V.A.Morgan document1 15 March 2016 USING WEIGHTS WITH SHIP DATA Version: 05 July 2012 Contents USING WEIGHTS WITH SHIP DATA .................................................................... 1 General introduction ................................................................................ 2 Weights in Stata...................................................................................... 3 Weights in SPSS ...................................................................................... 4 Weights in SAS ........................................................................................ 5 APPENDICES ............................................................................................. 6 Stata example output ............................................................................... 6 Stata: Mean ........................................................................................ 6 Stata: Tabulate .................................................................................... 6 Stata: Logistic regression ........................................................................ 7 SPSS example output ................................................................................ 9 SPSS: Complex Samples Analysis Plan .......................................................... 9 SPSS: Complex Samples Descriptives .......................................................... 10 SPSS: Complex Samples Tables ................................................................. 11 SPSS: Complex Samples Logistic regression .................................................. 15 SAS example output ................................................................................ 17 SAS: The SURVEYMEANS Procedure ............................................................ 17 SAS: The SURVEYFREQ Procedure .............................................................. 18 SAS: The SURVEYLOGISTIC Procedure ......................................................... 19 1 V.A.Morgan document1 15 March 2016 General introduction 1. Weighted analyses should be only be done using appropriate statistical software: Stata svy or pweight commands SPSS Complex Samples SAS Survey Procedures 2. The weights to use in Stata, SPSS Complex Samples and SAS Survey Procedures are in the variable: weightsa 3. Weightsa may also be used for subgroup analyses using Stata commands, SPSS Complex Samples and SAS Survey Procedures. 4. If working with the full SHIP dataset based on 1825 survey participants, using weights results in a population N of 7955. For SHIP, ‘population’ refers to the individuals ascertained in the screening phase This N will be smaller for subsamples (e.g. those meeting full ICD-10 criteria for psychosis; schizophrenia only etc.). 5. It is very important that weightsa be used only as a sampling (or probably) weight. Using it as a weight in a Stata [fweight=…] option, the SPSS “Weight by…” command or as a WEIGHT or FREQ variable in a normal SAS proc will lead the program to believe that 7955 participants were actually interviewed leading to extremely serious errors in many statistics. 6. When relevant strata information is included in a model, the ‘need’ for weighting is dramatically reduced. Site strata information is not available, but our experience to date is that when age is included in models (e.g., regression), the difference in estimates and standard errors between weighted and unweighted data is relatively small. Nevertheless, if it possible to use weighting, it should be done for accuracy and for uniformity across analyses. 7. For information only: there is a variable that Andrew has created called weightsa_nrm which gives an approximation for those using basic SPSS and basic SAS. However we are not recommending that people work in basic SPSS or basic SAS with this variable and will not be providing it unless the person requesting to use this variable can justify their request. 2 V.A.Morgan document1 15 March 2016 Weights in Stata In Stata, there are two approaches, depending on the procedure. (a) For many procedures (e.g. mean; logistic), you can use the [pweight=weightsa] option (b) Other procedures (e.g. tabulate) need to be run in conjunction with the survey (svy) feature. The survey feature can also be used with procedures that permit the [pweight=weightsa] option. The survey structure must be first specified in svyset if you need to use the survey procedure: * Specifying the survey structure before using the survey (svy) feature. svyset [pweight=weightsa] This will produce the following information output pweight: weightsa VCE: linearized Single unit: missing Strata 1: <one> SU 1: <observations> FPC 1: <zero> * Means using (a) pweight and (b) the survey feature. Note that (b) assumes you have specified the survey structure as above. (a) mean age_calc [pweight = weightsa] (b) svy: mean age_calc * Tables using the survey feature. Note that this assumes you have specified the survey structure as above. svy: tabulate sex DIAGicd10, count ci * Logistic Regression a) pweight and (b) the survey feature. Note that (b) assumes you have specified the survey structure as above.. (a) logit anyIP_rev age_calc [pweight = weightsa] (b) svy: logit anyIP_rev age_calc 3 V.A.Morgan document1 15 March 2016 Weights in SPSS The appropriate SPSS Complex Samples analysis plan is in a file called SHIP_SPSSweighting.csaplan which is created by running the syntax below: CSPLAN ANALYSIS /PLAN FILE='D:\SHIPdata_files\SHIP_SPSSweighting.csaplan' /PLANVARS ANALYSISWEIGHT=weightsa /SRSESTIMATOR TYPE=WR /PRINT PLAN /DESIGN /ESTIMATOR TYPE=WR. You must use the SPSS Complex Samples battery of statistics with this plan. For example: * Complex Samples Descriptives. CSDESCRIPTIVES /PLAN FILE='D:\SHIPdata_files\SHIP_SPSSweighting.csaplan' /SUMMARY VARIABLES=age_calc /MEAN /STATISTICS SE COUNT POPSIZE CIN(95) /MISSING SCOPE=ANALYSIS CLASSMISSING=EXCLUDE. * Complex Samples Tabulate. CSTABULATE /PLAN FILE='D:\SHIPdata_files\SHIP_SPSSweighting.csaplan' /TABLES VARIABLES=sex BY DIAGicd10 /CELLS POPSIZE COLPCT /STATISTICS CIN(95) /MISSING SCOPE=TABLE CLASSMISSING=EXCLUDE. * Complex Samples Logistic Regression. CSLOGISTIC anyIP WITH age_calc /PLAN FILE='D:\SHIPdata_files\SHIP_SPSSweighting.csaplan' /MODEL age_calc /INTERCEPT INCLUDE=YES SHOW=YES /STATISTICS PARAMETER EXP CINTERVAL /TEST TYPE=F PADJUST=LSD /ODDSRATIOS COVARIATE=[age_calc(1)] /CRITERIA MXITER=100 MXSTEP=5 PCONVERGE=[1E-006 RELATIVE] LCONVERGE=[0] CHKSEP=20 CILEVEL=95 /PRINT SUMMARY CLASSTABLE VARIABLEINFO SAMPLEINFO. 4 V.A.Morgan document1 15 March 2016 Weights in SAS For SAS Complex Survey Design, SAS uses specific commands when using weight variable. For example: for Means, SAS uses Proc Surveymeans as the main command instead of 'Proc Means'. Similarly for frequency and logistic procedures, SAS uses Proc Surveyfreq and Proc Surveylogistic respectively SAS SURVEY PROCEDURES COMMANDS. For example: *Complex survey design: MEAN PROCEDURE. Proc Surveymeans data=xxx.xxxxxx; weight weightsa; var age_calc; Title 'MEAN PROCEDURE FOR AGE'; run; *Complex survey design: CROSS TABULATION. Proc Surveyfreq data=xxx.xxxxxx ; weight weightsa; tables sex*DIAGICD10; Title 'Cross tabulation of Sex by Psychotic illness'; run; *Complex survey design: LOGISTIC REGRESSION. Proc Surveylogistic data= xxx.xxxxxx order=internal; weight weightsa; Model anyIP (ref=last order=internal)= age_calc /link=glogit; Title 'Effect of age on inpatient admission'; run; w 5 V.A.Morgan document1 15 March 2016 APPENDICES Stata example output Stata: Mean Stata: Tabulate 6 V.A.Morgan document1 15 March 2016 Stata: Logistic regression 7 V.A.Morgan document1 15 March 2016 8 V.A.Morgan document1 15 March 2016 SPSS example output SPSS: Complex Samples Analysis Plan * SHIP_SPSSweighting_egs.sps. * Analysis Preparation Wizard. CSPLAN ANALYSIS /PLAN FILE='D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan' /PLANVARS ANALYSISWEIGHT=weightsa /SRSESTIMATOR TYPE=WR /PRINT PLAN /DESIGN /ESTIMATOR TYPE=WR. Complex Samples: Plan Summary Analysis Information Estimator Assumption Stage 1 Sampling with replacement Plan File: D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan Weight Variable: weightsa Sampling weight by site and age stratum SRS Estimator: Sampling with replacement CSPLAN VIEW /PLAN FILE='D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan' . Complex Samples: Plan Analysis Plan Summary Analysis Information Estimator Assumption Stage 1 Sampling with replacement Plan File: D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan Weight Variable: weightsa Sampling weight by site and age stratum SRS Estimator: Sampling with replacement 9 V.A.Morgan document1 15 March 2016 SPSS: Complex Samples Descriptives * Complex Samples Descriptives. CSDESCRIPTIVES /PLAN FILE='D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan' /SUMMARY VARIABLES=age_calc /MEAN /STATISTICS SE COUNT POPSIZE CIN(95) /MISSING SCOPE=ANALYSIS CLASSMISSING=EXCLUDE. Complex Samples: Descriptives Univariate Statistics Estimate Standard Error Mean age_calc Age (calculated) in years at time of interview 39.47 95% Confidence Interval Lower Upper .273 38.93 40.00 Univariate Statistics Population Size Mean age_calc Age (calculated) in years at time of interview 7955.000 Unweighted Count 1825 10 V.A.Morgan document1 15 March 2016 SPSS: Complex Samples Tables * Complex Samples Tables. CSTABULATE /PLAN FILE='D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan' /TABLES VARIABLES=sex BY DIAGicd10 /CELLS POPSIZE COLPCT /STATISTICS CIN(95) /MISSING SCOPE=TABLE CLASSMISSING=EXCLUDE. Complex Samples: Tables sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 95% Confidence Interval 0 male Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval 1 female Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval Total Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 1 schizophrenia 2650.360 2464.343 2836.378 70.4% 67.1% 73.6% 1112.866 974.388 1251.344 29.6% 26.4% 32.9% 3763.226 3559.907 3966.545 100.0% 100.0% 100.0% sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 0 male 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 1 female 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size Total 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 2 schizoaffective 623.920 524.582 723.257 53.6% 47.5% 59.5% 540.863 447.709 634.017 46.4% 40.5% 52.5% 1164.782 1033.930 1295.635 100.0% 100.0% 100.0% 11 V.A.Morgan document1 15 March 2016 sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 95% Confidence Interval 0 male Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval 1 female Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval Total Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 3 bipolar, mania 621.156 516.721 725.591 44.5% 38.8% 50.3% 774.091 658.974 889.208 55.5% 49.7% 61.2% 1395.247 1246.483 1544.011 100.0% 100.0% 100.0% sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 0 male 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 1 female 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size Total 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 4 depressive psychosis 146.112 93.097 199.128 39.7% 29.1% 51.3% 221.883 156.582 287.185 60.3% 48.7% 70.9% 367.996 284.700 451.292 100.0% 100.0% 100.0% 12 V.A.Morgan document1 15 March 2016 sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 95% Confidence Interval 0 male Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval 1 female Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval Total Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 5 delusional disorders and other non-organic psychosis 303.380 225.754 381.007 72.0% 61.4% 80.7% 117.849 70.240 165.459 28.0% 19.3% 38.6% 421.230 330.999 511.461 100.0% 100.0% 100.0% sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 0 male 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 1 female 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size Total 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 6 severe depression without psychosis 288.228 212.997 363.459 40.2% 32.4% 48.6% 427.942 338.840 517.044 59.8% 51.4% 67.6% 716.170 601.809 830.532 100.0% 100.0% 100.0% 13 V.A.Morgan document1 15 March 2016 sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 95% Confidence Interval 0 male Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval 1 female Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 95% Confidence Interval Total Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 7 screen-positive for psychosis but did not meet full criteria for ICD-10 psychosis 79.350 40.256 118.444 62.8% 41.5% 80.1% 46.999 13.491 80.507 37.2% 19.9% 58.5% 126.349 75.013 177.685 100.0% 100.0% 100.0% sex sex * DIAGicd10 DIP ICD-10 sex sex Estimate Population Size 0 male 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size 1 female 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper Estimate Population Size Total 95% Confidence Interval Lower Upper Estimate % within DIAGicd10 DIP ICD-10 95% Confidence Interval Lower Upper DIAGicd10 DIP ICD-10 Total 4712.507 4508.928 4916.086 59.2% 56.8% 61.6% 3242.493 3043.639 3441.347 40.8% 38.4% 43.2% 7955.000 7826.394 8083.606 100.0% 100.0% 100.0% 14 V.A.Morgan document1 15 March 2016 SPSS: Complex Samples Logistic regression * Complex Samples Logistic Regression. CSLOGISTIC anyIP WITH age_calc /PLAN FILE='D:\SHIPdata_files\DataMain\weighting\SPSS\SHIP_SPSSweighting.csaplan' /MODEL age_calc /INTERCEPT INCLUDE=YES SHOW=YES /STATISTICS PARAMETER EXP CINTERVAL /TEST TYPE=F PADJUST=LSD /ODDSRATIOS COVARIATE=[age_calc(1)] /CRITERIA MXITER=100 MXSTEP=5 PCONVERGE=[1E-006 RELATIVE] LCONVERGE=[0] CHKSEP=20 CILEVEL=95 /PRINT SUMMARY CLASSTABLE VARIABLEINFO SAMPLEINFO. Complex Samples: Logistic Regression Sample Design Information N Unweighted Cases Valid Invalid Total 1825 0 1825 7955.000 1 1825 1824 Population Size Strata Units Sampling Design Degrees of Freedom Stage 1 Categorical Variable Information Weighted Count Weighted Percent 0 no/na/dk 4552.607 57.2% anyIP Any inpatient admissions - past yeara 1 yesb 3402.393 42.8% Population Size 7955.000 100.0% a. Dependent Variable b. Reference Category Covariate Information Mean age_calc Age (calculated) in years at time of interview 39.47 Pseudo R Squares Cox and Snell .002 Nagelkerke .003 McFadden .002 Dependent Variable: anyIP Any inpatient admissions past year (reference category = 1 yes) Model: (Intercept), age_calc Source (Corrected Model) (Intercept) age_calc Tests of Model Effects df1 df2 1.000 1824.000 1.000 1824.000 1.000 1824.000 Wald F 3.883 .101 3.883 Sig. .049 .751 .049 Dependent Variable: anyIP Any inpatient admissions - past year (reference category = 1 yes) Model: (Intercept), age_calc Parameter Estimates 15 V.A.Morgan document1 15 March 2016 anyIP Any inpatient admissions - past year 0 no/na/dk Parameter B (Intercept) age_calc -.057 .009 anyIP Any inpatient admissions - past year Parameter Estimates Parameter (Intercept) age_calc 0 no/na/dk 95% Confidence Interval Lower Upper -.407 .293 4.109E-005 .018 Exp(B) .945 1.009 95% Confidence Interval for Exp(B) Lower Upper .666 1.341 1.000 1.018 Dependent Variable: anyIP Any inpatient admissions - past year (reference category = 1 yes) Model: (Intercept), age_calc Observed 0 no/na/dk 1 yes Overall Percent Classification Predicted 0 no/na/dk 1 yes Percent Correct 4552.607 .000 100.0% 3402.393 .000 0.0% 100.0% 0.0% 57.2% Dependent Variable: anyIP Any inpatient admissions - past year (reference category = 1 yes) Model: (Intercept), age_calc Odds Ratiosa anyIP Any inpatient admissions - past year Units of Change age_calc Age (calculated) in years at time of interview 1.000 Units of Change age_calc Age (calculated) in years at time of 1.000 interview Units of Change age_calc Age (calculated) in years at time of 1.000 interview Odds Ratio 0 no/na/dk Odds Ratiosa anyIP Any inpatient admissions - past year 0 no/na/dk Odds Ratiosa anyIP Any inpatient admissions - past year 0 no/na/dk 1.009 95% Confidence Interval Lower 1.000 95% Confidence Interval Upper 1.018 Dependent Variable: anyIP Any inpatient admissions - past year (reference category = 1 yes) Model: (Intercept), age_calca a. Factors and covariates used in the computation are fixed at the following values: age_calc Age (calculated) in years at time of interview=39.47 16 V.A.Morgan document1 15 March 2016 SAS example output SAS: The SURVEYMEANS Procedure MEAN PROCEDURE FOR AGE Data Summary Number of Observations 1825 Sum of Weights 7955 Statistics Variable Label AGE_CALC Age (calculated) in years at time of interview N Mean 182 39.46561 5 7 Std Error of Mean 95% CL for Mean 0.272877 38.930432 40.000801 2 2 17 V.A.Morgan document1 15 March 2016 SAS: The SURVEYFREQ Procedure Cross tabulation of Sex by Psychotic illness Data Summary Number of Observations 1825 Sum of Weights 7955 Table of SEX by DIAGICD10 Weighted Frequency Frequency Std Dev of Wgt Freq Percent Std Err of Percent 94.84571 33.3169 1.1668 623.91958 50.64980 7.8431 0.6389 141 621.15590 53.24893 7.8084 0.6662 depressive psychosis 33 146.11248 27.03131 1.8367 0.3392 delusional disorders and other nonorganic psychosis 65 303.38049 39.57960 3.8137 0.4950 severe depression without psychosis 62 288.22831 38.35848 3.6232 0.4800 screen-positive for psychosis but did not meet full criteria for ICD-10 psychosis 17 79.34982 19.93295 0.9975 0.2504 Total 1087 4713 103.79991 59.2396 1.2216 female schizophrenia 245 1113 70.60636 13.9895 0.8748 schizoaffective 136 540.86291 47.49690 6.7990 0.5987 bipolar, mania 178 774.09087 58.69530 9.7309 0.7339 depressive psychosis 48 221.88316 33.29560 2.7892 0.4174 delusional disorders and other nonorganic psychosis 27 117.84948 24.27508 1.4815 0.3048 severe depression without psychosis 96 427.94215 45.43072 5.3795 0.5684 8 46.99893 17.08476 0.5908 0.2144 Total 738 3242 101.39056 40.7604 1.2216 schizophrenia 857 3763 103.66711 47.3064 1.2396 schizoaffective 293 1165 66.71838 14.6421 0.8442 bipolar, mania 319 1395 75.85090 17.5392 0.9430 depressive psychosis 81 367.99565 42.47047 4.6260 0.5315 delusional disorders and other nonorganic psychosis 92 421.22997 46.00675 5.2952 0.5745 severe depression without psychosis 158 716.17047 58.31017 9.0028 0.7259 25 126.34875 26.17484 1.5883 0.3282 1825 7955 SEX DIAGICD10 male schizophrenia 612 2650 schizoaffective 157 bipolar, mania screen-positive for psychosis but did not meet full criteria for ICD-10 psychosis Total screen-positive for psychosis but did not meet full criteria for ICD-10 psychosis Total 65.57273 100.000 18 V.A.Morgan document1 15 March 2016 SAS: The SURVEYLOGISTIC Procedure Logistic regression Effect of age on inpatient admission Model Information Data Set SAHA.SHIPDATA Response Variable ANYIP Number of Response Levels 2 Weight Variable weightsa Model Generalized Logit Optimization Technique Newton-Raphson Variance Adjustment Degrees of Freedom (DF) Any inpatient admissions - past year weightsa Variance Estimation Method Taylor Series Variance Adjustment Degrees of Freedom (DF) Number of Observations Read 1825 Number of Observations Used 1825 Sum of Weights Read 7955 Sum of Weights Used 7955 Response Profile Ordered Value ANYIP Total Frequency 1 no/na/ dk 2 yes Total Weight 1028 4552.607 1 797 3402.392 9 Logits modeled use ANYIP='yes' as the reference category. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied. Model Fit Statistics Criterion AIC Intercept Intercept and Only Covariates 10863.07 8 10846.426 19 V.A.Morgan document1 15 March 2016 Model Fit Statistics Criterion Intercept Intercept and Only Covariates SC 10868.58 7 10857.445 -2 Log L 10861.07 8 10842.426 Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 18.6517 1 <.0001 Score 18.6230 1 <.0001 Wald 3.8805 1 0.0489 Type 3 Analysis of Effects Effect AGE_CALC DF Wald Chi-Square Pr > ChiSq 1 3.8805 0.0489 Analysis of Maximum Likelihood Estimates Parameter ANYIP Intercept no/na/dk 1 AGE_CALC no/na/dk 1 Standard Error Wald Chi-Square Pr > ChiSq -0.0568 0.1786 0.1011 0.7506 0.00884 0.00449 3.8805 0.0489 DF Estimate Odds Ratio Estimates Effect ANYIP AGE_CALC no/na/dk Point Estimate 1.009 95% Wald Confidence Limits 1.000 1.018 Association of Predicted Probabilities and Observed Responses Percent Concordant 51.8 Somers' D 0.06 2 Percent Discordant 45.7 Gamma 0.06 3 2.5 Tau-a 0.03 0 Percent Tied Pairs 81931 c 6 0.53 1 20