15.2.6 Algorithms (Fit Linear with X Error)RefLinearXErr
The Fitting Model
For given dataset , where X is the independent variable and Y is the dependent variable, and are Errors for X, Y, respectively.  Fit Linear with X Error fits the data to a model of the following form:
Fit Control
Computation Method
 York Method
 York Method is the computation method of D. York, described in Unified equations for the slope, intercept, and standard error of the best straight line
 FV Method
 FV Method is the computation method of Giovanni Fasano & Roberto Vio, described in Fittng a Straight Line with Errors on Both Coordinates.
 Deming Method
 Deming regression is the maximum likelihood estimation of an errorsinvariables model, the X/Y errors are assumed to be independent identically distributed.
 Correlation Between X and Y Errors
 Correlation Between X and Y Errors (For York method only)
 Standard Deviation of X/Y
 Standard Deviation of X/Y (For Deming method only)
Quantities (York Method)
When you perform a linear fit, you generate an analysis report sheet listing computed quantities. The Parameters table reports model slope and intercept (numbers in parentheses show how the quantities are derived):
Fit Parameters
Fitted Value and Standard Errors
Define which involves the weight (error) for both x and y;
Therein, are weights of , is Correlation between X and Y Errors (i.e. and ), and .
The slope of the fitted line for with no weighting (errors) is the initial value for . They should be solved iteratively, until successive estimates of agree within desired tolerance.
The concise equations which estimate parameters and for the bestfit line with X_Y errors are:
where .
U and V are the deviation for X and Y:
and
The corresponding variation and standard error for parameter is:
where , is the expectation value of , and .
The standard error for parameters is final given by:
where is:
tValue and Confidence Level
If the regression assumptions hold, we have:
The ttest can be used to examine whether the fitting parameters are significantly different from zero, which means that we can test whether (if true, this means that the fitted line passes through the origin) or . The hypotheses of the ttests are:


The tvalues can be computed by:
With the computed tvalue, we can decide whether or not to reject the corresponding null hypothesis. Usually, for a given confidence level , we can reject when . Additionally, the pvalue, or significance level, is reported with a ttest. We also reject the null hypothesis if the pvalue is less than .
Prob>t
The probability that in the t test above is true.
where tcdf(t, df) computes the lower tail probability for the Student's t distribution with df degree of freedom.
LCL and UCL
From the tvalue, we can calculate the Confidence Interval for each parameter by:
where and is short for the Upper Confidence Interval and Lower Confidence Interval, respectively.
CI Half Width
The Confidence Interval Half Width is:
where UCL and LCL is the Upper Confidence Interval and Lower Confidence Interval, respectively.
For more information ,see Reference 1 (below).
Fit Statistics
Degrees of Freedom
n is total number of points
Residual Sum of Squares
Reduced ChiSqr
Pearson's r
In simple linear regression, the correlation coefficient between x and y, denoted by r, equals to:
can be computed as:
RootMSE (SD)
Root mean square of the error, or residual standard deviation, which equals to:
Covariance and Correlation Matrix
The Covariance matrix of linear regression is calculated by:
The correlation between any two parameters is:
Quantities (FV Method)
FV Method is the computation method of Giovanni Fasano & Roberto Vio, described in Fittng a Straight Line with Errors on Both Coordinates.
The weighting is defined as:
The slope of the fitted line for with no weighting (errors) is .
Let
by minimizing the sum , we can get the estimate value and by setting the partial derivatives to 0.
where
should be solved iteratively, until successive estimates of agree within desired tolerance.
For each parameter standard error, please refer to Linear Regression Model
For more information ,see Reference 2 (below).
Quantities (Deming Method)
When you perform a linear fit, you generate an analysis report sheet listing computed quantities. The Parameters table reports model slope and intercept (numbers in parentheses show how the quantities are derived):
Fit Parameters
Deming regression is used for situation where both x and y are subjected to measurement error.
Assume are independent identically distributed with , and that are independent identically distributed with , where denotes the normal distribution with mean 0 and standard deviation . If , it’s orthogonal regression.
The weighted sum of squared residuals of the model is minimized:
Fitted Value and Standard Errors
We can solve the parameters:
where:
and:
The corresponding variation for parameters is:
The standard error for parameters can be estimated by:
and
tValue and Confidence Level
If the regression assumptions hold, we have:
The ttest can be used to examine whether the fitting parameters are significantly different from zero, which means that we can test whether (if true, this means that the fitted line passes through the origin) or . The hypotheses of the ttests are:


The tvalues can be computed by:
With the computed tvalue, we can decide whether or not to reject the corresponding null hypothesis. Usually, for a given confidence level , we can reject when . Additionally, the pvalue, or significance level, is reported with a ttest. We also reject the null hypothesis if the pvalue is less than .
Prob>t
The probability that in the t test above is true.
where tcdf(t, df) computes the lower tail probability for the Student's t distribution with df degree of freedom.
LCL and UCL
From the tvalue, we can calculate the Confidence Interval for each parameter by:
where and is short for the Upper Confidence Interval and Lower Confidence Interval, respectively.
CI Half Width
The Confidence Interval Half Width is:
where UCL and LCL is the Upper Confidence Interval and Lower Confidence Interval, respectively.
For more information ,see Reference 1 (below).
Fit Statistics
Degrees of Freedom
n is total number of points
Residual Sum of Squares
See formula (33)
Reduced ChiSqr
Pearson's r
In simple linear regression, the correlation coefficient between x and y, denoted by r, equals to:
can be computed as:
RootMSE (SD)
Root mean square of the error, which equals to:
Covariance and Correlation Matrix
The Covariance matrix of linear regression is calculated by:
The correlation between any two parameters is:
Residual Plots
Residual vs. Independent
Scatter plot of residual vs. indenpendent variable , each plot is located in a seperate graphs.
Residual vs. Predicted Value
Scatter plot of residual vs. fitted results
Residual vs. Order of the Data
vs. sequence number
Histogram of the Residual
The Histogram plot of the Residual
Residual Lag Plot
Residuals vs. lagged residual .
Normal Probability Plot of Residuals
A normal probability plot of the residuals can be used to check whether the variance is normally distributed as well. If the resulting plot is approximately linear, we proceed to assume that the error terms are normally distributed. The plot is based on the percentiles versus ordered residual, and the percentiles is estimated by
where n is the total number of dataset and i is the i th data.
Also refer to Probability Plot and QQ Plot
Reference
 York D, Unified equations for the slope, intercept, and standard error of the best straight line, American Journal of Physics, Volume 72, Issue 3, pp. 367375 (2004).
 G. Fasano and R. Vio, "Fitting straight lines with errors on both coordinates", Newsletter of Working Group for Modern Astronomical Methodology, No. 7, 27, Sept. 1988.
