Risks and Risk Differences

The FREQ Procedure

Risks and Risk Differences

The RISKDIFF option in the TABLES statement provides estimates of risks (or binomial proportions) and risk differences for 2 ×2 tables. This analysis may be appropriate when comparing the proportion of some characteristic for two groups, where row 1 and row 2 correspond to the two groups, and the columns correspond to two possible characteristics or outcomes. For example, the row variable might be a treatment or dose, and the column variable might be the response. Refer to Collett (1991), Fleiss (1981), and Stokes, Davis, and Koch (1995).

Let the frequencies of the 2 ×2 table be represented as follows.

	Column 1	Column 2	Total
Row 1	n₁₁	n₁₂	n_{1 ·}
Row 2	n₂₁	n₂₂	n_{2 ·}
Total	n_·1	n_·2	n

The column 1 risk for row 1 is the proportion of row 1 observations classified in column 1,

$p_{1| 1} = n_{11} / n_{1 \cdot}$

This estimates the conditional probability of the column 1 response, given the first level of the row variable.

The column 1 risk for row 2 is the proportion of row 2 observations classified in column 1,

$p_{1| 2} = n_{21} / n_{2 \cdot}$

and the overall column 1 risk is the proportion of all observations classified in column 1,

p_·1 = n_·1 / n

The column 1 risk difference compares the risks for the two rows, and it is computed as the column 1 risk for row 1 minus the column 1 risk for row 2,

$(\mathit{pdiff})_{1} = p_{1| 1} - p_{1| 2}$

The risks and risk difference are defined similarly for column 2.

The standard error of the column 1 risk estimate for row i is computed as

$se(p_{1| i}) = \sqrt{ p_{1| i} ( 1 - p_{1| i} ) / n_{i \cdot} }$

The standard error of the overall column 1 risk estimate is computed as

$se(p_{\cdot 1}) = \sqrt{ p_{\cdot 1} ( 1 - p_{\cdot 1} ) / n }$

If the two rows represent independent binomial samples, the standard error for the column 1 risk difference is computed as

$se( (\mathit{pdiff})_{1} ) = \sqrt{var(p_{1| 1}) + var(p_{1| 2})}$

The standard errors are computed in a similar manner for the column 2 risks and risk difference.

Using the normal approximation to the binomial distribution, PROC FREQ constructs asymptotic confidence limits for the risks and risk differences according to

$est\pmz_{\alpha/2} \cdot se(est)$

where est is the estimate, $z_{\alpha/2}$ is the $100(1 - \alpha/2)$ percentile of the standard normal distribution, and se(est) is the standard error of the estimate. The confidence level $\alpha$ is determined from the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits.

PROC FREQ computes exact confidence limits for the column 1, column 2, and overall risks using the F distribution method given in Collett (1991) and also described by Leemis and Trivedi (1996). PROC FREQ does not provide exact confidence limits for the risk differences. Refer to Agresti (1992) for a discussion of issues involved in constructing exact confidence limits for differences of proportions.

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