The RISKDIFF option in the TABLES statement provides estimates of risks (binomial proportions) and risk differences for tables. This analysis might 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. See Collett (1991), Fleiss, Levin, and Paik (2003), and Stokes, Davis, and Koch (2000) for more information.
Let the frequencies of the table be represented as follows.
Column 1 
Column 2 
Total 

Row 1 



Row 2 



Total 



For column 1 and column 2, PROC FREQ provides estimates of the row 1 risk (proportion), the row 2 risk, the overall risk, and the risk difference. The risk difference is defined as the row 1 risk minus the row 2 risk. The risks are binomial proportions of their rows (row 1, row 2, or overall), and the computation of their standard errors and confidence limits follow the binomial proportion computations, which are described in the section Binomial Proportion.
The column 1 risk for row 1 is the proportion of row 1 observations classified in column 1,
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,
The overall column 1 risk is the proportion of all observations classified in column 1,
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,
The risks and risk difference are defined similarly for column 2.
The standard error of the column 1 risk for row is computed as
The standard error of the overall column 1 risk is computed as
If the two rows represent independent binomial samples, the standard error for the column 1 risk difference is computed as
The standard errors are computed in a similar manner for the column 2 risks and risk difference.
By default, the RISKDIFF option provides standard Wald asymptotic confidence limits for the risks (row 1, row 2, and overall) and the risk difference. The RISKDIFF option also provides other types of confidence limits and tests for the risk difference. See the sections Risk Difference Confidence Limits and Risk Difference Tests for details.
The risks are equivalent to binomial proportions of their corresponding rows. This section describes the Wald confidence limits for risks that are provided by the RISKDIFF option. The BINOMIAL option provides additional confidence limit types and tests for risks in the binomial proportion framework. See the sections Binomial Confidence Limits and Binomial Tests for details.
The Wald asymptotic confidence limits are based on the normal approximation to the binomial distribution. PROC FREQ computes the Wald confidence limits for the risks and risk differences as
where is the estimate, is the th percentile of the standard normal distribution, and is the standard error of the estimate. The confidence level is determined from the value of the ALPHA= option, which, by default, equals 0.05 and produces 95% confidence limits.
If you specify the CORRECT riskdiffoption or the RISKDIFFC option, PROC FREQ includes continuity corrections in the Wald asymptotic confidence limits for the risks and risk differences. The purpose of a continuity correction is to adjust for the difference between the normal approximation and the binomial distribution, which is discrete. See Fleiss, Levin, and Paik (2003) for more information. With the continuity correction, the asymptotic confidence limits are computed as
where is the continuity correction. For the row 1 risk, ; for the row 2 risk, ; for the overall risk, ; and for the risk difference, . The column 1 and column 2 risks use the same continuity corrections.
PROC FREQ also computes exact (ClopperPearson) confidence limits for the column 1, column 2, and overall risks. These confidence limits are constructed by inverting the equaltailed test based on the binomial distribution. PROC FREQ uses the distribution to compute the ClopperPearson confidence limits. See the section Exact (ClopperPearson) Confidence Limits for details.
You can request additional confidence limits for the risk difference by specifying the CL= riskdiffoption. Available confidence limit types include exact unconditional, FarringtonManning, HauckAnderson, Newcombe score, and Wald. Continuitycorrected versions of the Newcombe and Wald confidence limits are available. By default, the Wald confidence limits use a samplebased variance; alternatively, you can request a testbased variance and specify the null risk difference value.
The confidence coefficient for the confidence limits produced by the CL= riskdiffoption is %, where the value of is determined by the ALPHA= option. By default, ALPHA=0.05, which produces 95% confidence limits. This differs from the testbased confidence limits that are provided with the equivalence, noninferiority, and superiority tests, which have a confidence coefficient of % (Schuirmann 1999). See the section Risk Difference Tests for details.
The section Exact Unconditional Confidence Limits for the Risk Difference describes the computation of the exact confidence limits. The confidence limits are constructed by inverting two separate onesided exact tests (tail method). By default, the tests are based on the unstandardized risk difference. If you specify the RISKDIFF(METHOD=FMSCORE) option, the FarringtonManning score is used as the test statistic.
PROC FREQ computes the Newcombe confidence limits for the risk difference as described in the subsection Newcombe Score Confidence Limits in the section Noninferiority Tests, except that the Newcombe confidence limits produced by the CL= riskdiffoption have a confidence coefficient of %.
The following sections describe the computation of the FarringtonManning, HauckAnderson, and Wald confidence limits for the risk difference.
FarringtonManning Confidence Limits The FarringtonManning confidence limits for the risk difference are computed as
where , is the th percentile of the standard normal distribution, and the standard error is
where and are the maximum likelihood estimators of and under the null hypothesis that the risk difference equals .
The subsection FarringtonManning Test in the section Noninferiority Tests describes the computation of the maximum likelihood estimators and . See Farrington and Manning (1990) for details.
This computation uses a null hypothesis value of the risk difference, which you can specify in the CL=FM(NULL=value) riskdiff=option. By default, PROC FREQ uses a null value of 0. This differs from the FarringtonManning confidence limits that are produced in the noninferiority analysis, where the null value of the risk difference is based on the test margin (which is specified by the MARGIN= riskdiffoption).
HauckAnderson Confidence Limits The HauckAnderson confidence limits for the risk difference are computed as
where and is the th percentile of the standard normal distribution. The standard error is computed from the sample proportions as
The HauckAnderson continuity correction is computed as
See Hauck and Anderson (1986) for more information. The subsection HauckAnderson Test in the section Noninferiority Tests describes the corresponding noninferiority test.
Wald Confidence Limits The Wald confidence limits for the risk difference are computed as
where and is the th percentile of the standard normal distribution. By default, the standard error is computed from the sample proportions as
If you specify the CL=WALD(NULL=value) riskdiffoption, the standard error is based on the null hypothesis that the risk difference equals (Dunnett and Gent 1977). The standard error is computed as
where
If you specify the CORRECT riskdiffoption, the Wald confidence limits include a continuity correction ,
where .
The subsection Wald Test in the section Noninferiority Tests describes the corresponding noninferiority test.
You can specify riskdiffoptions to request tests of the risk (proportion) difference. You can request tests of equality, noninferiority, superiority, and equivalence for the risk difference. The test of equality is a standard Wald asymptotic test, available with or without a continuity correction. For noninferiority, superiority, and equivalence tests of the risk difference, the following test methods are provided: Wald (with and without continuity correction), HauckAnderson, FarringtonManning, and Newcombe score (with and without continuity correction). You can specify the test method with the METHOD= riskdiffoption. By default, PROC FREQ uses METHOD=WALD.
If you specify the EQUAL riskdiffoption, PROC FREQ computes a test of equality, or a test of the null hypothesis that the risk difference equals zero. For the column 1 (or 2) risk difference, this test can be expressed as versus the alternative , where denotes the column 1 (or 2) risk difference. PROC FREQ provides a Wald asymptotic test of equality. The test statistic is computed as
By default, the standard error is computed from the sample proportions as
If you specify the VAR=NULL riskdiffoption, the standard error is based on the null hypothesis that the row 1 and row 2 risks are equal,
where estimates the overall column 1 risk.
If you specify the CORRECT riskdiffoption or the RISKDIFFC option, PROC FREQ includes a continuity correction in the test statistic. If , the continuity correction is subtracted from in the numerator of the test statistic; otherwise, the continuity correction is added to the numerator. The value of the continuity correction is .
PROC FREQ computes onesided and twosided values for this test. When the test statistic is greater than 0, PROC FREQ displays the rightsided pvalue, which is the probability of a larger value occurring under the null hypothesis. The onesided pvalue can be expressed as
where has a standard normal distribution. The twosided pvalue is computed as .
If you specify the NONINF riskdiffoption, PROC FREQ provides a noninferiority test for the risk difference, or the difference between two proportions. The null hypothesis for the noninferiority test is
versus the alternative
where is the noninferiority margin. Rejection of the null hypothesis indicates that the row 1 risk is not inferior to the row 2 risk. See Chow, Shao, and Wang (2003) for more information.
You can specify the value of with the MARGIN= riskdiffoption. By default, . You can specify the test method with the METHOD= riskdiffoption. The following methods are available for the risk difference noninferiority analysis: Wald (with and without continuity correction), HauckAnderson, FarringtonManning, and Newcombe score (with and without continuity correction). The Wald, HauckAnderson, and FarringtonManning methods provide tests and corresponding testbased confidence limits; the Newcombe score method provides only confidence limits. If you do not specify METHOD=, PROC FREQ uses the Wald test by default.
The confidence coefficient for the testbased confidence limits is % (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are % confidence limits. You can compare the confidence limits to the noninferiority limit, –.
The following sections describe the noninferiority analysis methods for the risk difference.
Wald Test If you specify the METHOD=WALD riskdiffoption, PROC FREQ provides an asymptotic Wald test of noninferiority for the risk difference. This is also the default method. The Wald test statistic is computed as
where () estimates the risk difference and is the noninferiority margin.
By default, the standard error for the Wald test is computed from the sample proportions as
If you specify the VAR=NULL riskdiffoption, the standard error is based on the null hypothesis that the risk difference equals – (Dunnett and Gent 1977). The standard error is computed as
where
If you specify the CORRECT riskdiffoption or the RISKDIFFC option, a continuity correction is included in the test statistic. The continuity correction is subtracted from the numerator of the test statistic if the numerator is greater than zero; otherwise, the continuity correction is added to the numerator. The value of the continuity correction is .
The pvalue for the Wald noninferiority test is , where has a standard normal distribution.
HauckAnderson Test If you specify the METHOD=HA riskdiffoption, PROC FREQ provides the HauckAnderson test for noninferiority. The HauckAnderson test statistic is computed as
where and the standard error is computed from the sample proportions as
The HauckAnderson continuity correction is computed as
The pvalue for the HauckAnderson noninferiority test is , where has a standard normal distribution. See Hauck and Anderson (1986) and Schuirmann (1999) for more information.
FarringtonManning Test If you specify the METHOD=FM riskdiffoption, PROC FREQ provides the FarringtonManning test of noninferiority for the risk difference. The FarringtonManning test statistic is computed as
where and
where and are the maximum likelihood estimators of and under the null hypothesis that the risk difference equals –. The pvalue for the FarringtonManning noninferiority test is then , where has a standard normal distribution.
From Farrington and Manning (1990), the solution to the maximum likelihood equation is
where
Newcombe Score Confidence Limits If you specify the METHOD=NEWCOMBE riskdiffoption, PROC FREQ provides the Newcombe hybrid score (Wilson) confidence limits for the risk difference. The confidence coefficient for the confidence limits is % (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are % confidence limits. You can compare the confidence limits with the noninferiority limit, –.
The Newcombe score confidence limits for the risk difference are constructed from the Wilson score confidence limits for each of the two individual proportions. The confidence limits for the individual proportions are used in the standard error terms of the Wald confidence limits for the proportion difference. See Newcombe (1998) and Barker et al. (2001) for more information.
Wilson score confidence limits for and are the roots of
for . The confidence limits are computed as
See the section Wilson (Score) Confidence Limits for details.
Denote the lower and upper Wilson score confidence limits for as and , and denote the lower and upper confidence limits for as and . The Newcombe score confidence limits for the proportion difference () are computed as
If you specify the CORRECT riskdiffoption, PROC FREQ provides continuitycorrected Newcombe score confidence limits. By including a continuity correction of , the Wilson score confidence limits for the individual proportions are computed as the roots of
The continuitycorrected confidence limits for the individual proportions are then used to compute the proportion difference confidence limits and .
If you specify the SUP riskdiffoption, PROC FREQ provides a superiority test for the risk difference. The null hypothesis is
versus the alternative
where is the superiority margin. Rejection of the null hypothesis indicates that the row 1 proportion is superior to the row 2 proportion. You can specify the value of with the MARGIN= riskdiffoption. By default, .
The superiority analysis is identical to the noninferiority analysis but uses a positive value of the margin in the null hypothesis. The superiority computations follow those in the section Noninferiority Tests by replacing – by . See Chow, Shao, and Wang (2003) for more information.
If you specify the EQUIV riskdiffoption, PROC FREQ provides an equivalence test for the risk difference, or the difference between two proportions. The null hypothesis for the equivalence test is
versus the alternative
where is the lower margin and is the upper margin. Rejection of the null hypothesis indicates that the two binomial proportions are equivalent. See Chow, Shao, and Wang (2003) for more information.
You can specify the value of the margins and with the MARGIN= riskdiffoption. If you do not specify MARGIN=, PROC FREQ uses lower and upper margins of –0.2 and 0.2 by default. If you specify a single margin value , PROC FREQ uses lower and upper margins of – and . You can specify the test method with the METHOD= riskdiffoption. The following methods are available for the risk difference equivalence analysis: Wald (with and without continuity correction), HauckAnderson, FarringtonManning, and Newcombe’s score (with and without continuity correction). The Wald, HauckAnderson, and FarringtonManning methods provide tests and corresponding testbased confidence limits; the Newcombe score method provides only confidence limits. If you do not specify METHOD=, PROC FREQ uses the Wald test by default.
PROC FREQ computes two onesided tests (TOST) for equivalence analysis (Schuirmann 1987). The TOST approach includes a rightsided test for the lower margin and a leftsided test for the upper margin . The overall pvalue is taken to be the larger of the two pvalues from the lower and upper tests.
The section Noninferiority Tests gives details about the Wald, HauckAnderson, FarringtonManning and Newcombe score methods for the risk difference. The lower margin equivalence test statistic takes the same form as the noninferiority test statistic but uses the lower margin value in place of –. The upper margin equivalence test statistic take the same form as the noninferiority test statistic but uses the upper margin value in place of –.
The testbased confidence limits for the risk difference are computed according to the equivalence test method that you select. If you specify METHOD=WALD with VAR=NULL, or METHOD=FM, separate standard errors are computed for the lower and upper margin tests. In this case, the testbased confidence limits are computed by using the maximum of these two standard errors. The confidence limits have a confidence coefficient of % (Schuirmann 1999). By default, if you do not specify the ALPHA= option, these are % confidence limits. You can compare the confidence limits to the equivalence limits, .
If you specify the RISKDIFF option in the EXACT statement, PROC FREQ provides exact unconditional confidence limits for the risk difference. PROC FREQ computes the confidence limits by inverting two separate onesided tests (tail method), where the size of each test is at most and the confidence coefficient is at least ). Exact conditional methods, described in the section Exact Statistics, do not apply to the risk difference due to the presence of a nuisance parameter (Agresti 1992). The unconditional approach eliminates the nuisance parameter by maximizing the pvalue over all possible values of the parameter (Santner and Snell 1980).
By default, PROC FREQ uses the unstandardized risk difference as the test statistic in the confidence limit computations. If you specify the RISKDIFF(METHOD=FMSCORE) option, the procedure uses the FarringtonManning score statistic (Chan and Zhang 1999). The score statistic is a less discrete statistic than the raw risk difference and produces less conservative confidence limits (Agresti and Min 2001). See also Santner et al. (2007) for comparisons of methods for computing exact confidence limits for the risk difference.
PROC FREQ computes the confidence limits as follows. The risk difference is defined as the difference between the row 1 and row 2 risks (proportions), , and and denote the row totals of the table. The joint probability function for the table can be expressed in terms of the table cell frequencies, the risk difference, and the nuisance parameter as
The % confidence limits for the risk difference are computed as
where
The set includes all tables with row sums equal to and , and denotes the value of the test statistic for table in . To compute , the sum includes probabilities of those tables for which (), where is the value of the test statistic for the observed table. For a fixed value of , is taken to be the maximum sum over all possible values of .
By default, PROC FREQ uses the unstandardized risk difference as the test statistic . If you specify the RISKDIFF(METHOD=FMSCORE) option, the procedure uses the FarringtonManning risk difference score statistic as the test statistic. The computation of the risk difference score statistic is described in the subsection FarringtonManning Test in the section Noninferiority Tests. See Farrington and Manning (1990) and Miettinen and Nurminen (1985) for more information.