Comparing Two Quantitative Variables | STAT
The response variable measures the outcome of a study. In summarizing the relationship between two quantitative variables, we need to consider: We can interpret from either graph that there is a positive association between Quiz. Scatter Plot (also called scatter diagram) is used to investigate the possible relationship between two variables that both relate to the same event. A straight line. Scatter plots are similar to line graphs in that they use horizontal and vertical axes to The relationship between two variables is called their correlation. would be if you examined the number of hours students spent studying for an exam.
Scatter Plot: Is there a relationship between two variables?
The individual values can be read from the plot and an idea of the relationship between the variables across individuals is obtained. Even if the plot is not used in the final presentation, it may highlight outliers and will help to indicate the appropriate form of analysis to use. For example, the plot below comes from a study from the BMJ, looking at the association between age and ear length. They found a weak positive association, meaning higher values of 'age' are associated with higher values of 'ear length' the article was published in the Christmas edition, where more light-hearted articles are encouraged.
Why do old men have big ears?
Alternatively, the same variable may be measured in a matched pair of individuals. Any 'pairing' that is inherent because of the way in which the data was collected should be retained in both displaying and analysing the data.Two-variable linear equations and their graphs - Algebra I - Khan Academy
Line diagrams are often used whereas a scatterplot is generally more appropriate. Milliner et al, Results of long-term treatment with orthophosphate and pyridoxine in patients with primary hyperoxaluria, New England Journal of Medicine,;Vol ,No. Measurements were made of calcium oxalate inhibition in 12 patients, pre and post treatment.
The authors displayed the data as a line-plot. The values for each patient are shown pre and post treatment and are joined by lines to show the within person pairing of the measurements. Inhibition of the formation of calcium oxalate crystals during treatment with orthophosphate and pyridoxine in 12 patients with primary hyperoxaluria.
The line plot shows that all individuals have values that rise during treatment. One individual shows a very large increase from about 25 pre-treatment to about during treatment as illustrated by the steeply rising diagonal line.
Comparing Two Quantitative Variables
The same data is presented below as a scatterplot: The line of equality no change in values pre to during treatment is shown as a dashed line on the display. All points lie above the line of equality showing that values rose for each individual.
Whilst the same information is given by the two displays, the scatterplot uses only one point to represent each individual compared to 2 points and a line for the line diagram. The line diagram may be confusing to assess if there are changes in various directions, the scatterplot with the line of equality superimposed if necessary is easier to interpret.
No information is lost, the display clearly shows the relationship between the variables and also highlights possible outliers. We saw an example earlier of the times it takes for a scorpion to capture its prey presented as a dot plot. This means that every time that "x" number of people go, "y" amount of money is spent on tickets without variation.
An example of a situation where you might find a perfect negative correlation, as in the graph on the right above, would be if you were comparing the amount of time it takes to reach a destination with the distance of a car traveling at constant speed from that destination. On the other hand, a situation where you might find a strong but not perfect positive correlation would be if you examined the number of hours students spent studying for an exam versus the grade received.
Relationships Between Two Variables
This won't be a perfect correlation because two people could spend the same amount of time studying and get different grades. But in general the rule will hold true that as the amount of time studying increases so does the grade received. Let's take a look at some examples.
The graphs that were shown above each had a perfect correlation, so their values were 1 and The graphs below obviously do not have perfect correlations. Similar calculations are continued to produce the remaining fitted values and residuals.
What does the slope of 0. The slope tells us how y changes as x changes. That is, for this example, as x, Quiz Average, increases by one percentage point we would expect, on average, that the Final percentage would increase by 0.
Coefficient of Determination, R2 The values of the response variable vary in regression problems think of how not all people of the same height have the same weightin which we try to predict the value of y from the explanatory variable x. The amount of variation in the response variable that can be explained i. Since this value is in the output and is related to the correlation we mention R2 now; we will take a further look at this statistic in a future lesson.
Residuals or Prediction Error As with most predictions about anything you expect there to be some error, that is you expect the prediction to not be exactly correct e.