Power transforms are used in multiple fields, including multi-resolution and wavelet analysis,[1] statistical data analysis, medical research, modeling of physical processes,[2] geochemical data analysis,[3] epidemiology[4] and many other clinical, environmental and social research areas.
The power transformation is defined as a continuous function of power parameter λ, typically given in piece-wise form that makes it continuous at the point of singularity (λ = 0).
The inclusion of the (λ − 1)th power of the geometric mean in the denominator simplifies the scientific interpretation of any equation involving
produces an expression that establishes that minimizing the sum of squares of residuals from
is equivalent to maximizing the sum of the normal log likelihood of deviations from
Sometimes Y is a version of some other variable scaled to give Y = 1 at some sort of average value.
Box and Cox also proposed a more general form of the transformation that incorporates a shift parameter.
Bickel and Doksum eliminated the need to use a truncated distribution by extending the range of the transformation to all y, as follows: where sgn(.)
[5] Bickel and Doksum also proved that the parameter estimates are consistent and asymptotically normal under appropriate regularity conditions, though the standard Cramér–Rao lower bound can substantially underestimate the variance when parameter values are small relative to the noise variance.
is estimated using the profile likelihood function and using goodness-of-fit tests.
[10] Confidence interval for the Box–Cox transformation can be asymptotically constructed using Wilks's theorem on the profile likelihood function to find all the possible values of
The horizontal reference line is at a distance of χ12/2 from the maximum and can be used to read off an approximate 95% confidence interval for λ.
In this case, the maximum of the likelihood is close to zero suggesting that a shift parameter is not needed.
The final panel shows the transformed data with a superimposed regression line.
In the current example, the data are rather heavy-tailed so that the assumption of normality is not realistic and a robust regression approach leads to a more precise model.
Economists often characterize production relationships by some variant of the Box–Cox transformation.
[13] Consider a common representation of production Q as dependent on services provided by a capital stock K and by labor hours N: Solving for Q by inverting the Box–Cox transformation we find which is known as the constant elasticity of substitution (CES) production function.
When λ = 1, this produces the linear production function: When λ → 0 this produces the famous Cobb–Douglas production function: The SOCR resource pages contain a number of hands-on interactive activities[14] demonstrating the Box–Cox (power) transformation using Java applets and charts.
The transformation law reads: The Box-Tidwell transformation is a statistical technique used to assess and correct non-linearity between predictor variables and the logit in a generalized linear model, particularly in logistic regression.
This transformation is useful when the relationship between the independent variables and the outcome is non-linear and cannot be adequately captured by the standard model.
The Box-Tidwell test is typically performed by augmenting the regression model with terms like
If significant, this suggests that a transformation should be applied to achieve a linear relationship between the predictor and the logit.
The transformation is beneficial in logistic regression or proportional hazards models where non-linearity in continuous predictors can distort the relationship with the dependent variable.
It is a flexible tool that allows the researcher to fit a more appropriate model to the data without guessing the relationship's functional form in advance.
Violations of this assumption can lead to biased estimates and reduced model performance.
The Box-Tidwell transformation introduces an interaction term between each continuous variable Xi and its natural logarithm
This term is included in the logistic regression model to test whether the relationship between Xi and the logit is non-linear.
A statistically significant coefficient for this interaction term indicates a violation of the linearity assumption, suggesting the need for a transformation of the predictor.
One limitation of the Box-Tidwell transformation is that it only works for positive values of the independent variables.
If your data contains negative values, the transformation cannot be applied directly without modifying the variables (e.g., adding a constant).