Missing confidence intervals from orthogonal regression

[#1]

Hi,

I'm using an orthogonal regression analysis on some data I have modelling the covariance of deviations made the left (y-axis) and right (x-axis) hands during a bilateral reaching task. For the most part, I am able to use the confidence intervals for the slope that JMP provides but on some analysis either the lower, upper or both confidence intervals are not calculated (a period is provided instead of a numerical value) and I can't seem to work out why.

I checked the JMP documentation (here: https://www.jmp.com/support/help/en/17.0/index.shtml#page/jmp/statistical-details-for-the-fit-orthog...) which directs me to paper (https://www.jstor.org/stable/1270564) that was used to calculate the confidence intervals, but I'm struggling to find anything from it that would explain why in some situations the CIs cannot be found, partly due to my limited mathematics ability/knowledge. Its not necessarily a problem that they can't be calculated but I would like to be able to describe why in some cases its not possible.

I'm using JMP Pro 16 on a macbook with the Ventura 13.0 OS and have provided a screenshot of the output. The data includes ~500 points per fit but is poorly correlated in some conditions. However, in other cases I have tried this doesn't seem to be a limiting factor. An example dataset is attached that I used to create the example outputs in the screenshot too if thats helpful,

Thanks,

Nick

[#2 · Accepted solution]

Hi @OrderedMongoose ,

As you described in your question, JMP calculates a confidence interval as described in the Tan and Iglewicz 1999 reference.
You can use the following script to compute a confidence interval using a parameter that JMP generates (covariance and so on).

n = 500;
gamma = 1; //Ratio
t_dist = t Quantile( 1 - 0.025, n - 2 );
s_yy = 1.641302 ^ 2; //Variance of Y
s_xx = 1.336159 ^ 2; //Variance of X
s_xy = -0.16487; //Covariance of X and Y

phi = 1 / 2 * ArcSine(
	t_dist * (2 / Sqrt( n - 2 )) * Sqrt(
		gamma * (s_yy * s_xx - s_xy ^ 2) / ((s_yy - gamma * s_xx) ^ 2 + 4 * gamma * s_xy ^ 2)
	)
);

beta_hat = -5.68667; //Estimate of Slope
theta_hat = ArcTan( beta_hat / Sqrt( gamma ) );
lower_confidence = Sqrt( gamma ) * Tan( theta_hat - phi );
upper_confidence = Sqrt( gamma ) * Tan( theta_hat + phi );
Show( Eval List( {lower_confidence, upper_confidence} ) );
Show( theta_hat - phi );

When calculating a confidence interval, beta(slope) is calculated from theta (beta equals tan(theta)). Theta is described in the reference. When calculating a confidence interval with your example data, the value of theta is less than -pi/2. In this case, a lower confidence interval (32.44) is greater than an upper confidence interval (-2.51), which is not reasonable. I think this is why JMP doesn't display a result.

Another approach is to use boostraping. Please see https://www.jmp.com/support/help/en/16.2/#page/jmp/example-of-bootstrapping.shtml#. 

[#3]

Ahhh okay got it. That makes sense, thanks so much for your help @yuichi_katsumur !