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I'm working with time series data that has error in both the dependent and independent variables, so I'm analyzing each half hour of data with model II linear regression, specifically geometric mean regression (GMR, described here). In GMR, the slope estimate is the square root of the dependent data divided by the square root of the independent data. The intercept can be calculated based on the fact that the line passes through the point (mean(x), mean(y)).

Later in my analysis, I have to filter out half hours based on the standard error of the slope estimate, but I don't know how to calculate the standard error of the GMR slope.

Possible leads:

I've found a function in the lmodel2 R package that calculates the 97.5% confidence intervals for the GMR coefficients. Is there a way to transform that into standard error?

I read somewhere that the Standardized Major Axis method of computing coefficient standard errors can be used for GMR. This webpage presents that equation but doesn't explain the variables and I can't find an explanation in the source they cite. Does anyone know about this equation?

I know about the deming functions in R that calculate SE, but deming regression isn't behaving well with my data so I'd rather not use it.

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I found a good source that answered my question [1]. Standard error of the slope ($s_a$) can be calculated using $s_a = \frac{\sigma_y}{\sigma_x}\sqrt{\frac{1-r^2}{n}}$ where $\frac{\sigma_y}{\sigma_x}$ is the slope estimate, $r^2$ is the square of the correlation coefficient, and $n$ is the number of observations used in the regression.

The standard error of the intercept ($s_b$) can be calculated using $s_b=\sigma_y\sqrt{\frac{1-r^2}{n}(1+\frac{\bar{x}^2}{\sigma_x^2})}$, where $\bar{x}$ is the mean of the independent data.

Sources

1) Miller, R.L., and J.S. Kahn, 1962. Statistical Analysis in the Geological Sciences. Wiley: New York, 205 p

2) A brief history of model II regression usage

3) Steven Holland's lecture notes on model II regression

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