topic Re: How JMP calculate confidence interval for a linear fit model in Discussions

How JMP calculates confidence interval for a linear fit model

ILoveJMP — Wed, 03 May 2017 21:40:09 GMT

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!

Re: How JMP calculate confidence interval for a linear fit model

shoffmeister — Mon, 13 Mar 2017 13:10:18 GMT

Hi!

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

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

Re: How JMP calculate confidence interval for a linear fit model

ILoveJMP — Mon, 13 Mar 2017 17:51:05 GMT

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!

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Mon, 13 Mar 2017 18:08:26 GMT

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.

Re: How JMP calculate confidence interval for a linear fit model

ILoveJMP — Mon, 13 Mar 2017 18:23:31 GMT

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

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Mon, 13 Mar 2017 18:46:31 GMT

Second answer:

No, it is not more appropriate to use the prediction interval if you are estimating or testing the expected value (or mean) of the response with an interval.
Yes, it is more appropriate to use the prediction interval if you are estimating a proportion of the population.

They are both correct and both appropriate, depending on what you want to estimate or test. They answer different questions. They are not used for the same purpose. The former is for the statistic and the latter is for the data.

BTW, you can easily visualize either one or both of these intervals when you fit the line in Bivariate. Use same menu under the plot and select Confid Shaded Fit or Confid Shaded Indiv. One is related to the uncertainty in the sample statistic (regression line) and the other is related to the uncertainty of individual observations.

Re: How JMP calculate confidence interval for a linear fit model

ILoveJMP — Mon, 13 Mar 2017 18:50:48 GMT

Thanks for the detailed explanation!

Re: How JMP calculate confidence interval for a linear fit model

akrantasiwakoti — Tue, 15 Aug 2017 12:52:07 GMT

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

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Tue, 15 Aug 2017 15:17:54 GMT

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.

Re: How JMP calculate confidence interval for a linear fit model

Jeff_Perkinson — Tue, 15 Aug 2017 15:25:42 GMT

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.

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Tue, 15 Aug 2017 15:25:49 GMT

Yes, I can show you how JMP performs the Vec Quadratic() calculation. Here is a direct excerpt from the guide:

Select Help > Books > Scripting Guide. Then press CTRL-F and enter "quadratic(" for the search string. Very easy.

Using the same simple technique, I found the answer to your next question:

This entry is right above the one for Vec Quadratic().

Re: How JMP calculate confidence interval for a linear fit model

akrantasiwakoti — Tue, 15 Aug 2017 15:58:56 GMT

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.

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Tue, 15 Aug 2017 16:14:22 GMT

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.

Re: How JMP calculate confidence interval for a linear fit model

akrantasiwakoti — Thu, 17 Aug 2017 03:08:37 GMT

Thanks Mark! But where in the JMP report can i actually see these values (other than the formula box I showed here).

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Thu, 17 Aug 2017 15:14:07 GMT

The covariance matrix for the estimates is not provided in the JMP reports for Bivariate or Fit Least Squares. (Fit Least Squares provides an optional report about the correlation of the estimates, but that information is not what you want.) It is not available through a menu command, either. It is available as the first argument to the Vec Quadratic() function when you save the formula for confidence intervals, as you already discovered.

All of the textbooks about linear regression methods that I know of cover the linear model, parameter estimation, interval estimation, covariance of the estimates, and more. I am sure that much of the same information is available on reliable Web sites if you don't have or do not want to procure one of these textbooks.

Given that you are using JMP and JMP already provides these intervals graphically and numerically, I don't understand why you want to go through of the explicit computations.

Re: How JMP calculate confidence interval for a linear fit model

Mark_Bailey — Fri, 18 Aug 2017 15:46:34 GMT

Here is a script that demonstrates how the covariance of the estimates is computed using your sample of data.

Names Default to Here( 1 );

// make example into data table
dt = New Table( "Example for CI in Regression",
	Add Rows( 20 ),
	New Column( "X",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Values(
			[7.9, 13, 10.8, 11.6, 10.5, 11.1, 8.8, 11.9, 11.1, 9, 7.4, 6.9, 9.4, 7.2,
			8.8, 7.1, 12.4, 12.1, 15.1, 14.2]
		)
	),
	New Column( "Y",
		Numeric,
		"Continuous",
		Format( "Best", 12 ),
		Set Values(
			[115, 154, 156, 182, 124, 157, 129, 181, 164, 122, 100, 86.2, 118.9,
			94.2, 114.3, 104.9, 181.5, 166.1, 166.1, 157.7]
		)
	)
);

// perform regression analysis
fls = dt << Fit Model(
	Y( :Y ),
	Effects( :X ),
	Personality( "Standard Least Squares" ),
	Emphasis( "Minimal Report" ),
	Run(
		:Y << {Summary of Fit( 1 ), Analysis of Variance( 1 ),
		Parameter Estimates( 1 ), Plot Actual by Predicted( 0 ),
		Plot Residual by Predicted( 0 ), Plot Studentized Residuals( 0 ),
		Plot Effect Leverage( 0 ), Box Cox Y Transformation( 0 )}
	)
);

// obtain model matrix x
xx = fls << Get X Matrix;

// compute covariance matrix of parameter estimates
cov est = Inverse( xx` * xx );

// display covariance matrix
New Window( "Estimates Covariance",
	Outline Box( "Covariance of Estimates",
		Matrix Box( cov est )
	)
);

The first part of the script produces a data table with the data that you first provided. I use the Fit Least Squares platform to get the regression model matrix but it could also be made using matrix operations. Then it computes and displays the covariance matrix.

You should recognize this matrix as the first argument in the Vec Quadratic() function in the column formula that you saved.

The covariance matrix is the inverse of the information matrix for the parameters, which, in turn, is the Transpose( XX ) * XX, where XX is the model matrix. XX has a column for every term in the linear model. In your example, the frist column is 1 to estimate the intercept (constant response) and the second column is the X data column from the data table (your predictor). If you included the quadratic term, the third column of XX would be X^2, and so on. The response column Y is not involved.

I hope that this example illustrates what the covariance matrix is and where it comes from.

Re: How JMP calculate confidence interval for a linear fit model

papaya — Fri, 06 Oct 2017 22:43:01 GMT

I think my JMP is an old versiont (10.0) so that in my drop down menu it does not have ""Mean Confidence Limit formula" or "Indiv Confidence Limit Formula". Any suggestion how to get those Y+CI and Y-CI numbers? Thanks!

Re: How JMP calculate confidence interval for a linear fit model

Ressel — Fri, 04 Nov 2022 10:23:11 GMT

This forum is wonderful!