Transcript Chapter 4 Describing the Relation Between Two Variables

Chapter 4
Describing the Relation
Between Two Variables
4.3
Diagnostics on the Least-squares
Regression Line
The coefficient of determination, R2,
measures the percentage of total variation
in the response variable that is explained by
least-squares regression line.
The coefficient of determination is a number
between 0 and 1, inclusive. That is, 0 < R2 < 1.
If R2 = 0 the line has no explanatory value
If R2 = 1 means the line variable explains 100% of
the variation in the response variable.
The following data are based on a study for
drilling rock. The researchers wanted to
determine whether the time it takes to dry
drill a distance of 5 feet in rock increases
with the depth at which the drilling begins.
So, depth at which drilling begins is the
predictor variable, x, and time (in minutes)
to drill five feet is the response variable, y.
Source: Penner, R., and Watts, D.G. “Mining Information.” The American Statistician, Vol. 45, No. 1,
Feb. 1991, p. 6.
Sample Statistics
Mean
Standard Deviation
Depth
126.2
52.2
Time
6.99
0.781
Correlation Between Depth and Time: 0.773
Regression Analysis
The regression equation is
Time = 5.53 + 0.0116 Depth
Suppose we were asked to predict the time to
drill an additional 5 feet, but we did not know
the current depth of the drill. What would be
our best “guess”?
Suppose we were asked to predict the time to
drill an additional 5 feet, but we did not know
the current depth of the drill. What would be
our best “guess”?
ANSWER:
The mean time to drill an additional 5 feet:
6.99 minutes.
Now suppose that we are asked to predict the
time to drill an additional 5 feet if the current
depth of the drill is 160 feet?
Now suppose that we are asked to predict the
time to drill an additional 5 feet if the current
depth of the drill is 160 feet?
ANSWER:
Our “guess” increased from 6.99 minutes to 7.39
minutes based on the knowledge that drill depth
is positively associated with drill time.
The difference between the predicted drill time of
6.99 minutes and the predicted drill time of 7.39
minutes is due to the depth of the drill. In other
words, the difference in our “guess” is explained by
the depth of the drill.
The difference between the predicted value of 7.39
minutes and the observed drill time of 7.92
minutes is explained by factors other than drill
time.
The difference between the observed value
of the response variable and the mean
value of the response variable is called the
total deviation and is equal to
The difference between the predicted value
of the response variable and the mean value
of the response variable is called the
explained deviation and is equal to
The difference between the observed value
of the response variable and the predicted
value of the response variable is called the
unexplained deviation and is equal to
Total Deviation
= Unexplained Deviation + Explained Deviation
Total Variation
= Unexplained Variation + Explained Variation
Total Variation
= Unexplained Variation + Explained Variation
1=
Unexplained Variation
Total Variation
Explained Variation
Total Variation
+
Explained Variation
Total Variation
= 1 – Unexplained Variation
Total Variation
To determine R2 for the linear regression model
simply square the value of the linear correlation
coefficient.
EXAMPLE
Determining the Coefficient of
Determination
Find and interpret the coefficient of determination
for the drilling data.
EXAMPLE
Determining the Coefficient of
Determination
Find and interpret the coefficient of determination
for the drilling data.
Because the linear correlation coefficient, r, is
0.773, we have that R2 = 0.7732 = 0.5975 =
59.75%.
So, 59.75% of the variability in drilling time is
explained by the least-squares regression line.
Draw a scatter diagram for each of these data sets.
For each data set, the variance of y is 17.49.
Data Set A,
R2 = 100%
Data Set B,
R2 = 94.7%
Data Set C,
R2 = 9.4%
Residuals play an important role in determining the
adequacy of the linear model. In fact, residuals
can be used for the following purposes:
• To determine whether a linear model is
appropriate to describe the relation between the
predictor and response variables.
• To determine whether the variance of the
residuals is constant.
• To check for outliers.
If a plot of the residuals against the predictor
variable shows a discernable pattern, such as
curved, then the response and predictor
variable may not be linearly related.
A chemist as a 1000-gram sample of a
radioactive material. She records the amount of
radioactive material remaining in the sample
every day for a week and obtains the following
data.
Day Weight (in grams)
0
1
2
3
4
5
6
7
1000.0
897.1
802.5
719.8
651.1
583.4
521.7
468.3
Linear correlation coefficient: -0.994
Linear model not appropriate
If a plot of the residuals against the
predictor variable shows the spread of the
residuals increasing or decreasing as the
predictor increases, then a strict
requirement of the linear model is violated.
This requirement is called constant error
variance. The statistical term for constant
error variance is homoscedasticity
A plot of residuals against the predictor
variable may also reveal outliers. These
values will be easy to identify because the
residual will lie far from the rest of the plot.
0
-5
We can also use a boxplot of residuals to identify
outliers.
EXAMPLE Residual Analysis
Draw a residual plot of the drilling time data.
Comment on the appropriateness of the linear
least-squares regression model.
Boxplot of Residuals for the Drilling Data
An influential observation is one that has a
disproportionate affect on the value of the
slope and y-intercept in the least-squares
regression equation.
Case 1
(outlier)
Case 2
Case 3
(influential)
Influential observations typically exist when
the point is large relative to its X value.
EXAMPLE
Influential Observations
Suppose an additional data point is added to the
drilling data. At a depth of 300 feet, it took 12.49
minutes to drill 5 feet. Is this point influential?
With
influential
Without
influential
As with outliers, influential observations should be
removed only if there is justification to do so.
When an influential observation occurs in a data
set and its removal is not warranted, there are two
courses of action:
(1) Collect more data so that additional points
near the influential observation are obtained, or
(2) Use techniques that reduce the influence of
the influential observation (such as a
transformation or different method of estimation e.g. minimize absolute deviations).
Chapter Four
Describing the Relation
Between Two Variables
Section 4.4
Nonlinear Regression:
Transformations
EXAMPLE Using the Definition of a Logarithm
Rewrite the logarithmic expressions to an
equivalent expression involving an exponent.
Rewrite the exponential expressions to an
equivalent logarithmic expression.
(a) log315 = a
(b) 45 = z
In the following properties, M, N, and a are
positive real numbers, with a  1, and r is
any real number.
loga (MN) = loga M + loga N
loga Mr = r loga M
EXAMPLE
Simplifying Logarithms
Write the following logarithms as the sum of
logarithms. Express exponents as factors.
(a) log2 x4
(b) log5(a4b)
If a = 10 in the expression y = logax, the
resulting logarithm, y = log10x is called the
common logarithm. It is common
practice to omit the base, a, when it is
equal to 10 and write the common
logarithm as y = log x
EXAMPLE Evaluating Exponential and
Logarithmic Expressions
Evaluate the following expressions. Round your
answers to three decimal places.
(a) log 23
(b) 102.6
y = abx
Exponential Model
log y = log (abx) Take the common logarithm of
both sides
log y = log a + log bx
log y = log a + x log b
Y=A+Bx
b = 10B
where
a = 10A
EXAMPLE 4
Finding the Curve of Best Fit to
an Exponential Model
A chemist as a 1000gram sample of a
radioactive material. She
records the amount of
radioactive material
remaining in the sample
every day for a week and
obtains the following
data.
Day Weight (in grams)
0
1
2
3
4
5
6
7
1000.0
897.1
802.5
719.8
651.1
583.4
521.7
468.3
(a) Draw a scatter diagram of the data treating the
day, x, as the predictor variable.
(b) Determine Y = log y and draw a scatter diagram
treating the day, x, as the predictor variable and
Y = log y as the response variable. Comment on the
shape of the scatter diagram.
(c) Find the least-squares regression line of the
transformed data.
(d) Determine the exponential equation of best fit and
graph it on the scatter diagram obtained in part (a).
(e) Use the exponential equation of best fit to predict
the amount of radioactive material is left after 8 days.
y = axb
Power Model
log y = log (axb) Take the common logarithm
of both sides
log y = log a + log xb
log y = log a + b log x
Y=A+bX
where
a = 10A
EXAMPLE Finding the Curve of Best Fit to a
Power Model
Distance
Cathy wishes to measure 1.0
1.1
the relation between a
light bulb’s intensity and 1.2
the distance from some
1.3
light source. She
1.4
measures a 40-watt light 1.5
bulb’s intensity 1 meter
1.6
from the bulb and at 0.1- 1.7
meter intervals up to 2
1.8
meters from the bulb and
1.9
obtains the following data.
2.0
Intensity
0.0972
0.0804
0.0674
0.0572
0.0495
0.0433
0.0384
0.0339
0.0294
0.0268
0.0224
(a) Draw a scatter diagram of the data treating the
distance, x, as the predictor variable.
(b) Determine X = log x and Y = log y and draw a scatter
diagram treating the day, X = log x, as the predictor
variable and Y = log y as the response variable.
Comment on the shape of the scatter diagram.
(c) Find the least-squares regression line of the
transformed data.
(d) Determine the power equation of best fit and graph it
on the scatter diagram obtained in part (a).
(e) Use the power equation of best fit to predict the
intensity of the light if you stand 2.3 meters away from
the bulb.
Modeling is not only a science but also an art
form. Selecting an appropriate model requires
experience and skill in the field in which you are
modeling. For example, knowledge of
economics is imperative when trying to determine
a model to predict unemployment. The main
reason for this is that there are theories in the
field that can help the modeler to select
appropriate relations and variables.