1 / 30

Statistics

Learn how to calculate and interpret odds and odds ratios for different groups in a 2x2 table. Discover the significance and implications of odds ratios in statistical analysis.

frita
Download Presentation

Statistics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Statistics April 23, 2009 Tim Bunnell, Ph.D. & Jobayer Hossain, Ph.D. Nemours Bioinformatics Core Facility Nemours Biomedical Research

  2. Odds • The odds in favor of an event (e.g. occurrence of a disease) is the ratio of the probabilities of the event occurring to that of not occurring, i.e., p/(1-p) • where p is the probability of the event occurring • If probability of an event is 0.6 (i.e., it is observed in 60% of the cases) then the probability of it not occurring is (1 - 0.6) = 0.4 • The odds in favor of the event occurring is thus 0.6/0.4 = 1.5 • The greater the odds of an event, the greater it’s probability Nemours Biomedical Research

  3. Calculating odds for a 2x2 table The proportion of smokers with cancer: p = a/(a+b) (this is the likelihood of smokers having cancer) The proportion of smokers without cancer: 1-p = b/(a+b) (this is the likelihood of smokers not having cancer) The odds of smokers having cancer: p / (1 - p) = (a/(a+b))/(b/(a+b)) = a/b Nemours Biomedical Research

  4. Calculating odds for a 2x2 table The proportion of non-smokers with cancer: p = c/(c+d) The proportion of non-smokers without cancer: 1-p = d/(d+d) The odds of cancer among non-smokers: (c/(c+d)) / (d/(c+d)) = c/d Nemours Biomedical Research

  5. Odds Ratio • The odds ratio of an event is the ratio of the odds of the event occurring in one group to the odds of it occurring in another group. • Let p1 be the probability of an event in group 1 and p2 be the probability of the same event in group 2. Then the odds ratio (OR) of the event in these two groups is: • The odds ratio is a measure of effect size Nemours Biomedical Research

  6. The odds of cancer among smokers is a/b The odds of cancer among non-smokers is c/d So, the odds ratio of cancer among smokers vs non-smokers Odds Ratio • In the previous example, Nemours Biomedical Research

  7. The odds of cancer in smokers is 120/80 = 1.5 and the odds of cancer in non-smokers is 60/140 = 0.4286 and the ratio of two odds is, Interpretation: The risk of cancer is 3.5 times greater for smokers compared to non-smokers (in this sample) Odds Ratio Estimating odds ratio in a 2x2 table Nemours Biomedical Research

  8. Odds Ratio • The odds ratio must be greater than or equal to zero. • As the odds of the first group approaches to zero, the odds ratio approaches to zero. • As the odds of the second group approaches to zero, the odds ratio approaches to positive infinity • An odds ratio of 1 indicates that the condition or event under study is equally likely in both groups. In our example, that would mean no association between cancer and smoking was observed. Nemours Biomedical Research

  9. Odds Ratio • An odds ratio greater than 1 indicates that the condition or event is more likely in the first group. In the previous example, an odds ratio of 2 means that cancer is 2 times more likely in smokers compared to non-smokers • An odds ratio less than 1 indicates that the condition or event is less likely in the first group. In our example, an odds ratio of 0.8 would mean that the cancer is 20% (i.e., 1 - 0.8) less likely in smokers compared to non-smokers Nemours Biomedical Research

  10. R-demo: Odds ratio (from 2 x 2 contingency table) • Install Package epitools for odds ratio • Load epitools: >library(“epitools”) • Input data in previous example: • x<-matrix(c(120,60,80,140), 2, 2) • oddsratio(x, method="wald") Nemours Biomedical Research

  11. R-demo: Odds ratio - Output • > oddsratio(x, method="wald") • $data • Outcome • Predictor Disease1 Disease2 Total • Exposed1 120 80 200 • Exposed2 60 140 200 • Total 180 220 400 • $measure • odds ratio with 95% C.I. • Predictor estimate lower upper • Exposed1 1.0 NA NA • Exposed2 3.5 2.313230 5.295625 • $p.value • two-sided • Predictor midp.exact fisher.exact chi.square • Exposed1 NA NA NA • Exposed2 1.467097e-09 2.298090e-09 1.637296e-09 • $correction • [1] FALSE • attr(,"method") • [1] "Unconditional MLE & normal approximation (Wald) CI" Nemours Biomedical Research

  12. R-demo: Odds ratio (from raw data) • Categorize the outcome variable PLUC.post in two groups (high and low PLUC groups) . • Data management -> Manage variable in active data set -> Bin numeric variable ->From this window: pick variable to bin (e.g. PLUC.post ), Name the new variable (e.g. Newvar), Select number of bins (e.g. 2), select binning method (e.g. k-means clustering) • oddsratio(data$Ped, data$Newvar, method="wald"), where Newvar is the categorized outcome variable and Ped is also a categorical variable with pediatrician 1 and 2. Nemours Biomedical Research

  13. R-demo: Odds ratio - Output • > oddsratio(data$Ped, data$Newvar, method="wald") • $data • Outcome • Predictor 0 1 Total • 0 12 18 30 • 1 16 14 30 • Total 28 32 60 • $measure • odds ratio with 95% C.I. • Predictor estimate lower upper • 0 1.0000000 NA NA • 1 0.5833333 0.2095645 1.623737 • $p.value • two-sided • Predictor midp.exact fisher.exact chi.square • 0 NA NA NA • 1 0.3166258 0.4378954 0.300623 • $correction • [1] FALSE • attr(,"method") • [1] "Unconditional MLE & normal approximation (Wald) CI" Nemours Biomedical Research

  14. Logit of p • The logit of a number P between 0 and 1 is log(P/1-P). It is defined by logit(P) • If P is the probability of an event, then P/1-P is odds of thatevent and logit(P) is the log(odds) = log(P/1-P). • The difference between the logits of two probabilities is the log of the odds ratio (OR). • log (OR) = • The logit scale is linear and functions much like a z-score scale. Nemours Biomedical Research

  15. Logit of p • Logit is a continuous score in the range - to  • p = 0.50, then logit = 0.0p = 0.70, then logit = 0.85p = 0.30, then logit = -0.85 • The standard deviation of logit is sqrt(1/a + 1/b + 1/c + 1/d). Nemours Biomedical Research

  16. Logistic Regression • Recall: For a categorical variable, we focus on number or proportion for each category. • Proportion of a category simply says about how likely to happen that category • Suppose, y is a variable that represent occurrence or not occurrence of cancer (two categories only). • And y=1 indicates occurrence and y=0 indicates (not occurrence). • Let, p=likelihood of the event (y=1), so 1-p = likelihood of the event (y=0). • We want to relate p or (1-p) i.e. likelihood of happening or not happening, instead the response y itself), with an independent variable Nemours Biomedical Research

  17. Logistic Regression Plot of two variables: An outcome variable (say cancer happening/not happening ) and an independent variable (say age) Could we use a linear regression? Nemours Biomedical Research

  18. Logistic Regression Plot of proportions for different ages Likelihood of cancer increases with an increasing age Likelihood of cancer Relationship: not linear Age (in years) Nemours Biomedical Research

  19. Linear vs. Logistic Function

  20. Logistic Regression • Recall: Simple linear regression: • y = b0+ b1x, where y is a continuous quantitative outcome variable, x is a quantitative/categorical variable. • Like y, logit is a quantitative variable, and we can replace y by logit of p where p is the likelihood of an event • That is, log(p/(1-p)) = log (odds) = b0+ b1x, which is simple linear regression between log(p/(1-p)) =log(odds) and the independent variable x (say age). • Association patterns with log(odds) are the same as the patterns with odds itself. Nemours Biomedical Research

  21. Logistic Regression • log(p/1-p) = log (odds of cancer) = b0 + b1*age • Interpretation of b1: change of log odds of cancer for 1 year change of age. • Let us consider two persons of ages 55 and 56, then, • log (odds of cancer at age 55) = b0 + b1*55 • log (odds of cancer at age 56) = b0 + b1*56 • The difference, log(odds at 56) – log (55) = b1 Nemours Biomedical Research

  22. Logistic Regression • b1=0 (or equivalently odds ratio exp(b1) =1), indicates no association of log(Odds of cancer) with the variable age (X). • b1>0, indicates a positive association of log(Odds of cancer) with the variable age (X). • b1<0, indicates a negative association of log(Odds of cancer) with the variable age (X). • If 95% confidence interval (CI) of b1 does not contains 0 (the null hypothesis), it indicates that the independent variable has an significant influence on the response variable at 5% level of significance. • b0 is the intercept Nemours Biomedical Research

  23. Logistic Regression • exp(b1)= 1, No association of response with predictor. For categorical predictor, an event is equally likely in both reference as well as comparative group. • exp(b1)>1, indicates that an event is more likely to the comparative group compare to the reference group. • exp(b1)<1, indicates that an event is less likely to the comparative group compare to the reference group. • If 95% CI of odds ratio contains 1, it indicates that likelihood of an event two groups are significantly different at 5% level of significance. Nemours Biomedical Research

  24. Rcmdr demo: Logistic Regression • Statistics -> Fit models -> Generalized linear model -> from this window ( under the model formula, select dependent variable (e.g. Newvar) ~ independent (e.g. age), family – binomial, link – logit. • Newvar is the categorical form of the variable PLUC.post (Details in the example for odds ratio) Nemours Biomedical Research

  25. Simple Logistic Regression: Rcmdr output • Call: • glm(formula = Newvar ~ age, family = binomial(logit), data = data) • Deviance Residuals: • Min 1Q Median 3Q Max • -1.246 -1.233 1.109 1.123 1.131 • Coefficients: • Estimate Std. Error z value Pr(>|z|) • (Intercept) 0.0818711 0.8222250 0.100 0.921 • age 0.0005715 0.0086349 0.066 0.947 • (Dispersion parameter for binomial family taken to be 1) • Null deviance: 82.911 on 59 degrees of freedom • Residual deviance: 82.906 on 58 degrees of freedom • AIC: 86.906 95% CI for the coefficient of age is 0.0005715  2*0.0086349 = (-0.0166983, 0.0178413) In terms odds ratio: Odds ratio = exp (0.0005715) = 1.000572 95% CI for the odds ratio is (exp(-0.0080634), exp(0.0092064)) = (0.991969, 1.009249) Nemours Biomedical Research

  26. Simple logistic regression: Rcmdr output • Call: • glm(formula = Newvar ~ Ped, family = binomial(logit), data = data) • Deviance Residuals: • Min 1Q Median 3Q Max • -1.354 -1.121 1.011 1.011 1.235 • Coefficients: • Estimate Std. Error z value Pr(>|z|) • (Intercept) 0.4055 0.3727 1.088 0.277 • Ped[T.1] -0.5390 0.5223 -1.032 0.302 • (Dispersion parameter for binomial family taken to be 1) • Null deviance: 82.911 on 59 degrees of freedom • Residual deviance: 81.836 on 58 degrees of freedom • AIC: 85.836 Nemours Biomedical Research

  27. Logistic Regression • Outcome (response) variable is binary • Independent variable (predictor) can be either categorical or quantitative • Relationship of outcome variable and predictor (s) is not linear Nemours Biomedical Research

  28. Multiple Logistic Regression • More than one independent variables in the model i.e. • log(p/1-p) = b0 + bx1 + bx2 • Interpretation: the same as it is for the simple logistic regression • Response variable is binary Nemours Biomedical Research

  29. Multinomial and Ordinal Logistic Regressions • Multinomial: The response (outcome) variable is multicategorical (e.g. race- Caucasian, African American, Hispanic, Asian etc). • Ordinal: Categories of the response (outcome) variable can be ranked or order (e.g. disease condition: mild, moderate, and severe) Nemours Biomedical Research

  30. Thank you Nemours Biomedical Research

More Related