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Logistic Regression Using SAS

For this handout we will examine a dataset that is part of the data collected from “A study of preventive lifestyles and women’s health” conducted by a group of students in School of Public Health, at the University of Michigan during the1997 winter term. There are 370 women in this study aged 40 to 91 years.

Description of variables:

Variable Name Description Column Location

IDNUM Identification number 1-4 STOPMENS 1= Yes, 2= NO, 9= Missing 5

AGESTOP1 88=NA (haven't stopped) 99= Missing 6-7NUMPREG1 88=NA (no births) 99= Missing 8-9AGEBIRTH 88=NA (no births) 99= Missing 10-11MAMFREQ4 1= Every 6 months 12

2= Every year3= Every 2 years4= Every 5 years 5= Never 6= Other9= Missing

DOB 01/01/00 to 12/31/57 13-2099/99/99= Missing

EDUC 1= No formal school 21-222= Grade school3= Some high school 4= High school graduate/ Diploma equivalent5= Some college education/ Associate’s degree 6= College graduate 7= Some graduate school 8= Graduate school or professional degree9= Other 99= Missing

TOTINCOM 1= Less than $10,000 232= $10.000 to 24,9993= $25,000 to 39,9994= $40.000 to 54,9995= More than $55,0008= Don’t know

9= Missing

SMOKER 1= Yes, 2= No, 9= Missing 24WEIGHT1 999= Missing 25-27

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The yearcutoff option is used, which defines the 100-year window SAS will use for a two-digit year. We set yearcutoff=1900 so that a date of birth of 12/21/05 will be read as Dec 21, 1905, rather than as Dec 21, 2005 (the default yearcutoff for SAS 9.2 is 1920).

options yearcutoff=1900;

The data step commands read in the raw data and set up the missing value codes. We set up the missing value code for DOB to be 09/09/99, using a SAS date constant ("09SEP99"D). We also create some new variables: MENOPAUSE (a 0,1 dummy variable), YEARBIRTH, AGE (age in years), EDCAT (a 3-level categorical variable), AGECAT (a 4-level categorical variable), OVER50 (a 0, 1 dummy variable), and HIGHAGE (a categorical variable with values 1 and 2).

data bcancer; infile "brca.dat" lrecl=300; input idnum 1-4 stopmens 5 agestop1 6-7 numpreg1 8-9 agebirth 10-11 mamfreq4 12 @13 dob mmddyy8. educ 21-22

totincom 23 smoker 24 weight1 25-27; format dob mmddyy10.; if dob = "09SEP99"D then dob=.; if stopmens=9 then stopmens=.; if agestop1 = 88 or agestop1=99 then agestop1=.; if agebirth =99 then agebirth=.; if numpreg1=99 then numpreg1=.; if mamfreq4=9 then mamfreq4=.; if educ=99 then educ=.; if totincom=8 or totincom=9 then totincom=.; if smoker=9 then smoker=.; if weight1=999 then weight1=.;

if stopmens = 1 then menopause=1; if stopmens = 2 then menopause=0;

yearbirth = year(dob); age = int(("01JAN1997"d - dob)/365.25);

if educ not=. then do; if educ in (1,2,3,4) then edcat = 1; if educ in (5,6) then edcat = 2; if educ in (7,8) then edcat = 3; highed = (educ in (6,7,8)); end;

if age not=. then do; if age <50 then agecat=1; if age >=50 and age < 60 then agecat=2; if age >=60 and age < 70 then agecat=3; if age >=70 then agecat=4;

if age < 50 then over50 = 0; if age >=50 then over50 = 1;

if age >= 50 then highage = 1; if age < 50 then highage = 2; end;run;

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Descriptives and FrequenciesWe first get descriptive statistics for all the numerical variables in the dataset. We request specific statistics, including nmiss, to stress the number of missing values for each variable.

title "Descriptive Statistics";proc means data=bcancer n nmiss min max mean std;run; Descriptive Statistics The MEANS Procedure N Variable N Miss Minimum Maximum Mean Std Dev ---------------------------------------------------------------------------------------- idnum 370 0 1008.00 2448.00 1761.69 412.7290352 stopmens 369 1 1.0000000 2.0000000 1.1598916 0.3670031 agestop1 297 73 27.0000000 61.0000000 47.1818182 6.3101650 numpreg1 366 4 0 12.0000000 2.9480874 1.8726683 agebirth 359 11 9.0000000 88.0000000 30.2228412 19.5615468 mamfreq4 328 42 1.0000000 6.0000000 2.9420732 1.3812853 dob 361 9 -19734.00 -1248.00 -7899.50 4007.12 educ 365 5 1.0000000 9.0000000 5.6410959 1.6374595 totincom 325 45 1.0000000 5.0000000 3.8276923 1.3080364 smoker 364 6 1.0000000 2.0000000 1.4862637 0.5004993 weight1 360 10 86.0000000 295.0000000 148.3527778 31.1093049 menopause 369 1 0 1.0000000 0.8401084 0.3670031 yearbirth 361 9 1905.00 1956.00 1937.86 10.9836177 age 361 9 40.0000000 91.0000000 58.1440443 10.9899588 edcat 364 6 1.0000000 3.0000000 2.0137363 0.7694786 highed 365 5 0 1.0000000 0.4383562 0.4968666 agecat 361 9 1.0000000 4.0000000 2.3296399 1.0798313 over50 361 9 0 1.0000000 0.7257618 0.4467488 highage 361 9 1.0000000 2.0000000 1.2742382 0.4467488 ----------------------------------------------------------------------------------------

Next, we examine oneway frequencies for selected variables. Note that these could all have been requested in a single tables statement. We will carefully check the Frequency Missing for each variable.

title "Oneway Frequencies";proc freq data=bcancer; tables dob stopmens menopause educ edcat age agecat over50 highage;run;

The FREQ Procedure Cumulative Cumulative dob Frequency Percent Frequency Percent --------------------------------------------------------------- 12/21/1905 1 0.28 1 0.28 09/11/1909 1 0.28 2 0.55 12/04/1909 1 0.28 3 0.83 07/15/1911 1 0.28 4 1.11 04/01/1913 1 0.28 5 1.39 07/28/1913 1 0.28 6 1.66 .... 11/18/1955 1 0.28 358 99.17 11/22/1955 1 0.28 359 99.45 02/24/1956 1 0.28 360 99.72 08/01/1956 1 0.28 361 100.00 Frequency Missing = 9

Cumulative Cumulative stopmens Frequency Percent Frequency Percent ------------------------------------------------------------- 1 310 84.01 310 84.01 2 59 15.99 369 100.00

Frequency Missing = 1

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Cumulative Cumulative menopause Frequency Percent Frequency Percent -------------------------------------------------------------- 0 59 15.99 59 15.99 1 310 84.01 369 100.00


Cumulative Cumulative educ Frequency Percent Frequency Percent --------------------------------------------------------- 1 1 0.27 1 0.27 2 4 1.10 5 1.37 3 11 3.01 16 4.38 4 89 24.38 105 28.77 5 99 27.12 204 55.89 6 50 13.70 254 69.59 7 23 6.30 277 75.89 8 87 23.84 364 99.73 9 1 0.27 365 100.00


Cumulative Cumulative edcat Frequency Percent Frequency Percent ---------------------------------------------------------- 1 105 28.85 105 28.85 2 149 40.93 254 69.78 3 110 30.22 364 100.00


Cumulative Cumulative age Frequency Percent Frequency Percent -------------------------------------------------------- 40 2 0.55 2 0.55 41 5 1.39 7 1.94 42 7 1.94 14 3.88 43 11 3.05 25 6.93 44 7 1.94 32 8.86 45 11 3.05 43 11.91 46 10 2.77 53 14.68 47 16 4.43 69 19.11 48 13 3.60 82 22.71 49 17 4.71 99 27.42 50 12 3.32 111 30.75 51 9 2.49 120 33.24 52 14 3.88 134 37.12 53 13 3.60 147 40.72 54 13 3.60 160 44.32 55 10 2.77 170 47.09 56 9 2.49 179 49.58 57 10 2.77 189 52.35 58 11 3.05 200 55.40 59 14 3.88 214 59.28 60 10 2.77 224 62.05 61 8 2.22 232 64.27 62 11 3.05 243 67.31 63 5 1.39 248 68.70 64 4 1.11 252 69.81 65 8 2.22 260 72.02 66 8 2.22 268 74.24 67 8 2.22 276 76.45 68 7 1.94 283 78.39 69 7 1.94 290 80.33 70 9 2.49 299 82.83 71 10 2.77 309 85.60 72 13 3.60 322 89.20 73 5 1.39 327 90.58 74 4 1.11 331 91.69 75 5 1.39 336 93.07

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76 4 1.11 340 94.18 77 5 1.39 345 95.57 78 2 0.55 347 96.12 79 2 0.55 349 96.68 80 2 0.55 351 97.23 81 3 0.83 354 98.06 82 1 0.28 355 98.34 83 2 0.55 357 98.89 85 1 0.28 358 99.17 87 2 0.55 360 99.72 91 1 0.28 361 100.00


Cumulative Cumulative agecat Frequency Percent Frequency Percent ----------------------------------------------------------- 1 99 27.42 99 27.42 2 115 31.86 214 59.28 3 76 21.05 290 80.33 4 71 19.67 361 100.00 Frequency Missing = 9

Cumulative Cumulative over50 Frequency Percent Frequency Percent ----------------------------------------------------------- 0 99 27.42 99 27.42 1 262 72.58 361 100.00


Cumulative Cumulative highage Frequency Percent Frequency Percent ------------------------------------------------------------ 1 262 72.58 262 72.58 2 99 27.42 361 100.00


Crosstabulation

Prior to fitting a logistic regression model, we check a crosstabulation to understand the relationship between menopause and high age. In this 2 by 2 table both the predictor variable, HIGHAGE, and the outcome variable, STOPMENS, are coded as 1 and 2. For HIGHAGE, the value 1 represents the high risk group (those whose age is greater than or equal to 50 years), and for STOPMENS, the value 1 represents the outcome of interest (those who are in menopause). Notice also that HIGHAGE is considered to be the risk factor so it is listed first (the row variable) in the tables statement and STOPMENS is the outcome of interest so it is listed second (the column variable).

We request the relative risk and the odds ratio.

/*Crosstabs of HIGHAGE by STOPMENS*/title "2 x 2 Table";title2 "HIGHAGE Coded as 1, 2";proc freq data=bcancer; tables highage*stopmens / relrisk chisq;run; 2 x 2 Table HIGHAGE Coded as 1, 2

The FREQ Procedure Table of highage by stopmens

highage stopmens Frequency|

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Percent | Row Pct | Col Pct | 1| 2| Total ---------+--------+--------+ 1 | 251 | 10 | 261 | 69.72 | 2.78 | 72.50 | 96.17 | 3.83 | | 83.39 | 16.95 | ---------+--------+--------+ 2 | 50 | 49 | 99 | 13.89 | 13.61 | 27.50 | 50.51 | 49.49 | | 16.61 | 83.05 | ---------+--------+--------+ Total 301 59 360 83.61 16.39 100.00


Statistics for Table of highage by stopmens Statistic DF Value Prob ------------------------------------------------------ Chi-Square 1 109.2191 <.0001 Likelihood Ratio Chi-Square 1 99.0815 <.0001 Continuity Adj. Chi-Square 1 105.9122 <.0001 Mantel-Haenszel Chi-Square 1 108.9157 <.0001 Phi Coefficient 0.5508 Contingency Coefficient 0.4825 Cramer's V 0.5508

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Fisher's Exact Test ---------------------------------- Cell (1,1) Frequency (F) 251 Left-sided Pr <= F 1.0000 Right-sided Pr >= F 5.719E-23

Table Probability (P) 1.204E-21 Two-sided Pr <= P 5.719E-23

The output below says "Estimates of the Relative Risk (Row1/Row2)". This is what we want: the risk of menopause for those who are high age (ROW1) divided by the risk of menopause for those who are not high age (ROW2). To get the relative risk, we read the Cohort (Col 1 Risk) because we are interested in the relative risk for being in menopause (Column 1 of STOPMENS). Notice that the odds ratio (24.6) is not a good estimate of the risk ratio (1.90), because the outcome is not rare in this group of older women. Estimates of the Relative Risk (Row1/Row2) Type of Study Value 95% Confidence Limits ----------------------------------------------------------------- Case-Control (Odds Ratio) 24.5980 11.6802 51.8021 Cohort (Col1 Risk) 1.9041 1.5644 2.3176 Cohort (Col2 Risk) 0.0774 0.0408 0.1467

Effective Sample Size = 360 Frequency Missing = 10

Logistic Regression Model with a dummy variable predictor

We now fit a logistic regression model, but using two different variables: OVER50 (coded as 0, 1) is used as the predictor, and MENOPAUSE (also coded as 0,1) is used as the outcome. We use the descending option so SAS will fit the probability of being a 1, rather than of being a zero.

proc logistic data=bcancer descending; model menopause = over50/ rsquare;run;

Alternatively, the same model can be fitted by specifying the level of menopause that is to be considered the "event", as shown below:

proc logistic data=bcancer; model menopause(event="1") = over50/ rsquare;run;

You can check that the correct probability is being modeled by looking at the portion of the output that lists it. In this case, SAS reports "Probability modeled is menopause=1." so we know the model is set up correctly. We use the predictor variable in this model as OVER50, which is a binary dummy variable coded as 0, 1. Just as in a linear regression model, we can use a dummy variable as a predictor in a logistic regression model.

The value of the parameter estimate for OVER50 (3.2) tells us that the log-odds of being in menopause increase (because the estimate is positive) by 3.2 units for those in menopause compared to those women who are not. This result is significant, Wald chi-square (1 df) = 71.04, p< 0.0001. The odds ratio (24.6) is easier to interpret. It tells us that the odds of being in menopause are 24.6 times higher for a woman who is over 50 than for someone who is not. We can see that the 95% CI for the odds ratio does not include 1, so we can be pretty confident that there is a strong relationship between being over 50 and being in menopause. Logistic Regression with Dummy Variable Predictor Over50 Coded as 0, 1 The LOGISTIC Procedure

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Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring

Number of Observations Read 370 Number of Observations Used 360

Response Profile Ordered Total Value menopause Frequency 1 1 301 2 0 59

Probability modeled is menopause=1.NOTE: 10 observations were deleted due to missing values for the response or explanatory variables. Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 323.165 226.084 SC 327.051 233.856 -2 Log L 321.165 222.084 R-Square 0.2406 Max-rescaled R-Square 0.4076

Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 99.0815 1 <.0001 Score 109.2191 1 <.0001 Wald 71.0363 1 <.0001

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 0.0202 0.2010 0.0101 0.9199 over50 1 3.2026 0.3800 71.0363 <.0001

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits over50 24.596 11.680 51.798

Association of Predicted Probabilities and Observed Responses Percent Concordant 69.3 Somers' D 0.664 Percent Discordant 2.8 Gamma 0.922 Percent Tied 27.9 Tau-a 0.183 Pairs 17759 c 0.832

Logistic Regression Model with a class variable as predictor

We now fit the same model, but using a class variable, HIGHAGE (coded as 1=Highage and 2=Not Highage), as the predictor. In this case, we want to use a class statement, because this predictor is not a dummy (0,1) variable as in the previous example. Note that in the class statement, we specify the reference level for HIGHAGE (ref="2"), and we also use the option param=ref after a forward slash. So, we will be fitting a model in which we are comparing the odds of being in menopause for those women who are over 50 (HIGHAGE=1) to those who are not over 50 (HIGHAGE=2, the reference category).

Note that the results of this model fit are the same as in the previous model, but with some minor modifications.

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proc logistic data=bcancer; class highage(ref="2") / param=ref; model menopause(event="1") = highage/ rsquare;run;

Logistic Regression with a Class Statement Highage used as Predictor Reference Category is Not-Highage (HighAge=2)

The LOGISTIC Procedure Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring


Response Profile Ordered Total Value menopause Frequency

1 1 301 2 0 59

Probability modeled is menopause=1.NOTE: 10 observations were deleted due to missing values for the response or explanatory variables.

Check the Class Level Information to be sure SAS has set up the coding for HIGHAGE correctly. We see that the Design Variables are set up so that HIGHAGE=1 is coded as 1, and HIGHAGE=2 is coded as 0, as we intended.

Class Level Information Design Class Value Variables highage 1 1 2 0

Model Fit Statistics Intercept Intercept and Criterion Only Covariates AIC 323.165 226.084 SC 327.051 233.856 -2 Log L 321.165 222.084

R-Square 0.2406 Max-rescaled R-Square 0.4076


Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq highage 1 71.0363 <.0001

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The output for the parameter estimate is slightly different than for the previous model. In this case, we see highage 1, to emphasize that this is for HIGHAGE=1 compared to the reference category (HIGHAGE=2).

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 0.0202 0.2010 0.0101 0.9199 highage 1 1 3.2026 0.3800 71.0363 <.0001

The output for the odds ratio also emphasizes that we are looking at the odds ratio for HIGHAGE=1 vs. 2.

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits highage 1 vs 2 24.596 11.680 51.798

Logistic Regression Model with a class predictor with more than two categories

We now look at the relationship of education categories to menopause. Again, we begin by checking the cross-tabulation between education and menopause, using the variable EDCAT as the "exposure" and STOPMENS as the "outcome" or event. Because we are interested in the probability of STOPMENS = 1, for each level of EDCAT, we really need only the row percents, so we suppress the display of column and total percent by using the nocol nopercent options. We see in the output that the proportion of women in menopause decreases with increasing education level. title "Relationship of Education Categories to Menopause";proc freq data=bcancer; tables edcat*stopmens / chisq nocol nopercent;run;

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Table of edcat by stopmens

edcat stopmens Frequency| Row Pct | 1| 2| Total ---------+--------+--------+ 1 | 96 | 9 | 105 | 91.43 | 8.57 | ---------+--------+--------+ 2 | 125 | 23 | 148 | 84.46 | 15.54 | ---------+--------+--------+ 3 | 84 | 26 | 110 | 76.36 | 23.64 | ---------+--------+--------+ Total 305 58 363


Statistics for Table of edcat by stopmens

Statistic DF Value Prob ------------------------------------------------------ Chi-Square 2 9.1172 0.0105 Likelihood Ratio Chi-Square 2 9.3370 0.0094 Mantel-Haenszel Chi-Square 1 9.0715 0.0026 Phi Coefficient 0.1585 Contingency Coefficient 0.1565 Cramer's V 0.1585

We now fit a logistic regression model, using EDCAT as a predictor, by including it in the class statement. The reference category is EDCAT=1.

proc logistic data=bcancer; class edcat(ref="1") / param = ref; model menopause(event="1") = edcat/ rsquare;run;

Logistic Regression to Predict Menopause From Education The LOGISTIC Procedure Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring



Probability modeled is menopause=1.NOTE: 7 observations were deleted due to missing values for the response or explanatory variables. We can look at the class level information below to see that EDCAT=1 is the reference category, because it has a value of 0 for all of the design variables.

Class Level Information Design Class Value Variables edcat 1 0 0

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2 1 0 3 0 1

Model Convergence Status Convergence criterion (GCONV=1E-8) satisfied.


The portion of the output on Testing the Global Null Hypothesis provides an overall test for all parameters in the model. Thus, we can see that there is a likelihood ratio chi-square test of whether there is any effect of EDCAT, Χ2 (2df) = 9.337, p=.0094. The Wald chi-square test is slightly smaller Χ2 (2df) = 8.63, p=.0134, but gives similar results.

Testing Global Null Hypothesis: BETA=0 Test Chi-Square DF Pr > ChiSq Likelihood Ratio 9.3370 2 0.0094 Score 9.1172 2 0.0105 Wald 8.6314 2 0.0134

Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq edcat 2 8.6314 0.0134

The parmeter estimate for EDCAT 2 shows that the log-odds of menopause for someone with EDCAT=2 are smaller than for someone with EDCAT=1, but this difference is not significant (p=0.1050). The parameter estimate for EDCAT 3 are negative, indicating that someone with EDCAT=3 has a lower log-odds of menopause than a person with EDCAT=1, and this difference is significant (p=0.004) .

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 2.3671 0.3486 46.1069 <.0001 edcat 2 1 -0.6743 0.4159 2.6279 0.1050 edcat 3 1 -1.1944 0.4146 8.2990 0.0040

The odds ratio estimate for EDCAT 3 vs 1 is .303, indicating that the odds of being in menopause for a person with EDCAT=3 are only 30% of the odds of being in menopause for a person with EDCAT=1. Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits edcat 2 vs 1 0.510 0.225 1.151 edcat 3 vs 1 0.303 0.134 0.683


Logistic Regression Model with a continuous predictor

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We now look at a logistic regression model, but this time with a single continuous predictor (AGE). We also request ods graphics to obtain a plot showing the relationship between AGE and the estimated probability of being in menopause. The units statement allows us to see the effect of a 1-year, 5-year, and 10-year increase in age on the odds of being in menopause.

The parameter estimate for AGE is positive (0.2829) telling us that the log-odds of being in menopause increase by .28 units for a woman who is one year older compared to her counterpart who is one year younger. The odds ratio (1.33) tells us that the odds of being in menopause for a woman who is one year older are 1.33 times greater. The last part of the output shows us the odds ratio for a one-year, five-year and 10-year increase in age. We estimate that the odds of being in menopause for a woman who is five years older than her counterpart are 4.1 times greater, and for a 10-year increase in age, the odds of being in menopause are almost 17 times greater.

ods graphics on;proc logistic data=bcancer plots=(effect); model menopause(event="1") = age / rsquare; units age = 1 5 10;run;ods graphics off;

Logistic Regression with a Continuous Predictor The LOGISTIC Procedure

Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring

Number of Observations Read 370 Number of Observations Used 360 Response Profile Ordered Total Value menopause Frequency 1 1 301 2 0 59 Probability modeled is menopause=1.NOTE: 10 observations were deleted due to missing values for the response or explanatory variables.





Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -12.8675 1.9360 44.1735 <.0001

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age 1 0.2829 0.0401 49.7646 <.0001

Odds Ratio Estimates

Point 95% Wald Effect Estimate Confidence Limits age 1.327 1.227 1.436


Adjusted Odds Ratios Effect Unit Estimate age 1.0000 1.327 age 5.0000 4.115 age 10.0000 16.935

Quasi-Complete Separation in a Logistic Regression Model

One fairly common occurrence in a logistic regression model is that the model fails to converge. This often happens when you have a categorical predictor that is too perfect, that is, there may be a category with no variability in the response (all subjects in one category of the predictor have the same response). This is called quasi-complete separation. When this happens, SAS will give a warning message in the output. These warnings

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should be taken seriously, and the model should be refitted, perhaps by combining some categories of the predictor.

Even if there is not quasi-complete separation, separation may be nearly complete, so the standard error for a parameter estimate can become very large. It is good practice to examine the parameter estimates and their standard errors carefully for any logistic regression output.

We now examine a situation where quasi-complete separation occurs, using the variable AGECAT as a predictor in a logistic regression. However, first we check the crosstabulation between AGECAT and STOPMENS, using Proc Freq. Notice that in the highest age category, all 71 women are in menopause (not surprisingly).

title "Relationship of AGECAT to MENOPAUSE";proc freq data=bcancer; tables agecat*stopmens/ chisq nocol nopercent;run;

Relationship of AGECAT to MENOPAUSE The FREQ Procedure Table of agecat by stopmens agecat stopmens Frequency| Row Pct | 1| 2| Total ---------+--------+--------+ 1 | 50 | 49 | 99 | 50.51 | 49.49 | ---------+--------+--------+ 2 | 106 | 9 | 115 | 92.17 | 7.83 | ---------+--------+--------+ 3 | 74 | 1 | 75 | 98.67 | 1.33 | ---------+--------+--------+ 4 | 71 | 0 | 71 | 100.00 | 0.00 | ---------+--------+--------+ Total 301 59 360 Frequency Missing = 10

Statistics for Table of agecat by stopmens Statistic DF Value Prob ------------------------------------------------------ Chi-Square 3 111.6605 <.0001 Likelihood Ratio Chi-Square 3 110.1752 <.0001 Mantel-Haenszel Chi-Square 1 78.6978 <.0001 Phi Coefficient 0.5569 Contingency Coefficient 0.4866 Cramer's V 0.5569

Effective Sample Size = 360 Frequency Missing = 10

We now fit the corresponding logistic regression model, using AGECAT in a class statement, with AGECAT=1 as the reference category.

proc logistic data=bcancer; class agecat(ref="1") / param = ref; model menopause(event="1") = agecat/ rsquare; run;

Logistic Regression with AGECAT as Predictor

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This Analysis Does not Work Check out the Parameter Estimates and Standard Errors The LOGISTIC Procedure Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring Number of Observations Read 370 Number of Observations Used 360

Response Profile Ordered Total Value menopause Frequency 1 1 301 2 0 59 Probability modeled is menopause=1.

NOTE: 10 observations were deleted due to missing values for the response or explanatory variables.

We check out the Class Level Information to see that the design variables are set up correctly to have AGECAT=1 as the reference category.

Class Level Information Class Value Design Variables agecat 1 0 0 0 2 1 0 0 3 0 1 0 4 0 0 1

The notes in the output below alert us to the problem of quasi-complete separation of data points.

Model Convergence Status Quasi-complete separation of data points detected.

WARNING: The maximum likelihood estimate may not exist.WARNING: The LOGISTIC procedure continues in spite of the above warning. Results shown are based on the last maximum likelihood iteration. Validity of the model fit is questionable.WARNING: The validity of the model fit is questionable.




Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq agecat 3 50.0793 <.0001

Analysis of Maximum Likelihood Estimates

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Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 0.0202 0.2010 0.0101 0.9199 agecat 2 1 2.4460 0.4012 37.1721 <.0001 agecat 3 1 4.2839 1.0266 17.4126 <.0001 agecat 4 1 14.8969 205.9 0.0052 0.9423

WARNING: The validity of the model fit is questionable.

Note that the point estimate of the odds ratio for AGECAT=4 is >999.999 and the 95% confidence interval is huge. This alerts us that there is a problem with AGECAT=4.

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits agecat 2 vs 1 11.542 5.258 25.339 agecat 3 vs 1 72.520 9.696 542.384 agecat 4 vs 1 >999.999 <0.001 >999.999


Based on the information that we saw in the crosstabulation, we will create a new variable AGECAT3, with 3 age categories, collapsing category 3 and category 4.

/*Recode Agecat into AGECAT3 with 3 categories*/data bcancer2; set bcancer; if age not=. then do; if age < 50 then agecat3 = 1; if age >=50 and age < 60 then agecat3 = 2; if age >=60 then agecat3 = 3; end;run;

We now fit a new logistic regression, with AGECAT3 as a categorical predictor. Note in the output for this model, we do not have a problem with quasi-complete separation, but we do have a very large confidence interval for the odds ratio for AGECAT=3, owing to the fact that there was only one person who was not in menopause in this category.

proc logistic data=bcancer2; class agecat3(ref="1") / param = ref; model menopause(event="1") = agecat3/ rsquare;run;

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Logistic Regression with Recoded Age Categories This Analysis Works

The LOGISTIC Procedure Model Information

Data Set WORK.BCANCER2 Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring


Response Profile Ordered Total Value menopause Frequency

1 1 301 2 0 59

Probability modeled is menopause=1.


Class Level Information Design Class Value Variables agecat3 1 0 0 2 1 0 3 0 1





Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq agecat3 2 55.3535 <.0001

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 0.0202 0.2010 0.0101 0.9199 agecat3 2 1 2.4460 0.4012 37.1721 <.0001 agecat3 3 1 4.9565 1.0234 23.4578 <.0001

Odds Ratio Estimates Point 95% Wald

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Effect Estimate Confidence Limits agecat3 2 vs 1 11.542 5.258 25.339 agecat3 3 vs 1 142.097 19.120 >999.999


Logistic Regression Model with Several Predictors

We now fit a logistic regression model with several predictors, both continuous and categorical. Note especially the global test for the model, which has 6 degrees of freedom, due to the 6 parameters that are estimated for the predictors in the model. There are two parameters for EDCAT and one each for AGE, SMOKER, TOTINCOM, and NUMPREG1. The only predictor that is significant in this model is AGE (p<0.0001)..

proc logistic data=bcancer; class edcat(ref="1") / param = ref; model menopause(event="1") = age edcat smoker totincom numpreg1 / rsquare;run;

Logistic Regression with Several Predictors The LOGISTIC Procedure Model Information Data Set WORK.BCANCER Response Variable menopause Number of Response Levels 2 Model binary logit Optimization Technique Fisher's scoring


Response Profile Ordered Total Value menopause Frequency 1 1 259 2 0 54 Probability modeled is menopause=1.


Class Level Information Design Class Value Variables edcat 1 0 0 2 1 0 3 0 1



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Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq age 1 40.6094 <.0001 edcat 2 2.3776 0.3046 smoker 1 2.9092 0.0881 totincom 1 0.3032 0.5819 numpreg1 1 0.0025 0.9605

Analysis of Maximum Likelihood Estimates Standard Wald Parameter DF Estimate Error Chi-Square Pr > ChiSq Intercept 1 -10.8151 2.2132 23.8788 <.0001 age 1 0.2797 0.0439 40.6094 <.0001 edcat 2 1 -0.4356 0.5524 0.6219 0.4304 edcat 3 1 -0.8401 0.5636 2.2214 0.1361 smoker 1 -0.6543 0.3836 2.9092 0.0881 totincom 1 -0.0927 0.1683 0.3032 0.5819 numpreg1 1 0.00646 0.1305 0.0025 0.9605

Odds Ratio Estimates Point 95% Wald Effect Estimate Confidence Limits age 1.323 1.214 1.442 edcat 2 vs 1 0.647 0.219 1.910 edcat 3 vs 1 0.432 0.143 1.303 smoker 0.520 0.245 1.102 totincom 0.911 0.655 1.268 numpreg1 1.006 0.779 1.300


Logistic Regression Model Using Proc Genmod

Logistic regression models, along with several other types of models, can be fitted using Proc Genmod. You need to supply the distribution that the dependent variable has (in this case we use dist=bin), and you can also specify a link function. In this case we use the default link (for a binary distribution, the default link is logit), so we will be fitting the same model as in Proc Logistic. The type3 option in the model statement gives us the type3 test for each predictor in the model. However, unlike Proc Logistic, which gives Wald tests in the type3 output, we get Likelihood Ratio tests, which are preferable.

proc genmod data=bcancer descending; class edcat(ref="1") / param = ref; model menopause = age edcat smoker totincom numpreg1 / dist=bin type3;run;

Logistic Regression Using Proc Genmod The GENMOD Procedure Model Information Data Set WORK.BCANCER Distribution Binomial Link Function Logit

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Dependent Variable menopause

Number of Observations Read 370 Number of Observations Used 313 Number of Events 259 Number of Trials 313 Missing Values 57

Class Level Information Design Class Value Variables edcat 1 0 0 2 1 0 3 0 1


PROC GENMOD is modeling the probability that menopause='1'.

Criteria For Assessing Goodness Of Fit Criterion DF Value Value/DF Deviance 306 177.5102 0.5801 Scaled Deviance 306 177.5102 0.5801 Pearson Chi-Square 306 297.4367 0.9720 Scaled Pearson X2 306 297.4367 0.9720 Log Likelihood -88.7551

Algorithm converged.

Analysis Of Parameter Estimates Standard Wald 95% Confidence Chi- Parameter DF Estimate Error Limits Square Pr > ChiSq Intercept 1 -10.8151 2.2132 -15.1530 -6.4773 23.88 <.0001 age 1 0.2797 0.0439 0.1937 0.3658 40.61 <.0001 edcat 2 1 -0.4356 0.5524 -1.5182 0.6470 0.62 0.4304 edcat 3 1 -0.8401 0.5636 -1.9448 0.2647 2.22 0.1361 smoker 1 -0.6543 0.3836 -1.4062 0.0976 2.91 0.0881 totincom 1 -0.0927 0.1683 -0.4226 0.2372 0.30 0.5819 numpreg1 1 0.0065 0.1305 -0.2494 0.2623 0.00 0.9605 Scale 0 1.0000 0.0000 1.0000 1.0000

NOTE: The scale parameter was held fixed.

LR Statistics For Type 3 Analysis Chi- Source DF Square Pr > ChiSq age 1 89.12 <.0001 edcat 2 2.45 0.2932 smoker 1 2.96 0.0852 totincom 1 0.31 0.5794 numpreg1 1 0.00 0.9605

Compare these LR (Likelihood Ratio) statistics in the Type 3 Analysis with the Type 3 tests from Proc Logistic shown below. Those given below show more significant digits, and are in some cases quite different.

Type 3 Analysis of Effects Wald Effect DF Chi-Square Pr > ChiSq age 1 40.6094 <.0001 edcat 2 2.3776 0.3046 smoker 1 2.9092 0.0881 totincom 1 0.3032 0.5819 numpreg1 1 0.0025 0.9605

Estimating the Risk Ratio Using Proc Genmod

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Because we can specify a large range of link functions using Proc Genmod, we have the opportunity to fit a model using the Log link with the binary distribution. When we exponentiate this estimate, we get an estimate of the Risk Ratio, rather than the Odds Ratio. Here is the code for this example, using HIGHAGE as the predictor:

proc genmod data=bcancer descending; class highage(ref="2") / param = ref; model menopause = highage / dist=bin type3 link=log; estimate "Effect of High Age" highage 1 / exp;run;

Partial output from this model is shown below. Note that the Risk Ratio (1.90) matches the one we calculated for HIGHAGE in the Proc Freq output at the beginning of this handout:

Criteria For Assessing Goodness Of Fit

Criterion DF Value Value/DF Log Likelihood -111.0418 Full Log Likelihood -111.0418 AIC (smaller is better) 226.0836 AICC (smaller is better) 226.1172 BIC (smaller is better) 233.8558

Algorithm converged.

Analysis Of Maximum Likelihood Parameter Estimates

Standard Wald 95% Confidence Wald Parameter DF Estimate Error Limits Chi-Square Pr > ChiSq

Intercept 1 -0.6831 0.0995 -0.8781 -0.4881 47.14 <.0001 highage 1 1 0.6440 0.1003 0.4475 0.8405 41.26 <.0001 Scale 0 1.0000 0.0000 1.0000 1.0000

NOTE: The scale parameter was held fixed.

LR Statistics For Type 3 Analysis

Chi- Source DF Square Pr > ChiSq highage 1 99.08 <.0001

Contrast Estimate Results Mean Mean L'Beta Standard L'BetaLabel Estimate Confidence Limits Estimate Error Alpha Confidence LimitsEffect of High Age 1.9041 1.5644 2.3176 0.6440 0.1003 0.05 0.4475 0.8405Exp(Effect of High Age) 1.9041 0.1909 0.05 1.5644 2.3176

Contrast Estimate Results Chi- Label Square Pr > ChiSq Effect of High Age 41.26 <.0001 Exp(Effect of High Age)

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