Chapter 4Chapter 4More on Two-Variable DataMore on Two-Variable Data

YMS 4.1YMS 4.1

Transforming RelationshipsTransforming Relationships

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

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

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

– 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

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

YMS 4.2YMS 4.2

Cautions about Correlation Cautions about Correlation and Regressionand Regression

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

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

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

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

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

?

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

YMS 4.3YMS 4.3

Relations in Categorical Relations in Categorical DataData

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)

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