How JMP calculates confidence interval for a linear fit model

[#1]

For the following set of data, 

X	Y
7.9	115
13	154
10.8	156
11.6	182
10.5	124
11.1	157
8.8	129
11.9	181
11.1	164
9	122
7.4	100
6.9	86.2
9.4	118.9
7.2	94.2
8.8	114.3
7.1	104.9
12.4	181.5
12.1	166.1
15.1	166.1
14.2	157.7

one can generate a linear fit model  as shown below:

Linear Fit
Y = 24.522739 + 11.068566*X

Summary of Fit

RSquare 0.743712
RSquare Adj 0.729474
Root Mean Square Error 16.17783
Mean of Response 138.695
Observations (or Sum Wgts) 20

The question I have is: how should I go about and generate the 99% confidence intervals for the projected Y when X = 11.8 and 12.7? Or even better if you could elaborate the details on how JMP calculates the confidence intervals (such as, 95%, 99%) for a linear fit model.

Many thanks in advance!

[#2]

Hi!

to get the predicted values and confidence intervals you would do the following: 

1. Use the hotspot (red triangle) next to "Linear Fit". 
2. You will find a menu entry "Set alpha Level". Set this to 1% if you are interested in 99% confidence.
3. Now use the same menu and choose "Save predicteds" and "Mean Confidence Limit formula" (or "Indiv Confidence Limit Formula).
4. As a result you will find 3 new columns in the data table. 

   To see how jmp calculates the prediction/CIs you could just klick on the "+"-symbol on the left hand side (next to the column names).

5. To get the prediction and CI for specific values of X you would just add new rows (like it did in the picture above: Row 21 and 22). JMP will automatically generate the predictions and CI-values for the X-values you enter.

Hope that helps.

[#3]

Thanks for the reply! This was what I did. However, I was puzzled by the Vec Quadratic term in the formula as shown belown and also I was expecting 2.576 multiplying the standard error (for 99% confidence interval calculation instead of 2.878 or 1.96 for 95% confidence interval calculation)

(24.5227390655596 + 11.0685662563684 * :X) - 2.87844047273861 *
Sqrt(
Vec Quadratic(
[1.00352196297906 -0.0924403260280234, -0.0924403260280234
0.00896173786020585],
[1] || :X
) * 261.722077754035
)

2nd quesition: is it more appropriate to use Indiv Confidence Limit Formula than Mean Confidence Limit Formula? If not, why so?

Thanks!

[#4]

The Vec Quadratic() function is an optimized way to perform the matrix computations before applying the scalar multipliers.

You want a two-sided 99% confidence interval, so the probability cutoff on each tail is 0.005%. This calculation with the given 18 degrees of freedom for the error indicates the correct multiplier:

t Quantile( 0.995, 18 ) -> 2.87844047273861

Note that you can select Graph > Profiler and enter all three of the formula columns. You can now change X by dragging or typing over the current value (in red) and entering an exact value. The profilers will give the point estimates, lower and upper bounds.

[#5]

Thanks for the speedy feedback, Mark! BTW, could you please comment on my 2nd question: indiv or mean?

[#6]

I can't answer your second question about which kind of interval is appropriate because you  have not stated how you would use or interpret the interval. They are both appropriate but for different questions. It depends on the alternative hypothesis you are trying to test.

[#7]

HI Mark!

Can you show how JMP performs Vec Quadratic calculation? I am looking for a generic equation that JMP uses to calculate prediction interval for OLS. Thanks

Ram

[#8]

Help->Scripting Index will tell you that Vec Quadratic(S, X) is a evaluated as Vec Diag(X *S * X`).

Vec Diag(X) returns the diagonal elements of the square matrix as a vector.

[#9]

Thanks Jeff and Mark for your very quick reply. I got that part. However, in my formula, I have (see attached) 0.395, 0.051, and 0.007 inside first term in Vec Quadratic. I was wondering what these values represent? 0.051 and 0.007 have something to do with the slope, but not very sure. Again, thanks for your assistance. 

[#10]

Taken directly from the description and explanation I just posted, it is a "symmetric matrix," the "inverse covariance matrix." It refers to the covariance of the parameter estimates, of course.