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37 Pages

### 20081105PARVIN

Course: K 30, Fall 2009
School: Washington University...
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Word Count: 1151

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Regression DOCR Logistic Course November 5, 2008 Curtis A. Parvin, Ph.D. Department of Pathology &amp; Immunology Phone: 454-8699 email: parvin@wustl.edu Regression Relate one or more independent (predictor) variables to a dependent (outcome) variable Ordinary linear regression Continuous outcome variable Determine the relationship between a continuous outcome variable and the predictor variable(s)...

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Regression DOCR Logistic Course November 5, 2008 Curtis A. Parvin, Ph.D. Department of Pathology & Immunology Phone: 454-8699 email: parvin@wustl.edu Regression Relate one or more independent (predictor) variables to a dependent (outcome) variable Ordinary linear regression Continuous outcome variable Determine the relationship between a continuous outcome variable and the predictor variable(s) Logistic regression Binary outcome variable Determine the relationship between the probability of the outcome occurring and the predictor variable(s) Other Flavors of Logistic Regression Conditional Logistic Regression Matched pairs data (1:1, 1:k, k1:k2 matching) Ordinal Logistic Regression More than two ordered groups for outcome Multinomial Logistic Regression More than two unordered groups for outcome Example: Relationship between gestational age at birth and whether an infant is breast feeding at time of hospital discharge GA 28 29 30 31 32 33 No 4 3 2 2 4 1 Yes 2 2 7 7 16 14 Total 6 5 9 9 20 15 P .33 .40 .78 .78 .80 .93 Ordinary Linear Regression Logistic Regression How do we get an "S-shaped" curve? Rather than using probability as our outcome variable, we use a transformation that is a function of probability We choose our transformation so that it ranges between (-,) as probability ranges between (0,1) We will use the logit transform Fitting a straight line using the logit transform as the outcome variable is called logistic regression After we estimate the straight line we can transform back to get our S-shaped curve Probability, Odds, and the Logit Transform Probability, P, ranges between 0 and 1 Define Odds = P/(1-P) Odds range between 0 and Note: P = Odds/(1+Odds) The Logit transform is the logarithm of the Odds Logit = log(Odds) = log[P/(1-P)] Logit ranges between - and Note: Odds = eLogit Note: P = eLogit/(1+eLogit) Odds and Logit for Breast Feeding Example GA 28 29 30 31 No 4 3 2 2 Yes 2 2 7 7 P .33 .40 .78 .78 Odds .50 .67 3.5 3.5 Logit -.69 -.41 1.25 1.25 32 33 4 1 16 14 .80 .93 4.0 14.0 1.39 2.64 Log(Odds) = -16.72 + 0.577*GA Logistic Regression Model the logarithm of the odds of an outcome as a linear combination of predictor variables Logit = log(Odds) = a+bX+cY+. . . Estimate the coefficients a, b, c based on a random sample of subjects' data Determine which of the predictors are "good" Assess model fit Use the model to predict future cases Logistic Regression Coefficients For a single predictor variable, logistic regression fits a straight line to the log of the Odds log(Odds) = a + bX b is the slope coefficient for X Each 1 unit change in X, changes the log(Odds) by b units Logistic Regression Coefficients b = log[Odds(X+1)] log[Odds(X)] Note: log(A) log(B) = log(A/B) b = log[Odds(X+1)/Odds(X)] Note: Odds(X+1)/Odds(X) is called an Odds ratio Odds and Odds Ratios Odds defines the probability that an event occurs divided by the probability that the event doesn't occur An Odds ratio is the ratio of two Odds An Odds ratio could represent the ratio of the odds in two different groups An Odds ratio could represent the ratio of the odds at two different values for a risk variable Breast Feeding Example GA 28 29 30 No 4 3 2 Yes 2 2 7 P .33 .40 .78 Odds .50 .67 3.5 Logit -.69 -.41 1.25 31 32 33 2 4 1 7 16 14 .78 .80 .93 3.5 4.0 14.0 1.25 1.39 2.64 The Odds ratio for breast feeding at hospital discharge for GA=32 compared to GA=28 is 4.0/0.5 = 8.0 Logistic Regression Coefficients and Odds Ratios b = log[Odds(X+1)/Odds(X)] b estimates the log of the Odds ratio associated with a 1 unit increase in X eb estimates the the odds ratio for a 1 unit increase in X For the breast feeding example log(Odds) = -16.72 + 0.577*GA the odds of breast feeding at hospital discharge increase by a factor of e0.577 = 1.78 for each additional week of GA Logistic Regression Odds Ratios GA 28 29 30 31 32 33 No 4 3 2 2 4 1 Yes 2 2 7 7 16 14 Odds .50 .67 3.5 3.5 4.0 14.0 OR 1 1.33 7.0 7.0 8.0 28.0 Logistic OR 1 1.78 3.17 5.64 10.05 17.89 Logistic Regression When There is Only One Binary Predictor This situation can be handled as a classic case-control study Disease Cases Controls Risk Yes a b Factor No c d Odds Ratio (OR)= a/c = a/b = ad b/d c/d bc Example: CHD and Age (Dichotomized at 55 Years) CHD Present 55 years <55 years 21 22 Absent 6 51 Total 27 73 Total 43 57 100 2X2 table calculation: OR = (21/22)/(6/51) = 8.11 Example: CHD and Age (Dichotomized at 55 Years) CHD Present 55 years <55 years 21 22 Absent 6 51 Total 27 73 Total 43 57 100 Logistic Regression: log(Odds) = -0.841 + 2.094 * Age OR = exp(2.094) = 8.11 Logistic regression produces the exact same Odds ratio estimate as the 2X2 table calculation The Real Strength of Logistic Regression is When There are Multiple Predictor Variables The independent variables (predictors, risk factors) can be categorical or continuous Example: TDx-FLM II and gestational age as predictors of risk for respiratory distress syndrome (RDS) TDx-FLM II measures mg surfactant/g of albumin in amniotic fluid The Data (some of it) TDxFLM 75 7 14.8 18.3 27 22 29 135 4 15 16.5 25 44.2 35.5 41 48 49 55.8 59 59 GA 30 31 31 31 31 31 31 31 32 32 32 32 32 32 32 32 32 32 32 32 RDS 0 1 1 1 1 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 Logistic Regression Parameter Estimates -----------------------------------------------------------------------------rds | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------tdxflm | -.1121656 .0163848 0.000 -7.11 -.1442792 -.0800520 ga | -.3661113 .1192559 -2.58 0.010 -.5998486 -.1323740 _cons | 15.68597 4.322678 3.63 0.000 7.213680 24.15827 ------------------------------------------------------------------------------ log(Odds) = 15.69 - 0.112*TDxFLM - 0.366*GA Odds Ratio for a 1 g/mg increase in TDxFLM: e-0.112 = 0.894 Odds Ratio for a 1 week increase in GA: e-0.366 = 0.693 -----------------------------------------------------------------------------rds | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------tdxflm | .893896 .0154269 -7.11 0.000 .8636601 .9241324 ga | .6934256 .0871025 -2.58 0.010 .5227078 .8641434 ------------------------------------------------------------------------------ Using the Logistic Model to Predict Risk of RDS We can use the logistic model equation to; Identify variables that are significant predictors calculate the absolute risk (probability) of RDS (may give biased estimates) calculate the relative risk (odds ratio) of RDS develop a classifier for diagnosing RDS Logistic Regression Parameter Estimates Significant coefficients mean significantly different from zero -----------------------------------------------------------------------------rds | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------tdxflm | -.1121656 .0163848 -7.11 0.000 -.1442792 -.0800520 ga | -.3661113 .1192559 -2.58 0.010 -.5998486 -.1323740 _cons | 15.68597 4.322678 3.63 0.000 7.213680 24.15827 ------------------------------------------------------------------------------ Significant Odds ratios mean significantly different from one -----------------------------------------------------------------------------rds | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------tdxflm | .893896 .0154269 -7.11 0.000 .8636601 .9241324 ga | .6934256 .0871025 -2.58 0.010 .5227078 .8641434 ------------------------------------------------------------------------------ Absolute Risk of RDS based on TDX FLM II and gestational age (for RDS prevalence of 8.5%) Gestational Age (weeks) FLM 10 20 30 40 29 98% 94% 85% 64% 30 97% 92% 79% 55% 31 96% 89% 73% 46% 32 95% 85% 65% 37% 33 92% 80% 56% 29% 34 89% 73% 47% 22% 35 85% 65% 38% 17% 36 80% 57% 30% 12% 37 73% 47% 23% 8.7% 38% 17% 6.2% 12% 4.4% 38 39 50 60 70 80 90 100 37% 16% 5.8% 2.0% 29% 12% 4.1% 1.4% 0.45% 22% 8.4% 2.9% 0.96% 0.31% 0.10% 16% 6.0% 2.0% 0.67% 0.22% 0.07% 12% 4.2% 1.4% 0.46% 0.15% 0.05% 8.5% 3.0% 0.98% 0.32% 0.11% 0.03% 6.1% 2.1% 0.68% 0.22% 0.07% 0.02% 4.3% 1.4% 0.47% 0.16% 0.05% 0.02% 3.0% 1.0% 0.33% 0.11% 0.04% 0.01% 2.1% 0.70% 0.23% 0.07% 0.02% <0.01% 1.5% 0.49% 0.16% 0.05% 0.02% <0.01% Risk e(15.69 0.112*TDxFLM 0.366*GA) 1 e(15.69 0.112*TDxFLM 0.366*GA) Odds ratios for RDS relative to a TDX FLM II ratio of 70 mg/g at 37 weeks gestational age Gestational Age (weeks) FLM 29 30 >1000 >1000 541 >1000 375 31 >1000 799 260 32 >1000 554 180 33 34 35 36 37 38 189 61.6 20.1 42.7 13.9 39 10 20 30 40 >1000 >1000 384 125 >1000 818 266 86.8 >1000 567 185 60.2 >1000 393 128 41.7 837 273 88.8 28.9 50 60 70 80 176 57.4 18.7 6.09 122 39.8 13.0 4.23 84.8 27.6 9.00 2.93 58.8 19.1 6.24 2.03 40.8 13.3 4...

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