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 Equation: Line through origin

Introduction

Prism's linear regression analysis fits a straight line through your data, and lets you force the line to go through the origin. This is useful when you are sure that the line must begin at the origin (X=0 and Y=0).

Prism's nonlinear regression offers the equation Line through origin. It offers more options than its linear regression analysis, such as the ability to compare two models, apply weighting, automatically exclude outliers and perform normality tests on the residuals. See a longer discussion of the advantages of using the nonlinear regression analysis to fit a straight line.

When to force the line through the origin

In many scientific situations, it just makes sense that when X=0, Y must also equal 0, so the line should be forced to go through the origin (X=0, Y=0). But even in these situations, it can make sense to fit an ordinary linear regression line that also fits the intercept. The data you are analyzing may be far from the origin, and you may get a better fit through the points (what you care about) when you don't force the line to go through the origin. This can happen when the true model is curved (beginning at the origin), so a line through the data points that is not forced to the origin may fit a whole lot better than line forced to go through the origin.

Prism makes it easy to compare the fit that goes through the origin with one that doesn't.

Step by step

Create an XY data table. There is one X column, and many Y columns. If you have several experimental conditions, place the first into column A, the second into column B, etc.

After entering data, click Analyze, choose nonlinear regression, choose the panel equations for lines, and choose Line Through Origin

Model

Y=  Slope*X

Slope is the slope of the line, expressed in Y units divided by X units. It estimates the ratio of Y/X in the entire population.

Weighting

In situations where linear regression through the origin is appropriate, it is common for the variation among replicate Y values increases as X (and Y) increase. Prism provides two weighting choices (in the Weights tab) that are used in this situation.  Weight by 1/X2 when you think the variance in Y is proportional to the square of X, which means the SD among Y values is proportional to X.Weight by 1/X if you think the variance in Y is proportional to X.

R2 when fitting a line through the origin

When you constrain a line to go through a point, there are two possible ways to compute R2:

Compare the fit of the best-fit line with the fit of a horizontal line at the mean Y value. But that null hypothesis (horizontal line through the Ymean) doesn't obey the constraint that it go through the origin.

Compare the best-fit line with a horizontal line at Y=0. This obeys the constraint, but often fits the data really badly, pushing up the R2 value.

When you use nonlinear regression to fit a line through the origin, Prism uses the first definition above. If you use linear regression, and ask Prism to constrain the line to go through the origin, it simply will not report R2 at all.

Reference

 1 J. G. Eisenhauer, Regression through the Origin. Teaching Statistics 25, 76–80 (2003).