Computing Confidence Interval Functions with cifunction

A Stata Package for Visualizing Statistical Uncertainty

statistics

stata

confidence intervals

data visualization

statistical methods

Learn how to compute and visualize confidence interval functions (confidence curves, p-value functions) using the cifunction Stata package. Includes practical examples and implementation guide.

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Author

Zad R

Published

March 23, 2024

Last updated

September 13, 2026

Abstract

The cifunction package for Stata computes and graphically displays all possible confidence intervals around a point estimate, creating what’s known as a confidence interval function, confidence curve, p-value function, or consonance interval. This approach conveys more information than traditional single-level confidence intervals by showing the full range of values compatible with the data at all confidence levels.

Keywords

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

1 Overview

Traditional confidence intervals present a single level of uncertainty (typically 95%), but this dichotomous approach discards valuable information about the data. The confidence interval function (CI-function) provides a more complete picture by displaying all possible confidence intervals simultaneously.

Alternative Names

The CI-function has been described in the literature under various names:

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

2 Installation

Install 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

A CI-function graphically represents the relationship between confidence levels and confidence intervals:

Peak of curve: Indicates the point estimate
Concentration around peak: Reflects precision of estimate
Narrow curve: High precision (large sample size)
Broad curve: Low precision (small sample size)

4.1 Advantages Over Single-Level CIs

Complete information: Shows compatibility of all values with data
Precision visualization: Curve width directly shows estimate precision
Multiple comparisons: Reveals whether data provide reassurance about null effects
Nuanced interpretation: Moves beyond binary “significant/non-significant” thinking

Interpretation Guide

Values toward the peak are more compatible with the observed data (assuming the statistical model is correct). The curve explicitly shows the gradual nature of statistical evidence rather than suggesting arbitrary cutoffs.

5 Practical Examples

5.1 Example 1: Basic Risk Ratio

Reproducing 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)"))
## file ()print() not found
## r(601);
## 
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |          2    .707306    -1.96   0.050            1           4
## ------------------------------------------------------------------------------

5.2 Example 2: Age-Stratified Rate Ratios

Using 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)"))
## file ()print() not found
## r(601);
## 
## 
## (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. interval]
## -------------+----------------------------------------------------------------
##       smokes |   5.736638   4.181258     2.40   0.017     1.374811    23.93712
##        _cons |   .0001064   .0000753   -12.94   0.000     .0000266    .0004256
##   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. interval]
## -------------+----------------------------------------------------------------
##       smokes |   2.138812   .6520701     2.49   0.013     1.176691    3.887609
##        _cons |   .0011243   .0003246   -23.52   0.000     .0006385    .0019798
##   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. interval]
## -------------+----------------------------------------------------------------
##       smokes |    1.46824    .295728     1.91   0.057     .9893522     2.17893
##        _cons |   .0049037   .0009267   -28.14   0.000     .0033858    .0071021
##   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. interval]
## -------------+----------------------------------------------------------------
##       smokes |    1.35606   .2748845     1.50   0.133     .9114509    2.017551
##        _cons |   .0108317    .002047   -23.95   0.000     .0074789    .0156877
##   ln(pyears) |          1  (exposure)
## ------------------------------------------------------------------------------
## Note: _cons estimates baseline incidence rate.

This example demonstrates how CI-functions can reveal patterns across subgroups—in this case, showing how the smoking-mortality relationship varies by age.

5.3 Example 3: Crude vs Adjusted Estimates

Comparing 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)"))
## file ()print() not found
## r(601);
## 
## 
## 
##                                                          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 (Woolf)
##  Attr. frac. ex. |              .55       |      -.4414    .8595116 (Woolf)
##  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 (Woolf)
##  Attr. frac. ex. |         .6472868       |    .1102698    .8601749 (Woolf)
##  Attr. frac. pop |         .1213663       |
##                  +-------------------------------------------------
##                                chi2(1) =     5.18  Pr>chi2 = 0.0228
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |   2.835165   1.338429     2.21   0.027     1.123936    7.151794
## ------------------------------------------------------------------------------
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |   2.222222   1.319891     1.34   0.179     .6937698    7.118025
## ------------------------------------------------------------------------------

5.4 Example 4: Multi-Study Comparison

Comparing 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)"))
## file ()print() not found
## r(601);
## 
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |         .7   .4474236    -0.56   0.577           .2        2.45
## ------------------------------------------------------------------------------
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |        1.2   .2482485     0.88   0.378           .8         1.8
## ------------------------------------------------------------------------------
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |        1.3   .6337692     0.54   0.590           .5        3.38
## ------------------------------------------------------------------------------
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Risk Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |        3.6   1.870876     2.46   0.014          1.3    9.969231
## ------------------------------------------------------------------------------

This visualization makes it immediately apparent which studies had greater precision and how their point estimates compare.

5.5 Example 5: Meta-Analysis Context

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)
## file ()print() not found
## r(601);
## 
## 
## 
## 
## ------------------------------------------------------------------------------
##              | Odds Ratio   Std. Err.      z    P>|z|     [95% Conf. Interval]
## -------------+----------------------------------------------------------------
##    Estimates |        .87   .0965065    -1.26   0.209           .7    1.081286
## ------------------------------------------------------------------------------

6 Output Variables

When using saving(), cifunction stores these 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 #

S-Values

S-values represent the information content against a null hypothesis, measured in bits. An S-value of 4.32 corresponds to p=0.05 (roughly 4.3 bits of information against the null). See⁶ for details.

7 Best Practices

Always use log scale for ratio measures (OR, RR, HR, IRR)
Label curves clearly when plotting multiple estimates
Save output data for further analysis or custom plotting
Consider degrees of freedom when using regression-based estimates
Interpret curves holistically rather than focusing on single cutoffs

8 Theoretical Foundation

CI-functions rest on the principle that statistical evidence is continuous rather than dichotomous. 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.”

The approach aligns with modern calls for moving beyond p-value thresholds⁷ by presenting 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

Assumes the statistical model is correctly specified
Does not account for unmeasured confounding or bias
Cannot compensate for poor study design or execution
Interpretation requires understanding of continuous evidence

Model Assumptions

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

11 Citation

cifunction is free research software. 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...

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:

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

This ensures symmetric intervals on the log scale and proper geometric interpretation.

Last updated: 2026-09-13 Source code: Available on request License: Free for academic and non-commercial use

Citation

BibTeX citation:

@article{rafi2024,
  author = {Rafi, Zad and R, Zad},
  title = {Computing {Confidence} {Interval} {Functions} with
    Cifunction},
  journal = {Less Likely},
  date = {2024-03-23},
  url = {https://lesslikely.com/statistics/cifunction},
  langid = {en-US},
  abstract = {The cifunction package for Stata computes and graphically
    displays all possible confidence intervals around a point estimate,
    creating what’s known as a confidence interval function, confidence
    curve, p-value function, or consonance interval. This approach
    conveys more information than traditional single-level confidence
    intervals by showing the full range of values compatible with the
    data at all confidence levels.}
}

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Computing Confidence Interval Functions with cifunction

1 Overview

2 Installation

3 Syntax

3.1 Required Arguments

3.2 Optional Arguments

4 Understanding CI-Functions

4.1 Advantages Over Single-Level CIs

5 Practical Examples

5.1 Example 1: Basic Risk Ratio

5.2 Example 2: Age-Stratified Rate Ratios

5.3 Example 3: Crude vs Adjusted Estimates

5.4 Example 4: Multi-Study Comparison

5.5 Example 5: Meta-Analysis Context

6 Output Variables

7 Best Practices

8 Theoretical Foundation

9 Comparison with Other Approaches

10 Limitations

11 Citation

12 Author

13 See Also

14 References

15 Appendix: Technical Details

15.1 Mathematical Formulation

15.2 Exponentiated Coefficients

Citation

Linked from (6)

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