The SURVEYMEANS Procedure

For a stratified clustered sample design, together with the sampling weights, the sample can be represented by an matrix

where

For a categorical variable C, let l denote the number of levels of C, and denote the level values as . Let be an indicator variable for the category with the observed value in unit j in cluster i of stratum h:

Note that the indicator variable is set to missing when is missing. Therefore, the total number of analysis variables, P, is the total number of numerical variables plus the total number of levels of all categorical variables.

The sampling rate for stratum h, which is used in Taylor series and bootstrap variance estimation, is the fraction of first-stage units (PSUs) that are selected for the sample. You can use the TOTAL= or RATE= option to input population totals or sampling rates. For more information, see the section Specification of Population Totals and Sampling Rates. If you input stratum totals, PROC SURVEYMEANS computes as the ratio of the stratum sample size to the stratum total. If you input stratum sampling rates, PROC SURVEYMEANS uses these values directly for . If you do not specify the TOTAL= or RATE= option, then the procedure assumes that the stratum sampling rates are negligible, and a finite population correction is not used when computing variances. Replication methods that are specified by the VARMETHOD=BRR or VARMETHOD=JACKKNIFE option do not use this finite population correction .

When you specify the keyword MEAN, the procedure computes the estimate of the mean (mean per element) from the survey data. Also, the procedure computes the mean by default if you do not specify any statistic-keywords in the PROC SURVEYMEANS statement.

PROC SURVEYMEANS computes the estimate of the mean as

where

is the sum of the weights over all observations in the sample.

When you specify the keyword STDERR, the procedure computes the standard error of the mean. Also, the procedure computes the standard error by default if you specify the keyword MEAN, or if you do not specify any statistic-keywords in the PROC SURVEYMEANS statement. The keyword VAR requests the variance of the mean.

Taylor Series Method

When you use VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the mean . The procedure computes the estimated variance as

where, if , then

and if , then

Standard Error

The standard error of the mean is the square root of the estimated variance.

If you specify the keyword T, PROC SURVEYMEANS computes the t-value for testing that the population mean equals zero, . The test statistic equals

The two-sided p-value for this test is

where T is a random variable with the t distribution with df degrees of freedom.

PROC SURVEYMEANS computes the degrees of freedom df to obtain the t distribution’s percentile to construct the % confidence limits for means, proportions, totals, ratios, and other statistics, and for hypothesis testing. The degrees of freedom computation depends on the variance estimation method that you request. Missing values can affect the degrees of freedom computation. For more information, see the section Missing Values.

If you specify the keyword CLM, the procedure computes two-sided confidence limits for the mean. Also, the procedure includes the confidence limits by default if you do not specify any statistic-keywords in the PROC SURVEYMEANS statement.

The confidence coefficient is determined by the value of the ALPHA= option, which by default equals 0.05 and produces 95% confidence limits. The confidence limits are computed as

where is the estimate of the mean, is the standard error of the mean, and is the th percentile of the t distribution with df calculated as in the section t Test for the Mean.

If you specify the keyword UCLM, the procedure computes the one-sided upper % confidence limit for the mean:

If you specify the keyword LCLM, the procedure computes the one-sided lower % confidence limit for the mean:

If you specify the keyword CV, PROC SURVEYMEANS computes the coefficient of variation, which is the ratio of the standard error of the mean to the estimated mean:

If you specify the keyword CVSUM, PROC SURVEYMEANS computes the coefficient of variation for the estimated total, which is the ratio of the standard deviation of the sum to the estimated total:

If you specify the keyword MEAN for a categorical variable, PROC SURVEYMEANS estimates the proportion, or relative frequency, for each level of the categorical variable. If you do not specify any statistic-keywords in the PROC SURVEYMEANS statement, the procedure estimates the proportions for levels of the categorical variables, together with their standard errors and confidence limits.

The procedure estimates the proportion in level for variable C as

where is the value of the indicator function for level , defined in the section Definitions and Notation, and equals 1 if the observed value of variable C equals , and equals 0 otherwise. Since the proportion estimator is actually an estimator of the mean for an indicator variable, the procedure computes its variance and standard error according to the method outlined in the section Variance and Standard Error of the Mean. Similarly, the procedure computes confidence limits for proportions as in the section Confidence Limits for the Mean.

If you specify the keyword SUM, the procedure computes the estimate of the population total from the survey data. The estimate of the total is the weighted sum over the sample:

For a categorical variable level, estimates its total frequency in the population.

When you specify the keyword STD or the keyword SUM, the procedure estimates the standard deviation of the total. The keyword VARSUM requests the variance of the total.

Taylor Series Method

When you use VARMETHOD=TAYLOR, or by default, PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the total as

where, if , then

and if , then

Standard Deviation

The standard deviation of the total equals

If you specify the keyword CLSUM, the procedure computes confidence limits for the total. The confidence coefficient is determined by the value of the ALPHA= option, which by default equals 0.05 and produces 95% confidence limits. The confidence limits are computed as

where is the estimate of the total, is the estimated standard deviation, and is the th percentile of the t distribution with df calculated as described in the section t Test for the Mean.

If you specify the keyword UCLSUM, the procedure computes the one-sided upper % confidence limit for the sum:

If you specify the keyword LCLSUM, the procedure computes the one-sided lower % confidence limit for the sum:

If you specify the keyword DEFF in the PROC SURVEYMEANS statement, the procedure calculates the design effects for the mean. The design effect is the ratio of the variance computed under the sample design to the variance computed under the assumption of simple random sampling:

For more information, see Kish (1965, p. 258).

In the numerator, PROC SURVEYMEANS computes the variance of the mean according to the method outlined in the section Variance and Standard Error of the Mean. And the denominator is computed under the assumption that the sample design is simple random sampling, with no stratification and no clustering.

For Taylor series or bootstrap variance estimation, PROC SURVEYMEANS computes the overall sampling fraction in the simple random sampling variance by using the value of the RATE= or TOTAL= option.

If you do not specify either of these options, PROC SURVEYMEANS assumes that the value of is negligible and does not use a finite population correction in the analysis, as described in the section Specification of Population Totals and Sampling Rates.

If you specify RATE=value, PROC SURVEYMEANS uses this value as the overall sampling fraction . If you specify TOTAL=value, PROC SURVEYMEANS computes as the ratio of the number of PSUs in the sample to the specified total.

If you specify stratum sampling rates by using the RATE=SAS-data-set option, then PROC SURVEYMEANS computes stratum totals based on these stratum sampling rates and the number of sample PSUs in each stratum. The procedure sums the stratum totals to form the overall total, and it computes as the ratio of the number of sample PSUs to the overall total. Alternatively, if you specify stratum totals by using the TOTAL=SAS-data-set option, then PROC SURVEYMEANS sums these totals to compute the overall total. The overall sampling fraction is then computed as the ratio of the number of sample PSUs to the overall total.

When you use a RATIO statement, the procedure produces statistics requested by the statistic-keywords in the PROC SURVEYMEANS statement.

Suppose that you want to calculate the ratio of variable Y to variable X. Let be the value of variable X for the jth member in cluster i in the hth stratum.

The ratio of Y to X is

PROC SURVEYMEANS uses the Taylor series method to estimate the variance of the ratio as

where, if , then

and if , then

The standard error of the ratio is the square root of the estimated variance:

When the denominator for a ratio is zero, then the value of the ratio is displayed as '–Infty', 'Infty', or a missing value, depending on whether the numerator is negative, positive, or zero, respectively; and the corresponding internal value is the special missing value '.M', the special missing value '.I', or the usual missing value, respectively.

When you use a DOMAIN statement to request a domain analysis, the procedure computes the requested statistics for each domain level.

For a domain D, let be the corresponding indicator variable:

Let

Let

The requested statistics for variable y in domain D are computed by using the new weights v.

Note that is set to missing if represents a level of a categorical variable and is missing.

Domain Mean

The estimated mean of Y in the domain D is

where

When you use the Taylor series method, the variance of is estimated by

where, if , then

and if , then

If you use replication methods to estimate the variance (by specifying the VARMETHOD=BOOTSTRAP, VARMETHOD=BRR, or VARMETHOD=JACKKNIFE option), PROC SURVEYMEANS estimates the variance of by using the variability among replicate estimates to estimate the overall variance. For more information, see the section Replication Methods for Variance Estimation.

Difference of Domain Means

If you specify the DIFFMEANS option in a DOMAIN statement, PROC SURVEYMEANS compares the means of a continuous analysis variable at different domain levels.

Let D be a domain definition that is specified in a DOMAIN statement. Let be the r levels of D, and let the corresponding indicator variables be

for

For variable Y, the difference between the means for domain levels and () can be expressed as

The estimated variance for this difference is

where the estimated variances and for means at corresponding domain levels (in addition to the covariance between these two domain means) are described as in the section Domain Mean.

For variable Y, PROC SURVEYMEANS computes the t statistics to test the significance of the difference of the domain means between two levels and as the following ratio with df degrees of freedom:

For more information about df, see the section Degrees of Freedom. For more information about computing the these domain means and the variance and covariance of domain means with poststratification, see the section Variance of the Domain Mean and Sum. The corresponding t statistics are then computed accordingly.

Domain Total

The estimated total in domain D is

and its estimated variance is

where, if , then

and if , then

Domain Ratio

The estimated ratio of Y to X in domain D is

and its estimated variance is

where, if , then

and if , then

For domain analysis with poststratification, see the section Poststratification. For quantile estimation in a domain, see the section Domain Quantile. For quantile estimation in a domain with poststratification, see the section Domain Quantile Estimation with Poststratification.

Let Y be the variable of interest in a complex survey. Denote as the cumulative distribution function of Y. For , the pth quantile of the population cumulative distribution function is

Estimate of Quantile

Let be the observed values for variable Y that are associated with sampling weights, where are the stratum index, cluster index, and member index, respectively, as shown in the section Definitions and Notation. Let denote the sample order statistics for variable Y.

An estimate of quantile is

where is the estimated cumulative distribution for Y,

and is the indicator function.

Standard Error

How PROC SURVEYMEANS estimates the standard error of a quantile depends on which variance method you specify in the VARMETHOD= option.

Taylor Series Method

When you specify VARMETHOD=TAYLOR, or by default if you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses Woodruff’s method (Dorfman and Valliant 1993; Särndal, Swensson, and Wretman 1992; Francisco and Fuller 1991) to estimate the variances of quantiles. This method first constructs a confidence interval on a quantile . Then it uses the width of the confidence interval to estimate the standard error of . To construct the confidence interval, PROC SURVEYMEANS first estimates the variance of the estimated distribution function by

where

Then % confidence limits for can be constructed by

where is the th percentile of the t distribution with df degrees of freedom, described in the section Degrees of Freedom.

When is out of the range of [0,1], the procedure does not compute the standard error of .

The th quantile is defined as

and the th quantile is defined as

The standard error of is then estimated by

where is the th percentile of the t distribution with df degrees of freedom.

Replication Methods

When you use the jackknife replication method as described in the section Replication Methods for Variance Estimation, the following naive replication variance estimate for a quantile can have poor properties:

Here is a coefficient that corresponds to each replicate, R is the number of replicates, and is the estimated quantile in the rth replicate.

Fuller (2009) proposes using a smoothing method similar to the following to modify the quantile estimates for each replicate before using this naive variance estimate for a quantile .

In the rth replicate (), denote to be the observed values of variable Y that are associated with the replicate weights. Let be the sample order statistics for variable Y in the rth replicate, where is the total number of observations whose replicate weights .

Let be the estimated pth quantile and be the estimated cumulative distribution for Y,

and

where is the indicator function.

Choose a segment in the distribution function by two points, and ,

where is the replicate weight that corresponds to in the rth replicate. Then a modified pth quantile in the rth replicate is defined by

PROC SURVEYMEANS then uses as the quantile estimate in the rth replicate and its deviation from the mean of to estimate the variance of :

If you want to use the naive replication variance estimates by using instead of the smoothed to estimate the variance of , you can specify the method-option NAIVEQVAR as VARMETHOD=BRR(NAIVEQVAR) or VARMETHOD=JACKKNIFE(NAIVEQVAR).

Confidence Limits

Symmetric % confidence limits are computed as

If you specify the NONSYMCL option in the PROC SURVEYMEANS statement, when you use the VARMETHOD=TAYLOR option, the procedure computes % nonsymmetric confidence limits:

Quantile Estimation with Poststratification

When you specify a POSTSTRATA statement, the quantile estimation and its variance estimation incorporate poststratification. For more information about poststratification, see the section Poststratification.

For a selected sample, let be the poststratum index; let be the population totals for each corresponding poststratum, and let be the indicator variable for the poststratum r that is defined by

Denote the total sum of original weights in the sample for each poststratum as

Assume that the observation (h, i, j) belongs to the rth poststratum. Then the poststratification weight for the observation (h, i, j) is

Then the estimated cumulative distribution function of Y, and the estimated pth quantile estimation can be computed as in the section Estimate of Quantile by replacing the original weights, , with the poststratification weights, .

When you specify VARMETHOD=TAYLOR (or by default), the variance of is estimated as in the section Standard Error, except that the variance of the estimated distribution function is computed as follows.

For each poststratum , define

where is the indicator function.

Assume that the observation (h, i, j) belongs to the rth poststratum. Let

PROC SURVEYMEANS estimates the variance of the estimated distribution function with poststratification by

where

Domain Quantile

Let Y be the variable of interest in a complex survey, and let a subpopulation of interest be domain D. Denote as the cumulative distribution function of Y in domain D. For , the pth quantile of the population cumulative distribution function is

Let be the corresponding indicator variable:

Assume that there are a total of d observations among the n observations in the entire sample that belong to domain D. Let denote the order statistics of variable Y for these d observations that fall in domain D.

The cumulative distribution function of Y in domain D is estimated by

and is the indicator function. Then the estimated quantile in domain D is

In order to estimate the variance for , PROC SURVEYMEANS first estimates the variance of the estimated distribution function in domain D. When you specify VARMETHOD=TAYLOR (or by default), the variance of is estimated by

where

Then % confidence limits for can be constructed by , where

and is the th percentile of the t distribution with df degrees of freedom, described in the section Degrees of Freedom. When is out of the range of [0,1], PROC SURVEYMEANS does not compute the standard error of .

The th quantile is then estimated as

The th quantile is then estimated as

The standard error of is then estimated by

where is the th percentile of the t distribution with df degrees of freedom.

Symmetric % confidence limits for are computed as

If you specify the NONSYMCL option in the PROC SURVEYMEANS statement, the procedure displays % nonsymmetric confidence limits as

Domain Quantile Estimation with Poststratification

When you specify both a POSTSTRATA statement and a DOMAIN statement, the domain quantile estimation and its variance estimation incorporate poststratification. For more information about poststratification, see the section Poststratification.

For a selected sample, let be the poststratum index, let be the population totals for each corresponding poststratum, and let be the indicator variable for the poststratum r:

The poststratification weights, , are defined as in the section Quantile Estimation with Poststratification.

For domain D, let be the corresponding indicator variable:

With poststratification, for variable Y, the estimated cumulative distribution in domain D, , and its pth quantile estimation, , can be computed as in the section Domain Quantile by replacing the original weights, , with the poststratification weights, . However, the variance of , which is described in the section Domain Quantile, is computed as follows when you specify the VARMETHOD=TAYLOR option (or by default).

Define

Assume that the observation (h, i, j) belongs to the rth poststratum. Then the variance of is estimated by

For a continuous variable Y that has positive values, the SURVEYMEANS procedure can compute its geometric mean and associated standard error and confidence limits. To request these statistics, you can specify statistic-keywords such as GEOMEAN, GMSTDERR, and GMCLM.

The geometric mean of Y from a sample is computed as

where

is the sum of the weights over all observations in the data set.

When you use the Taylor series method, the variance estimation for the geometric mean is computed as

where

The standard error of the geometric mean is the square root of the estimated variance:

The confidence limits for the geometric means are computed based on the confidence limits for the log transformation of the Y variable as

where

and is the th percentile of the t distribution, with df calculated as in the section t Test for the Mean.

If you use replication methods to estimate the variance by specifying the VARMETHOD=BOOTSTRAP, VARMETHOD=BRR, or VARMETHOD=JACKKNIFE option, the procedure computes the variance of a geometric means by using the variability among replicate estimates to estimate the overall variance. See the section Replication Methods for Variance Estimation for more information.

Then the standard error is the square root of the estimated variance:

The confidence limits for the geometric means are computed based on the confidence limits for the log transformation of the variable Y as

where

and is the th percentile of the t distribution, with df calculated as in the section t Test for the Mean.

After a probability sample is drawn and survey data are collected, researchers sometimes want to stratify the sample according to auxiliary information about the sampled population. This process is often called poststratification.

When poststratification is done properly, it can improve efficiency. It can also be used to adjust the sampling weights such that the marginal distribution of the sampling weights is in agreement with known auxiliary information from other resources, such as the census. The adjusted weight is often called the poststratification weight.

It is quite common for researchers to use poststratification techniques in survey data analysis.

Poststratification is also used by epidemiologists, who frequently analyze health survey data. They often compute statistics based on a process called direct standardization, a form of poststratification. For example, certain diseases, such as cancer, are more common among older populations. Therefore, to compare the prevalence rates among geographic regions that are populated with different age groups, it is necessary to make adjustments according to such demographic categories and to compute relative prevalence rates of the diseases.

For more information about poststratification, see Fuller (2009); Lohr (2010); Wolter (2007); Rao, Yung, and Hidiroglou (2002).

After you provide the population controls for each poststratum that is defined by the poststratification variables, the SURVEYMEANS procedure creates the poststratification weights accordingly. Then the procedure computes statistics that you request by using poststratification weights.

You can save the poststratification weights in an OUTPSWGT= data set to be used in subsequent analyses.

For a selected sample, let be the poststratum index; let be the population totals (or poststratum totals) for the corresponding poststrata, and let be a corresponding indicator variable for poststratum p defined by

Denote the total sum of original weights in the sample for each poststratum as

Then the poststratification weight for observation (h, i, j) is

The SURVEYMEANS procedure computes statistics by using the poststratification weights instead of the original weights .

The standard error and confidence intervals of computed statistics are based on the estimated variances, which are computed by using either a replication method or the Taylor series method.

Taylor Series Method

When you specify VARMETHOD=TAYLOR, or by default when you do not specify the VARMETHOD= option, PROC SURVEYMEANS uses the Taylor series method to estimate the variances of requested statistics.

Variance of the Mean and Sum

The sum and mean of variable Y under poststratification are

where

is the sum of the poststratification weights over all observations in the sample.

For each poststratum , let the mean of variable Y be

where is the total of the poststratification weights in poststratum p.

For observation (h, i, j), assume that it belongs to the pth poststratum. Let

PROC SURVEYMEANS estimates the variance of as

where, if , then

and if , then

PROC SURVEYMEANS estimates the variance of as

Variance of the Domain Mean and Sum

For a domain D, let be the corresponding indicator variable:

Let

The sum and mean of variable Y under poststratification in domain D are

where

is the sum of the poststratification weights over all observations in the sample in domain D. For each poststratum , let the mean of variable Y and the mean of the domain indicator variable in each poststratum be

Assume that the observation (h, i, j) belongs to the pth poststratum. Let

Then PROC SURVEYMEANS estimates the variance of domain sum as

where, if , then

and if , then

Then PROC SURVEYMEANS estimates the variance of domain mean as

where, if , then

and if , then

Variance of the Ratio

Suppose you want to calculate the ratio of variable Y to variable X. Let and be the values of variable X and variable Y, respectively, for observation (h, i, j).

The ratio of Y to X after poststratification is

where is the poststratification weight for observation .

Assume that the observation (h, i, j) belongs to the pth poststratum. Let

where and are the means of variable Y and variable X, respectively, in poststratum p.

The variance of is estimated by

where, if , then

and if , then

Variance of the Domain Ratio

For a domain D, let be the corresponding indicator variable:

Let

The ratio of variable Y to variable X in domain D after poststratification is estimated by

For each poststratum , let the mean of variable X and Y in each poststratum be

Assume that the observation (h, i, j) belongs to the pth poststratum. Let

Then PROC SURVEYMEANS estimates the variance of domain ratio after poststratification as

where, if , then

and if , then