Download - Chapter 4 More on Two-Variable Data YMS 4.1
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Chapter 4Chapter 4More on Two-Variable DataMore on Two-Variable Data
YMS 4.1YMS 4.1
Transforming RelationshipsTransforming Relationships
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BasicsBasics Transforming dataTransforming data
– Changing the scale of measurement used Changing the scale of measurement used when the data was collectedwhen the data was collected
Ch 4 Transforming Ch 4 Transforming – Choose a power or logarithmic Choose a power or logarithmic
transformation that straightens the datatransformation that straightens the data– Why? We know how to analyze linear Why? We know how to analyze linear
relationships! relationships! Monotonic FunctionMonotonic Function
– f(t) moves in one direction as t increasesf(t) moves in one direction as t increases
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Algebraic Properties of LogarithmsAlgebraic Properties of Logarithms
loglogbbx = y if and only if bx = y if and only if byy = x = x Multiply/addMultiply/add
– Log (AB) = Log A + Log BLog (AB) = Log A + Log B Divide/subtractDivide/subtract
– Log (A/B) = Log A – Log BLog (A/B) = Log A – Log B Power to front Power to front
– Log (x)Log (x)AA = A*Log x = A*Log x
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GrowthGrowth LinearLinear
– Increases by a fixed amount in each Increases by a fixed amount in each equal time periodequal time period
ExponentialExponential– Increases by a fixed percentage of the Increases by a fixed percentage of the
previous totalprevious total– y=aby=abxx
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– Plot Plot log y vs. xlog y vs. x– If a variable grows exponentially, its If a variable grows exponentially, its
logarithm grows linearlylogarithm grows linearly
log log y y = log = log ababxx
log log yy = log = log aa + log + log b bxx
log log yy = log = log aa + + xxlog log bb
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Power ModelsPower Models Ladder of Power Functions p201Ladder of Power Functions p201 y = axy = axpp
Take logarithm of both sides Take logarithm of both sides straightens the datastraightens the data
log log y y = log (= log (axaxpp))
log log yy = log = log aa + log + logxxpp
log log yy = log = log aa + + pploglogxx
p213 #4.10-4.11p213 #4.10-4.11
Homework: p222 #4.17 to 4.20Homework: p222 #4.17 to 4.20
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YMS 4.2YMS 4.2
Cautions about Correlation Cautions about Correlation and Regressionand Regression
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Some VocabularySome Vocabulary ExtrapolationExtrapolation
– Predicting outside the domain of values Predicting outside the domain of values of of xx used to obtain the line or curve used to obtain the line or curve
Lurking variableLurking variable– Is not among the explanatory or Is not among the explanatory or
response variables but can influence the response variables but can influence the interpretation of relationships among interpretation of relationships among those variablesthose variables
– Can dramatically change the Can dramatically change the conclusionsconclusions
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Reminders!Reminders!
Correlation and regression only Correlation and regression only describe linear relationships and describe linear relationships and neither one is resistant!neither one is resistant!
Using averaged dataUsing averaged data– Correlations based on averages are Correlations based on averages are
usually too high when applied to usually too high when applied to individualsindividuals
p230 #4.28 and 4.31p230 #4.28 and 4.31
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Explaining AssociationExplaining Association
CausationCausation– May not generalize to other May not generalize to other
settingssettings– A direct causation is rarely A direct causation is rarely
the complete explanationthe complete explanation– Is established by an Is established by an
experiment where lurking experiment where lurking variables are controlledvariables are controlled
x y
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Common Response Common Response – The observed The observed
association between association between xx and and yy is explained by a is explained by a lurking variable lurking variable zz
– An association is An association is created even though created even though there may be no direct there may be no direct causal linkcausal link
x y
z
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ConfoundingConfounding– Two variables whose Two variables whose
effects on a response effects on a response variable are variable are undistinguishableundistinguishable
– May be either May be either explanatory or lurking explanatory or lurking variablesvariables
p237 #4.33 to 4.37p237 #4.33 to 4.37
x y
z
?
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Establishing CausationEstablishing Causation StrengthStrength
– There is a strong association between variablesThere is a strong association between variables Consistency Consistency
– Many different studies show the same resultsMany different studies show the same results Response Response
– Higher explanatory values produce a higher Higher explanatory values produce a higher responseresponse
Temporal Relationship Temporal Relationship – Alleged cause precedes the effect in timeAlleged cause precedes the effect in time
CoherenceCoherence– The alleged cause is plausible/logical The alleged cause is plausible/logical
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YMS 4.3YMS 4.3
Relations in Categorical Relations in Categorical DataData
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Two-Way TablesTwo-Way Tables
Row variable/Column variableRow variable/Column variable Marginal DistributionsMarginal Distributions
– Found at the bottom or right Found at the bottom or right marginmargin– Are entire rows/columns over the totalAre entire rows/columns over the total
Conditional DistributionsConditional Distributions– Only a cell that satisfies a certain Only a cell that satisfies a certain
condition (given in the row/column) condition (given in the row/column)
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Simpson’s ParadoxSimpson’s Paradox
The reversal of the direction of a The reversal of the direction of a comparison or an association when comparison or an association when data from several groups are data from several groups are combined to form a single group combined to form a single group – Alaska Airlines vs. American WestAlaska Airlines vs. American West– Business vs. Law School AdmissionsBusiness vs. Law School Admissions
Workshop Statistics 7-2 and 7-4Workshop Statistics 7-2 and 7-4