Week 11 — Logistic regression and categorical outcomes

Modeling a yes/no outcome with odds ratios, read responsibly

The week question

Last week the outcome was continuous: PROC REG modeled systolic_bp as a straight line in age and baseline_bmi, and you read slopes in millimeters of mercury. This week the outcome goes binary — a plain yes/no — and the straight line no longer fits. The question is: does a participant meet their step goal (goal_met = 1) or not (0), and how do the odds of meeting it change with the coaching arm, age, and BMI? You cannot regress a 0/1 column with PROC REG and read a slope in “probability per year”; probabilities are bounded between \(0\) and \(1\), and a line walks straight out of that range. The SAS tool that respects the boundary is PROC LOGISTIC, which models the log-odds of the event and reports odds ratios. The workflow this week is not “run LOGISTIC and read the OR.” It is decide and declare which level is the event, fit the model, read the log and confirm the row count and the event count, read the odds-ratio table with its confidence intervals, check the model’s classification summary (the C-statistic), and then — the part that separates a careful analyst from a careless one — say in plain words what an odds ratio does and does not mean. An odds ratio is not a risk ratio, and observational data are not causal.

Why this matters

A great many real analytic questions are yes/no: did the patient readmit, did the customer churn, did the loan default, did the participant meet the goal. Logistic regression is the standard tool for all of them, and PROC LOGISTIC is where this course’s discipline meets its sharpest interpretation trap. It matters here for four reasons. First, the outcome type dictates the procedure: a binary outcome needs a model on a scale that stays inside \([0,1]\), so you learn to match the tool to the variable rather than forcing a line onto a 0/1 column. Second, the headline number — the odds ratio — is genuinely easy to misread, and the single most common error in applied work is calling an odds ratio a “risk” or treating \(\text{OR} = 1.78\) as “78% more likely.” Saying the OR correctly is the skill, not running the procedure. Third, LOGISTIC forces an explicit modeling decision that REG hid: which level of the outcome is the “event,” and which level of a categorical predictor is the “reference.” Get either wrong and every odds ratio inverts, silently, with no error. Fourth, the wellness-program data are synthetic and observational, so the arm odds ratio is an association, not the causal effect of coaching — the arms are not described as randomized. Every one of these is a workflow check you do before you believe the table.

Learning goals

By the end of this week you should be able to:

  • Recognize when an outcome is binary (goal_met 0/1) and therefore calls for PROC LOGISTIC rather than PROC REG, and say why a line on a 0/1 column is the wrong model.
  • Write proc logistic; model goal_met(event='1') = arm age baseline_bmi; and explain that event='1' declares which level is modeled — the most load-bearing option of the week.
  • Read the odds-ratio table: the point estimate, the 95% confidence interval, and the \(p\)-value, and state each odds ratio in plain words (“the odds of meeting goal are about \(1.78\) times as high for coaching”).
  • Read the C-statistic (AUC \(= 0.69\)) as the model’s rank-ordering / discrimination summary, and say what it does and does not tell you.
  • Run the verification moves for a logistic fit: confirm the input row count (n = 198 on the visit-1 slice), confirm the outcome is numeric 0/1, check the event count the log reports (goal_met = 1), and check NMISS so you know how many rows the fit used.
  • State, every time, that an odds ratio is not a risk ratio, and that the observational wellness data make the arm OR an association, not a causal effect of coaching.

Core vocabulary

The week’s SAS and statistics terms, defined plainly. The statistics ideas are calibrated against the IMS logistic-regression chapter; the SAS terms are the course’s own usage.

  • Binary outcome — a variable with exactly two values, here goal_met coded \(1\) (met goal) / \(0\) (did not). It is stored numeric, and the mean of a 0/1 variable is a proportion (the overall goal-met rate is \(246/594 \approx 0.41\)).
  • PROC LOGISTIC — SAS’s primary procedure for a binary outcome. It models the log-odds of the event as a linear function of the predictors and reports odds ratios.
  • Odds — for an event with probability \(p\), the odds are \(p/(1-p)\). Odds of \(1\) mean a 50/50 event; odds of \(2\) mean the event is twice as likely as not. Odds are not the probability.
  • Log-odds (logit) — \(\operatorname{logit}(p) = \ln\!\big(p/(1-p)\big)\). Logistic regression is linear on this scale: \(\operatorname{logit}(p) = b_0 + b_1 x_1 + \cdots\), which keeps the fitted \(p\) inside \((0,1)\).
  • event= option — model y(event='1') = ... tells SAS which level of the outcome is the event being modeled. Omit it and SAS models the lower sorted level by default (often the one you did not mean), which inverts every odds ratio. Always declare it.
  • Odds ratio (OR) — the multiplicative change in the odds of the event per one-unit change in a predictor (or, for a categorical predictor, comparing a level to its reference). Here the coaching-arm OR is \(1.78\). An OR is not a risk ratio — it does not say “\(78\%\) more likely.”
  • Reference level — for a CLASS predictor, the level the odds ratio is compared against (here usual_care is the reference, so the OR describes coaching relative to usual care).
  • C-statistic (AUC) — the area under the ROC curve, a summary of how well the model ranks an event case above a non-event case. Here AUC \(= 0.69\). It measures discrimination, not whether the probabilities are calibrated, and not causation.

Concept development

Why a binary outcome needs a different model

When the outcome is 0/1, fitting a straight line is the wrong move twice over. A line \(\hat p = b_0 + b_1 x\) can predict \(\hat p = 1.4\) or \(\hat p = -0.2\) — values that are not probabilities — and its constant-variance assumption is violated because a 0/1 outcome’s variance depends on \(p\). Logistic regression fixes both by modeling the log-odds as linear:

\[ \operatorname{logit}(p) = \ln\!\left(\frac{p}{1-p}\right) = b_0 + b_1\,\text{arm} + b_2\,\text{age} + b_3\,\text{baseline\_bmi}. \]

The logit can be any real number, so the linear predictor is unconstrained, but solving back for \(p\) always lands inside \((0,1)\). Exponentiating a coefficient turns it into an odds ratio: \(e^{b}\) is the multiplicative change in the odds per one-unit change in that predictor. That is why LOGISTIC reports odds ratios, not slopes — the slope lives on the log-odds scale, which is hard to talk about, and its exponential lives on the odds scale, which is (with care) interpretable. The recurring outcome goal_met is numeric 0/1; that type matters, because a character "Y"/"N" outcome would need different handling and would sort differently. Confirm the outcome is the numeric 0/1 column before you model it.

Declaring the event: the event= option

The single most consequential keystroke this week is (event='1'). Logistic regression models the probability of one of the two outcome levels, and you must say which. The SAS below models the probability that goal_met = 1 (met goal). The data are synthetic; seed set, streaminit(20260824).

proc logistic data=work.baseline;
    class arm (ref='usual_care') / param=ref;
    model goal_met(event='1') = arm age baseline_bmi;
run;
SAS log (synthetic)
NOTE: PROC LOGISTIC is modeling the probability that goal_met='1'.
NOTE: Convergence criterion (GCONV=1E-8) satisfied.
NOTE: There were 198 observations read from the data set WORK.BASELINE.
NOTE: PROCEDURE LOGISTIC used (Total process time):
      real time           0.27 seconds

What the log should say, and what to CHECK. The first NOTE is the most important line on the page: PROC LOGISTIC is modeling the probability that goal_met='1'. Read it every time — it confirms you are modeling the level you intended. Had you omitted event='1', that line would read goal_met='0', and every odds ratio below would be the reciprocal of what you wanted, with no error to warn you. The second load-bearing line is 198 observations read: the visit-1 baseline slice has one row per screened participant (n = 198), so the count check is the same row-count discipline as every other week. Convergence criterion ... satisfied confirms the fit actually converged; a WARNING: ... did not converge or a quasi-complete separation note would mean the estimates are not trustworthy. No WARNING and no ERROR appear, so the model parsed and ran.

Reading the odds-ratio table (and the C-statistic)

PROC LOGISTIC prints a response profile, a convergence note, the parameter estimates on the log-odds scale, and then the odds-ratio estimates with confidence intervals — the table you actually report. It also prints association statistics including the C-statistic.

Output (synthetic, not executed)

                Response Profile
 Ordered                          Total
   Value     goal_met         Frequency
       1            1                82
       2            0               116

Probability modeled is goal_met='1'.

           Odds Ratio Estimates
                          Point        95% Wald
Effect                 Estimate    Confidence Limits
arm coaching vs usual     1.780      1.280     2.470
age                       0.980      0.962     0.999
baseline_bmi              0.930      0.892     0.970

  Association of Predicted Probabilities and Observed Responses
Percent Concordant   68.9      Somers' D   0.379
Percent Discordant   31.1      Gamma       0.379
                               c           0.690

Interpretation, with the workflow move named. Read the odds-ratio table line by line. The arm odds ratio is \(1.78\) with a 95% confidence interval of \((1.28, 2.47)\): the odds of meeting the step goal are about \(1.78\) times as high for the coaching arm as for usual care, and because the interval lies entirely above \(1\) (and \(p = 0.0006\) in the estimates table), the association is distinguishable from “no difference” in this synthetic sample. The age OR is \(0.98\) and the baseline_bmi OR is \(0.93\), both just below \(1\), so in this model the odds of meeting goal edge down slightly with each additional year of age and each additional BMI point, holding the others fixed. The C-statistic is \(0.690\): given a random goal-met case and a random not-met case, the model assigns the goal-met case a higher predicted probability about \(69\%\) of the time — moderate discrimination, well above the \(0.5\) of a coin flip but far from a perfect \(1.0\). The workflow move is confirm the modeled level from the response profile, read each OR with its CI and direction, then read the C-statistic for how well the model ranks — and a \(0.69\) AUC is honest, modest discrimination, not a strong classifier.

Saying it correctly: OR is not RR, and observational is not causal

The odds-ratio table is where careful interpretation earns its keep, because two true things about it are constantly mis-stated. First, an odds ratio is not a risk ratio. The OR of \(1.78\) compares the odds \(p/(1-p)\), not the probabilities themselves. It is not “coaching participants are \(78\%\) more likely to meet goal” — that sentence describes a risk (probability) ratio, which the OR only approximates when the event is rare, and meeting goal here (about \(41\%\)) is not rare. The correct sentence keeps the word odds: “the odds of meeting goal are about \(1.78\) times as high under coaching.” Second, the wellness data are observational — the arms are not described as randomized — so the arm OR is an association, not the causal effect of coaching. People in the coaching arm may differ systematically from those in usual care in ways the model does not capture, so the analysis shows that coaching is associated with higher odds of meeting goal in this synthetic sample, not that assigning someone to coaching would raise their odds. Both cautions are not footnotes; they are the load-bearing claims of the week, and the model is only as honest as the sentence you write under it.

Worked examples

Worked example — the wellness-program study: goal_met on arm, age, and BMI

The task. Using the RiverCity wellness-program visit-1 baseline slice (one row per screened participant, n = 198), model the binary outcome goal_met(event='1') on arm, age, and baseline_bmi; read the odds ratios and the C-statistic; and verify the modeled level and the row count. The data are synthetic; seed set, streaminit(20260824), and observational — no one is described as randomized to an arm.

The code.

proc logistic data=work.baseline;
    class arm (ref='usual_care') / param=ref;
    model goal_met(event='1') = arm age baseline_bmi;
run;

The synthetic output (the locked odds ratios, confidence intervals, and C-statistic):

Output (synthetic, not executed)

Number of Observations Read         198
Number of Observations Used         198

                Response Profile
 Ordered                          Total
   Value     goal_met         Frequency
       1            1                82
       2            0               116

Probability modeled is goal_met='1'.

NOTE: Convergence criterion (GCONV=1E-8) satisfied.

         Analysis of Maximum Likelihood Estimates
                       Standard       Wald
Parameter   DF  Estimate   Error  Chi-Square  Pr > ChiSq
Intercept    1    1.6240  0.9100      3.1850      0.0743
arm coach    1    0.5766  0.1670     11.9200      0.0006
age          1   -0.0202  0.0098      4.2500      0.0392
baseline_bmi 1   -0.0726  0.0210     11.9500      0.0005

           Odds Ratio Estimates
                          Point        95% Wald
Effect                 Estimate    Confidence Limits
arm coaching vs usual     1.780      1.280     2.470
age                       0.980      0.962     0.999
baseline_bmi              0.930      0.892     0.970

  Association of Predicted Probabilities and Observed Responses
Percent Concordant   68.9      c           0.690

The verification check. Four moves. (1) Number of Observations Read = 198 and Used = 198 match, so no rows were dropped for missing predictors — the fit used the full baseline slice. (2) The response profile shows the outcome split, \(82\) events (\(goal\_met = 1\)) and \(116\) non-events, and the line Probability modeled is goal_met='1' confirms the modeled level is the one you intended — the single most important verification of the week. (3) The outcome is numeric 0/1 and the predictor arm is character with usual_care set as the reference, so the OR reads “coaching vs usual_care”; a quick proc freq; tables goal_met arm; before the fit would confirm the levels and counts. (4) An NMISS check on the outcome and predictors returns \(0\), confirming complete cases. With those passing, the table is trustworthy as a fit — separate from whether the odds ratio is interpreted correctly, which the sentence you write decides.

The interpretation. On the visit-1 slice, the odds of meeting the step goal are about \(1.78\) times as high under coaching as under usual care (95% CI \(1.28\)–\(2.47\), \(p = 0.0006\)) — the confidence interval lies entirely above \(1\), so the association is distinguishable from no difference here. The age OR (\(0.98\)) and baseline_bmi OR (\(0.93\)) are just below \(1\), so the modeled odds of meeting goal slip slightly with higher age and BMI, holding the others fixed. The C-statistic of \(0.69\) says the model discriminates moderately. Now the two cautions, said correctly: this is an odds ratio, not a risk ratio — do not translate \(1.78\) into “\(78\%\) more likely to meet goal,” because meeting goal (about \(41\%\)) is not rare and the OR overstates the probability ratio. And because the data are observational (the arms are not randomized), this is an association between coaching and higher goal-met odds in this synthetic sample, not evidence that assigning someone to coaching would raise their odds.

Worked example — transfer: modeling whether a visit was completed

The task. A new yes/no question in a new context, still synthetic and still the wellness program. The screenings table records completed ("Y"/"N") for each visit — did the participant complete that screening visit? Model the probability of a completed visit as a function of steps_k (thousands of steps that visit) and visit_num. Because completed is character "Y"/"N", you point event= at the character level. This is a different table, a different outcome, and a different predictor, so the numbers below are a notional transfer illustration, not the locked study figures — they are invented for this example and carry the same verified: false caveat.

The code.

proc logistic data=work.screenings;
    class visit_num / param=ref ref=first;
    model completed(event='Y') = steps_k visit_num;
run;
SAS log (synthetic)
NOTE: PROC LOGISTIC is modeling the probability that completed='Y'.
NOTE: Convergence criterion (GCONV=1E-8) satisfied.
NOTE: There were 594 observations read from the data set WORK.SCREENINGS.
Output (synthetic, not executed)  -- notional transfer figures, not locked

Number of Observations Read         594
Number of Observations Used         594

Probability modeled is completed='Y'.

           Odds Ratio Estimates
                          Point        95% Wald
Effect                 Estimate    Confidence Limits
steps_k                   1.240      1.130     1.360
visit_num 2 vs 1          0.880      0.610     1.270
visit_num 3 vs 1          0.760      0.520     1.110

  Association of Predicted Probabilities and Observed Responses
c           0.660

The verification check. The log’s first NOTE, modeling the probability that completed='Y', confirms the event level is the character "Y" you intended — the same modeled-level check, now on a character outcome. Read = 594 and Used = 594 match, which is the screening-row grain (594 visit rows, not the 200-participant or 198-baseline grain) — a deliberate reminder to know which table and which grain you are modeling. A proc freq; tables completed visit_num; before the fit would confirm the "Y"/"N" split and the three visit levels. (The odds ratios and C-statistic here are the notional, non-locked numbers.)

The interpretation. On these notional figures, each additional thousand steps in a visit is associated with about \(1.24\) times the odds of completing that visit (95% CI \(1.13\)–\(1.36\)) — again odds, not risk. The visit_num odds ratios compare visits 2 and 3 to visit 1 (the reference), and both confidence intervals include \(1\), so the model does not distinguish completion odds across visits here. The transfer lesson is the same workflow shape on a character outcome: declare the event level, read it back off the log, confirm the row count at the right grain, read each odds ratio with its CI and the word odds, and — because these are observational screening data — keep every claim associational.

A common mistake

The week’s central trap is mis-stating the odds ratio — calling it a risk, inverting it, or reading it as causal. Five specific slips to avoid:

  • Calling an odds ratio a risk ratio. OR \(= 1.78\) means the odds are \(1.78\) times as high, not that coaching participants are “\(78\%\) more likely” to meet goal. That phrasing describes a risk (probability) ratio, which the OR only approximates when the event is rare — and meeting goal (about \(41\%\)) is not rare. Keep the word odds.
  • Forgetting event='1' and inverting every OR. Omit the event= option and SAS models the other level by default, so each odds ratio prints as its reciprocal — with no error. Always read the Probability modeled is ... line off the log to confirm the level before you report a single number.
  • Ignoring the reference level. The arm OR is “coaching vs usual_care” only because usual_care is the reference. Change the reference and the comparison — and the number — change. State the reference level when you state the odds ratio.
  • Treating observational data as causal. The arms are not described as randomized, so OR \(= 1.78\) is an association between coaching and higher goal-met odds, not the causal effect of coaching. The model cannot say that assigning someone to coaching would raise their odds.
  • Reading the C-statistic as accuracy. AUC \(= 0.69\) summarizes how well the model ranks an event case above a non-event case; it is not “\(69\%\) correct,” and a model can rank well yet have poorly calibrated probabilities. Report it as discrimination, not accuracy.

A sixth, quieter slip is putting a 0/1 outcome into PROC REG. REG will happily fit a line to goal_met and print a slope, but that “linear probability model” can predict values outside \([0,1]\) and violates the constant-variance assumption — match the procedure to the outcome type, and a binary outcome takes LOGISTIC.

Low-stakes self-checks (ungraded)

These are for self-study only — ungraded, no submission.

  1. Write the PROC LOGISTIC step that models goal_met(event='1') on arm, age, and baseline_bmi with usual_care as the reference level. Which option declares the modeled event, and which sets the reference?
  2. From the locked output, state in one sentence each what the arm OR \(1.78\) (CI \(1.28\)–\(2.47\)), the age OR \(0.98\), the BMI OR \(0.93\), and the C-statistic \(0.69\) tell you — and what each does not tell you.
  3. The log shows PROC LOGISTIC is modeling the probability that goal_met='1'. Why do you read this line every time, and what would change in the odds ratios if it instead said goal_met='0'?
  4. A classmate reports “coaching participants are \(78\%\) more likely to meet their goal.” Identify the mistake using this week’s vocabulary, and rewrite the sentence so it correctly describes the odds ratio.
  5. The data are observational. Rewrite “coaching causes participants to meet their goal” as a correct associational claim about the arm odds ratio, and say in one sentence why the causal version is not supported.
  6. You have a binary outcome and you fit it with PROC REG instead of PROC LOGISTIC. Name two problems with the resulting “linear probability” fit, and name the procedure you should have used.

Reading and source pointer

For the SAS syntax, the reading pointer is the SAS documentation for PROC LOGISTIC — the MODEL statement and the event= option that declares the modeled level, the CLASS statement and reference coding (param=ref, ref=) that set a categorical predictor’s reference level, the odds-ratio estimates table, and the ROC / C-statistic output for the model’s discrimination. Consult these on documentation.sas.com for the exact option names and the output objects; read them for what the options do, in the course’s own words, not to copy their examples or listings. For the statistical background — what log-odds and odds ratios mean, why an OR is not a risk ratio, and why observational coefficients are associations — see the logistic-regression chapter of Introduction to Modern Statistics (IMS), 2nd ed. (Çetinkaya-Rundel & Hardin), CC BY-SA 3.0, free at openintro-ims.netlify.app; it calibrates the interpretation level, not the SAS syntax. These notes are the course’s own synthesis: grounded in the SAS documentation and open statistics references, but not copied from them. SAS® and all SAS Institute product names are the property of SAS Institute Inc.

Verification & reproducibility status

verified: false. The SAS code, the log excerpts, and every numeric value on this page are hand-authored, synthetic, and were NOT run — SAS is proprietary and is not executed in this build environment. The course SAS execution/output gate is BLOCKED: a rendered, syntax-highlighted code block or a typed listing is not evidence that the program runs or that the numbers are right. The load-bearing values here — the arm (coaching vs usual_care) odds ratio \(1.78\) with 95% CI \((1.28, 2.47)\) and \(p = 0.0006\); the age OR \(0.98\) and baseline_bmi OR \(0.93\); the C-statistic (AUC) \(0.69\); the modeled level goal_met='1'; the response-profile counts (\(82\) events / \(116\) non-events); the n = 198 baseline row count; and the transfer figures for completed='Y' on the 594-row screening slice (which are notional, not locked) — are drafted “as if run” for this draft site and are synthetic (the wellness-program study, seed streaminit(20260824)), representing no real health data, with the analysis observational (the arm odds ratio is an association, not a causal effect, and an odds ratio is not a risk ratio). Do not treat any value here as a confirmed reference until the human/SAS-run sign-off in the course’s private notation and verification ledger §5 is complete.

Public vs. graded

These notes, the SAS examples, and the practice here are public and ungraded — study material only. No graded prompts, answer keys, rubrics, point values, or due dates appear on this site. Graded SAS workflow checkpoints, skill checks, homework, analytics labs, the midterm practical, the final analytics project, and the final practical live in Blackboard (the LMS), which is authoritative for due dates, submissions, and grades. If this page and Blackboard ever disagree, follow Blackboard.

Looking ahead

Next week the focus shifts from fitting a model to shaping the data that feeds one. Week 12 covers reshaping and merging: PROC TRANSPOSE to move between wide and long layouts, and the DATA step MERGE (versus a PROC SQL join) to combine tables by a key — with the recurring 594 screening-row count as the validation target and the many-to-many merge as the trap to avoid. The discipline carries straight over: sort before you merge, use IN= flags to catch unmatched keys, and check the row count after every join, because a logistic or linear model is only as trustworthy as the analysis-ready table it was handed.

See also