Report

Statistics for Health Research Assessing Binary Outcomes: Logistic Regression Peter T. Donnan Professor of Epidemiology and Biostatistics Objectives of Session • Understand what is meant by a binary outcome • How analyses of binary outcomes implemented in logistic regression model • Understand when a logistic model is appropriate • Be able to implement in SPSS and • Interpret logistic model output Binary Outcome Extremely common in health research: •Dead / Alive •Hospitalisation (Yes / No) •Diagnosis of diabetes (Yes / No) •Met target e.g. total cholesterol < 5.0 mmol/l (Yes / No) n.b. Can use any code such as 1 / 2 but mathematically easier to use 0 / 1 How is relationship formulated? For linear simplest equation is : y a bx ei y is the outcome; a is the intercept; b is the slope related to x the explanatory variable and; e is the error term or random ‘noise’ Can we fit y as a probability range 0 to 1? y a bx ei Not quite! Y as continuous can take any value from -∞ to + ∞ Outcome is a probability of event, Π (or p) on scale 0 – 1 Certain transformations of p can give the required scale Probit is a normal transformation of p But not easy to interpret results The logit transformation works! We can now fit p as a probability range 0 to 1 And y in range -∞ to + ∞ y log it (p) a bx e i p log a bx e 1 p i Logistic Regression Model p log a bx e 1 p i This has very useful properties The term p/(1-p) is called the ‘Odds’ of an event Note: not the same as the probability of an event p If x is binary coded 0/1 then - exp (b) = ODDS RATIO for the outcome in those coded 1 relative to code 0 e.g. Odds of death in men (1) vs. women (0) Logistic Regression Model Consider the LDL data. It has two binary outcomes – 1) LDL target achieved 2) Chol target achieved For example consider gender as a predictor – Male = 1 & Female = 2 For a binary x we can express results as odds ratios (available in crosstabs) LDL target achieved Gender No Male Female 140 149 Yes 563 Odds yes = 563/140 531 Odds yes = 531/149 Odds ratio = 4.02 / 3.56 OR = 0.886 Female cf Male LDL target achieved No Gender Male Female 140 149 Yes 563 531 Odds yes = 563/140 = 4.02 Odds yes = 531/149 = 3.56 N.b. Odds is different to prob – Men p = 563/(140+563) = 0.80 or 80% Odds ratio from Crosstabs Obtain odds ratios for 2 x 2 tables from crosstabs and select option ‘risk’ Results from Crosstabs Odds ratios for achieving LDL target in females vs. males n.b. OR given for Female vs male = 0.886 Fit Logistic Regression Model Dependent is binary outcome – LDL target met (Yes = 1, No = 0) Independent – Gender 1 = M, 2 = F Should get same as the crosstabs result Select Analyze / Regression / Binary Logistic Select option of 95% CI for exp (b) Regression / Binary logistic….. Odds ratio from logistic model results for a binary predictor EXP (B) = Odds ratio F vs. M Note that OR for Men vs Women = 1/0.886 = 1.13 Fit Logistic Regression Model – continuous predictor Dependent is binary outcome – LDL target met Independent – Continuous predictor – Adherence B represents the change in the ODDS RATIO for a 1 unit increase in adherence B x 10 represents the change in the ODDS RATIO for a 10 unit increase in adherence Odds ratio from logistic model results for a continuous EXP (B) = Odds ratio for 1% increase in Adherence OR for 10% increase is exp(10 x 0.010) = 1.105 i.e. a 10.5% increase in odds of meeting LDL target for each 10% increase in adherence Fit Logistic Regression Model – categorical predictor Dependent is binary outcome – LDL target met Independent – APOE genotype (1 – 6) Choose a reference category, in this case worst outcome is genotype 6 so choose 6 to give ORs > 1 B represents the OR for each category relative to the reference category Regression / Binary logistic….. Choose Categorical Odds ratios from logistic model results for a categorical predictor EXP (B) = Odds ratio for APOE (2) vs APOE (6) OR = 4.381 (95% CI 1.742, 11.021) Epidemiological Designs • Logistic model common in epidemiological research • In case-control designs, case is coded 1 and controls as 0 and used as dependent variable • In cohort study outcome (e.g. death) is used as binary outcome in logistic model • Note in cohort study exp(b) is Relative Risk (RR) rather than OR Definition- Clinical Prediction Rule • Clinical tool that quantifies contribution of: – History – Examination – Diagnostic tests • Stratify patients according to probability of having target disorder • Outcome can be in terms of diagnosis, prognosis, referral or treatment Thresholds for decision making 100% Treatment Diagnosis / test threshold Derived Probability of disease Further diagnostic testing Test / reassurance threshold 0% Reassurance Ottawa ankle rule Risk Stratification Kaiser-Permanente Pyramid Identify high risk through ‘risk stratification’ and Intervene through case management at highest risk Framingham Risk Algorithm • Prediction of risk: Cardiovascular (Framingham) 55 yr-old woman 15-20% 5 yr risk Increasing appearance of “prediction models” in literature (ISI Web of Knowledge v3) Stages of development and assessment of a CPR Step 1 Derivation Identification of factors with predictive power Cross Sectional or Cohort Step 2 Validation Evidence of reproducible accuracy Application of a rule in similar clinical settings and population or better still multiple clinical settings and different populations with varying prevalence and outcomes of disease Cross Sectional or Cohort Step 3 Impact Analysis Evidence that rule changes physician behaviour and improves patient outcomes and /or reduces costs Randomized Controlled Trial How to derive a CPR? 1. Toss a coin to make decision? 2. Individual opinion and experience? 3. Huddle of wise ones – Delphi technique to reach consensus? 4. Statistical prediction models ! Regression Models for prediction • In all of these models we combine a set of factors: Usually between 2-20 predictors Occam’s razor suggests smaller is better • Fit a multiple regression model • Extract probabilities of outcome or diagnosis • Create CPR Regression Models for prediction • Linear if outcome continuous • Binary Outcomes Logistic regression model Survival models – Cox PH, Weibull, log logistic, etc • Ordinal or nominal outcomes Ordinal logistic regression The logit transformation We can now fit p as a probability range 0 to 1 And y in range -∞ to + ∞ y log it (p) a bx e i p log a bx e 1 p i Statistical prediction Models Logistic regression model: p log( ) = β0 + β1x1 + β2 x2 + ..... 1- p p= probability of the Event and effect of factors (x) increase or decrease risk of this event Derivation of probability of events Logistic regression model: p log( ) = β0 + β1x1 + β2 x2 + ..... 1- p Call X β β x β x ..... 0 1 1 2 2 Linear Predictor as a linear function of the predictors x1, x2, x3, etc…. Derivation of probability of events Then: p log( ) X 1- p Take exp of both sides : p ( ) exp( X) 1- p Derivation of probability of events Then rearrange: Or: exp( X) p 1 exp( X) 1 p 1 exp( X) Risk Stratification based on derived probabilities Example: PEONY model to predict risk of emergency admission to hospital over the next year Now implemented in NHS Tayside as part of Virtual Wards management of LTC PEONY II model developed – watch this space! Donnan et al Arch Int Med 2008 Other binary models The logistic model is only applicable whenever the length of follow-up is same for each individual e.g. 5-yr follow-up of a cohort For binary outcomes where censoring occurs i.e. people leave the cohort from death or migration then length of followup varies and need to use survival models such as Cox Proportional Hazards model Summary • Logistic model easily fitted in SPSS • Clear link with ODDS RATIOS • Common model for case-control, cohort studies as well as development of clinical prediction models General References • Campbell MJ, Machin D. Medical Statistics. A commonsense approach. 3rd ed. Wiley, New York, 1999. • Hosmer DW and Lemeshow S. Applied logistic regression. John Wiley& sons, New Jersey, 2000. • Altman DG. Practical statistics for medical research. London: Chapman and Hall, 1991. • Armitage P and Berry G. Statistical Methods in Medical research. 3rd ed. Oxford: Blackwell Scientific, 1994. • Agresti A. An introduction to Categorical Data Analysis. Wiley, New York, 1996. Practical: Fit Multiple Logistic Regression Model Dependent is binary outcome – LDL target met (Yes = 1, No = 0) Independent – Gender 1 = M, 2 = F, add APOE, adherence, etc Remember Select Analyze / Regression / Binary Logistic Select option of 95% CI for exp (b) 3) Screening for variables to eliminate • Consider screening procedures to eliminate a number of variables under consideration • Test each variable separately • If p > 0.3 then they would have to be very strong confounders to become significant on adjustment in a multiple regression so could be discarded • Hosmer-Lemeshow criteria 4) A mixture of automatic procedures and self selection • Use automatic procedures as a guide • Compare stepwise and backward elimination • Think about what factors are important • Add ‘important’ factors • Do not follow blindly statistical significance Remember Occam’s Razor ‘Entia non sunt multiplicanda praeter necessitatem’ ‘Entities must not be multiplied beyond necessity’ William of Ockham 14th century Friar and logician 1288-1347