Computing Confidence Interval Functions with cifunction

A Stata Package for Visualizing Statistical Uncertainty

How to compute and plot confidence interval functions (confidence curves, P-value functions) in Stata with the cifunction package, with worked examples.

Abstract

The cifunction package for Stata computes and plots all possible confidence intervals around a point estimate, which gives us what is known as a confidence interval function (also called a confidence curve, P-value function, or consonance interval). Here, we go through how to use it, since the curve tells us more than a single interval at one level by showing the full range of values that are compatible with the data at every confidence level.

Keywords

confidence interval function, confidence curve, p-value function, consonance interval, Sullivan and Foster, Stata statistical software, statistical visualization, uncertainty quantification

1 Overview

Most of the time, we only report a confidence interval at a single level (typically 95%). However, that one interval throws away a fair amount of information about the data, since it invites us to sort values into those inside the interval and those outside of it. The confidence interval function (CI-function) gives us a more complete picture by displaying all possible confidence intervals at once. In this post, I’d like to go through cifunction, a Stata package written by Ariel Linden that computes and plots these functions.

NoteAlternative Names

The CI-function has gone by several names in the literature:

• Confidence curve¹
• P-value function^{2, 3}
• Consonance interval⁴
• Confidence density⁵

2 Installation

Let’s start by installing cifunction from SSC (Statistical Software Components):

3 Syntax

{stata, echo=TRUE, collectcode=TRUE, statapath="/Applications/StataNow/StataMP.app/Contents/MacOS/StataMP"}cifunction #b, se(#) [options]

3.1 Required Arguments

Required parameters for cifunction

Argument	Description
#b	Point estimate (coefficient or exponentiated value like OR, HR, IRR)
se(#)	Standard error of the estimate

3.2 Optional Arguments

Optional parameters for cifunction

Option	Description
df(#)	Degrees of freedom (for t-distributed estimates)
eform	Indicates coefficient is exponentiated (OR, RR, IRR)
figure[(options)]	Produces CI-function plot with optional twoway options
saving(filename, replace)	Saves results to file

4 Understanding CI-Functions

So what are we looking at when we plot one of these? A CI-function shows the relationship between confidence levels and their corresponding confidence intervals, and there are a few features that we can read off the curve:

• Peak of the curve: this is where the point estimate sits
• Concentration around the peak: this reflects the precision of the estimate
• Narrow curve: high precision (usually a large sample size)
• Broad curve: low precision (usually a small sample size)

4.1 Advantages Over Single-Level CIs

1. Complete information: we can see how compatible every value is with the data
2. Precision visualization: the width of the curve directly shows how precise the estimate is
3. Multiple comparisons: we can see whether the data provide any reassurance about null effects
4. Graded interpretation: it moves us away from binary “significant/non-significant” thinking

TipInterpretation Guide

Values toward the peak are more compatible with the observed data (assuming that the statistical model is correct). Thus, the curve shows us that statistical evidence changes gradually, and it does not suggest any particular cutoff.

5 Practical Examples

5.1 Example 1: Basic Risk Ratio

First, we’ll reproduce Figure 1 from Sullivan and Foster (1990):

* Get standard error for RR=2.0 with p=0.05
getregstats 2.0, p(0.05) mod(rr)

* Create CI-function with log-scaled x-axis
cifunction 2.0, se(.707306) eform ///
    fig(xscale(log) xlab(0.5 1 2 5 10) ///
    xtitle("Risk Ratio (Log Scale)"))
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |          2    .707306    -1.96   0.050            1       
## >     4
## -------------------------------------------------------------------------
## > -----

5.2 Example 2: Age-Stratified Rate Ratios

Next, we’ll use the Doll-Hill smoking data, stratified by age group:

* Load data and fit models
webuse dollhill3, clear

* Fit Poisson models for each age category
forvalues i = 1/4 {
    poisson deaths smokes if agecat==`i', exposure(pyears) irr
}

* Plot all four age-stratified CI-functions
cifunction 5.736638 2.138812 1.46824 1.35606, ///
    se(4.181258 .6520701 .295728 .2748845) eform ///
    fig(xscale(log) xlabel(.3 .5 1 2 5 10 20 50 100) ///
    legend(label(1 "35-44") label(2 "45-54") ///
           label(3 "55-64") label(4 "65-74") ///
           title("Age Group", size(small))) ///
    xtitle("Rate Ratio (Log Scale)"))
## (Doll and Hill (1966))
## 
## 
## Iteration 0:  Log likelihood = -4.0705386  
## Iteration 1:  Log likelihood = -3.9613823  
## Iteration 2:  Log likelihood = -3.9612634  
## Iteration 3:  Log likelihood = -3.9612634  
## 
## Poisson regression                                      Number of obs =  
## >     2
##                                                         LR chi2(1)    =  
## >  9.73
##                                                         Prob > chi2   = 0
## > .0018
## Log likelihood = -3.9612634                             Pseudo R2     = 0
## > .5511
## 
## -------------------------------------------------------------------------
## > -----
##       deaths |        IRR   Std. err.      z    P>|z|     [95% conf. inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##       smokes |   5.736638   4.181258     2.40   0.017     1.374811    23.
## > 93712
##        _cons |   .0001064   .0000753   -12.94   0.000     .0000266    .00
## > 04256
##   ln(pyears) |          1  (exposure)
## -------------------------------------------------------------------------
## > -----
## Note: _cons estimates baseline incidence rate.
## 
## Iteration 0:  Log likelihood = -5.4793565  
## Iteration 1:  Log likelihood = -5.4104407  
## Iteration 2:  Log likelihood =   -5.41027  
## Iteration 3:  Log likelihood =   -5.41027  
## 
## Poisson regression                                      Number of obs =  
## >     2
##                                                         LR chi2(1)    =  
## >  7.59
##                                                         Prob > chi2   = 0
## > .0059
## Log likelihood = -5.41027                               Pseudo R2     = 0
## > .4123
## 
## -------------------------------------------------------------------------
## > -----
##       deaths |        IRR   Std. err.      z    P>|z|     [95% conf. inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##       smokes |   2.138812   .6520701     2.49   0.013     1.176691    3.8
## > 87609
##        _cons |   .0011243   .0003246   -23.52   0.000     .0006385    .00
## > 19798
##   ln(pyears) |          1  (exposure)
## -------------------------------------------------------------------------
## > -----
## Note: _cons estimates baseline incidence rate.
## 
## Iteration 0:  Log likelihood = -6.2456963  
## Iteration 1:  Log likelihood = -6.1713863  
## Iteration 2:  Log likelihood =  -6.171298  
## Iteration 3:  Log likelihood =  -6.171298  
## 
## Poisson regression                                      Number of obs =  
## >     2
##                                                         LR chi2(1)    =  
## >  4.01
##                                                         Prob > chi2   = 0
## > .0453
## Log likelihood = -6.171298                              Pseudo R2     = 0
## > .2450
## 
## -------------------------------------------------------------------------
## > -----
##       deaths |        IRR   Std. err.      z    P>|z|     [95% conf. inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##       smokes |    1.46824    .295728     1.91   0.057     .9893522     2.
## > 17893
##        _cons |   .0049037   .0009267   -28.14   0.000     .0033858    .00
## > 71021
##   ln(pyears) |          1  (exposure)
## -------------------------------------------------------------------------
## > -----
## Note: _cons estimates baseline incidence rate.
## 
## Iteration 0:  Log likelihood = -6.1640322  
## Iteration 1:  Log likelihood = -6.1203079  
## Iteration 2:  Log likelihood = -6.1202768  
## Iteration 3:  Log likelihood = -6.1202768  
## 
## Poisson regression                                      Number of obs =  
## >     2
##                                                         LR chi2(1)    =  
## >  2.43
##                                                         Prob > chi2   = 0
## > .1189
## Log likelihood = -6.1202768                             Pseudo R2     = 0
## > .1658
## 
## -------------------------------------------------------------------------
## > -----
##       deaths |        IRR   Std. err.      z    P>|z|     [95% conf. inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##       smokes |    1.35606   .2748845     1.50   0.133     .9114509    2.0
## > 17551
##        _cons |   .0108317    .002047   -23.95   0.000     .0074789    .01
## > 56877
##   ln(pyears) |          1  (exposure)
## -------------------------------------------------------------------------
## > -----
## Note: _cons estimates baseline incidence rate.

Here, the CI-functions let us see patterns across subgroups (in this case, how the smoking-mortality relationship varies by age).

5.3 Example 3: Crude vs Adjusted Estimates

Now, let’s compare the crude and pooled odds ratios from Mann et al. (1976):

* Case-control analysis
cci 10 36 5 40, woolf
cci 18 78 7 86, woolf

* Get standard errors
getregstats 2.835165, ucl(7.151794) mod(or)
getregstats 2.222222, ucl(7.118025) mod(or)

* Compare crude vs pooled estimates
cifunction 2.222222 2.835165, se(1.319891 1.33843) eform ///
    fig(xscale(log) xlabel(.2 .5 1 2 5 10 20) ///
    legend(label(1 Crude) label(2 Pooled)) ///
    xtitle("Odds Ratio (Log Scale)"))
##                                                          Proportion
##                  |   Exposed   Unexposed  |      Total      exposed
## -----------------+------------------------+------------------------
##            Cases |        10          36  |         46       0.2174
##         Controls |         5          40  |         45       0.1111
## -----------------+------------------------+------------------------
##            Total |        15          76  |         91       0.1648
##                  |                        |
##                  |      Point estimate    |    [95% conf. interval]
##                  |------------------------+------------------------
##       Odds ratio |         2.222222       |      .69377    7.118025 (Wool
## > f)
##  Attr. frac. ex. |              .55       |      -.4414    .8595116 (Wool
## > f)
##  Attr. frac. pop |         .1195652       |
##                  +-------------------------------------------------
##                                chi2(1) =     1.87  Pr>chi2 = 0.1719
## 
## 
##                                                          Proportion
##                  |   Exposed   Unexposed  |      Total      exposed
## -----------------+------------------------+------------------------
##            Cases |        18          78  |         96       0.1875
##         Controls |         7          86  |         93       0.0753
## -----------------+------------------------+------------------------
##            Total |        25         164  |        189       0.1323
##                  |                        |
##                  |      Point estimate    |    [95% conf. interval]
##                  |------------------------+------------------------
##       Odds ratio |         2.835165       |    1.123936    7.151794 (Wool
## > f)
##  Attr. frac. ex. |         .6472868       |    .1102698    .8601749 (Wool
## > f)
##  Attr. frac. pop |         .1213663       |
##                  +-------------------------------------------------
##                                chi2(1) =     5.18  Pr>chi2 = 0.0228
## 
## 
## 
## -------------------------------------------------------------------------
## > -----
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |   2.835165   1.338429     2.21   0.027     1.123936    7.1
## > 51794
## -------------------------------------------------------------------------
## > -----
## 
## 
## 
## -------------------------------------------------------------------------
## > -----
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |   2.222222   1.319891     1.34   0.179     .6937698    7.1
## > 18025
## -------------------------------------------------------------------------
## > -----

5.4 Example 4: Multi-Study Comparison

We can also compare results across four geographic regions (Gufferman et al. 1984):

* Get standard errors from confidence limits
getregstats 0.7, l(.2) mod(rr)
getregstats 1.2, l(.8) mod(rr)
getregstats 1.3, l(.5) mod(rr)
getregstats 3.6, l(1.3) mod(rr)

* Plot all studies
cifunction .7 1.2 1.3 3.6, ///
    se(.4474236 .2482485 .6337692 1.870876) eform ///
    fig(xscale(log) xlabel(.05 .1 .2 .5 1 2 5 10 30) ///
    legend(label(1 "Finland") label(2 "NO/LA") ///
           label(3 "Denmark") label(4 "Upstate NY")) ///
    xtitle("Relative Risk (Log Scale)"))
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |         .7   .4474236    -0.56   0.577           .2       
## >  2.45
## -------------------------------------------------------------------------
## > -----
## 
## 
## 
## -------------------------------------------------------------------------
## > -----
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |        1.2   .2482485     0.88   0.378           .8       
## >   1.8
## -------------------------------------------------------------------------
## > -----
## 
## 
## 
## -------------------------------------------------------------------------
## > -----
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |        1.3   .6337692     0.54   0.590           .5       
## >  3.38
## -------------------------------------------------------------------------
## > -----
## 
## 
## 
## -------------------------------------------------------------------------
## > -----
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |        3.6   1.870876     2.46   0.014          1.3    9.9
## > 69231
## -------------------------------------------------------------------------
## > -----

As we can see from the plot, it becomes fairly easy to tell which studies had greater precision and how their point estimates compare.

5.5 Example 5: Meta-Analysis Context

Lastly, here’s an example taken from Rothman et al. (2008), Figure 10.5:

* Get standard errors
getregstats 0.87, l(0.70) model(or)

* Compare individual study vs meta-analysis
cifunction 0.87 0.88, se(.0965065 .0658232) eform ///
    fig(xlab(.5(.1)1.3) xtitle("Odds Ratio") ///
    legend(label(1 "Eisenberger et al (1998)") ///
           label(2 "Summary of 10 studies") ///
           ring(0) position(11))) ///
    sav(figdata, replace)
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Inte
## > rval]
## -------------+-----------------------------------------------------------
## > -----
##    Estimates |        .87   .0965065    -1.26   0.209           .7    1.0
## > 81286
## -------------------------------------------------------------------------
## > -----

6 Output Variables

When we use saving(), cifunction stores the following variables:

Variables saved in output dataset

Variable	Description
cilev	Confidence levels (0 to 99.99)
plev	P-values (0.0001 to 1.0)
sval	S-values (Shannon information, see6)
lcl#	Lower confidence limits for estimate #
ucl#	Upper confidence limits for estimate #

NoteS-Values

S-values measure the information against a null hypothesis (or any other test hypothesis) in bits. An S-value of 4.32 corresponds to p=0.05 (roughly 4.3 bits of information against the null). See⁶ for the details.

7 Best Practices

1. Use the log scale for ratio measures (OR, RR, HR, IRR)
2. Label the curves clearly when plotting more than one estimate
3. Save the output data so that we can analyze it further or build custom plots
4. Consider the degrees of freedom when the estimates come from a regression
5. Look at the whole curve instead of focusing on any single cutoff

8 Theoretical Foundation

CI-functions rest on the idea that statistical evidence is continuous, and that we lose something when we dichotomize it. As² argues:

“The P-value function provides a convenient way to present the information in the data concerning the magnitude of an effect… [It] avoids the arbitrariness of any single significance level.”

This also seems to fit with more recent calls to move beyond P-value thresholds⁷, since the curve presents the full gradient of statistical evidence.

9 Comparison with Other Approaches

Comparison of uncertainty visualization methods

Approach	Information Provided	Computational Complexity
Single CI	One interval at one level	Low
Multiple CIs	Discrete intervals	Low
CI-function	Continuous curve of all intervals	Low
Bootstrap distribution	Empirical sampling distribution	High
Likelihood profile	Likelihood across parameter space	Medium-High

10 Limitations

However, there are a few things that a CI-function cannot do for us:

• It assumes that the statistical model is correctly specified
• It does not account for unmeasured confounding or bias
• It cannot compensate for poor study design or execution
• Interpreting it requires some understanding of continuous evidence

WarningModel Assumptions

CI-functions, like all frequentist intervals, are only valid to the extent that the underlying statistical model and its assumptions are appropriate for the data.

11 Citation

cifunction is free research software. If you use it, please cite it as:

Linden, A. (2019). CIFUNCTION: Stata module for computing and graphically displaying all possible confidence intervals around a point estimate. Statistical Software Components, Boston College Department of Economics.

12 Author

Ariel Linden President, Linden Consulting Group, LLC Email: alinden@lindenconsulting.org

13 See Also

• getregstats (SSC): Helper for obtaining standard errors
• niceloglabels (SSC): Improved logarithmic axis labels
• concurve (R package): Similar functionality in R

14 References

1. Birnbaum A. (1961). “Confidence curves: An omnibus technique for estimation and testing statistical hypotheses.” Journal of the American Statistical Association. 56:246–249.

2. Poole C. (1987). “Beyond the confidence interval.” American Journal of Public Health. 77:195–199.

3. Miettinen OS. (1985). “Theoretical epidemiology: Principles of occurrence research in medicine.” New York: John Wiley & Sons.

4. Folks JL. (1981). “Ideas of statistics.” New York: John Wiley & Sons.

5. Xie M, Singh K. (2013). “Confidence Distribution, the Frequentist Distribution Estimator of a Parameter: A Review.” International Statistical Review. 81:3–39. doi: 10.1111/insr.12000.

Abstract

R\'esum\'e Il est courant, en inf\'erence fr\'equentielle, d'utiliser un point unique (une estimation ponctuelle) ou un intervalle (intervalle de confiance) dans le but d'estimer un param\`etre d'int\'er\^t. Une question tr\`es simple se pose: peut-on \'egalement utiliser, dans le m\^eme but, et dans la m\^eme optique fr\'equentielle, \`a la facon dont les Bay\'esiens utilisent une loi a posteriori, une distribution de probabilit\'e? La r\'eponse est affirmative, et les distributions de confiance apparaissent comme un choix naturel dans ce contexte. Le concept de distribution de confiance a une longue histoire, longtemps associ\'ee, \`a tort, aux th\'eories d'inf\'erence fiducielle, ce qui a compromis son d\'eveloppement dans l'optique fr\'equentielle. Les distributions de confiance ont r\'ecemment attir\'e un regain d'int\'er\^et, et plusieurs r\'esultats ont mis en \'evidence leur potentiel consid\'erable en tant qu'outil inf\'erentiel. Cet article pr\'esente une d\'efinition moderne du concept, et examine les ses \'evolutions r\'ecentes. Il aborde les m\'ethodes d'inf\'erence, les probl\`emes d'optimalit\'e, et les applications …

6. Greenland S. (2019). “Valid p-values behave exactly as they should: Some misleading criticisms of p-values and their resolution with s-values.” The American Statistician. 73:106–114. doi: 10.1080/00031305.2018.1529625.

Abstract

The present note explores sources of misplaced criticisms of P-values, such as conflicting definitions of “significance levels” and “P-values” in authoritative sources, and the consequent misinterpretation of P-values as error probabilities. It then discusses several properties of P-values that have been presented as fatal flaws: That P-values exhibit extreme variation across samples (and thus are “unreliable”), confound effect size with sample size, are sensitive to sample size, and depend on investigator sampling intentions. These properties are often criticized from a likelihood or Bayesian framework, yet they are exactly the properties P-values should exhibit when they are constructed and interpreted correctly within their originating framework. Other common criticisms are that P-values force users to focus on irrelevant hypotheses and overstate evidence against those hypotheses. These problems are not however properties of P-values but are faults of researchers who focus on null hypotheses and overstate evidence based on misperceptions that p = 0.05 represents enough evidence to reject hypotheses …

7. Wasserstein RL, Schirm AL, Lazar NA. (2019). “Moving to a world beyond "p < 0.05".” The American Statistician. 73:1–19. doi: 10.1080/00031305.2019.1583913.

Abstract

EDITORIAL: The editorial was written by the three editors acting as individuals and reflects their scientific views not an endorsed position of the American Statistical Association.Some of you expl...

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15 Appendix: Technical Details

15.1 Mathematical Formulation

For a normally distributed estimator $\hat{\theta}$ with standard error $SE(\hat{\theta})$, the confidence limits at level $(1-\alpha)$ are:

$$\hat{\theta} \pm z_{1-\alpha/2} \cdot SE(\hat{\theta})$$

The CI-function plots these limits across all values of $\alpha \in [0, 1]$.

For t-distributed estimators (when degrees of freedom are specified):

$$\hat{\theta} \pm t_{1-\alpha/2, df} \cdot SE(\hat{\theta})$$

15.2 Exponentiated Coefficients

When eform is specified, computations occur on the log scale:

1. Transform: $\log(\hat{\theta})$ with $SE(\log(\hat{\theta}))$
2. Compute CI limits on log scale
3. Back-transform: $\exp(limits)$

Thus, the intervals are symmetric on the log scale and keep their proper geometric interpretation.

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Last updated: 2026-10-01 Source code: Available on request License: Free for academic and non-commercial use

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