Relations Between Variables Scientists are forever trying to find relations between quantities:
In each case above, the scientists run experiments and collect pairs of numbers for the quantities that they are trying to relate to each other: · (weekly exercise minutes, systolic blood pressure) , e.g. (35, 136), (0, 155), (200, 121), … · (angle of ramp (deg.), time of ball (sec.)), e.g. (5, 17), (20, 6), (45, 3), … · (cc fertilizer/sq. meter garden, plant height m.), e.g. (.5, .33), (.77, .02), (.01,.54), … They might plot these number pairs on a graph and examine the graph for a trend. For example, in repeating the ball-rolling experiment pictured below the student records the pairs of numbers representing the angle of the ramp and the corresponding “rolling times. She then plots these numbers on a graph as follows: What can you conclude about a relationship between the angle of the ramp and the time the ball rolls? Can you explain this? Another student does a very careful experiment to relate the growth of plants with the amount of fertilizer used and comes up with the following graph:
Is there a relationship between plant height and the amount of fertilizer used? Can you explain this? These examples of seeking a relationship between variables can be quantified by using methods of statistical analysis called Correlation and Regression. Correlation analysis seeks to identify (by a single number) the degree to which there is a (linear) relation between the numbers in sets of data pairs. The correlation coefficient of a set of data pairs You don’t need to worry about computing this number; it’s easy to use a computer to calculate it. The interpretation of this number is more important – it is somewhere between –1 and 1. The closer r is to 1, the more positively correlated are the sets of numbers in the sense that an increase in x corresponds to a proportional increase in y; similarly with decreases in x corresponding to proportional decreases in y. On the other hand, if r is close to –1, then increases in x correspond to decreases in y and decreases in x correspond to increases in y, so we say that x and y are negatively correlated. Finally, if r is close to zero, there is little if any relationship between the variables – we say they are uncorrelated. Consider the earlier graphs from the “ball-rolling” and “fertilizer” experiments:
As the height of players increases, does the weight generally
i. -.75 ii. .03 iii. .73 iv. .99
As the length of the lobster increases, does the number of eggs produced generally
i. -.89 ii. -.13 iii. .25 iv. .91 RegressionRegression analysis is used to determine if a relationship exists between two variables. To do this a line is created that best fits a set of data pairs. We will use linear regression which seeks a line with equation 1)Generation of the regression line and equation for the line: For example, if a computer program for doing regression is applied to the data from the “Ball rolling experiment” the best fitting line is shown on the graph below: It will turn out that any other line will give a larger overall distance to the points than this line does. You can frequently estimate the equation of the regression line (y= mx + b) by estimating its slope (m) From the graph above, we could estimate that the line has y-intercept close to 6 because if you continue to draw the line out, it crosses the y-axis near 6. To determine the slope you must first choose two points on the line—these are not existing data points, but points of your choice. The easiest and usually more accurate method is to use the grid lines as your guide in choosing your values on the x-axis and then estimate your y-values. So, for the above graph choose these two sets of points (20,4.5) and (60, 1.25). It is best to spread out the points you choose, one from either end of your line. It is especially important to remember to choose two points ON THE LINE because you are trying to estimate an equation for the line itself, NOT your data points. Using the points we chose, plug the numbers into the equation for the slope. y=-0.081 x+ 6. 2) Generation of R2 value When you do regression analysis using a computer program, you’ll sometime see some indication of the coefficient of determination or “goodness of fit”, 3) Generation of a p-value Using the computer will allow you to calculate a p-value for your relationship. The p-value allows you to decide whether to accept or reject your null hypothesis. If your p-value is greater than 0.05 there is NO significant relationship and you would accept your null hypothesis. If your p-value is less than 0.05 there IS a significant relationship and you would reject your null hypothesis. Which of the other examples displayed this causal relationship?
y=3758.525 x + -106704 and the graph (with regression line) is drawn below: Homework on Correlation and Regression: Turn in answers to problems 1—4 from earlier and also for the following:
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