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.

Ladder of Powers

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