**LO**cally

**WE**ighted

**S**catter plot

**S**moothing) "fits polynomials (usually linear or quadratic) to local subsets of the data, using weighted least squares so as to pay less attention to distant points" (

*Oxford Dictionary of Statistics*, entry for

**loess**), In other words, each observation $(x_i, y_i)$ is fitted to a separate linear regression based on nearby observations with the points weighted such that the further away $x$ is from $x_i$, the less it is weighted (

*Statistical Modeling for Biomedical Researchers*, Dupont). The strength of lowess smoothing is that it can reveal trends in the data and is a locally based smoother: it follows the data. Most lowess smoothing involves just two variables, a y-variable and a single x-variable, but this methodology has been extended to multiple x-variables and can be executed in Stata with the -mlowess- command. According to the help file, "mlowess computes lowess smooths of

*yvar*on all predictors in

*xvarlist*simultaneously; that is, each smooth is adjusted for the others." The authors of mlowess, however, caution that multiple variable lowess smoothing should be reserved primarily for exploratory graphics, not inferential model fitting.

Consider the Stata system data set,

*auto.dta*, and the variables

*mpg, price, weight, length*, and

*gear ratio*. In the bivariate case between

*mpg*and

*price*, there is some hint of a slight quadratic relationship but this could be due to a couple of outlying observations. The inclusion of the ordinary least squares fitted regression line illustrates the mild departure from the linear model for the cheapest and most expensive vehicles.

In the multiple-variable case (controlling for

*weight, length*, and

*gear ratio*), the mild non-linear curve between

*mpg*and

*price*is dampened although the inexpensive vehicle with super high gas mileage appears to be largely responsible for the departure from linearity on the left side of the graph. The remaining three graphs ---

*mpg*versus

*weight, length*, and

*gear ratio*, respectively --- all reveal mild non-linear associations even after for controlling for the other variables.

The Stata code used to generate the graphs above follows:

capture log close

log using lowess-01, replace

datetime

// program: lowess-01.do

// task: lowess demo

// project: n/a

// author: cjt

// born on date: 20130329

// #0

// program setup

version 11.2

clear all

macro drop _all

set more off

// #1

// read in auto

sysuse auto

// #2

// bivariate lowess smoother between mpg (yvar) and price w/ fitted values line

lowess mpg price, addplot(lfit mpg price) xtick(2000(2000)18000) xlabel(2000(2000)18000) ///

scheme(sj) legend(row(1)) title("Lowess Smoother and Fitted Regression Line")

* **save graph

gr save mpgXprice-01, replace

* **export graph for inclusion into blog

gr export mpgXprice-01.png, replace

// #3

// multiple lowess smoother

mlowess mpg price weight length gear_ratio, scatter(msymbol(o)) scheme(sj)

* **export graph

gr export mpgXall-01.png, replace

log close

exit

log using lowess-01, replace

datetime

// program: lowess-01.do

// task: lowess demo

// project: n/a

// author: cjt

// born on date: 20130329

// #0

// program setup

version 11.2

clear all

macro drop _all

set more off

// #1

// read in auto

sysuse auto

// #2

// bivariate lowess smoother between mpg (yvar) and price w/ fitted values line

lowess mpg price, addplot(lfit mpg price) xtick(2000(2000)18000) xlabel(2000(2000)18000) ///

scheme(sj) legend(row(1)) title("Lowess Smoother and Fitted Regression Line")

* **save graph

gr save mpgXprice-01, replace

* **export graph for inclusion into blog

gr export mpgXprice-01.png, replace

// #3

// multiple lowess smoother

mlowess mpg price weight length gear_ratio, scatter(msymbol(o)) scheme(sj)

* **export graph

gr export mpgXall-01.png, replace

log close

exit

Lowess smoothing, although not the most rigorous and complicated statistical technique, is great for exploratory analysis and can help reveal relationships between variables that may have otherwise gone unnoticed.