Nonlinear Relationships

Students in a physics class are studying free-fall to determine the relationship between the distance an object has fallen and the amount of time since release.

 Time (sec) Distance (cm) Time (sec) Distance (cm) .16 12.1 .57 150.2 .24 29.8 .61 182.2 .25 32.7 .61 189.4 .30 42.8 .68 220.4 .30 44.2 .72 250.4 .32 55.8 .72 261.0 .36 63.5 .83 334.5 .36 65.1 .88 375.5 .50 124.6 .89 399.1 .50 129.7

1.     Draw a scatter plot of the data.

2.     Even though it doesn’t look linear find the regression line.

3.     Find the median-median line.

4.     Look at the residuals for both equations and find the standard error.

In a residual plot, positive and negative values should occur randomly.

The relationship between y and x need not be linear.  We can examine the shape of the relationship with a scatter plot and look for more detailed information by plotting the residuals from the median-median line.  If either the original or residual plot shows a bend and if the y-versus-x plot shows a generally consistent trend either up or down rather than a cup shape, we may be able to straighten the y-versus-x relationship by re-expressing one or both variables.

From the summary points of the median-median line, we compute the half-slopes:

and then find the half-slope ratio: .  If the half-slopes are equal, then the relationship is linear and the half-slope ratio is 1.  If the half-slope ratio is not close to one, then re-expressing x or y or both may help.  If the half-slope ratio is negative, the half-slopes have different signs, and re-expression will not help.

If the half-slopes are not equal, the two line segments will meet and form an angle.  We can think of the angle as an arrowhead that points toward re-expressions on the ladder of powers that might make the relationship straighter.  To determine how we might re-express y, we need to know if the arrow points more upward—towards higher values of y—or more downward—toward lower y-values.  To determine how we might re-express x, we need to know if the arrow points more to the right—toward higher x values—or more to the left—toward lower x values.

The rule for selecting a re-expression to straighten a plot is that we consider moving the expression of y or x in the direction the arrow points.  That is if the arrow points down toward lower y we might try re-expressions of y lower in the ladder of powers.  If the arrow points to the right, toward higher x, we might try re-expressions of x higher on the ladder of powers.

The half-slopes will suggest re-expressions for both x and y.  We may choose to re-express either y or x or both.

1.     Do this for our data.

 p Re-expression Name Notes Higher powers can be used. 3 Cube 2 Square One of the most commonly used powers 1 “Raw” No re-expression necessary ½ Square root A commonly used power, especially for counts (0) Logarithm –½ Reciprocal root The minus sign preserves order. -1 Reciprocal -2 Reciprocal square Lower powers can be used.

1.     Let’s re-express our data.

2.     Re try the fit.

3.     There are other ways to think of re-expression.

Gas Mileage and Displacement for Some 1976 Automobiles

 Automobile mpg Displacement Mazda RX-4 21.0 160.0 Mazda RX-4 Wagon 21.0 160.0 Datsun 710 22.8 108.0 Hornet 4-Drive 21.4 258.0 Hornet Sportabout 18.7 360 Valiant 18.1 225 Plymouth Duster 14.3 360 Mercedes 240D 24.4 146.7 Mercedes 230 22.8 140.8 Mercedes 280 19.2 167.6 Mercedes 280C 17.8 167.6 Mercedes 450SE 16.4 275.8 Mercedes 450SL 17.3 275.8 Mercedes 450SLC 15.2 275.8 Cadillac Fleetwood 10.4 472 Lincoln Continental 10.4 460 Chrysler Imperial 14.7 440 Fiat 128 32.4 78.7 Honda Civic 30.4 75.7 Toyota Corolla 33.9 71.1 Toyota Corona 21.5 120.1 Dodge Challenger 15.5 318.0 AMC Javelin 15.2 304 Camaro Z28 13.3 350 Pontiac Firebird 19.2 400 Fiat X1-9 27.3 79 Porsche 914-2 26.0 120.3 Lotus Europa 30.4 95.1 Ford Pantera L 15.8 351 Ferrari Dina 1973 19.7 145 Maserati Bora 15.0 301 Volvo 142E 21.4 121