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Multiple Regression - Overview Multiple Regression in statistics is the science and art of creating an equation that relates an outcome Y, to one or more predictors, X1, X2, X3, .. Xk. Linear Regression Y = a + b1X1 + b2 X2 + b3 X3 + ... + bk Xk + e = Ŷ + e where "e" is the residual error between the observed Y and the prediction (Ŷ). In linear regression, bi is the average change in Y for a single unit change in Xi. “Performance” stats: R2, SDe
Logistic Regression In multiple logistic regression, Y is 0 or 1 with mean P, the “risk”. The logit of P, not P, is assumed a linear function of the Xs Logit(P) = ln(P/(1-P)) = a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk (“Logit” = log of the odds since P/(1-P) is the odds) odds=exp(a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk) risk = P = odds/(odds+1) “Performance” stats: Sensitivity, Specificity, Accuracy, Concordance (C), mean deviance Poisson Regression When Y is a positive integer (0,1,2,3…), we model the log of Y so Y can never be negative. This is the multiple Poisson regression model. ln(mean Y) = a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk mean Y = exp(a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk) mean Y cannot be negative. “Performance” stats: R2, mean deviance
Logistic regression Predictors of in hospital infection Characteristic Odds Ratio (95% CI) pvalue Incr APACHE score 1.15 (1.11-1.18) <.001 Transfusion (y/n) 4.15 (2.46-6.99) <.001 Increasing age (yr) 1.03 (1.02-1.05) <.001 Malignancy 2.60 (1.62-4.17) <.001 Max Temperature 0.70 (0.58-0.85) <.001 Adm to treat>7 d 1.66 (1.05-2.61) 0.03 Female (y/n) 1.32 (0.90-1.94) 0.16 *APACHE = Acute Physiology & Chronic Health Evaluation Score
Multiple Proportional Hazards Regr (Cox model) For time dependent outcomes (ie time to death), we model the hazard rate,h , the event rate per unit time (for death, it is the mortality rate). Since h > 0, we model the log of the hazard as a linear function of the Xs so h can never be zero (similar to Poisson regression) . ln(h) = a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk so h = exp(a + b1 X1 + b2 X2 + b3 X3 + ... + bk Xk) > 0 If h0=exp(a) is the ‘baseline’ hazard, (that is, a=log(h0)) the hazard ratio is HR = h/h0 = exp(b1 X1 + b2 X2 + b3 X3 + ... + bk Xk) no intercept ‘a’. If S0(t) is the ‘baseline’ survival curve corresponding to the baseline hazard, then the survival curve for a given combination of X1, X2, … Xk is given by S(t) = S0(t)HR where HR is computed with the equation above. exp(bi) is the hazard rate ratio for a one unit change in Xi. “Performance” stats: Harrell’s Concordance (C) (0.5 < C < 1.0)
Cox HRs for donor age(Busuttil 2005) Harrell C= 0.70
Multiple Linear Regression Example Consider predictors of Y=Bilirubin (mg/dl) in liver transplant candidates. Two predictors are X1=Prothombin time (PT) in seconds X2=ALT (alanine aminotransferase in U/L). A multiple regression equation (on the log scale) is Ŷ = (predicted) log Bilirubin = -3.96 + 3.47 log PT + 0.21 log ALT
Regression output - equation (Equation) Parameter estimates term estimate SE t ratio p value Intercept -3.96 0.257 -15.4 < 0.001 log PT 3.47 0.214 16.2 < 0.001 log ALT 0.21 0.055 3.8 0.0002 equation: Log Bili=-3.96 + 3.47 log PT + 0.21 log ALT
Regression-analysis of variance table Source df sum squares mean square F Model 2 37.76 18.88 147.2 Error 363 46.56 0.1283=SDe2 -- Total 365 84.32 F = 18.88/0.1283 ->Screening F test that none of the predictors are related to Y. R2 = 37.76/84.32 = 0.448 = 44.8% R2 = model sum squares/total sum squares
Residual error plotResidual Log Bilirubin by Predicted When the model is valid, this plot should look like a circular cloud if the errors have constant variance. The example above is a “good” result.
Example of a “good” residual error histogramerrors have a Gaussian distribution about zero Moments - errors Mean 0.000 Std Dev (SDe) 0.357 Std Err Mean 0.0187 n 366 Quantiles - errors 100.0% maximum 1.711 97.5% 0.834 90.0% 0.454 75.0% quartile 0.177 50.0% median -0.025 25.0% quartile -0.213 10.0% -0.355 2.5% -0.630 0.0% minimum -1.191
Normal Quantile Plot(Should be approximately a straight line if the residual error data are Gaussian) residual error (e)
Interpreting multiple regression coefficients (cont.) The multiple regression coefficients will not in general be the same as the individual regression coefficient for each variable one at a time, even though the same Y is being modeled. Simple regression Multiple (simultaneous) variable (one Y, one X) (b1X1+b2X2) Log PT 3.560 3.470 Log ALT 0.310 0.211 Log Bilirubin = - 3.70 + 3.56 log PT, R2 = 0.425 Log Bilirubin = -.105 + 0.310 log ALT, R2 = 0.049 Log Bilirubin = -3.96 + 3.47 log PT + 0.211 log ALT , R2 = 0.448 Log PT coefficients don’t match
Orthogonally In general, regression coefficients from simple and multiple regression are not the same ↔ controlling for covariates does not give the same answer as ignoring covariates. Only when all the X variables have correlation zero with each other will the simple and multiple regression coefficients be the same orthogonally
Log PT (X1) vs log ALT (X2)since correlation is low, unadjusted and adjusted regression results are similar r12 = 0.111, R2 = 0.0123
Interaction Effects & subgroups The model Y = 0 + 1X1 + 2X2 + implies that change in Y due to X1 (=1) is the same (constant) for all values of X2. An ADDITIVE model. In the model Y = 0 + 1X1 + 2X2 + 3 X1X2 + the 3 term is an interaction term. Change in Y for a unit change in X1 is (1+3X2) and is therefore not constant. Positive 3 is often termed a “synergism” Negative 3 is often termed an “antagonism” Additive only if β3=0 How to implement? Make new variable W = X1X2.
Interaction example Response: Y= log HOMA IR (MESA study) R2=0.280, Root Mean Square Error=SDe=0.623 Mean Response= 0.395, n= 6782 Parameter Estimates Term Estimate Std Error t Ratio p value Intercept -1.388 0.049 -28.16 <.0001 Gender -0.669 0.085 -7.83 <.0001 BMI 0.061 0.00168 36.45 <.0001 gender*BMI 0.028 0.00299 9.38 <.0001 Predicted log HOMA IR = -1.39 – 0.669 gender + 0.061 BMI + 0.028 gender * BMI (gender is coded 0 for female and 1 for male)
gender x BMI interaction- non additivity In Females Log HOMA IR = -1.39 +0.061 BMI In Males Log HOMA IR = -2.06 + 0.089 BMI
Hierarchically well formulated (WWF) regression models HWF Rule – To correctly evaluate the X1*X2 interaction, must also have X1 and X2 in the model. In general, one must include the lower order terms in order to correctly evaluate the higher order terms.
HWF example- NON HWF Model:chol = a0 + a1 smoke x age = SMOKEAGE 0, 1 (dummy) coding: smoke=0 or 1, smokeage = smoke x age Variable DF Estimate std error t p value INTERCEPT 1 156.863 3.993 39.286 0.0001 SMOKEAGE 1 0.361 0.182 1.988 0.0567 ---------------------------------------------------------------------------- -1, 1 (effect) coding: smoke=-1 or 1, smokeage =smoke x age Variable DF Estimate std error t p value INTERCEPT 1 162.278 3.103 52.320 0.0001 SMOKEAGE 1 0.055 0.100 0.548 0.5881 p value changes just because coding changes!
HWF example (cont)- HWF HWF:Model: chol = b0 + b1 smoke + b2 age + b3 smoke x age For HWF models, significance is the same regardless of coding 0, 1 (dummy) coding: smoke=0 or 1, smokeage = smoke x age Variable DF Estimate std error t p value INTERCEPT 1 100.221 1.110 90.304 0.0001 SMOKE 1 3.812 1.570 2.429 0.0224 AGE 1 2.010 0.036 56.297 0.0001 SMOKEAGE 1 -0.009 0.050 -0.178 0.8599 -1, 1 (effect) coding:smoke=-1 or 1, smokeage=smoke x age Variable DF Estimate std error t p value INTERCEPT 1 102.127 0.785 130.138 0.0001 SMOKE 1 1.906 0.785 2.429 0.0224 AGE 1 2.005 0.025 79.437 0.0001 SMOKEAGE 1 -0.004 0.025 -0.178 0.8599
Regression assumptions Regression can simultaneously evaluate all factors and thus reduce confounding. But must check two critical assumptions. • If X is continuous/interval, check if relation of X & Y is linear on some scale. (otherwise polychotomize X) 2. Check if effects of X1, X2, X3 … are additive by adding interaction terms. (Ex: X4 = X1 x X2). Not additive if interactions are significant.
3 Also, in linear regression, prefer residual errors (e) to have a Gaussian distribution with a constant variance that is independent of Y. But additivity and linearity are more important since lack of additivity and linearity lead to bias and is more misleading.
Nonlinear Regression Log(Bilirubin)= -3.96 + 3.47 log(PT) + 0.211 log(ALT) is a nonlinear model in terms of PT and ALT but is a linear model in terms of log PT, log ALT and the regression coefficients b0=-3.96, b1=3.47 and b2=0.211. Consider model of the form: Ŷ= Drug concentration = b1 10 b2 X This is nonlinear in b2 but can be made linear with a transformation: log10(concentration)=log10(b1) + b2 x What about: Drug concentration = b0+ b1 10 b2 X This model is nonlinear in b2 and cannot be transformed. Nonlinear regression software is needed to estimate b0, b1, & b2.
Nonlinear ExampleCompartmental drug models Model of how drug (or any chemical) is metabolized by an organism. Y1=concentration in serum, Y2=concentration in organ, x=time d (Y1)/dx = -b1 Y1 d (Y2)/dx = b1Y1 - b2Y2 b1 > b2 > 0 solutions: Y1 = constant e -b1 x Y2 = (b1/(b1-b2)) [e -b2x – e -b1x] < - model Y2 takes on a maximum value when x = ln(b1/b2)/[b1-b2] Y2 is zero when x=0 or x is very large The constants b1 and b2 are rates. They are in units of 1/x (i.e 1/time). Y1 serum Y2 organ
Nonlinear equationY = (b1/(b1-b2)) [e -b2x – e -b1x] Y= [0.0967/(0.0967-0.0506)]*[exp(-0.0506*t)-exp(-0.0967*t)] at peak, t = 14, Ŷ= 0.49
Residual diagnostics & “model criticism” Assumptions of linear regression: 1. Linear relation between Y and each X except for random “noise” (but can transform X). 2. Effect of each X is additive (but can make interaction terms) 3. Errors (e) have constant variance and come from a Gaussian distribution 4. All observations from the same population 5. All observations independent (usually ok) A plot of Ŷ versus e, called a residual error (diagnostic) plot, can help verify if these assumptions are met.
Regression diagnostics Problem-outliers Solution – Find on residual plot & eliminate Problem-Curvilinear (non linear) Solution- try nonlinear trans formation (x2, 1/x, log(x), ex) Problem-Errors not Gaussian Solution-Robust regression (future class) Problem-Non constant SDe or Var(e) Solution-Weights (future class)
Ex: Meditation & change in pct body fat Overweight persons chose a meditation program or a “sham” (lectures) as part of a weight loss effort. They were NOT randomized. Change in percent body fat by treatment group (mediation or sham) over three months Unadjusted Means Level n Mean pct body SEM Mean dietary fat (gm) fat change (before study start) 1-meditate 439 -7.51% 0.47% 32.7 g 2-sham 704 1.34% 0.35% 67.1 g Unadjusted Mean difference (sham - meditation) = 8.85% SE of the difference = SEd = √0.47222 + 0.35322 = 0.59% t = mean diff/SEd = 8.85% / 0.59% = 15.1, p < 0.0001 Overall unweighted dietary fat = 49.9 g
Result via “regression” Y= change in body fat, X = 1 if sham, 0 if meditation Y = a + b X + error = a + b sham + error term estimate SE t p value a -7.51 0.46 -16.4 < 0.001 b 8.85 0.59 15.2 < 0.001 Ŷ = -7.51 + 8.85 sham
Regr-control for dietary fat pct body fat= Y = a + b1 X1 + b2 X2 + error X1= 1 if sham, 0 if meditation X2 = dietary fat in g term estimate SE t p value a -14.5 0.54 -26.6 < 0.001 b11.51 0.56 2.7 0.007 b2 0.21 0.011 18.9 <0.001 body fat chg=-14.5 +1.51 sham +0.21 diet fat
Adjusted means Plug into equation for X1=1=sham or X1=0. Hold X2 = diet fat= 49.9 g – overall mean X2 Med: -14.5 + 1.51(0)+0.213(49.9)=-3.84% Sham:-14.5 + 1.51(1)+0.213(49.9)=-2.33% Adj mean difference =2.33-(-3.84) =1.51% (Unadjusted was 8.85%)
General procedure Adjusted means for X1,controlling for (confounders) X2, X3, X4 … • Estimate model regression coefficients • Plug in different values for X1, and values for all other Xs held constant at their overall means. Gives adjusted means and their SEs. Assumes ????