In this section, degrees of freedom (df) refers to the denominator degrees of freedom for F statistics in hypothesis testing. It also refers to the degrees of freedom in t tests for parameter estimates and odds ratio estimates, and for computing t distribution percentiles for confidence limits of these estimates. The value of df is determined by the design degrees of freedom f and by what you specify in the DF= option in the MODEL statement.
The default df is determined as
where f is the design degrees of freedom and r is the rank of the contrast of model parameters to be tested.
The design degrees of freedom f is determined by the survey design and the variance estimation method.
For Taylor series variance estimation, the design degrees of freedom f can depend on the number of clusters, the number of strata, and the number of observations. These numbers are based on the observations that are included in the analysis; they do not count observations that are excluded from the analysis because of missing values. If all values in a stratum are excluded from the analysis as missing values, then that stratum is called an empty stratum. Empty strata are not counted in the total number of strata for the analysis. Similarly, empty clusters and missing observations are not included in the totals counts of clusters and observations that are used to compute the f for the analysis.
If you specify the MISSING option in the CLASS statement, missing values are treated as valid nonmissing levels and are included in determining the f. If you specify the NOMCAR option for Taylor series variance estimation, observations that have missing values for variables in the regression model are included. For more information about missing values, see the section Missing Values.
Using the notation that is defined in the section Notation, let be the total number of clusters if the design has a CLUSTER statement; let n be the total sample size; and let H be the number of strata if there is a STRATA statement, or 1 otherwise. Then for Taylor series variance estimation, the design degrees of freedom is
For the replication variance estimation method, the design degrees of freedom f depends on the replication method that you use or whether you use replication weights.
If you provide replicate weights but you do not specify DF=value in the REPWEIGHTS statement, f is the number of replicates.
If you specify the DF=value option in a REPWEIGHTS statement, then f=value.
If you do not provide replicate weights and use BRR (including Fay’s method ) method, then f=H, which is the number of strata.
If you do not provide replicate weights and use the jackknife method, then , where R is the number of replicates and H is the number of strata if you specify a STRATA statement or H = 1 otherwise.
If you do not want to use the default design degrees of freedom, then you can specify the DF=DESIGN(value) or DF=PARMADJ(value) (for Taylor method only) option in the MODEL statement, where value is a positive number. Then f=value.
However, if you specify the value in the DF= option in the MODEL statement as well as with the DF= option in a REPWEIGHTS statement, then the df is determined by the value in the MODEL statement, and the DF= option in the REPWEIGHTS statement is ignored.
If you specify DF=INFINITY in the MODEL statement, then the df is set to be infinite.
As the denominator degrees of freedom grows, an F distribution approaches a chisquare distribution, and similarly a t distribution approaches a normal distribution. Therefore, when you specify DF=INFINITY in the MODEL statement), PROC SURVEYLOGISTIC uses chisquare tests and normal distribution percentiles to construct confidence intervals.
When you use Taylor series variance estimation (by default or when you specify VARMETHOD=TAYLOR in the MODEL statement), and you are fitting a model that has many parameters relative to the design degrees of freedom, it is appropriate to modify the design degrees of freedom by using the number of nonsingular parameters p in the model (Korn and Graubard (1999, section 5.2), Rao, Scott, and Skinner (1998)). You can specify DF=PARMADJ or DF=PARMADJ(value) in the MODEL statement to request this modification only for Taylor series variance estimation method; and this option does not apply to the replication variance estimation method.
Let f be the design degrees of freedom that is described in the section Design Degrees of Freedom f for the Taylor Series Method. By default, or if you specify the DF=PARMADJ option, the df is modified as .
The global null hypothesis refers to the null hypothesis that all the explanatory effects can be eliminated and the model can contain only intercepts. By using the notations in the section Logistic Regression Models, the global null hypothesis is defined as follows:
If you have a cumulative model whose model parameters are , where are the parameters for the intercepts and are the parameters for the explanatory effects, then . The number of restrictions r that are imposed on is the number of parameters in slope parameter : .
If you have a generalized logit model whose model parameters are and , then . The number of restrictions r that are imposed on is the total number of slope parameters in : .
PROC SURVEYLOGISTIC displays these tests in the "Testing Global Null Hypothesis: BETA=0" table.
For complex survey design, you can use a designadjusted RaoScott likelihood ratio chisquare test to test the global null hypothesis. For information about designadjusted chisquare tests, see Lohr (2010, Section 10.3.2), Rao and Scott (1981), Rao and Scott (1984), Rao and Scott (1987), Thomas and Rao (1987), Rao and Thomas (1989), and Thomas, Singh, and Roberts (1996).
If you specify the CHISQ(NOADJUST) option, PROC SURVEYLOGISTIC computes the likelihood ratio chisquare test without the RaoScott design correction. If you specify the CHISQ(FIRSTORDER) option, PROC SURVEYLOGISTIC performs a firstorder RaoScott likelihood ratio chisquare test. If you specify the CHISQ(SECONDORDER) option, PROC SURVEYLOGISTIC performs a secondorder RaoScott likelihood ratio chisquare test.
If you do not specify the CHISQ option, the default test depends on the design and the model. By default, PROC SURVEYLOGISTIC performs a firstorder or secondorder RaoScott likelihood ratio chisquare (Satterthwaite) test if your design is not simple random sampling or when you provide replicate weights. Otherwise, if your design is simple random sampling and you do not provide replicate weights, PROC SURVEYLOGISTIC does not make any adjustment for the likelihood ratio test. In other words:
If your design does not contain stratification nor clustering, and you do not provide replicate weights, then by default PROC SURVEYLOGISTIC performs a likelihood ratio chisquare test without any adjustment.
If your design contains either stratification or clustering, or if you provide replicate weights, then by default PROC SURVEYLOGISTIC performs a likelihood ratio chisquare test with RaoScott adjustment. However, the default order of the adjustment depends on the number of model parameters excluding the intercepts.
If there is more than one nonintercept parameter in the model, the default is the secondorder RaoScott likelihood ratio test.
If there is only one nonintercept parameter in the model, there is no need to compute the secondorder adjustment. Therefore, the default is the firstorder RaoScott likelihood ratio test.
Let be the estimated parameters, let be the estimated parameters under the global null hypothesis, and let r be the restrictions imposed on under the global null hypothesis . Let be the loglikelihood function.
Denote the estimated covariance matrix of under simple random sampling as , and its partition corresponding to the r slope parameters as . Similarly, denote the estimated covariance matrix of under the sample design as , and its partition corresponding to the r slope parameters as .
Define the design effect matrix E as
Denote as the rank of E and the positive eigenvalues of the design matrix E as .
Without the RaoScott design correction, the global null hypothesis is tested using either the chisquare statistics,
with r degrees of freedom, or an equivalent F statistics,
with degrees of freedom.
To address the impact of a complex survey design on the significance level of the likelihood ratio test, Rao and Scott (1984) proposed a firstorder correction to the chisquare statistics as
where the firstorder design correction,
is the average of positive eigenvalues of the design effect matrix E.
Under the null hypothesis, the firstorder RaoScott chisquare approximately follows a chisquare distribution with degrees of freedom.
The corresponding F statistic is
which has an F distribution with and degrees of freedom under the null hypothesis (Thomas and Rao 1984, 1987), and df is the design degrees of freedom as described in the section Design Degrees of Freedom f for the Taylor Series Method.
Rao and Scott (1987) further proposed the secondorder (Satterthwaite) RaoScott chisquare statistic as
where is the firstorder RaoScott chisquare statistic and the secondorder design correction is computed from the coefficient of variation of the eigenvalues of the design effect matrix E as
Under the null hypothesis, the secondorder RaoScott chisquare approximately follows a chisquare distribution with degrees of freedom.
The corresponding F statistic is
which has an F distribution with and degrees of freedom under the null hypothesis (Thomas and Rao 1984, 1987), and df is the design degrees of freedom as described in the section Design Degrees of Freedom f for the Taylor Series Method.
To express the general form of the score statistic, let be the parameter vector you want to estimate and let be the vector of first partial derivatives (gradient vector) of the log likelihood with respect to the parameter vector .
Consider a null hypothesis that has r restrictions imposed on . Let be the MLE of under , let be the gradient vector evaluated at , and let be the estimated covariance matrix for , which is described in the section Variance Estimation.
For the Taylor series variance estimation method, by default (or when DF=PARMADJ ), PROC SURVEYLOGISTIC computes the score test statistic for the null hypothesis as
where f is the design degrees of freedom as described in the section Design Degrees of Freedom f for the Taylor Series Method.
If you specify DF=DESIGN option or if you use the replication variance estimation method, PROC SURVEYLOGISTIC computes the score test statistic for the null hypothesis as
Under , has an F distribution with degrees of freedom, where the denominator degrees of freedom df is described in the section Degrees of Freedom.
As the denominator degrees of freedom grows, an F distribution approaches a chisquare distribution, and similarly a t distribution approaches a normal distribution. If you specify DF=INFINITY in the MODEL statement, the score test statistic for both Taylor series and replication methods for testing the null hypothesis can be expressed as
has a chisquare distribution with r degrees of freedom under the null hypothesis .
For a model that has an ordinal response, the parallel lines assumption depends on the link function, which you can specify in the LINK= option in the MODEL statement. When the link function is probit or complementary loglog, the parallel lines assumption is the equal slopes assumption; PROC SURVEYLOGISTIC displays the corresponding test in the "Score Test for the Equal Slopes Assumption" table. When the link function is logit, the parallel lines assumption is the proportional odds assumption; PROC SURVEYLOGISTIC displays the corresponding test in the "Score Test for the Proportional Odds Assumption" table. This section describes the computation of the score tests of these assumptions.
For this test, the number of response levels, , is assumed to be strictly greater than 2. Let Y be the response variable taking values . Suppose there are k explanatory variables. Consider the general cumulative model without making the parallel lines assumption:
where is the link function, and is a vector of unknown parameters consisting of an intercept and k slope parameters . The parameter vector for this general cumulative model is
Under the null hypothesis of parallelism , there is a single common slope parameter for each of the k explanatory variables. Let be the common slope parameters. Let and be the MLEs of the intercept parameters and the common slope parameters. Then, under , the MLE of is
and the chisquare score statistic has an asymptotic chisquare distribution with degrees of freedom. This tests the parallel lines assumption by testing the equality of separate slope parameters simultaneously for all explanatory variables.
Note that this test is the same as what PROC LOGISTIC produces. It does not compute a score test that uses the covariance matrix incorporated with the survey design information as described in the section Score Statistics and Tests.
Wald confidence intervals are sometimes called normal confidence intervals. They are based on the asymptotic normality of the parameter estimators. The % Wald confidence interval for is given by
where is the th percentile of the standard normal distribution, is the pseudoestimate of , and is the standard error estimate of in the section Variance Estimation.
Linear hypotheses for can be expressed in matrix form as
where is a matrix of coefficients for the linear hypotheses and is a vector of constants whose rank is r. The vector of regression coefficients includes both slope parameters and intercept parameters.
Let be the MLE of , and let be the estimated covariance matrix that is described in the section Variance Estimation.
For the Taylor series variance estimation method, PROC SURVEYLOGISTIC computes the test statistic for the null hypothesis as
where f is the design degrees of freedom as described in the section Design Degrees of Freedom f for the Taylor Series Method.
For the replication variance estimation method, PROC SURVEYLOGISTIC computes the test statistic for the null hypothesis as
Under the , has an F distribution with degrees of freedom, and the denominator degrees of freedom df is described in the section Degrees of Freedom.
As the denominator degrees of freedom grows, an F distribution approaches a chisquare distribution, and similarly a t distribution approaches a normal distribution. If you specify DF=INFINITY in the MODEL statement, PROC SURVEYLOGISTIC computes the test statistic for both Taylor series and replication methods for testing the null hypothesis as
Under , has an asymptotic chisquare distribution with r degrees of freedom.
For models that use lessthanfullrank parameterization (as specified by the PARAM=GLM option in the CLASS statement), a Type 3 test of an effect of interest (main effect or interaction) is a test of the Type III estimable functions that are defined for that effect. When the model contains no missing cells, the Type 3 test of a main effect corresponds to testing the hypothesis of equal marginal means. For more information about Type III estimable functions, see Chapter 46: The GLM Procedure, and Chapter 15: The Four Types of Estimable Functions. Also see Littell, Freund, and Spector (1991).
For models that use fullrank parameterization, all parameters are estimable when there are no missing cells, so it is unnecessary to define estimable functions. The standard test of an effect of interest in this case is the joint test that the values of the parameters associated with that effect are zero. For a model that uses effects parameterization (as specified by the PARAM=EFFECT option in the CLASS statement), the joint test for a main effect is equivalent to testing the equality of marginal means. For a model that uses reference parameterization (as specified by the PARAM=REF option in the CLASS statement), the joint test is equivalent to testing the equality of cell means at the reference level of the other model effects. For more information about the coding scheme and the associated interpretation of results, see Muller and Fetterman (2002, Chapter 14).
If there is no interaction term, the Type 3 test of an effect for a model that uses GLM parameterization is the same as the joint test of the effect for the model that uses fullrank parameterization. In this situation, the joint test is also called the Type 3 test. For a model that contains an interaction term and no missing cells, the Type 3 test of a component main effect under GLM parameterization is the same as the joint test of the component main effect under effect parameterization. Both test the equality of cell means. But this Type 3 test differs from the joint test under reference parameterization, which tests the equality of cell means at the reference level of the other component main effect. If some cells are missing, you can obtain meaningful tests only by testing a Type III estimation function, so in this case you should use GLM parameterization.
The results of a Type 3 test or a joint test do not depend on the order in which you specify the terms in the MODEL statement.
Consider a dichotomous response variable with outcomes event and nonevent. Let a dichotomous risk factor variable X take the value 1 if the risk factor is present and 0 if the risk factor is absent. According to the logistic model, the log odds function, , is given by
The odds ratio is defined as the ratio of the odds for those with the risk factor (X = 1) to the odds for those without the risk factor (X = 0). The log of the odds ratio is given by
The parameter, , associated with X represents the change in the log odds from X = 0 to X = 1. So the odds ratio is obtained by simply exponentiating the value of the parameter associated with the risk factor. The odds ratio indicates how the odds of event change as you change X from 0 to 1. For instance, means that the odds of an event when X = 1 are twice the odds of an event when X = 0.
Suppose the values of the dichotomous risk factor are coded as constants a and b instead of 0 and 1. The odds when become , and the odds when become . The odds ratio corresponding to an increase in X from a to b is
Note that for any a and b such that . So the odds ratio can be interpreted as the change in the odds for any increase of one unit in the corresponding risk factor. However, the change in odds for some amount other than one unit is often of greater interest. For example, a change of one pound in body weight might be too small to be considered important, while a change of 10 pounds might be more meaningful. The odds ratio for a change in X from a to b is estimated by raising the odds ratio estimate for a unit change in X to the power of , as shown previously.
For a polytomous risk factor, the computation of odds ratios depends on how the risk factor is parameterized. For illustration,
suppose that Race
is a risk factor with four categories: White, Black, Hispanic, and Other.
For the effect parameterization scheme (PARAM=EFFECT) with White as the reference group, the design variables for Race
are as follows.
Design Variables 


Race 



Black 
1 
0 
0 
Hispanic 
0 
1 
0 
Other 
0 
0 
1 
White 
–1 
–1 
–1 
The log odds for Black is
The log odds for White is
Therefore, the log odds ratio of Black versus White becomes
For the reference cell parameterization scheme (PARAM=REF) with White as the reference cell, the design variables for race are as follows.
Design Variables 


Race 



Black 
1 
0 
0 
Hispanic 
0 
1 
0 
Other 
0 
0 
1 
White 
0 
0 
0 
The log odds ratio of Black versus White is given by
For the GLM parameterization scheme (PARAM=GLM), the design variables are as follows.
Design Variables 


Race 




Black 
1 
0 
0 
0 
Hispanic 
0 
1 
0 
0 
Other 
0 
0 
1 
0 
White 
0 
0 
0 
1 
The log odds ratio of Black versus White is
Consider the hypothetical example of heart disease among race in Hosmer and Lemeshow (2000, p. 51). The entries in the following contingency table represent counts.
Race 


Disease Status 
White 
Black 
Hispanic 
Other 
Present 
5 
20 
15 
10 
Absent 
20 
10 
10 
10 
The computation of odds ratio of Black versus White for various parameterization schemes is shown in Table 111.9.
Table 111.9: Odds Ratio of Heart Disease Comparing Black to White
Parameter Estimates 


PARAM= 




Odds Ratio Estimates 
EFFECT 
0.7651 
0.4774 
0.0719 


REF 
2.0794 
1.7917 
1.3863 


GLM 
2.0794 
1.7917 
1.3863 
0.0000 

Since the log odds ratio () is a linear function of the parameters, the Wald confidence interval for can be derived from the parameter estimates and the estimated covariance matrix. Confidence intervals for the odds ratios are obtained by exponentiating the corresponding confidence intervals for the log odd ratios. In the displayed output of PROC SURVEYLOGISTIC, the "Odds Ratio Estimates" table contains the odds ratio estimates and the corresponding t or Wald confidence intervals computed by using the covariance matrix in the section Variance Estimation. For continuous explanatory variables, these odds ratios correspond to a unit increase in the risk factors.
To customize odds ratios for specific units of change for a continuous risk factor, you can use the UNITS statement to specify a list of relevant units for each explanatory variable in the model. Estimates of these customized odds ratios are given in a separate table. Let be a confidence interval for . The corresponding lower and upper confidence limits for the customized odds ratio are and , respectively, (for ); or and , respectively, (for c < 0). You use the CLODDS option in the MODEL statement to request confidence intervals for the odds ratios.
For a generalized logit model, odds ratios are computed similarly, except D odds ratios are computed for each effect, corresponding to the D logits in the model.
The predicted mean score of an observation is the sum of the ordered values (shown in the "Response Profile" table) minus one, weighted by the corresponding predicted probabilities for that observation; that is, the predicted means score is , where D + 1 is the number of response levels and is the predicted probability of the dth (ordered) response.
A pair of observations with different observed responses is said to be concordant if the observation with the lowerordered response value has a lower predicted mean score than the observation with the higherordered response value. If the observation with the lowerordered response value has a higher predicted mean score than the observation with the higherordered response value, then the pair is discordant. If the pair is neither concordant nor discordant, it is a tie. Enumeration of the total numbers of concordant and discordant pairs is carried out by categorizing the predicted mean score into intervals of length and accumulating the corresponding frequencies of observations.
Let N be the sum of observation frequencies in the data. Suppose there are a total of t pairs with different responses, of them are concordant, of them are discordant, and of them are tied. PROC SURVEYLOGISTIC computes the following four indices of rank correlation for assessing the predictive ability of a model:
Note that c also gives an estimate of the area under the receiver operating characteristic (ROC) curve when the response is binary (Hanley and McNeil 1982).
For binary responses, the predicted mean score is equal to the predicted probability for Ordered Value 2. As such, the preceding definition of concordance is consistent with the definition used in previous releases for the binary response model.