3Straight to the PointWe cannot use a linear model unless the relationship between the two variables is linear. Often re-expression (transformation) can save the day, straightening bent relationships so that we can fit and use a simple linear model.Two simple ways to re-express data are with logarithms and reciprocals.Re-expressions can be seen in everyday life—everybody does it.
4Straight to the PointThe relationship between fuel efficiency (in miles per gallon) and weight (in pounds) for late model cars looks fairly linear at first:
5Straight to the PointA look at the residuals plot shows a problem:
6Straight to the PointWe can re-express fuel efficiency as gallons per hundred miles (a reciprocal) and eliminate the bend in the original scatterplot:
7Straight to the PointA look at the residuals plot for the new model seems more reasonable:
8Goals of Re-expression Goal 1: Make the distribution of a variable (as seen in its histogram, for example) more symmetric.It’s easier to summarize the center of a symmetric distribution, we can use the mean and standard deviation.If the distribution is unimodal also, we can analysis using the normal model.Here taking the log of the explanatory variable.
9Goals of Re-expression Goal 2: Make the spread of several groups (as seen in side-by-side boxplots) more alike, even if their centers differ.Groups that share a common spread are easier to compare.Here taking the log makes the individual boxplots more symmetric and gives them spreads that are more nearly equal.
10Goals of Re-expression Goal 3: Make the form of a scatterplot more nearly linear.Linear scatterplots are easier to model.By re-expressing to straighten the scatterplot relationship we can fit a linear model and use linear techniques to analysis.Here taking the log of the response variable.
11Goals of Re-expression Goal 4: Make the scatter in a scatterplot spread out evenly rather than thickening at one end.Having an even scatter is a condition of many methods of Statistics, as we will see later.This is closely related to goal 2, but often comes along with goal 3, as seen below. When taking the log to straighten the data, it also evened out the spread.
12The Ladder of PowersThere is a family of simple re-expressions that move data toward our goals in a consistent way. This collection of re-expressions is called the Ladder of Powers.The Ladder of Powers orders the effects that the re-expressions have on data.
13The Ladder of Powers 2 1 ½ “0” –1/2 –1 Ratios of two quantities (e.g., mph) often benefit from a reciprocal.The reciprocal of the data–1An uncommon re-expression, but sometimes useful.Reciprocal square root–1/2Measurements that cannot be negative often benefit from a log re-expression.We’ll use logarithms here“0”Counts often benefit from a square root re-expression.Square root of data valuesData with positive and negative values and no bounds are less likely to benefit from re-expression.Raw data1Try with unimodal distributions that are skewed to the left.Square of data values2CommentNamePower
14The Ladder of PowersThe Ladder of Powers orders the effects that the re-expressions have on data.How it works.If you try taking the square root of all the values in a variable and it helps, but not enough, then move further down the ladder to the log or reciprocal root. Those re-expressions will have a similar, but even stronger, effect on your data.If you go too far, you can always back up.Remember, when you take a negative power, the direction of the relationship will change. This is OK, you can always change the sign of the response variable if you want to keep the same direction.
15Plan B: Attack of the Logarithms When none of the data values is zero or negative, logarithms can be a helpful ally in the search for a useful model.Try taking the logs of both the x- and y-variable.Then re-express the data using some combination of x or log(x) vs. y or log(y).
17Multiple BenefitsWe often choose a re-expression for one reason and then discover that it has helped other aspects of an analysis.For example, a re-expression that makes a histogram more symmetric might also straighten a scatterplot or stabilize variance.
18Why Not Just Use a Curve?If there’s a curve in the scatterplot, why not just fit a curve to the data?
19Why Not Just Use a Curve?The mathematics and calculations for “curves of best fit” are considerably more difficult than “lines of best fit.”Besides, straight lines are easy to understand.We know how to think about the slope and the y-intercept.
20More Plan B: Modeling Nonlinear Data - Logarithms Two specific types of nonlinear growth.Exponential function (form y = abx)Power function (form y = axb)Equations of both forms can be transformed into linear forms.Can then use linear regression to model and analyze the transformed data.Can also perform an inverse transformation to obtain a model of the original data.
22Linear vs. Exponential Growth Linear Growth – A variable grows linearly over time if it adds a fixed increment in each equal time period.Arthmetic Sequence – common difference(yn-yn-1)Exponential Growth – A variable grows exponentially if it is multiplied by a fixed number greater than 1 in each equal time period. Exponential decay occurs when the factor is less than 1.Geometric Sequence – common ratio (yn/yn-1)
23To Transform the exponential Function use its Inverse the Logarithmic Function Properties of Logarithms
24Using Logarithms to Transform Data Logarithms can be useful in straightening a scatterplot whose data values are greater than zero.Remember, you cannot take the logarithm of a nonpositive number.When you use transformed data to create a linear model, your regression equation is not in terms of (x,y) but in terms of the transformed variable(s) (log ŷ or log x).
26Test for Exponential Functions View the scatterplot, does it look exponential?Calculate the common ratio between successive response values – yn/yn-1.Can only be used if the explanatory values (x) change in equal increments.
27Example: Testing for Exponential Association Data
28View ScatterplotLooks like it has a curved pattern, could possibly be an exponential relationship.
29Verify Exponential Association Density (y)22.214.171.124.126.96.36.1990.613.416.921.225.631.035.6188.8.131.520.657.464.070.3r = yn/yn-16.1/4.5 = 1.364.3/6.1 = .705.5/4.3 = 1.287.4/5.5 = 1.359.8/7.4 = 1.327.9/9.8 = .8110.6/7.9 = 1.3413.4/10.6 = 1.2616.9/13.4 = 1.2621.2/16.9 = 1.2525.6/21.2 = 1.2131.0/25.6 = 1.2135.6/31.0 = 1.1541.2/35.6 = 1.1644.2/41.2 = 1.0750.7/44.2 = 1.1550.6/50.7 = 1.0057.4/50.6 = 1.1364.0/57.4 = 1.1170.3/64.0 = 1.10Is there a common ratio?YES, r≈1.2(mean r=1.16 and median r=1.19)
30Your Turn: Is the following data exponential & if so, what is r? Yes, it is exponential and r ≈ 1.45
31Your Turn: Is the following data (Hours vs Your Turn: Is the following data (Hours vs. Number) exponential & if so, what is r?No, it is not exponential.
32Exponential Regression Procedure Verify data is exponential.Graph scatterplot & calculate common ratioTransform data to linear by taking the log of the response variable.Calculate the LSRL for the transformed data; log ŷ =b0+b1x (linear model). Analyze using linear techniques, LSRL, r, r2, and residuals.Find exponential model for the original data by inverse transformation of the LSRL, exponentiating both sides of the LSRL equation to base 10; ŷ = C • 10kx (exponential model).
33Example: Data Year Mbbl 1880 1890 1900 1910 1920 1930 1940 1950 1960 Annual crude oil production from 1880 to 1970Year1880189019001910192019301940195019601970Mbbl30771493286891,4122,1503,8037,67416,690
34What to do: Graph scatterplot. Calculate common ratio. Transform data to linear (take the log of y).Calculate LSRL of transformed data & graph.Analyze transformed data (r, r2, residual plot).Perform inverse transformation (exponentiate LSRL to base 10).Graph exponential model.
35Back to the Data Year Mbbl 1880 1890 1900 1910 1920 1930 1940 1950 Annual crude oil production from 1880 to 1970Year1880189019001910192019301940195019601970Mbbl30771493286891,4122,1503,8037,67416,690
36Models of DataData is exponential (scatterplot curved pattern and constant common ratio ≈ 2.1)Linear modellog ŷ= xExponential modelŷ=( ) • xUse model on the calculate to make predictions, not the exponential model equation.Predict oil production for 1956.6564 MbblPredict oil production for 1992.75027 Mbbl – extrapolation, be careful.
40Models for DataData is exponential (scatterplot curved pattern and constant common ratio ≈ 1.04)Linear ModelLog ŷ = xExponential Modelŷ = (101.89) • ( x)If comparing Height vs Weight, could a common ratio be calculated?NO, because the explanatory variable Height does not in crease in equal increments.Have to calculate different models and see which best fits the data.
42Power Function Model Power Function general form: y = axb When we apply the log transformation to the response variable y in an exponential growth model, we produce a linear relationship. To produce a linear relationship from a power function model, we apply the log transformation to both variables (x & y).Here is how it is done.Power function model: y = axbTake the log of both sides of the equation:log y = log (axb)Using the product and power properties of logs, this results in a linear relationship between log y and log x.log y = log a + log xblog y = log a + b log xThe power b in the power function model becomes the slope of the straight line that links log y to log x.
43Inverse Transformation Obtaining a power function model for the original data from the LSRL on the transformed data.LSRL will have the form:log ŷ = a + b log xInverse transform the LSRL by exponentiating both sides of the equation to base 10.10log ŷ = 10(a + b log x)ŷ = (10a)(10b log x)ŷ = (10a)(10log x)bŷ = (10a)(xb) which is in the form y = C · xbA Power Function (can not be done on the calulator, must be done by hand).
44Power Function Procedure Graph scatterplot.Determine it is a power function (ie. not exponential).Transform data to linear (take the log of y & x).Calculate LSRL of transformed data & graph.Analyze transformed data (r, r2, residual plot).Perform inverse transformation (exponentiate LSRL to base 10).Graph power model.Make predictions based on the power model.
45Example 1The table shows the temperature of an instrument measured as its distance from a heat source is varied. Find a suitable model for Dist. vs Temp.LSRL: log(Temp.) = log(Dist.)log ŷ = log xPower model: Temp. = (104.84)·(Dist.)-.255ŷ = · x-.255
46Your Turn:The owner of a Video Game Store records the business costs and revenue for different years with the results listed. Find the best model.LSRL: log ŷ = log xPower model: ŷ = · x.4 or ŷ = (1995)x.4
47What Can Go Wrong? Don’t expect your model to be perfect. Don’t stray too far from the ladder.Don’t choose a model based on R2 alone:
48What Can Go Wrong? Beware of multiple modes. Re-expression cannot pull separate modes together.Watch out for scatterplots that turn around.Re-expression can straighten many bent relationships, but not those that go up then down, or down then up.
49What Can Go Wrong? Watch out for negative data values. It’s impossible to re-express negative values by any power that is not a whole number on the Ladder of Powers or to re-express values that are zero for negative powers.Watch for data far from 1.Data values that are all very far from 1 may not be much affected by re-expression unless the range is very large. If all the data values are large (e.g., years), consider subtracting a constant to bring them back near 1.
50What have we learned?When the conditions for regression are not met, a simple re-expression of the data may help.A re-expression may make the:Distribution of a variable more symmetric.Spread across different groups more similar.Form of a scatterplot straighter.Scatter around the line in a scatterplot more consistent.
51What have we learned?Taking logs is often a good, simple starting point.To search further, the Ladder of Powers or the log-log approach can help us find a good re-expression.Our models won’t be perfect, but re-expression can lead us to a useful model.