Chapter 6. Transformation of Variables

Transcription

1 6.1 Chapter 6. Transformation of Variables 1. Need for transformation 2. Power transformations: Transformation to achieve linearity Transformation to stabilize variance Logarithmic transformation MACT 427: Applied Regression Methods, Prof Hadi, Need for Transformation Data do not always come in a form that is immediately suitable for analysis. We often have to transform the variables before carrying out the analysis. It often becomes necessary to fit a linear regression model to the transformed rather than to the original variables. This is common practice. MACT 427: Applied Regression Methods, Prof Hadi, 6.2 1

2 6.3 Need for Transformation The necessity for transforming the data arises because the original variables, or the model in terms of the original variables, violate one or more of the standard regression assumptions. Two of the most commonly violated assumptions are the linearity of the model and the constancy of the error variance. MACT 427: Applied Regression Methods, Prof Hadi, Need for Transformation Transformations are applied to accomplish certain objectives such as to: To ensure linearity To achieve normality To stabilize the variance We illustrate transformation mainly using simple regression. In multiple regression where there are several predictors, some may require transformation and others may not. MACT 427: Applied Regression Methods, Prof Hadi, 6.4 2

3 6.5 As we mentioned before, models can be linear or nonlinear. Some of the nonlinear models can be transformed to linear models. Those models are called linearizable (nonlinear) models. Note that a regression model is linear when the parameters present in the model occur linearly even if the predictor variables occur nonlinearly. MACT 427: Applied Regression Methods, Prof Hadi, Example: Each of the following models is linear because the parameters enter linearly: y 0 1 x y 0 1x 1x 2 y 0 1 log( x) y 0 1 x MACT 427: Applied Regression Methods, Prof Hadi, 6.6 3

4 6.7 Example: But the model x e 1 y 0 Is nonlinear in the parameters. To satisfy the assumptions of the standard regression model, instead of working with the original variables, we sometimes work with transformed variables. MACT 427: Applied Regression Methods, Prof Hadi, Theoretical considerations may specify that the relationship between two variables is nonlinear. An appropriate transformation of the variables can make the relationship between the transformed variables linear. MACT 427: Applied Regression Methods, Prof Hadi, 6.8 4

5 6.9 Example: From experimental psychology, a learning model that is widely used states that the time taken to perform a task on the i-th occasion is T i i, 0, 0 1 Which is nonlinear. Taking the logarithm of both sides we obtain log( T i ) log ilog( ), Which is linear. Why? 0, 0 1 MACT 427: Applied Regression Methods, Prof Hadi, Figure 6.1: y x log( y) log( ) log( x) y' 0 1 x' MACT 427: Applied Regression Methods, Prof Hadi,

6 6.11 Figure 6.2: y e x log( y) log( ) x y' x 0 1 MACT 427: Applied Regression Methods, Prof Hadi, Figure 6.3: y y log( x) 0 1 x' MACT 427: Applied Regression Methods, Prof Hadi,

7 6.13 Figure 6.4: x y x 1 1 y x exp( x) y 1 exp( x) y' x MACT 427: Applied Regression Methods, Prof Hadi, Transformation to Stabilize the Error Variance The constancy of error variance is one of the standard assumptions of least squares theory. It is often referred to as the Homogeneity or Homoscedasticity or simply the constant variance assumption. The variance of the response variable Y may be related to its mean. This is usually the case when the distribution of Y is not normal. MACT 427: Applied Regression Methods, Prof Hadi,

8 6.15 Transformation to Stabilize the Error Variance When this assumption does not hold, we say we have the Heteroscedasticity, Heterogeneity or simply the non-constant variance problem. This can be seen for example in the plot of the standardized residuals versus each X or versus the fitted values. MACT 427: Applied Regression Methods, Prof Hadi, 6.15 Residuals X 6.16 Transformation to Stabilize the Error Variance Transformations are also used to stabilize the error variance, that is, to make the error variance constant for all the observations. All these transformations can be achieved by Power Transformation. MACT 427: Applied Regression Methods, Prof Hadi,

9 6.17 Power Transformations Power Transformations are best explained by examples: Example 1: Brain Weight Data Example 2: Enzyme Data Use Data Desk MACT 427: Applied Regression Methods, Prof Hadi,