An influential point is an outlier that greatly affects the slope of the regression line. Thus for the ith point in the sample, where each h ij only depends on the x values in the sample. where: r i is the i th residual; p is the number of coefficients in the regression model; MSE is the mean squared error; h ii is the i th leverage value Q: The term "Freshman 15" is an expression commonly used in the United States that refers to the amount of weight gained during a student's first year at college. Leverage – By Property 1 of Method of Least Squares for Multiple Regression, Y-hat = HY where H is the n × n hat matrix = [h ij]. (1991) ‘Statistics’ refers to the percapita consumption of cigarettes in various countries in 1930 and the death rates (number of deaths per million people) from lung cancer for 1950. - have no effect of the regression coefficients as it lies on the same line passing through the remaining observations. A bewilderingly large number of statistical quantities have been proposed to study outliers and influence of individual observations in regression analysis. While the high leverage observation corresponding to Bobby Scales in the previous exercise is influential, the three observations for players with OBP and SLG values of 0 are not influential. The following statements use the population example in the section Polynomial Regression. We want the model to be a representative of the whole population. Sometimes a small group of influential points can have an unduly large impact on the fit of the model. But it's something that's very strongly changing the data set. Points with large residuals are potential outliers. C) (10 Points) Additional Diagnostic Plots For The Transformed Regression In Question 4 Are Included On The Following Two Pages. The formula for Cook’s distance is: D i = (r i 2 / p*MSE) * (h ii / (1-h ii) 2). Second, points with high leverage may be influential: that is, deleting them would change the model a lot. Influential points in simple linear regression are points that, when removed from the calculation, cause a ‘great’ change in the regression line.The term ‘influential points’ is typically applied when assessing outliers.Influential points tipically have high leverage (extreme in X) and/or high residual (extreme in Y). An example of a low leverage point would be pushing on the side of a ship to change its course. Viewed 518 times 2 $\begingroup$ Do we look at the absolute value of the leverage or the relative value? Points with a large residual and high leverage have the most influence. A) (6 Points) Briefly Describe Each Of: Outliers, Leverage, And Influential Points. To simulate a linear regression dataset, we generate the explanatory variable by randomly choosing 20 points between 0 and 5. This is because they happen to lie right near the regression anyway. In the following figure Xi yi A the point A - will have a large hat diagonal and is surely a leverage point. Simulated Data. Cook’s distance was introduced by American statistician R Dennis Cook in 1977. The fact that an observation is an outlier or has high leverage is not necessarily a problem in regression. Including data points like C generally leads to more precise estimates of the slope and intercept, and such data points are also called good leverage points (Rousseeuw and Leroy 1987:63; Wilcox 2001: pp. Neither plot suggests concerns relative to influential points or multicollinearity. Cook’s distance, often denoted D i, is used in regression analysis to identify influential data points that may negatively affect your regression model.. In this article we describe the inter-relationships which For example, an observation with a value equal to the mean on the predictor variable has no influence on the slope of the regression line regardless of its value on the criterion variable. This point is prepended to the 100 points generated earlier. ; Know how to detect potentially influential data points by way of DFFITS and Cook's distance. Therefore it is important to identify the data points which impact the model significantly. So it could change the mean. Outlier, Leverage, and Influential Points An observation could be unusual with respect to its y-value or x-value. One way to test the influence of an outlier is to compute the regression equation with and without the outlier. It could change the slope of the regression line, which we'll learn about a little bit later. There is a wide and somewhat confusing range of measures for detecting influential points, and a good summary of what is available is given by Chatterjee and Hadi [25] and the ensuing discussion.Some measures highlight problems with y (outliers), others highlight problems with the x-variables (high leverage), while some focus on both. B) (4 Points) Are All Outliers Influential? Specifically I want to remove studentized residuals larger than 3 and data points with cooks D > 4/n. The DFFITS statistic is a measure of how the predicted value at the i_th observation changes when the i_th observation is deleted. Influential Points. This would require a large amount of force to have the intended effect. It might be obvious that influential observations are typically also leverage points. Experts answer in as little as 30 minutes. How could I perform that in the sample data and do the same analysi swithout the influential points? This simple Shiny App demonstrates the concepts of leverage and influence, displays the linear model coefficients and some of the influence measures for a point with adjustable coordinates. Influential points vs Outliers. Outliers, leverage and influential data points In general, unusual data points will impact the model and need to be identified. These leverage points can have an effect on the estimate of regression coefficients. Outliers, Leverage Points and Influential Points. Influence¶. 1 Outliers Are Data Points Which Break a Pat-tern Consider Figure 1. Leverage - influential points. Cook’s distance is the dotted red line here, and points outside the dotted line have high influence. A common measure of influence is Cook’s Distance, which is defined as \[ D_i = \frac{1}{p}r_i^2\frac{h_i}{1-{h_i}}. Outliers, Leverage & Influential points in regression A famous data set found in Freedman et al. I want to identify data points with high leverage and large residuals. For this we can look at Cook’s distance, which measures the effect of deleting a point on the combined parameter vector. The influence of a point is a combination its leverage and its discrepancy. My aim is to remove them and repeat linear regression analyses. But if the high leverage point of pushing on the rudder is used instead, it takes only a small amount of force to achieve the same effect.. Easy problems can be solved by pushing on low leverage points. All leverage points are not influential on the regression coefficients. The greater an observation's leverage, the more potential it has to be an influential observation. It is used to identify influential data points. 4.11.4. Briefly Justify Your Answer. Question: [20 Points] Answer The Following Questions. \] Notice that this is a function of both leverage … Leverage is a measure of how far an observation deviates from the mean of that variable. ; Understand leverage, and know how to detect extreme x values using leverages. Observations that fall into the latter category, points with (some combination of) high leverage and large residual, we will call influential. Know how to detect outlying y values by way of standardized residuals or studentized residuals. Figure 3.58 Whole Model and Effect Leverage Plots The Leverage Plot for height, on the right, also shows that height is significant, even with age and sex in the model. The influence of each data point can be quantified by seeing how much the model changes when we omit that data point. Cook’s D measures how much the model coefficient estimates would change if an observation were to be removed from the data set. Q&A related to Outliers And Influential Points. Leverage, outliers, and influence •Leverage: measures how far away x iis from the other xvalues [goes from 0 to 1, from “average x” to “very unusual x”] •High leverage: unusual value of x i, which may or may not be well predicted by our line High-leverage points tend to pull the regression surface towards the response at that point, so the change in the predicted value at that point is a good indication of how influential the observation is. Bar Plot of Cook’s distance to detect observations that strongly influence fitted values of the model. Practice thinking about how influential points can impact a least-squares regression line and what makes a point “influential.” In model A, the square point had large discrepancy but low leverage, so its influence on the model parameters (slope and intercept) was small. 218–19, 2005: 417). Ask Question Asked 6 years, 1 month ago. The scatterplots are identical, except that one plot includes an outlier. ... A statistic referred to as Cook’s D, or Cook’s Distance, helps us identify influential points. Belsley, Kuh, and Welsch (1980) recommend 2 as a general cutoff value to indicate influential observations and as a size-adjusted cutoff. ... h or leverage is a measure of distance between x value of i-th data point and mean of x values for all n data points. And, when detected as outliers and influential points, to investigate and eliminate their effect in the fitted model, analytic procedures; leverage value, studentized residuals and cook's distance This type of analysis is illustrated below. Keywords Influence leverage outliers regression diagnostics residuals Citation Chatterjee, Samprit; Hadi, Ali S. Influential Observations, High Leverage Points, and Outliers in Linear Regression. They can have an adverse effect on (perturb) the model if they are changed or excluded, making the model less robust. Activate the analysis report worksheet. Active 4 years, 5 months ago. Then you can see how the regression line is affected and how the displayed values change. Not all points of high leverage are influential. Identifying outliers and other influential points Plot measures to identify cases with large outliers, high leverage, or major influence on the fitted model. In general, large values of DFBETAS indicate observations that are influential in estimating a given parameter. Key Learning Goals for this Lesson: Understand the concept of an influential data point. Influential points are points that when removed significantly change a statistical measure.