Strategies for Quantitative Research by Grant S. McCall

Author:Grant S. McCall [McCall, Grant S.]
Language: eng
Format: epub
Tags: Social Science, Archaeology
ISBN: 9781351802949
Google: 8T1MDwAAQBAJ
Barnesnoble:
Goodreads: 34566656
Publisher: Routledge
Published: 2018-02-13T00:00:00+00:00

Graphical approaches to linear regression and the statistical uses of residuals

Archaeologists are generally visual learners, and bivariate plots are a striking graphical method of representing the relationship between two variables. In addition, graphing the relationship between two variables can often reveal nuances that regression and correlation statistics cannot, in the same way that the graphical representation of the distribution of univariate data may reveal patterning that simple descriptive statistics may not. Furthermore, graphs are just more fun to look at than a series of correlation coefficients, T statistics, and p-values. The simplest way of graphing the relationship between two variables is through the use of a scatterplot. This technique employs the same logical basis as the Cartesian plane and plots the values for one variable along the x-axis and the values of the other variable along the y-axis.

Scatterplots are capable of detecting some forms of data distribution that may be potentially problematic from the perspective of linear regression. One potential problem is that of multimodality. I discussed this problem already relative to univariate data. Specifically, multimodality defies the assumptions of normality, since a univariate distribution with multiple peaks obviously does not fit the normal distribution. Beyond this, multimodality renders the calculation of descriptive statistics, such as mean, median, and standard deviation, fatally problematic.

Multimodality has a similarly misleading effect on linear regression analyses. The most common and problematic form of this situation is a bimodal distribution (Figure 6.4). For bivariate data, this means that there are two clusters, or clouds, of data points that are separated from one another in bivariate graphical space. These two clouds of data effectively form two points, and as we all know, two points determine a line. Therefore, a bimodal distribution of bivariate data can often lead to spuriously strong results in terms of the strength of the relationship between two variables.

I am often asked whether there is some formal statistical method for examining the potential bimodality of data. Unfortunately, my usual answer is, not really. In the same way that graphically inspecting one’s univariate data is a fundamental precursor to further statistical analysis, the examination of bivariate data using a scatterplot should be a first step in any linear regression analysis. There are aspects of the patterning of bivariate data that cannot be recognized or assessed any other way. The best way of dealing with this kind of multimodality is to deal with each cluster of data points separately.

Figure 6.4 Hypothetical relationship between two bimodally distributed variables

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