Confidence Interval

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Default confidence intervals

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

\bar x \pm t_{\alpha/2, df}s_{\bar x} or \bar x \pm z_{\alpha/2}s_{\bar x}

where:

\bar x is the observed Average, %, Column %, Row %, Probability %, Total % or Coefficient,
s_{\bar x} is the estimated or computed Standard Error which includes any computer and/or specified design effects,
t_{\alpha/2, df} is the \alpha/2th quantile of student's t-distribution,
df is n - 1, and
z_{\alpha/2} is the \alpha/2th 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:

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

where:

\tilde{n} = n + z_{\alpha/2}^2,
n is Base n,
\tilde{\bar x} = \frac{x + \frac{{z}_{\alpha/2}^2}{2}}{\tilde{n}}, and
x 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:

\bar x \pm t_{\alpha/2, n - 1}s_{\bar x}

where:

s_{\bar x} = \sqrt{\frac{d_{eff} \bar x (1 - \bar x)}{n - b}} if \bar x represents a proportion,
s_{\bar x} = s_ x \sqrt{\frac{d_{eff}}{n}} otherwise,
s_x is the Standard Deviation,
d_{eff} is Extra deff,
b 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, s is used in the formulas above where it is more conventional to use \sigma.
  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.