Hi everyone,
I would like to know how I can determine if a point significantly deviates from the linear regression when I plot the data in a scatter plot. I am using excel and I made a scatter plot of two variables. Now the general trend is definitely linear but since I have some points that are a bit far from the line, I would like to determine if these specific points significantly deviate from the linear trend I see for all the other points. Is that possible with excel and/or GraphPad?
Thanks.
I don't think you need to worry about specific points. Just rely on your r or r2 values and also p values. The higher the r values, the more that the model explains the outcome :)
Outliers can dramatically impact correlations so they’re something worth considering (see image link below). In your case it seems like outliers are impacting your results since there’s a generally linear trend with a few odd points away from the trend. Exactly how we declare data outliers is tricky, as is what to do about them. Most often researchers identify outliers just by looking at their graphs. The first thing I would do is check there’s not some kind of data “goof” that created an outlier (e.g., mistyped data, participant misunderstands item and enters impossible value, a code for missing data being treated as a data value). If you find any “goofs,” just correct them. If an outlier is genuine data, you need to decide if it’s okay to delete. Deleting a point is like saying that the process that led it to be the way it is, is qualitatively different from the process behind the other data points. For example, if you correlated health-related variables, you might have an outlier because of disease. If it’s iffy, and you’re looking for a statistical justification, sometimes researchers will say data 3 standard deviations from the norm can be removed as outliers; you could calculate SD for your x, y, and the difference of each y from the expected y given x. Finally, if you’re worried an outlier is changing your result in an invalid way, but you don’t feel justified removing it, you could use a non-parametric correlation because those are less susceptible to outliers (e.g., Spearman's rank-order correlation). Hope this helps some with your project, Silvia. ~ Kevin
https://statistics.laerd.com/statistical-guides/img/pearson-6.png
Thank you both!! Kevin, the approach you explained about calculating SD for x y, and the difference of each y from the expected y given x was exactly what I was trying to understand. I mean, I was wondering if there is a method to give a statistical justification why one point is defined as outlier...because I have doubts on some points..Could you please give me some more details of how to do it? Do I need to use the equation of the curve?Thanks so much
To get the difference of each y from the expected y given x: (1) Calculate the best-fit line for all the non-outlier items. You can do this by adding a trend line in an Excel scatterplot and choosing the option to display the equation. You can also get this by using the =SLOPE() and =INTERCEPT() formulas. (2) For each x, calculate the y in a new column, using a formula like =[slope_num]*A23[whatever_cell_contains_x]+[intercept_num], and pasting it down the new column, (3) In another column, calculate the difference with a formula like =B2-D2 (if D were the column with the calculated y and B were the column with the actual y. Paste it down the column. (4) Calculate the average and SD for this column of differences. The mean has to be 0 and the SD will depend on how strong your correlation, (5) For each of your potential outliers, use the x to calculate y with the same formula and then find the difference with the same formula, (6) Divide this difference by the SD and you’ll have the number of SD from the projected value along the line. Assuming it’s a big number, and more SD from the norm than values you’re keeping, you could argue you excluded the case because it was more than 3 SD of the expected value given x. I’ve seen some people call it an outlier with 2 SD and others say 2.67 SD. I’m not sure of a citation. But this would be a way to quantify just how much any particular point is an outlier from your best-fit line. Best wishes with your project, Silvia. ~ Kevin
Thank you very much Kevin!This was really really helpful and illuminated me!Really thank you for the detailed explanation. Just one last question. I suppose this method is the "Z-score" one (defined as the number of standard deviations away from the mean).I found online that this z-score is defined as z = (x - mean) ÷ standard Deviation. So, in the step you described in 5) and 6), according to this formula, shouldn't I subtract the mean from the x value of the outlier as well and then divide everything by the standard deviation?
Thank you very much again and sorry for bothering..Everything is completely new for me.
Silvia
Thank you very much Kevin!This was really really helpful and illuminated me!Really thank you for the detailed explanation. Just one last question. I suppose this method is the "Z-score" one (defined as the number of standard deviations away from the mean).I found online that this z-score is defined as z = (x - mean) ÷ standard Deviation. So, in the step you described in 5) and 6), according to this formula, shouldn't I subtract the mean from the x value of the outlier as well and then divide everything by the standard deviation?
Thank you very much again and sorry for bothering..Everything is completely new for me.
Silvia
Yes, that’s exactly it. Let’s say your average difference was 0 (which it should be) and the standard deviation of differences is 4. Then if your outlier value’s difference is 12, then it would be 3 standard deviations above the expected value (i.e., 3 = (12-0)/4). Good luck with your project, Silvia! ~ Kevin
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