# Confidence interval of multi-step calibration

I need to perform a two-step calibration and I need you to tell me if I am doing that correctly. First I want to calibrate a force sensor with masses. I measure several weights and perform a linear regression of my data such as: $$F_1 = a_1 s_1+b_1$$ with $F_1$ the applied force in N, $s_1$ the sensor value and $a_1$ and $b_1$ the regression parameters.

It is easy to estimate a confidence interval for $F_1$ from: $$\sigma_{F_1}=\sqrt{\frac{\sum{(F_1-(a_1s_1+b_1))^2}}{N-1}}$$ But then it starts to get tricky for me. I use the force sensor to calibrate another force sensor on a machine. I perform the same type of experiment with different loads, and I want to perform a linear regression such as: $$F_1 = a_2 s_2+b_2$$ with $F_1$ the measured force with the force sensor, $s_2$ the sensor value of the machine and $a_2$ ans $b_2$ the regression parameters. So I already know $\sigma_{F_1}$, how can I get a confidence interval of a single $s_2$ measure?

• Is it safe to assume that sensor 2 has the same error variance as sensor 1? It seems that if you are independently fitting a second regression for the new sensor then you are assuming the calabration coefficients might differ. If the calibration coefficients differ then error variance might also differ. You could test the variances to see if they differ significantly. – Michael R. Chernick Aug 31 '12 at 14:59
• What exactly is your problem? Are you trying to etimate an unknown s$_2$ given the force F$_1$ and fitted coefficients b$_2$ and a$_2$ in which case the estimate for s$_2$ is (F$_1$-b$_2$)/a$_2$ and the confidence interval would depend on the variance covariance matrix for (a$_2$, b$_2$). Or are you going to use a$_1$ and b$_1$ form the first model to estimate s$_2$. The former approach requires no assumption about the relationship between sensor 1 and sensor 2. But the latter approach assumes that the same calibration can be used for both sensors. – Michael R. Chernick Aug 31 '12 at 15:05
• Ultimately I want to estimate the force $F_1$ on the machine given $s_2$. But I use the sensor to get the regression coefficients $a_2$ and $b_2$. – touste Aug 31 '12 at 15:23
• Sensor 2 is not likely to have the same variance as sensor 1, as they are two completely different sensors. – touste Aug 31 '12 at 15:24
• I was referring to the model residual variance which is different from the measurement variance. If you expect a different residual variance and different coefficients for the regression. the first sensor should not come into play. – Michael R. Chernick Aug 31 '12 at 16:06

As I interpret your problem sensors 1 and 2 are different and so 1 isn't giving you information to calibrate 2. The confidence interval for the force F$_1$ is the standard confidence interval for an estimated response given the observed value of the covariate (in this case s$_2$).