Solved – How to interpret linear regression in scatter plot

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I am trying to understand the relationship between $2$ variables in $4$ different subgroups.

I would like to decide if the variables are correlated in a different way depending on the subgroup. For this reason I plotted $4$ scatter plots with a linear regression. Each scatter plot represent how the $2$ variables behave in a different subgroup (from left to right).

I would like to understand, from the scatter plot, if it is possible to assess that:

The outliers seem to not be affecting the linear regression
The homoscedasticity of the distribution seems good
I may use a Pearson's r also if one of the two variables has a bimodal distribution (y-value)
The 4th group has a positive correlation while the others have a negative correlation

Do these assessments hold? Are there more insights that I can get from these plots, or can I use different plots in this case? Would you use the Pearson's r correlation in this case or refer to another kind of correlation? Any other comments?

Best Answer

Most of your independent variable (x) is concentrated in a short interval; so what you call outliers (samples that are not in the interval) have a high impact on your regression. I would try to redo the analysis by deleting these outliers only to see the impact of them over the regression (not only in the slope but also maybe in its sign).

According to these plots, both variables don't seem to be correlated, there is not a clear tendency between them. It seems like there were two different groups according to response variable, but the most important thing in your data I think it is the fact that the x variable has no enough variability to make a regression model.

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Solved – Exploring a scatter-plot matrix for many variables

Explore relations between variables is quite vague, but two of the more general goals of examining scatterplots like this I would guess are;

Identify underlying latent groups (of variables or cases).
Identify outliers (in univariate, bivariate or multivariate space).

Both reduce the data into more manageble summaries, but have different goals. Identify latent groups one typically reduces the dimensions in the data (e.g. via PCA) and then explores whether variables or cases cluster together in this reduced space. See for example Friendly (2002) or Cook et al. (1995).

Identifying outliers can either mean fitting a model and plotting the deviations from the model (e.g. plotting residuals from a regression model) or reducing the data into its main components and only highlight points that deviate from either the model or the main body of data. E.g. boxplots in one or two dimensions typically only show individual points that are outside of the hinges (Wickham & Stryjewski, 2013). Plotting residuals has the nice property that it should flatten plots (Tukey, 1977), so any evidence of relationships in the remaining point cloud is "interesting". This question on CV has some excellent suggestions of identifying multivariate outliers.

A common way to explore such large SPLOMS is to not plot all of the individual points, but some type of simplified summary and then maybe points that deviate largely from this summary, e.g. confidence ellipses, scagnostic summaries (Wilkinson & Wills, 2008), bivariate box-plots, contour plots. Below is an example of plotting ellipses that define the covariance and superimposing a loess smoother to describe the linear association.

_{(source: statmethods.net)}

Either way, a real sucessful, interactive plot with so many variables would likely need intelligent sorting (Wilkinson, 2005) and a simple way to filter out variables (in addition to brushing/linking capabilities). Also any realistic dataset would need to have the capabilities to transform axis (e.g. plot the data on logarithmic scale, transform the data by taking roots etc.). Good luck, and don't stick with only one plot!

Citations

Cook, Dianne, Andreas Buja, Javier Cabrera & Catherine Hurley. 1995. Grand tour and projection pursuit. Journal of Computational and Graphical Statistics 4(3):155-172.
Friendly, Michael. 2002. Corrgrams: Exploratory displays for correlation matrices. The American Statistician 56(4): 316-324. PDF preprint.
Tukey, John. 1977. Exploratory Data Analysis. Addison-Wesley. Reading, Mass.
Wickham, Hadley & Lisa Stryjewski. 2013. 40 years of boxplots.
Wilkinson, Leland & Graham Wills. 2008. Scagnostic Distributions. Journal of Computational and Graphical Statistics 17(2): 473-491.
Wilkinson, Leland. 2005. The Grammar of Graphics. Springer. New York, NY.

Solved – Partial correlation plot, split by groups SPSS

Here is some code for how you could do it in R:

#Psuedo Data
group1 = rnorm(100)
group2 = rnorm(100)
response = rnorm(100)

#Plotting
plot(group1,response,xlim=c(min(group1,group2),max(group1,group2)),pch=21,bg="deepskyblue",
     ylab="Response Variable",xlab="Independent Variable")
points(group2,response,pch=21,bg="red")

#Adding a legend if desired
legend("topright",c("Group 1","Group 2"),pch=19,col=c("deepskyblue","red"),bg="white")

Which then generates the following plot: enter image description here

You could obviously add your own aesthetic touched but this should give you the general idea.

Best Answer

Related Solutions

Solved – Exploring a scatter-plot matrix for many variables

Citations

Solved – Partial correlation plot, split by groups SPSS

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