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Funky confidence intervals

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carey's picture
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Joined: 10/19/2009 - 15:38
Funky confidence intervals
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Are the default confidences intervals for mxCI 95% or 90%? Or is there something funky in the documentation of mxCI? My understanding of a 95% confidence interval is that it is equivalent to a two-tailed test with alpha = .05. That is, it delimits the 2.5% most extreme values at the low end and the 2.5% most extreme values at the high end, leaving the 95% most likely values between the confidence intervals.

The documentation seems to agree with this at one point. "In all cases, a two-sided hypothesis test is assumed. Therefore, the upper bound will exclude 2.5% (for interval=0.95) even though only one bound is requested. " Yet later, it states "Thus, a requested confidence interval for a parameter will first determine the corresponding quantile from the chi-squared distribution with one degree of freedom (a value of 3.841459 when a 95 percent confidence interval is requested)." The correct critical value of chi square for a two-tailed, 95% confidence interval is 5.024.

If I dummy up some data to test this (see attached), I get default 90% two-tailed confidence intervals. Here is the result of fitting a model with three parameters--a mean, a variance, and a covariance--to simulated "twin" data with one zygosity. I fixed a parameter at the lower (or upper) confidence interval given by the general model and computed the LR chi square and its p value.

parm CI chisq pValue
1 mean lower 3.854471 0.04961358
2 mean upper 3.854471 0.04961358
3 V lower 3.854562 0.04961087
4 V upper 3.867740 0.04922277
5 C lower 3.854694 0.04960697
6 C upper 3.867859 0.04921927

This from OpenMx 2.0.1-4157 on a Mac. Version 2.2.2 in Linux gave similar results.

mhunter's picture
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Joined: 07/31/2009 - 15:26
Quote

"Often reported are the 95% confidence intervals; these would be obtained by finding the point at which the Chi-Square reaches the .05 level of significance (approx 3.84) in each direction. Although it might be thought that the two 5% tails (one in each direction) sum to give 10% of the distribution outside the interval and hence a 90% confidence interval, this is not the case. The Chi-Square test in each direction is conditional on the value being on one side of the distribution, so it is still a 95% interval when both sides are considered jointly." (Neale & Miller, 1997, p. 115-116)

Neale, M. C. & Miller, M. B. (1997). The Use of Likelihood-Based Confidence Intervals in Genetic Models Behavior Genetics, 27, 113-120.

Empirically, the 95% CI is often close to [Estimate +/- 1.95*SE], but feel free to check with a simulation.

jpritikin's picture
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Joined: 05/24/2012 - 00:35
it's part of our test suite

Also see models/passing/simplestCI.R, in particular this section:

# For multivariate normal means, SEs match likelihood-based CIs
omxCheckCloseEnough(fit2$output$estimate['m1'] + fit1$output$standardErrors['m1',] * qnorm(.025),
                    fit2$output$confidenceIntervals['m1', 'lbound'], .0001)
omxCheckCloseEnough(fit2$output$estimate['m1'] - fit1$output$standardErrors['m1',] * qnorm(.025),
                    fit2$output$confidenceIntervals['m1', 'ubound'], .0001)
carey's picture
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Joined: 10/19/2009 - 15:38
Thanks, All My mistake--I

Thanks, All
My mistake--I should have realized that a chi square statistic is non directional.
Greg