Download - SC08EdProgK12

Teaching Basic Modeling Skills Using Real
Data
Steven Gordon
Senior Director of Education and Client Services
[email protected]
Goals of the Session
• Overview of statistical methods that can be used to
build a model
• Sources of real data to test hypotheses about
statistical relationships
• Example exercise(s) and techniques for
downloading and extracting data, testing statistical
relationships, and building a model based on the
results
Measuring the Strength of a Relationship
• Correlation
– Statistical relationship between two variables
– Goes between -1.0 and 1.0
– Zero means no relationship
• http://argyll.epsb.ca/jreed/math9/strand4/scatterPl
ot.htm
Linear Regression
• Assumes a cause and effect between one dependent
variable and one or more independent variables
• Solution of the linear equation with a best fit to data
• Y = aX + b
where:
Y = the dependent variable
a = a coefficient equivalent to the slope of the line
b = the Y intercept of the line (the place where it crosses
the Y axis)
• Y = a1X1 + a2X2 + a3X3 + b for multiple causes
•Use the vertical offsets to estimate Y for a given X
•Square the differences so all numbers are positive
•Sum of the squared offsets or deviations is a
measure of the goodness of fit
•R2 or coefficient of determination (0 to 1.0)
Using Excel for Regression Analysis
• Open the file Regress_Data.xls
– File shows a dataset with 40 observations showing
Income, Education, and Age
– What would we hypothesize to be the relationships?
Which is the dependent variable? Independent
variables?
• Need to activate the data analysis toolpak
– Click on Tools from main menu in Excel – if Data
Analysis is a submenu choice – good to go
– Otherwise – Select Tools> Add-Ins
– Mark the box next to the Analysis Toolpak and click OK
Do a Correlation Analysis
• Click on Tools – Data Analysis
• Choose Correlation from the pull-down menu
• Click in the input range box
– Choose all three columns including the labels by
clicking and dragging across the entire dataset
• Click the Labels box to indicate the first row is
labels
• Click New Worksheet and give it a name correlate
• What are the results?
Do a Regression
• Click Tools, Data Analysis
• Select Regression from the pull down menu
• Put your cursor in the field for the input Y Range
– Select the range for the dependent variable, including its label
• Put the cursor in the Input X Range
– Select the other two columns of number
• Mark the check boxes
– Labels
– Confidence Level
• Click on the radio button for New Worksheet and give it a name
• Click ok
Interpreting Regression Outputs in Excel
Regression Statistics
Multiple R
0.638081096
R Square
0.407147485
Adjusted R
Square
0.375944721
Standard Error
7556.02062
Observations
41
37.6 percent
of variance
explained
after
adjusting for
degrees of
freedom
More interpretation
ANOVA
df
Regression
Residual
Total
2
38
40
SS
1489961186
2169551009
3659512195
MS
744980593.1
57093447.61
F
13.04844294
Significance F
4.85274E-05
Significance of the
equation.
Probability of a false
positive is less than
0.004 %
Coefficients and Their Significance
• Equation: Income = -17954.1 + 440.71*Age+
1542.41*Education
Coefficients Standard Error
t Stat
Intercept
-17954.1
9658.880575 -1.858822842
Age (years) 440.7319
99.06262364 4.449022886
Ed. (yrs)
1542.411
617.4511538 2.498028499
P-value
0.070809952
7.29681E-05
0.016933884
Significance of
individual
coefficients
Fitting Non-Linear Data
• Same principle and measurement of the deviations
• Choice of curve to fit not automatic
– Individual choice with possibility of error
– Real relationship may not be fully represented by
experimental data
Potential Errors
• Chose the wrong function for non-linear data
• Experiments did not measure all possible
circumstances
– Behavior may change in areas outside the sample data
– E.G. physical limitations of system lead to failure
• Variables may be highly correlated and have
strong relationship in regression but one is not the
cause of the other
Exercise Using Real Data