Chapter 5: Data Transformation
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1 Chapter 5: Data Transformation The circle of transformations The x-squared transformation The log transformation The reciprocal transformation Regression analysis choosing the best transformation TEXT: Essential Further Mathematics TOPIC DATE SET PAGE(S) WORK The circle of transformations 167 Exercise 5A: Q1 The squared transformation 172 Exercise 5B: Q1 Q7 The log transformation 179 Exercise 5C: Q1 Q7 The reciprocal transformation 185 Exercise 5D: Q1 Q6 Chapter Review Multiple-choice questions: Q1 10 Extended response questions: Q1, Q2, Q3 1
2 Transforming Data Data transformation used to linearise a scatterplot When a scatterplot appears to be curved or non-linear we transform (change) the data to straighten out the graph. A residual plot that has some sort of pattern, also indicates that data may be non-linear. We transform either the x-variable or the y-variable so that the relationship becomes closer to a straight line. It is only when the relationship can be described as linear that linear regression analysis should be performed. How do data transformations work? Data transformations work by either stretching out or compressing the scale of measurement on an axis. The result of this is that a non-linear scatterplot can be made linear. Response variable (Y) Explanatory variable (X) 3 types of Transformations Squared transformations x-squared and y-squared Log transformations log x and log y Reciprocal transformations 1/x and 1/y Expands (or stretches out) the x-scale Expands (or stretches out) the y-scale Compresses the x-scale Compresses the y-scale Compresses the x-scale Compresses the y-scale 2
3 The squared transformation stretches a scale transformations Stretch right. Stretches x values to the right so that they meet a straight line drawn from the start of the data. transformations Stretch up. Stretches y values up so that they meet a straight line drawn from the start of the data. The log transformation - compresses (pushes back) a scale transformations Compress left. Compresses x values to the left so that they meet a straight line drawn from the start of the data. transformations Compress down. Compresses y values down so that they meet a straight line drawn from the start of the data. 3
4 The reciprocal transformation - compresses (pushes back) a scale transformations Compresses larger values of x relative to lower values of x. transformations Compresses larger values of y relative to lower values of y. Summarising all transformations: 4
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21 Which transformation is the best? The best transformation is the one that results in the best linear model. To choose the best linear model, the following must be considered for each transformation applied: the residual plot as a means of evaluating the linearity of the transformed relationship the value of the coefficient of determination ( r 2 ), a higher value indicates a better fit Performing a regression analysis Example 1 A recent study showed that Cigarette consumption (per day) is related to Cost per pack. Some data drawn from the study is given in the table. Cost ($) Cigarette consumption (a) Taking Cost as the explanatory variable, verify that the scatter plot below displays the relationship between the two variables. Cigarette consumption Cost (b) Find the coefficient of determination, giving the answer as a percentage to two decimal places. Interpret this value. (c) Write down the regression equation in terms of the variables, giving coefficients to two decimal places. 21
22 (d) Obtain a residual plot. Sketch a clearly labelled diagram of the residual plot. Interpret the residual plot. (e) What are the transformations that can be applied to linearise the relationship? Cigarette consumption The possible transformations are: Cost (f) Applying the Log X transformation (i) Construct a scatter plot using Cigarette consumption as the RV and log(cost) as the EV. Cigarette consumption Log ( Cost ) 22
23 (ii) Find the coefficient of determination as a percentage correct to two decimal places, and interpret its value. (iii) Write down the least squares regression line in terms of the variables, giving the coefficients correct to two decimal places. (iv) Construct the residual plot, labelling the axes very clearly. Interpret the residual plot. (v) In concluding this exercise in regression analysis at this point, 23
24 SKILLS CHECK Recognise which transformation might be used to linearise a bivariate relationship Apply the appropriate transformation to the data set Assess the transformation for the best linear model using a residual plot and value of r 2 Perform a regression analysis with the transformed variable Be able to use the CAS calculator to apply and assess the transformation in the regression exercise 24
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