Chapter 4 More on Two-Variable Data YMS 4.1 Transforming Relationships

Basics  Transforming data –Changing the scale of measurement used when the data was collected  Ch 4 Transforming –Choose a power or logarithmic transformation that straightens the data –Why? We know how to analyze linear relationships!  Monotonic Function –f(t) moves in one direction as t increases

Algebraic Properties of Logarithms  log b x = y if and only if b y = x  Multiply/add –Log (AB) = Log A + Log B  Divide/subtract –Log (A/B) = Log A – Log B  Power to front –Log (x) A = A*Log x

Growth  Linear –Increases by a fixed amount in each equal time period  Exponential –Increases by a fixed percentage of the previous total –y=ab x

–Plot log y vs. x –If a variable grows exponentially, its logarithm grows linearly log y = log ab x log y = log a + log b x log y = log a + xlog b

Power Models  Ladder of Power Functions p201  y = ax p  Take logarithm of both sides straightens the data log y = log (ax p ) log y = log a + logx p log y = log a + plogx p213 # Homework: p222 #4.17 to 4.20

YMS 4.2 Cautions about Correlation and Regression

Some Vocabulary  Extrapolation –Predicting outside the domain of values of x used to obtain the line or curve  Lurking variable –Is not among the explanatory or response variables but can influence the interpretation of relationships among those variables –Can dramatically change the conclusions

Reminders!  Correlation and regression only describe linear relationships and neither one is resistant!  Using averaged data –Correlations based on averages are usually too high when applied to individuals p230 #4.28 and 4.31

Explaining Association  Causation –May not generalize to other settings –A direct causation is rarely the complete explanation –Is established by an experiment where lurking variables are controlled xy

 Common Response –The observed association between x and y is explained by a lurking variable z –An association is created even though there may be no direct causal link xy z

 Confounding –Two variables whose effects on a response variable are undistinguishable –May be either explanatory or lurking variables p237 #4.33 to 4.37 xy z ?

Establishing Causation  Strength –There is a strong association between variables  Consistency –Many different studies show the same results  Response –Higher explanatory values produce a higher response  Temporal Relationship –Alleged cause precedes the effect in time  Coherence –The alleged cause is plausible/logical

YMS 4.3 Relations in Categorical Data

Two-Way Tables  Row variable/Column variable  Marginal Distributions –Found at the bottom or right margin –Are entire rows/columns over the total  Conditional Distributions –Only a cell that satisfies a certain condition (given in the row/column)

Simpson’s Paradox  The reversal of the direction of a comparison or an association when data from several groups are combined to form a single group –Alaska Airlines vs. American West –Business vs. Law School Admissions Workshop Statistics 7-2 and 7-4