Correlation and Causality Chapter 7 Correlation and Causality

GED111/CDS111 Statistics in Modern Society Correlation It exists between 2 variables when higher values of one variable consistently go with higher values of another variable or when higher values of one variable consistently go with lower values of another variable Height vs weight Demand for apples vs price of apples Practice time vs skill Inflation vs unemployment GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Types of correlation Positive correlation Negative correlation No correlation Nonlinear correlation Fig 7.3 p289 GED111/CDS111 Statistics in Modern Society

Measuring the strength of a correlation Correlation coefficient, r Rising straight line, r>0, positive correlation Descending straight line, r<0, negative correlation r ≈ 0, no correlation GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Q5-8 p295 Q9-13 p295 GED111/CDS111 Statistics in Modern Society

Interpreting Correlations Once a correlation between 2 variables has been established, we can then proceed to identify the nature of the relationship so that useful predictions can be made. Beware of outliners Correlation can be very sensitive to outliners Should NOT remove outliners unless we confirm our suspicion that they were invalid data points GED111/CDS111 Statistics in Modern Society

Correlation does not imply causality Possible explanation for a correlation Coincidence Both variables might be directly influenced by some common underlying cause. One of the correlated variables may actually be a cause of the other. But note that, even in this case, it may be just one of several causes. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Exercise Q9-13 p305 GED111/CDS111 Statistics in Modern Society

Best-fit lines and Prediction The best-fit line (or regression line) on a scatter diagram is a line that lies closer to the data point than any other possible line (according to a standard statistical measure of closeness). Using this Regression line, one variable could be used to predict the other variable GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Caution Don’t expect a best-fit line to give a good prediction unless the correlation is significant. Don’t use the best-fit line to make predictions beyond the bounds of the data points to which the line was fit. A best-fit line based on past data is not necessarily valid now and might not result in valid predictions of the future. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Caution (cont.) Don’t make predictions about a population that is different from the population from which the sample data were drawn. Remember that a best-fit line should not be used for predictions when there is no significant correlation or when the relationship is nonlinear. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Multiple Regression Multiple Regression allows the calculation of a best-fit equation that represents the best fit between one variable (such as price) and a combination of 2 or more other variables (such as weight and colour). The coefficient of determination R2, tells us the proportion of the scatter in the data accounted for by the best-fit equation. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Causality The fact that one factor causes another Correlation may suggest causality but by itself never establishes causality. Correlation can be explained by Coincidence Common underlying cause (for the 2 variables) One variable is actually having a direct influence on the other Causality is to eliminate the 1st 2 reasons for correlation GED111/CDS111 Statistics in Modern Society

Guidelines for Causality Look for situations in which the effect is correlated with the suspected cause even while other factors vary. Among groups that differ only in the presence or absence of the suspected cause, check that the effect is similarly present or absent. GED111/CDS111 Statistics in Modern Society

Guidelines for Causality (cont.) Look for evidence that larger amounts of the suspected cause produce larger amounts of the effect. If the effect might be produced by other potential causes (besides your suspected cause), make sure that the effect still remains after accounting for these other potential causes. GED111/CDS111 Statistics in Modern Society

Guidelines for Causality (cont.) If possible, test suspected cause with an experiment. If the experiment cannot be performed with humans for ethical reasons, consider doing the experiment with animals, cell cultures, or computer models. Try to determine the physical mechanism by which the suspected cause produces the effect. GED111/CDS111 Statistics in Modern Society

Broad Levels of Confidence in Causality Possible cause We have discovered a correlation, but cannot yet determine whether the correlation implies causality. In the legal system, possible cause is often the reason for starting an investigation GED111/CDS111 Statistics in Modern Society

Broad Levels of Confidence in Causality (cont.) Probable cause We have good reason to suspect that the correlation involves cause, perhaps because some of the guidelines for establishing causality are satisfied. In the legal system, probable cause is the general standard for getting a judge to grant a warrant for a search or wiretap GED111/CDS111 Statistics in Modern Society

Broad Levels of Confidence in Causality (cont.) Cause beyond reasonable doubt We have found a physical model that is so successful in explaining how one thing causes another that it seems unreasonable doubt is the usual standard for convictions and generally demands that the prosecution have shown how and why the suspect committed the crime. Note that beyond reasonable doubt does not mean beyond all doubt. GED111/CDS111 Statistics in Modern Society

GED111/CDS111 Statistics in Modern Society Focus on Education What Helps Children Learn to Read? pp325-327 GED111/CDS111 Statistics in Modern Society