The SURVEYFREQ Procedure

Risks and Risk Difference

The RISK option provides estimates of risks (binomial proportions) and risk differences for $2 \times 2$ tables, together with their standard errors and confidence limits. Risk statistics include the row 1 risk, row 2 risk, overall risk, and risk difference. If you specify the RISK option, PROC SURVEYFREQ provides both column 1 and column 2 risks. You can request only column 1 (or only column 2) risks by specifying the RISK1 (or RISK2 ) option.

The column 1 risk for row 1 is the row 1 proportion for table cell (1,1). The column 1 risk estimate is computed as the ratio of the estimated total for table cell (1,1) to the estimated total for row 1,

\[ \widehat{P}_{11}^{~ (1)} = \widehat{N}_{11} ~ / ~ \widehat{N}_{1 \cdot } \]

where the total estimates are computed as described in the section Totals. The column 1 risk for row 2 is the row 2 proportion for table cell (2,1), which is estimated as

\[ \widehat{P}_{21}^{~ (2)} = \widehat{N}_{21} ~ / ~ \widehat{N}_{2 \cdot } \]

The overall column 1 risk is the overall proportion in column 1, and its estimate is computed as

\[ \widehat{P}_{\cdot 1} = \widehat{N}_{\cdot 1} ~ / ~ \widehat{N} \]

The column 2 risk estimates are computed similarly.

The row 1 and row 2 risks are the same as the row proportions for a $2 \times 2$ table, and their variances are computed as described in the section Row and Column Proportions. The overall risk is the overall proportion in the column, and its variance computation is described in the section Proportions. Confidence limits for the column 1 risk for row 1 are computed as

\[ \widehat{P}_{11}^{~ (1)} \pm \left( t_{\mi{df}, \alpha /2} \times \mr{StdErr}(\widehat{P}_{11}^{~ (1)}) \right) \]

where $\mr{StdErr}(\widehat{P}_{11}^{~ (1)})$ is the standard error of the risk estimate and $t_{\mi{df}, \alpha /2}$ is the $100(1-\alpha /2)$ percentile of the t distribution with df degrees of freedom. (For more information, see the section Degrees of Freedom.) The value of the confidence coefficient $\alpha $ is determined by the ALPHA= option; by default, ALPHA=0.05, which produces 95% confidence limits. Confidence limits for the other risks are computed similarly.

The risk difference is defined as the row 1 risk minus the row 2 risk. The estimate of the column 1 risk difference $\widehat{\mathit{RD}}_1$ is computed as

\begin{eqnarray*} \widehat{\mathit{RD}}_1 & = & \widehat{P}_{11}^{~ (1)} - \widehat{P}_{21}^{~ (2)} \\[0.1in]& = & \left( \widehat{N}_{11} ~ / ~ \widehat{N}_{1 \cdot } \right) ~ - ~ \left( \widehat{N}_{21} ~ / ~ \widehat{N}_{2 \cdot } \right) \end{eqnarray*}

The column 2 risk difference is computed similarly.

PROC SURVEYFREQ estimates the variance of the risk difference by using the variance estimation method that you request. If you request BRR variance estimation (VARMETHOD=BRR ), the procedure estimates the variance as described in the section Balanced Repeated Replication (BRR). If you request jackknife variance estimation (VARMETHOD=JACKKNIFE ), the procedure estimates the variance as described in the section The Jackknife Method.

If you do not specify the VARMETHOD= option or a REPWEIGHTS statement, the default variance estimation method is Taylor series (VARMETHOD=TAYLOR ). By using Taylor series linearization, the variance estimate for the column 1 risk difference $\widehat{\mr{Var}}(\widehat{\mathit{RD}}_1)$ can be expressed as

\[ \widehat{\mr{Var}}(\widehat{\mathit{RD}}_1) = \widehat{\mb{D}} ~ \widehat{\mb{V}}(\widehat{\mb{X}}) ~ \widehat{\mb{D}}’ \]

where $\widehat{\mb{V}}(\widehat{\mb{X}})$ is the covariance matrix of $\widehat{\mb{X}}$,

\[ \widehat{\mb{X}} = \left( ~ \widehat{N}_{11}, ~ ~ \widehat{N}_{1 \cdot }, ~ ~ \widehat{N}_{21}, ~ ~ \widehat{N}_{2 \cdot } ~ \right) \]

and $\widehat{\mb{D}}$ is an array that contains the partial derivatives of the risk difference with respect to the elements of $\widehat{\mb{X}}$,

\[ \widehat{\mb{D}} = \left( ~ 1/\widehat{N}_{1 \cdot }, ~ ~ ~ -\widehat{N}_{11} / \widehat{N}_{1 \cdot }^{~ 2}, ~ ~ ~ -1/\widehat{N}_{2 \cdot }, ~ ~ ~ -\widehat{N}_{21} / \widehat{N}_{2 \cdot }^{~ 2} ~ \right) \]

For more information, see Wolter (1985, pp. 239–242). The variance estimate for the column 2 risk difference is computed similarly.

The standard error of the column 1 risk difference is

\[ \mr{StdErr}(\widehat{\mathit{RD}}_1) = \sqrt { \widehat{\mr{Var}}(\widehat{\mathit{RD}}_1) } \]

Confidence limits for the column 1 risk difference are computed as

\[ \widehat{\mathit{RD}}_1 \pm \left( t_{\mi{df}, \alpha /2} \times \mr{StdErr}(\widehat{\mathit{RD}}_1) \right) \]

where $t_{\mi{df}, \alpha /2}$ is the $100(1-\alpha /2)$ percentile of the t distribution with df degrees of freedom. (For more information, see the section Degrees of Freedom.) The value of the confidence coefficient $\alpha $ is determined by the ALPHA= option; by default, ALPHA=0.05, which produces 95% confidence limits. Confidence limits for the column 2 risk difference are computed in the same way.