The figures below are reports displayed using JMP's "Bivariate Relationships." In both graphs, we can see that there are values for "triglycerides" on the Y-axis that are significantly larger than the other data (values that can be considered outliers). The green line shows the result of a "robust fit" that reduces the influence of these outliers.
JMP's Bivariate Relationships tool offers a Robust Fit option in the Bivariate report when Y is continuous and X is continuous, and in the Oneway report when Y is continuous and X is nominal or ordinal.
In this article, we will show how to deal with cases where data contains outliers, and will explain one method, robust estimation, and provide examples while comparing it with conventional estimation methods.
A JMP customer previously commented that JMP has this robust estimation function, and I believe that there may be some aspects of it that are useful to other users as well.
What to do when there are outliers
When fitting a model to data, there are three typical approaches to reducing the influence of outliers:
- Remove outliers
- Perform variable transformation (e.g. logarithmic transformation)
- Perform fitting with less influence of outliers (robust estimation)
Graphs are the first display for many reports in JMP, so it's important to carefully review the graphs for any outliers.
The method for removing outliers in step 1 must be carefully considered, including from a practical standpoint. Arbitrarily excluding data may introduce bias into the results.
2. Variable transformation can be useful depending on the distribution and nature of the data. In the case of clinical test values (cholesterol, triglycerides) like in this example, logarithmic transformation may be performed. However, it can be difficult to decide what type of transformation to use, and there is also the disadvantage that it cannot be applied when there are zero or negative values, as with logarithmic transformation.
Therefore, the most effective option is to use the "robust estimation" method (3), which allows model estimation while including outliers.There are two options for robust estimation: Robust Fit and Cauthy Fit . Here we will explain the Robust Fit option.
Robust regression
The scatter plot of the relationship between triglycerides and cholesterol shown at the beginning isLet's think about it. The normal regression analysis (using the least squares method) that can be performed with the [Fit Line] option draws a straight line so that the sum of the squares of the vertical distances (residuals) between the data points and the regression line is minimized. In the figure below, the vertical line (residuals) drawn on the red straight line shows that distance.
In this case, the points circled in blue (outliers) have very large residuals, and we can see that such values tend to pull the straight line calculated by the least squares method .
Therefore, a method of drawing a straight line while minimizing the influence of these outliers is called "robust regression."
The Robust Fit option uses Huber's M-estimation, which reduces the influence of outliers by reducing the weighting of observations with large residuals.
Below is a report when the Robust Fit and Linear Fit options are applied. The red dotted line is the regression line obtained using the least squares method, and the green solid line is the line obtained using robust regression. The slope of the robust regression (=0.471) is smaller than the slope of the least squares regression (=0.684). This indicates that the influence of outliers has been reduced, resulting in a smaller estimated range of variation in Y relative to the variation in X.
Robust Means Comparison
The One-Way Analysis report can perform an Analysis of Variance (ANOVA) to compare the means of Y across levels of the variable specified as X, but this is also sensitive to outliers in Y.
Therefore, by implementing the [Robust Fit] option, we can use Huber's M estimation to calculate "robust means" for each level that reduce the influence of outliers.
Below is the report with the Robust fit and Means/ANOVA/Pooled t-test options applied, the dotted red line shows the mean levels from the ANOVA, and the solid green line shows the robust mean.
At the "female" level, the lines overlap because the ANOVA mean and the robust mean are equal (=77.375), whereas at the "male" level, there are several points with anomalously high triglyceride values, so the robust mean is smaller than the ANOVA mean.
The "ANOVA" report shows a p-value of 0.0641, so with a significance level of 0.05, we cannot conclude that "the triglyceride levels are significantly different between men and women." On the other hand, the "Robust Fit" report shows a p-value of 0.0205, so we can conclude that "the triglyceride levels are significantly different between men and women."
Reference: Robust Estimation Using the Response Screening Platform
Robust fitting can also be performed using the Response Screening platform, which allows testing of many combinations at once.
![「応答のスクリーニング」にて[ロバスト] のオプションを指定]](https://us.v-cdn.net/6038925/uploads/images/78020iEC3731A140D45F2D/Masukawa_Nao_0-1752121600747.png "Masukawa_Nao_0-1752121600747.png")
「応答のスクリーニング」にて[ロバスト] のオプションを指定]Select the Robust option in Response Screening.
ロバスト推定のp値、対数価値などを出力Outputs p-values, log-worth, etc. for robust estimation
*This article uses the JMP sample data "Lipid Data.jmp."
by Naohiro Masukawa (JMP Japan)
Naohiro Masukawa - JMP User Community
