Confidence Interval

On SUMMARY tables, the Statisics - Cells menu contains options to show the Upper Confidence Interval and Lower Confidence Interval. The level used for the measurement is determined by the overall level which is set under Edit > Project Options > Customize > Statistical Assumptions (default is 95%).

On a crosstab, there are no built-in options, but you can use the following rules to add confidence intervals to the table:

• Calculate Confidence Intervals for Crosstabs with Numeric Questions
• Calculate Confidence Intervals for Crosstabs with Categorical Questions

Default confidence intervals

In most instances in Q, the lower and upper bounds of confidence intervals are computed using whichever is appropriate of:

[math]\displaystyle{ \bar x \pm t_{\alpha/2, df}s_{\bar x} }[/math] or [math]\displaystyle{ \bar x \pm z_{\alpha/2}s_{\bar x} }[/math]

where:

[math]\displaystyle{ \bar x }[/math] is the observed Average, %, Column %, Row %, Probability %, Total % or Coefficient,

[math]\displaystyle{ s_{\bar x} }[/math] is the estimated or computed Standard Error which includes any computer and/or specified design effects,

[math]\displaystyle{ t_{\alpha/2, df} }[/math] is the [math]\displaystyle{ \alpha/2 }[/math]th quantile of student's t-distribution,

[math]\displaystyle{ df }[/math] is [math]\displaystyle{ n - 1 }[/math], and

[math]\displaystyle{ z_{\alpha/2} }[/math] is the [math]\displaystyle{ \alpha/2 }[/math]th quantile of the normal distribution.

Confidence intervals for percentages with unweighted samples

The Agresti-Coull interval is used to computed confidence intervals for categorical questions where there are no weights, except where Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests or when Extra deff is not 1. The Agresti-Coull interval is given by:

[math]\displaystyle{ \tilde{\bar x} \pm z_{\alpha/2} \sqrt{\frac{\tilde{\bar x}(1 - \tilde{\bar x})}{\tilde{n}}} }[/math]

where:

[math]\displaystyle{ \tilde{n} = n + z_{\alpha/2}^2 }[/math],

[math]\displaystyle{ n }[/math] is Base n,

[math]\displaystyle{ \tilde{\bar x} = \frac{x + \frac{{z}_{\alpha/2}^2}{2}}{\tilde{n}} }[/math], and

[math]\displaystyle{ x }[/math] is n.

Confidence intervals where Weights and significance has been set to Un-weighted sample size in tests

Where Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests, confidence intervals are computed using:

[math]\displaystyle{ \bar x \pm t_{\alpha/2, n - 1}s_{\bar x} }[/math]

where:

[math]\displaystyle{ s_{\bar x} = \sqrt{\frac{d_{eff} \bar x (1 - \bar x)}{n - b}} }[/math] if [math]\displaystyle{ \bar x }[/math] represents a proportion,

[math]\displaystyle{ s_{\bar x} = s_ x \sqrt{\frac{d_{eff}}{n}} }[/math] otherwise,

[math]\displaystyle{ s_x }[/math] is the Standard Deviation,

[math]\displaystyle{ d_{eff} }[/math] is Extra deff,

[math]\displaystyle{ b }[/math] is 1 if Bessel's correction is selected for Proportions in Statistical Assumptions and 0 otherwise.

Notes

1. In most situations, the statistical tests computed by Q will not correspond to conclusions drawn if attempting to construct tests from the confidence intervals. There are many reasons for this, including:
   • Multiple Comparison Corrections.
   • Use of non-parametric tests in Q.
   • The confidence intervals having statistical properties that make them sub-optimal from a testing perspective.
2. To keep this page relatively short, [math]\displaystyle{ s }[/math] is used in the formulas above where it is more conventional to use [math]\displaystyle{ \sigma }[/math].
3. Whereas the The Agresti-Coull interval is an improvement on the default formula for computing the confidence intervals, the formula used when Weights and significance in Statistical Assumptions has been set to Un-weighted sample size in tests, is generally inferior and is only included for the purposes of aiding comparison with results computed using this formula in other programs.