# leverage and influential points

Influence¶. These leverage points can have an effect on the estimate of regression coefficients. Influential points vs Outliers. Keywords Influence leverage outliers regression diagnostics residuals Citation Chatterjee, Samprit; Hadi, Ali S. Influential Observations, High Leverage Points, and Outliers in Linear Regression. Identifying outliers and other influential points Plot measures to identify cases with large outliers, high leverage, or major influence on the fitted model. Outliers, Leverage Points and Influential 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 4.11.4. Specifically I want to remove studentized residuals larger than 3 and data points with cooks D > 4/n. ; Know how to detect potentially influential data points by way of DFFITS and Cook's distance. 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. The DFFITS statistic is a measure of how the predicted value at the i_th observation changes when the i_th observation is deleted. 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. A common measure of influence is Cookâs Distance, which is defined as $D_i = \frac{1}{p}r_i^2\frac{h_i}{1-{h_i}}. Active 4 years, 5 months ago. Viewed 518 times 2 \begingroup Do we look at the absolute value of the leverage or the relative value? Figure 3.58 Whole Model and Effect Leverage Plots Influential Points. 1 Outliers Are Data Points Which Break a Pat-tern Consider Figure 1. Leverage â By Property 1 of Method of Least Squares for Multiple Regression, Y-hat = HY where H is the n × n hat matrix = [h ij]. B) (4 Points) Are All Outliers Influential? where: r i is the i th residual; p is the number of coefficients in the regression model; MSE is the mean squared error; h ii is the i th leverage value Not all points of high leverage are influential. An influential point is an outlier that greatly affects the slope of the regression line. Outliers, Leverage & Influential points in regression A famous data set found in Freedman et al. But if the high leverage point of pushing on the rudder is used instead, it takes only a small amount of force to achieve the same effect.. Easy problems can be solved by pushing on low leverage points. Cookâs distance is the dotted red line here, and points outside the dotted line have high influence.$ Notice that this is a function of both leverage â¦ I want to identify data points with high leverage and large residuals. The formula for Cookâs distance is: D i = (r i 2 / p*MSE) * (h ii / (1-h ii) 2). Cookâs distance was introduced by American statistician R Dennis Cook in 1977. ... A statistic referred to as Cookâs D, or Cookâs Distance, helps us identify influential points. Activate the analysis report worksheet. In model A, the square point had large discrepancy but low leverage, so its influence on the model parameters (slope and intercept) was small. The Leverage Plot for height, on the right, also shows that height is significant, even with age and sex in the model. Briefly Justify Your Answer. 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. 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. Q&A related to Outliers And Influential Points. Influential points in simple linear regression are points that, when removed from the calculation, cause a âgreatâ change in the regression line.The term âinfluential pointsâ is typically applied when assessing outliers.Influential points tipically have high leverage (extreme in X) and/or high residual (extreme in Y). Points with a large residual and high leverage have the most influence. 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. Second, points with high leverage may be influential: that is, deleting them would change the model a lot. A bewilderingly large number of statistical quantities have been proposed to study outliers and influence of individual observations in regression analysis. It might be obvious that influential observations are typically also leverage points. Influential Observations, High Leverage Points, and Outliers in Linear Regression Samprit Chatterjee and Ali S. Hadi Abstract. This point is prepended to the 100 points generated earlier. Observations that fall into the latter category, points with (some combination of) high leverage and large residual, we will call influential. Belsley, Kuh, and Welsch (1980) recommend 2 as a general cutoff value to indicate influential observations and as a size-adjusted cutoff. 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  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. 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 arenât related to the ycoordinates in the right way, they break a pattern. Simulated Data. The following statements use the population example in the section Polynomial Regression. For this we can look at Cookâs distance, which measures the effect of deleting a point on the combined parameter vector. Therefore it is important to identify the data points which impact the model significantly. Cookâs distance, often denoted D i, is used in regression analysis to identify influential data points that may negatively affect your regression model.. Thus for the ith point in the sample, where each h ij only depends on the x values in the sample. Cookâs D measures how much the model coefficient estimates would change if an observation were to be removed from the data set. Leverage is a measure of how far an observation deviates from the mean of that variable. My aim is to remove them and repeat linear regression analyses. Know how to detect outlying y values by way of standardized residuals or studentized residuals. 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. ; Understand leverage, and know how to detect extreme x values using leverages. The scatterplots are identical, except that one plot includes an outlier. Neither plot suggests concerns relative to influential points or multicollinearity. Leverage - influential points. Outliers, leverage and influential data points In general, unusual data points will impact the model and need to be identified. It could change the slope of the regression line, which we'll learn about a little bit later. This is because they happen to lie right near the regression anyway. The influence of each data point can be quantified by seeing how much the model changes when we omit that data point. - have no effect of the regression coefficients as it lies on the same line passing through the remaining observations. The greater an observation's leverage, the more potential it has to be an influential observation. Experts answer in as little as 30 minutes. A) (6 Points) Briefly Describe Each Of: Outliers, Leverage, And Influential Points. Practice thinking about how influential points can impact a least-squares regression line and what makes a point âinfluential.â But it's something that's very strongly changing the data set. This would require a large amount of force to have the intended effect. Influential points are points that when removed significantly change a statistical measure. ... 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. Sample data: To simulate a linear regression dataset, we generate the explanatory variable by randomly choosing 20 points between 0 and 5. So it could change the mean. Ask Question Asked 6 years, 1 month ago. (1991) âStatisticsâ 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. Then you can see how the regression line is affected and how the displayed values change. 218â19, 2005: 417). Sometimes a small group of influential points can have an unduly large impact on the fit of the model. We want the model to be a representative of the whole population. It is used to identify influential data points. Leverage, outliers, and influence â¢Leverage: measures how far away x iis from the other xvalues [goes from 0 to 1, from âaverage xâ to âvery unusual xâ] â¢High leverage: unusual value of x i, which may or may not be well predicted by our line They can have an adverse effect on (perturb) the model if they are changed or excluded, making the model less robust. In general, large values of DFBETAS indicate observations that are influential in estimating a given parameter. All leverage points are not influential on the regression coefficients. C) (10 Points) Additional Diagnostic Plots For The Transformed Regression In Question 4 Are Included On The Following Two Pages. Bar Plot of Cookâs distance to detect observations that strongly influence fitted values of the model. An example of a low leverage point would be pushing on the side of a ship to change its course. How could I perform that in the sample data and do the same analysi swithout the influential points? Key Learning Goals for this Lesson: Understand the concept of an influential data point. Question: [20 Points] Answer The Following Questions. Points with large residuals are potential outliers. 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. This type of analysis is illustrated below. One way to test the influence of an outlier is to compute the regression equation with and without the outlier. Outlier, Leverage, and Influential Points An observation could be unusual with respect to its y-value or x-value. 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