when
the model increases rapidly at first until it reaches its point of maximum growth rate,
At that point, growth steadily slows and the function becomes asymptotic to the upper bound
is the limiting value, sometimes called the
carrying capacity , of the model.
Logistic regression
Logistic regression is used to model situations where growth accelerates rapidly at first and then steadily slows to an upper limit. We use the command “Logistic” on a graphing utility to fit a logistic function to a set of data points. This returns an equation of the form
Note that
The initial value of the model is
Output values for the model grow closer and closer to
as time increases.
Given a set of data, perform logistic regression using a graphing utility.
Use the STAT then EDIT menu to enter given data.
Clear any existing data from the lists.
List the input values in the L1 column.
List the output values in the L2 column.
Graph and observe a scatter plot of the data using the STATPLOT feature.
Use ZOOM [9] to adjust axes to fit the data.
Verify the data follow a logistic pattern.
Find the equation that models the data.
Select “Logistic” from the STAT then CALC menu.
Use the values returned for
and
to record the model,
Graph the model in the same window as the scatterplot to verify it is a good fit for the data.
Using logistic regression to fit a model to data
Mobile telephone service has increased rapidly in America since the mid 1990s. Today, almost all residents have cellular service.
[link] shows the percentage of Americans with cellular service between the years 1995 and 2012
Source:
The World Bank, 2013 .
Year
Americans with Cellular Service (%)
Year
Americans with Cellular Service (%)
1995
12.69
2004
62.852
1996
16.35
2005
68.63
1997
20.29
2006
76.64
1998
25.08
2007
82.47
1999
30.81
2008
85.68
2000
38.75
2009
89.14
2001
45.00
2010
91.86
2002
49.16
2011
95.28
2003
55.15
2012
98.17
Let
represent time in years starting with
for the year 1995. Let
represent the corresponding percentage of residents with cellular service. Use logistic regression to fit a model to these data.
Use the model to calculate the percentage of Americans with cell service in the year 2013. Round to the nearest tenth of a percent.
Discuss the value returned for the upper limit,
What does this tell you about the model? What would the limiting value be if the model were exact?
Using the STAT then EDIT menu on a graphing utility, list the years using values 0–15 in L1 and the corresponding percentage in L2. Then use the STATPLOT feature to verify that the scatterplot follows a logistic pattern as shown in
[link] :
Use the “Logistic” command from the STAT then CALC menu to obtain the logistic model,
Next, graph the model in the same window as shown in
[link] the scatterplot to verify it is a good fit:
To approximate the percentage of Americans with cellular service in the year 2013, substitute
for the in the model and solve for
According to the model, about 98.8% of Americans had cellular service in 2013.
The model gives a limiting value of about 105. This means that the maximum possible percentage of Americans with cellular service would be 105%, which is impossible. (How could over 100% of a population have cellular service?) If the model were exact, the limiting value would be
and the model’s outputs would get very close to, but never actually reach 100%. After all, there will always be someone out there without cellular service!
